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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf,len=1533
+page_content='η-pairing on bipartite and non-bipartite lattices  Yutaro Misu1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Shun Tamura2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Yukio Tanaka2 and Shintaro Hoshino1  1Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Saitama University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Saitama 338-8570,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Japan  2Department of Applied Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Nagoya University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Nagoya 464-8603,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Japan  (Dated: January 23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' 2023)  The η-pairing is a type of Cooper pairing state in which the phase of the superconducting order  parameter is aligned in a staggered manner,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' in contrast to the usual BCS superconductors with a  spatially uniform phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' In this study, we search for a characteristic η-pairing state in a triangular  lattice where a simple staggered alignment of the phase is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' As an example, we consider  the attractive Hubbard model on both the square and triangular lattices under strong external  Zeeman field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  Using the mean-field approximation, we have identified several η-pairing states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  Additionally, we have examined the electromagnetic stability of the pairing state by calculating the  Meissner kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Odd-frequency pairing plays a crucial role in achieving diamagnetic response if the  electrons experience a staggered superconducting phase during the propagation of current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  INTRODUCTION  The diversity of superconducting phenomena has been  attracting continued attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  The superconducting  state of matter is characterized by the properties of  Cooper pairs, which can be classified based on their  space-time and spin structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  With regard to their  space structure, Cooper pairs are typically classified as  s-wave, p-wave, or d-wave pairs depending on their rel-  ative coordinate structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' As for their center-of-mass  coordinate, while it is usually assumed to be zero in  most superconductors, it is possible to consider the exis-  tence of a finite center-of-mass momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' One example  of this is the Flude-Ferrell-Larkin-Ovchinnikov (FFLO)  state [1, 2], in which the Cooper pair has a small but finite  center-of-mass momentum under the influence of a mag-  netic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' More generally, the magnitude of the center-  of-mass momentum can be larger and of the order of the  reciprocal lattice vector ∼ π/a, where a is a lattice con-  stant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' This type of pairing state is known as η-pairing,  a concept first proposed by C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Yang, which forms a  staggered alignment of the superconducting phase on a  bipartite lattice [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' The spatially modulating order pa-  rameter is known also as the pair density wave, and has  been discussed in relation to cuprate superconductors [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  The actual realization of the η-pairing has been pro-  posed for the correlated electron systems such as the at-  tractive Hubbard (AH) model with the magnetic field  [5], the single- and two-channel Kondo lattices [6, 7], the  Penson-Kolb model [8], and also the non-equilibrium sit-  uation [9–14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  Since the phase of the superconducting  order parameter can be regarded as the XY spin, the η-  pairing is analogous to an antiferromagnetic state of the  XY spin model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  Hence, the η-pairing state should be  strongly dependent on the underlying lattice structure  and we naively expect a variety of the η-pairing state  if we consider the geometrically frustrated lattice such  as the triangular lattice since the simple staggered state  cannot be realized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  In this paper, we deal with the AH model on the non-  bipartite lattice in order to search for possible new su-  perconducting states depending on the feature of the  non-bipartite lattice structure in equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  Already  in the normal state without superconductivity, it has  been pointed out that the non-bipartite lattice generates  a non-trivial state of matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' For example in the Kondo  lattice, a partial-Kondo-screening, which has a coexisting  feature of Kondo spin-singlet and antiferromagnetism, is  realized [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Also in the AH model at half-filling, charge-  density-wave (CDW) is suppressed due to the frustration  effect [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' The η-pairing that appears in a photodoped  Hubbard model on the triangular lattice has been studied  recently [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' In the equilibrium situation, the properties  of the AH model have been studied on bipartite lattices  [5], but the model on a non-bipartite lattice has not been  explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  As shown in the rest of this paper, there are several  types of η-pairings on the triangular lattice of the AH  model under the Zeeman field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  One of the η-pairing  states is regarded as a 120◦-N´eel state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  Since the rel-  ative phase between the nearest neighbor sites is neither  parallel nor anti-parallel, the inter-atomic Josephson cur-  rent is spontaneously generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' This state can also be  regarded as a staggered flux state, where the flux is cre-  ated by the atomic-scale superconducting loop current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  While the staggered flux state has been studied so far  [17–23], the staggered flux in this paper is induced by  the Josephson effect associated with superconductivity  and has a different origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  For the analysis of the AH model, we employ the mean-  field approximation in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' It has been suggested  that a simple η-pairing shows a paramagnetic Meissner  state [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Hence it is necessary to investigate the electro-  magnetic stability of the solution for superconductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  We evaluate the Meissner kernel whose sign corresponds  to the diamagnetic (minus) or paramagnetic (plus) re-  sponse of the whole system, where the physically sta-  ble state should show diamagnetism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' We confirm that  if the mean-field η-pairing state has the lowest energy  compared to the other ordered states, the calculation of  the Meissner kernel shows the diamagnetic response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' It  is also notable that the odd-frequency pairing amplitude,  which has an odd functional form with respect to the fre-  quency [6, 25–30], can contribute to the diamagnetism in  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='08426v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='supr-con]  20 Jan 2023  2  the η-pairing state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' This is in contrast to the usual super-  conductivity with the uniform phase where the conven-  tional even-frequency pairing contributes to the diamag-  netism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' It has been shown that the odd-frequency pairing  induced at the edge, interface or junctions [31–36] shows  a paramagnetic response [37–41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' In this paper, by con-  trast, we consider the odd-frequency pairing realized in  bulk, which shows a qualitatively different behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  We explain the  model and method for the AH model in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' II, and the  Meissner kernel in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' The numerical results for the  AH model are shown in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' IV, and we summarize the  paper in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  ATTRACTIVE HUBBARD MODEL  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  Hamiltonian  We consider the Hamiltonian of the AH model with  magnetic field h which induce Zeeman effect only (Zee-  man field) :  H = −t  �  ⟨i,j⟩σ  c†  iσcjσ + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' + U  �  i  ni↑ni↓  − µ  �  i  ni − h ·  �  i  si,  (1)  where c†  iσ and ciσ are the creation and annihilation op-  erators of the i-th site with spin σ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  The  symbol ⟨i, j⟩ represents a pair of the nearest-neighbor  sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  Here, the parameter t is the nearest-neighbor  single-electron hopping integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' U (= −|U|) is the on-  site attractive interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' The spin operator is defined  as si =  1  2  �  σσ′ c†  iστσσ′ciσ′, where τ is the Pauli ma-  trix, and the number operator of electrons is denoted as  ni = ni↑ + ni↓ = �  σ c†  iσciσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' The electron concentration  is controlled by adjusting the chemical potential µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  The AH model has been successfully used to elucidate  several important and fundamental issues in supercon-  ductors [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' The model on a bipartite lattice at half fill-  ing is theoretically mapped onto the repulsive Hubbard  model by the following partial particle-hole transforma-  tion [43]  c†  i↑ → c†  i↑, c†  i↓ → ci↓eiQ·Ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  (2)  The reciprocal vector Q satisfies the condition eiQ·Ri =  (−1)i that takes ±1 depending on Ri belonging to A or  B sublattice on the bipartite lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Then, the η-pairing  appears in the region that corresponds to a ferromagnet  with transverse magnetization in the repulsive model [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  In a mean-field theory, the phase diagram for the repul-  sive Hubbard model without the magnetic field is shown  in the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' 1 [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' From this figure, we find  that the ferromagnet is located in the regime where the  repulsive interaction U > 0 is large and the electron con-  centration is not half-filled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Hence, the η-pairing phase  nc  t  |U|  m  0  1  0  1  PM  AFM  FM  FF  BCS  pairing  η  Repulsive Hubbard (  )  U > 0  Attractive Hubbard (  )  U < 0  h = 0  nc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='0  Spin-polarized  normal state  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  Sketches of the phase diagrams for the repulsive  Hubbard model [44] (left panel) and AH model (right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  nc is the electron concentration and m is the magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  When the interaction |U| is large, the ground state in the re-  pulsive Hubbard model is ferromagnet (FM), while the ground  state in the AH model is η-pairing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  is located in the regime where the attractive interaction  U < 0 is large and the magnetization is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' The phase  diagram of the AH model at half filling is shown in the  right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' In principle, an attractive interac-  tion large enough to realize η-pairing could be realized in  artificial cold atom systems [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  The Cooper pair is formed by the two electrons  with (k ↑,  − k + q ↓) where q is the center-of-mass  momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' The FFLO state and the η-pairing are dis-  tinguished by the magnitude of |q|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  In η-pairing, the  center-of-mass momentum of the Cooper pair is the or-  der of the reciprocal lattice vector, while the momentum  of the FFLO state is much smaller and the spatial mod-  ulation is slowly-varying compared to the atomic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  Although the large center-of-mass momentum is usually  not energetically favorable, a strong attractive interac-  tion can make it stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  Mean-field theory  By applying the mean-field approximation, we obtain  the mean-field Hamiltonian  HMF = −t  �  ⟨i,j⟩σ  c†  iσcjσ + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' − µ  �  i  ni − h ·  �  i  si  −  �  i  �  vini + Hi · si − ∆ic†  i↑c†  i↓ − ∆∗  i ci↓ci↑  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  (3)  3  The order parameters are given by the self-consistent  equations  vi ≡ |U|  2 ⟨ni⟩,  (4)  ∆i ≡ −|U|⟨ci↓ci↑⟩,  (5)  mi = 1  2  �  σσ′  ⟨c†  iστσσ′ciσ′⟩,  Hi =  − 2|U|mi,  (6)  where ⟨A⟩ = Tr  �  Ae−HMF/T �  /Tr  �  e−HMF/T �  is a quan-  tum statistical average with the mean-field Hamiltonian  and T is temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  ∆i is the order parameter for  s-wave singlet superconductivity (pair potential).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  The  phase θi ∈ [0, 2π) of the pair potential ∆i = |∆i|eiθi is  dependent on the site index and will be represented by  the arrow in a two-dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' The mean-fields  for the charge and spin are given by vi and Hi, respec-  tively, at each site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' The derivation of the self-consistent  equations is summarized in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' We will consider  the AH model both on the two-dimensional square and  triangular lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  MEISSNER KERNEL FOR A GENERAL  TIGHT-BINDING LATTICE  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  Definition  As we explained in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' I, it is necessary to calculate  the Meissner kernel to determine whether the mean-field  solution for η-pairing is electromagnetically stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' In the  tight-binding model, the electromagnetic field appears as  Peierls phase:  Hkin = −t  �  ⟨i,j⟩σ  eiAijc†  iσcjσ + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='.  (7)  The Meissner effect is examined by the weak external or-  bital magnetic field applied perpendicular to the plane,  while the η-pairing is stabilized only under a strong Zee-  man field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' In order to make these compatible, we apply  the Zeeman field parallel to the plane h = (h, 0, 0), which  does not create the orbital motion of the tight-binding  electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  Thus, the weak magnetic field that triggers  the Meissner effect is applied perpendicular to the plane  in addition to the in-plane magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  While the  out-of-plane Zeeman effect is also induced by the weak  additional field, it is neglected since the dominant Zee-  man field already exists by the strong in-plane magnetic  field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  Let us formulate the Meissner response kernel on a  general tight-binding model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' We apply the formulation in  Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' [46–48] to the present case with sublattice degrees  of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' The current density operator between two  sites is defined as  jij = ∂Hkin  ∂Aij  ˆδij  = −it  �  σ  �  c†  iσcjσeiAij − c†  jσciσe−iAij�  ˆδij,  (8)  where δij = Ri − Rj is the inter-site lattice vector be-  tween i-th and j-th sites, and hat (ˆ) symbol means a unit  vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' In the linear response theory, the current oper-  ator which appears as a response to the static magnetic  field in equilibrium is written as  jij ≃ −it  �  σ  (c†  iσcjσ − c†  jσciσ)ˆδij  + t  �  σ  (c†  iσcjσ + c†  jσciσ)ˆδijAij  ≡ jpara  ij  + jdia  ij .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  (9)  The first term is called the paramagnetic term and the  second term is diamagnetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  The Fourier-transformed  paramagnetic and diamagnetic current density operators  are written as jpara(q) and jdia(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' The linear response  kernel is then defined by ⟨jν(q)⟩ = �  µ Kνµ(q)Aµ(q),  where ν, µ = x, y is the direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' We evaluate the ker-  nel Kνµ(q → 0) ≡ Kνµ when investigating the stability  of superconductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' This is called the Meissner kernel,  which is proportional to the superfluid density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  The Meissner kernel is separated into paramagnetic  and diamagnetic terms as Kνµ = (Kpara)νµ + (Kdia)νµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  The paramagnetic kernel is given by  (Kpara)νµ = 1  N  � 1/T  0  dτ⟨jpara  ν  (q = 0, τ)jpara  µ  (q = 0)⟩,  (10)  where N = �  i 1 is the number of sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' The Heisenberg  representation with the imaginary time τ is defined as  A(τ) = eHτAe−Hτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' The form of the diamagnetic kernel  is obvious from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  We note that if the sign of the Meissner kernel K is  negative, the superconducting state is electromagneti-  cally stable and is also called a diamagnetic Meissner  state, which expels magnetic flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' On the other hand, if  the sign is positive, the superconducting state is called  the paramagnetic Meissner state, which attracts mag-  netic flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' For a stable thermodynamic superconducting  state, the negative value of K is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  Method of evaluation  The actual evaluation of the kernels is performed based  on the wave-vector representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  Here, the physical  quantities are described by the operator cα  kσ where α dis-  tinguishes the sublattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Note that the Brillouin zone is  4  folded by �  α 1 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' The diamagnetic kernel is rewrit-  ten as  (Kdia)νµ = 1  N  �  α,β  �  kσ  �  m−1  kαβ  �  νµ ⟨cα†  kσcβ  kσ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  (11)  The inverse mass tensor m−1  kαβ, which reflects the char-  acteristics of the lattice shape, are given by  �  m−1  kαβ  �  νµ ≡ t  �  ⟨iα,jβ⟩  �  ˆδiαjβ  �  ν  �  ˆδiαjβ  �  µ e−ik·Riαjβ ,  (12)  where iα is the i-th unit cell with sublattice α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  The  symbol ⟨iα, jβ⟩ represents a pair of the nearest-neighbor  sites and Riαjβ is the vector between the unit lattice with  the i-th sublattice α and the unit lattice with the j-th  sublattice β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  The paramagnetic term has the form of a current-  current correlation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' We can calculate this term  by using the Green’s function matrix  ˇGk(τ) ≡ −⟨Tτψk(τ)ψ†  k⟩  (13)  where ψk = (cα  k↑, cα†  −k↓, · · · )T is the Nambu-spinor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Tτ is  time-ordering operator regrading τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Each component of  the Green’s function matrix is given by the diagonal and  off-diagonal Green’s functions:  Gαβ  σσ′(k, τ) ≡ −⟨Tτcα  kσ(τ)cβ†  kσ′⟩,  (14)  ¯Gαβ  σσ′(k, τ) ≡ −⟨Tτcα†  kσ(τ)cβ  k′σ′⟩,  (15)  F αβ  σσ′(k, τ) ≡ −⟨Tτcα  kσ(τ)cβ  −kσ′⟩,  (16)  F αβ†  σσ′ (k, τ) ≡ −⟨Tτcα†  −kσ(τ)cβ†  kσ′⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  (17)  The anomalous part of Green’s function [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' (16)] is also  called the pair amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' The paramagnetic kernel in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' (10) can be divided into the normal (G) and anoma-  lous (F) Green’s function contributions as  (Kpara)νµ = − 1  N  � � 1/T  0  dτ (vkαβ)ν · (vkα′β′)µ ×  �  ¯Gαβ′  σσ′(k, τ)Gα′β  σσ′(k, τ) + ¯Gαβ′  σσ′(−k, τ)Gα′β  σσ′(−k, τ)  �  − 1  N  � � 1/T  0  dτ (vkαβ)ν · (v−kα′β′)µ ×  �  F βα†  σ′σ (k, −τ)F α′β′  σ,σ′ (k, τ) + F βα†  σ′σ (−k, −τ)F α′β′  σ,σ′ (−k, τ)  �  ≡ KG  para + KF  para.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  (18)  The summation � is performed over the indices which appears only in the right-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' The velocity vector vkαβ  is defined by  (vkαβ)ν ≡ t  �  ⟨iα,jβ⟩  �  ˆδiαjβ  �  ν e−ik·Riαjβ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  (19)  In order to perform the integral with respect to τ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' (18), we define the Fourier-transformed Green’s function as  gk(iωn) ≡  � 1/T  0  dτgk(τ)eiωnτ,  (20)  where gk represents one of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' (14)-(17) and ωn = (2n + 1)πT is fermionic Mastubara frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Moreover, the  Fourier-transformed Green’s function matrix is given by using the matrix representation of mean-field Hamiltonian  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' (3) as  ˇGk(iωn) =  �  iωnˇ1 − ˇHMF  k  �−1 = ˇUk  �  iωnˇ1 − ˇΛk  �−1 ˇU †  k,  (21)  where ˇΛk and ˇUk are, respectively, a diagonal eigenvalue matrix and a unitary matrix satisfying ˇU † ˇHMF  k  ˇU = ˇΛk =  diag(λk1, λk2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' (21), Kpara can be calculated as  (Kpara)νµ = − 1  N  � �  (vkαβ)ν · (vkα′β′)µ Uβ′σ′,ασ  kp  Uα′σ,βσ′  kp′  + (vkαβ)ν · (v−kα′β′)µ Uβσ′,ασ  kp  Uα′σ,β′σ′  kp′  � f (λkp) − f (λkp′)  λkp − λkp′  + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  (22)  where f(λkp) =  1  eλkp/T +1 is the Fermi-Dirac distribution function and we have defined the coefficient Uασ,βσ′  kp  ≡  � ˇUk  �  ασ,p  �  ˇU †  k  �  p,βσ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  The anomalous part of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' (18) KF  para is further de-  composed into the contributions KEFP and KOFP from  5  the even-frequency pair (EFP) and odd-frequency pair  (OFP) amplitudes defined by  F EFP(k, iωn) ≡ F(k, iωn) + F(k, −iωn)  2  ,  (23)  F OFP(k, iωn) ≡ F(k, iωn) − F(k, −iωn)  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  (24)  Then, we obtain KEFP and KOFP by using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' (23) and  (24) as  KEFP,OFP  νµ  = − 1  2N  �  k  �  αβα′β′  (vkαβ)ν · (v−kα′β′)µ  ×  �  σσ′  �  pp′  Uβσ′,ασ  kp  Uα′σ,βσ′  kp′  ×  �f (λkp) − f (λkp′)  λkp − λkp′  ∓ f (λkp) − f (−λkp′)  λkp + λkp′  �  + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=',  (25)  where the minus (−) sign in the square bracket is taken  for EFP contribution and the plus (+) for OFP pairing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  These quantities are numerically calculated as shown in  the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Note that the cross term of the EFP  and OFP terms of Green’s functions vanishes after the  summation with respect to the Matsubara frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  Paramagnetic Meissner response of a simple  η-pairing state  Before we show the results of the AH model, let us show  that a simple η-pairing state leads to the paramagnetic  response which would not arise from thermodynamically  stable states [24, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' We consider the simple bipartite  lattice with staggered ordering vector Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' The anomalous  contribution to the Meissner kernel may be written as [49]  KF  para,xx = −T  �  nkk′σσ′  vx  kvx  k′F ∗  σ′σ(k′, k, iωn)Fσσ′(k, k′, iωn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  (26)  This contribution must be negative (diamagnetic re-  sponse) in order to dominate over the paramagnetic con-  tribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' For a purely η-pairing state, we assume the  relation Fσσ′(k, k′) = Fσσ′(k)δk′,−k−Q, and obtain  KF  para,xx = −T  �  nkσσ′  (vx  k)2F ∗  σ′σ(k, iωn)Fσσ′(k, iωn), (27)  where we have used vx  −k−Q = vx  k valid for square lat-  tice, which is in contrast to the relation vx  −k = −vx  k  for the uniform pairing with additional minus sign [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  We separate the spin-singlet and triplet parts as Fσσ′ =  Fsiτ y  σσ′ + Ft · (τiτ y)σσ′, and then obtain  KF  para,xx = 2T  �  nk  (vx  k)2�  |Fs(k, iωn)|2 − |Ft(k, iωn)|2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  (28)  If we consider the simple η-pairing with only spin-singlet  part (Ft = 0), it leads to the paramagnetic response  (positive).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  Thus, a simple s-wave spin-singlet η-pairing is unlikely  realized as a stable state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' On the other hand, in the AH  model with magnetic field, the spin-triplet pair contribu-  tion is substantially generated by the Zeeman field, which  plays an important role for the diamagnetic response as  shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  NUMERICAL RESULT FOR AH MODEL  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  Square lattice  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  Prerequisites  Let us begin with the analysis of the AH model on  the square lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' We consider the two-sublattice struc-  ture to describe the staggered ordered phase such as a  η-pairing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' While the superconducting states in the at-  tractive model are interpreted in terms of the magnetic  phases of the repulsive model by the particle-hole trans-  formation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' (2), the response functions such as the  Meissner kernel are specific to the attractive model and  have not been explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  In the following, we choose the band width W = 1  as the unit of energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  We fix the value of the attrac-  tive interaction U = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='375.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' The electron concentration  is fixed as nc = 1, and the temperature is taken to be  T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='0 × 10−3 unless otherwise specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' We will in-  vestigate the change of the Meissner kernel for η-pairing  as a function of magnetic field strength h = |h|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' In this  paper, the mean-field solutions are calculated using the  60 × 60 mesh in k-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' The result of the Meissner ker-  nel for η-pairings is calculated with the mesh 300 × 300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  We also checked that the behaviors remain qualitatively  unchanged when these numbers are increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' The self-  consistent equations in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' (4)-(6) are computed by  using an iterative method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  In the following subsec-  tion IV A 2, we restrict ourselves to the analysis of two-  sublattice mean-field solutions, and in IV A 3, we exam-  ine the solutions when the two-sublattice constraint is  relaxed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  Two-sublattice solution  Before investigating the electromagnetic stability, we  clarify the regime where the η-pairing becomes the  ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' In this paper, we assume that the inter-  nal energy in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' (1) is approximately equal to the free  energy in the low temperature region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' The upper panel  of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' 2 shows the internal energy of several ordered  states measured from the normal-state energy as a func-  tion of the Zeeman field h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Here, the η-pairing solution  is obtained by solving the self-consistent equation with  imposing the constraint of the staggered phase of the pair  6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='5  h  −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='0  −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='5  −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='0  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='5  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='0  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='0  Ei − Enormal  BCS  CDW  Normal  η-pairing  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='08  D0  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  (Upper panel) Magnetic-field dependence of the  internal energy for each state measured from the normal state  in the square lattice model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' (Lower plane) Density of state  (DOS) at zero energy D0 for each state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='5  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='0  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='5  ω  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='2  D(ω)  h = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='25  h = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='375  h = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='5  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Density of states for the η-pairing around magnetic  filed h = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='375 in the square lattice model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  Here D(ω) is  normalized as  �  dωD(ω) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' A constraint is also used for the calculation  of the other types of order parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Our calculations  have not found any ordered states other than the types  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' 2 even when a random initial condition is  employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  We determine the thermodynamically stable ground  state by comparing the internal energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' In low magnetic  fields, BCS and CDW are degenerated ground states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' On  the other hand, we find that the η-pairing becomes the  ground state in the magnetic field located in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='063 < h <  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='875.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' The η-pairing solution itself is found in the wider  regime although the internal energy is not the lowest one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  It has been known that the attractive Hubbard model  under a magnetic field also shows the FFLO state [50],  but this possibility cannot be considered when we take  the two-sublattice condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' This point will be revisited  in the next subsection where the two-sublattice condition  is relaxed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  The lower panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' 2 shows the density of  state (DOS) at the Fermi level for each state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' The re-  sult indicates that there is no energy gap in the η-pairing  state, in contrast to the conventional BCS pairing state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  There exists the regime where the DOS at the Fermi  level for η-pairing is larger than that of normal metal  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='25 ≲ h ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' This is due to the van-Hove singular-  ity of the square lattice model as shown in FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' We  also perform the calculation for the cubic lattice where  the van-Hove singularity is absent at zero energy and con-  firm in this case that the DOS is smaller than the normal  state (see Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  The stability of the η-pairing depends upon the mag-  nitude of the magnetic field as seen in the Meissner re-  sponse kernel K (= Kxx = Kyy) (green symbol) in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  The contributions from the paramagnetic  (Kpara, positive) and diamagnetic (Kdia, negative) parts  are also separately plotted in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' In the regime  with h ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='125 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='75 ≤ h, the η-pairing is electromag-  netically unstable, while it is stable in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='125 < h < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' 4, the yellow shaded rectangle indicates the regime  where the η-pairing becomes the ground state as seen  from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' We find a narrow region where η-pairing is  regarded as the ground state but is not an electromagnet-  ically stable state around h = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' From these results,  we see that the η-pairing is not necessarily electromag-  netically stable even if it becomes the ground state in  a two-sublattice calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' As we shall see later, the  simple η-pairing in this narrow regime does not necessar-  ily exist if we relax the two-sublattice condition of the  mean-field solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  We also show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' 4(a) the contributions from the  even- and odd-frequency pairs defined in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' (23) and  (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' The negative sign of the kernel, which means the re-  sponse is diamagnetic, is partly due to the odd-frequency  component of the pair amplitude, (KOFP < 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  This  is in contrast to the FFLO state whose Meissner ker-  nel is also negative due to the even-frequency component  [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Hence, it implies that the mechanism of the dia-  magnetism is different between the FFLO and η-pairing  states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  In  addition  to  the  Meissner  kernel,  we  calcu-  late the local pair amplitudes which are shown in  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  Here the left- and right-panels represent  the spin-triplet and spin-singlet components of the lo-  cal pair amplitude, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' The triplet component  �  σσ′(τ µiτ y)σσ′Fσσ′(iωn) with µ = x has a finite imagi-  nary part and zero real part, which represents the odd-  frequency pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' The other µ = y, z components are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  On the other hand, the singlet component has a finite real  part and zero imaginary part and is the even-frequency  pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' We can see that the maximum value of the spin-  triplet component of the pair amplitude is largest at the  magnetic field h = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='375, where the magnitude of KOFP  is largest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' It is also notable that the magnitude of the  odd-frequency pair amplitude correlates with the magni-  tude of DOS at zero energy as seen by comparing Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' 3  and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  We comment on the singular behavior of KOFP at the  magnetic field h = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='375, although it does not affect the  total Meissner kernel K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' This anomalous feature is re-  lated to the van Hove singularity of the DOS at zero  energy as shown in FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' 3, which shows a sharp peak at  the Fermi level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
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+page_content='4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
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+page_content='6  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='2  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='8  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='4  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='0  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='6  Im[F # #(i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='n) ° F " "(i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='n)]/  p  2  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  (a) Magnetic field dependence of the Meissner ker-  nel K(= Kxx = Kyy) for the η-pairing on the square lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  The yellow shaded rectangle indicates the range where the  η-pairing becomes the ground state in two-sublattice calcula-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' The number of the wavenumber k is taken as 300×300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  (b) Matsubara frequency dependence of the local pair ampli-  tude at several magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' The left panel represents the  imaginary part of [F↓↓(iωn) − F↑↑(iωn)] /  √  2, and the right  panel represents the real part of [F↑↓(iωn) − F↓↑(iωn)] /  √  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  The values of the pair amplitudes are shifted by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='6 at each  magnetic field for visual clarity, and the gray-dotted lines are  the zero axes for each magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  Beyond two-sublattice  In order to clarify the stable ordered state where the  Meissner kernel is positive (paramagnetic), we investi-  gate mean-field solutions on finite-sized lattice where the  two-sublattice condition is not imposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  We have nu-  merically solved the Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' (4)-(6) self-consistently by us-  ing the mean-field solutions of the η-pairing obtained for  two-sublattice as an initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  Figure 5 shows the spatial distribution of the phase of  the gap function when the number of sites is 8 × 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' At  h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='5 in (a), where the η-pairing is not a ground state,  the uniform BCS pairing state is realized as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  With increasing the magnetic field, the longer-periodicity  structures are found as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' 5(b), (c) and (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  At h = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='375 in (c), where the η-pairing solution has the  lowest energy and the electromagnetic response is well  diamagnetic, we obtain the staggered alignment of the  (a) h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='5  (d) h = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='875  (c) h = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='375  (b) h = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='125  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  Spatial distribution of the phase of the supercon-  ducting order parameter at several magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' The cal-  culation is performed on the finite-sized lattice (8 × 8) with  open boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Small black dots are lattice points  and red arrows indicate the phase of the pair potential for  each lattice point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' When the parameters are close to the edges of  the yellow-highlighted region in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' 4, the complex struc-  tures are formed as shown in (b) and (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' The behavior  in (b) is interpreted as due to the competing effect where  the simple uniform and staggered phases are energetically  close to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  We also investigate the case with the other choice of pa-  rameters: U = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='25 and h = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' In this case, we find  the staggered flux state where the phase of pair poten-  tial is characterized by 90◦-N´eel ordering as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' 6(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  This ordered state cannot be described in the mean-field  theory with two sublattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  Owing to a non-colinear  90◦-N´eel ordering vector, the spontaneous clockwise or  counterclockwise loop currents arise by the inter-atomic  Josephson effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' The current density is calculated by  jij = −it  �  σ  ⟨c†  iσcjσ − c†  jσciσ⟩  (29)  which is identical to the expression of the paramagnetic  current in the linear response theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' We can also evalu-  ate the flux for each plaquette, which is define by  Φ =  �  (i,j)∈plaquette  jij  (30)  This expression is similar to the flux  �  C  j ·ds =  �  S  b·dS  (j = ∇ × b) defined in a continuum system, where b is  a flux density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' The flux is aligned in a staggered manner  8  (a)  (b)  Current  Magnetic flux  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' (a) Spatial distribution of the phase of the supercon-  ducting order parameter for the η-pairing with 90◦-N´eel state  on the finite-sized lattice under open boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  (b) Spatial distributions of the spontaneous loop current and  the flux defined on each plaquette.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' The color of vectors dis-  plays the magnitude of current, and the color of dots in each  plaquette indicates the value of the magnetic flux defined in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  on a dual lattice as indicated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' 6(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' The staggered  flux originating from the normal part has been studied  before [20–23], while the staggered flux shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' 6(b)  has a different origin: it arises from the superconductiv-  ity associated with the off-diagonal part in the Nambu  representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  We also comment on a feedback effect to the electro-  magnetic field from the supercurrent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  Since the char-  acteristic length scale for the magnetic field in layered  superconductor becomes long [52], each magnetic flux on  the plaquette is smeared out with this length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Hence we  expect that the net magnetic field is not created from the  staggered superconducting flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  Triangular lattice  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  Mean-field solution  Now we search for the η-pairing reflecting the charac-  teristics of a geometrically frustrated triangular lattice  at the half-filling (nc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' We choose the parameters  U = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='83 and T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='0 × 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' We consider the cases of  two- and three-sublattice structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' For a usual antifer-  romagnet, the typical ordered state in the two-sublattice  case has a stripe pattern, while in the three-sublattice  case we expect a 120◦-N´eel state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Below we study the  superconducting η-pairing phases within the mean-field  theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  We have found the four types of superconducting states  reflecting the characteristics of the triangular lattice,  which are referred to as the η-pairing I, II, III, and IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  The schematic pictures for these four states are shown  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' 7(a), where the arrow indicates the phase of the  superconducting order parameter at each site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' We make  a few general remarks: the three-sublattice structure is  assumed for I, II, III, while the two sublattice is employed  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
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+page_content='0  ni, mx  i  x  y  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  (a) Schematics for the four η-pairings in the tri-  angular lattice model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' The arrows indicate the phase of the  pair potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' The size of the circles shows the amount of  the electron density for each sublattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' (b) Magnetic field  dependence of the internal energies measured from the nor-  mal state (upper panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' The lower panel shows the inter-  nal energy measured from the η-pairing I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' (c) Magnetic field  dependence of the number of electrons and magnetization on  each sublattice for the η-pairing II (upper panel) and IV (lower  panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  for IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' The type-I has a non-colinear structure, and in the  other η-pairings the vectors are aligned in a colinear man-  ner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' We also note that CDW accompanies the η-pairings  II and III, where the number of local filling is indicated  by the size of the filled circle symbols in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' 7(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  Figure 7(b) shows the internal energy of the ordered  states measured from the normal state (Upper panel) and  from the η-pairing I (Lower panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' From the lower panel  of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' 7(b), we can identify the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' With in-  creasing the magnetic field, the ground state changes as  BCS → η-pairing II→ η-pairing I → η-pairing IV→ η-  pairing I → normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Figure 7(c) shows the particle den-  9  (a)  (b)  Iloop  h  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
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+page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='6  h  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='1  Iloop  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' (a) Schematic picture of the staggered flux state on  the triangular lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' The straight arrows display the phase  of the pair potential at each site, and the circle arrows indicate  the staggered loop current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' (b) Magnetic field dependence of  the magnitude of loop current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' The yellow shaded rectangle  indicates the range where the η-pairing I becomes the ground  state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  sity and x-direction magnetization mx  i of each sublattice  for η-pairing II (Upper panel) and η-pairing IV (Lower  panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' The values of my  i and mz  i are zero because the  Zeeman field h is applied along the x-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Below,  we explain the characteristic features for each η-pairing  state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  η-pairing-I state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='— The η-pairing I has 120◦ N´eel or-  dering vector (Green pentagon in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' 7(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' The spon-  taneous supercurrent appears in this non-colinear state  as schematically shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' 8(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' This superconduct-  ing state forms a staggered flux state, where the flux is  aligned on a honeycomb dual lattice, which is similar to  the η-pairing with 90◦-N´eel ordering vector on the square  lattice shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' 6(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Figure 8(b) displays the val-  ues of spontaneous loop current density as a function of  the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  η-pairing-II state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='— The η-pairing II has the struc-  ture with up-up-down colinear phases plus CDW (Red  hexagon in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' 7(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' There is the relation nA = nB <  nC for the electron filling at each sublattice shown in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' 7(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  We note that this site-dependent feature is  characteristic for the II (and IV) state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' The phases of the  pair potential at A and B sublattices are “ferromagnetic”,  while the phase at C sublattice is “antiferromagnetic”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  The resulting ordered state is regarded as the emergence  of the honeycomb lattice formed by equivalent A and B  sublattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  η-pairing-III state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='— This is the η-pairing with a stag-  gered ordering vector and CDW (Magenta square in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' 7(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  The order parameter ∆ at C sublattice is  zero, but the others (A,B) are finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' The electron-rich  sublattices A and B form a simple bipartite η-pairing  state on an emergent honeycomb lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Since this state  does not become a ground state anywhere for the present  choice of U = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='83, we do not further investigate this  state in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  η-pairing-IV state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='— This is the η-pairing with a sim-  ple stripe alignment (Cyan rhombus in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' 7(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' This  η-pairing is accompanied by CDW around h = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
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+page_content='FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Magnetic field dependence of the Meissner kernels  Kxx and Kyy for the η-pairings I, II, IV on the triangular lat-  tice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' The yellow shaded rectangle indicates the regime where  each η-pairing becomes the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' The symbols are  the same as those in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' For the η-pairing IV, Kxx and  Kyy are separately plotted in (c1) and (c2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' 7(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' As shown below, this stripe phase show an  anisotropic behavior in linear response coefficients, while  the other η-pairing states are isotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  Meissner response  Now we discuss the Meissner response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Figure 9(a,b,c)  shows the Meissner kernels Kxx, Kyy for the η-pairing I,  II and IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' The yellow-highlighted parts indicate the re-  gion where each η-pairing becomes the ground state as  identified from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' 7(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  The result for the η-pairing  III is not shown because it does not become a ground  state at U = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' We confirm that the Meissner re-  sponse is basically diamagnetic if the η-pairing becomes  the ground state as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' 9(a,b,c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Thus the en-  ergetic stability and diamagnetic response are reasonably  correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' In the following, we discuss the properties of  the Meissner kernel for each state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  The Meissner kernels for both η-pairing I and η-pairing  II shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' 9(a) and (b) satisfy the relation Kxx =  Kyy, which means an isotropic linear response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' For the η-  pairing I, the Meissner kernel becomes positive in the re-  gions h < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='2, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='95 < h < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='12, while the kernel becomes  negative in the ground state region (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' 9(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Although  the local current density is finite for the η-pairing I state,  it does not affect the expression of the Meissner kernel in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' (10) since the total current j(q = 0) is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  Next we disucuss the η-pairing IV state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' The Meiss-  10  ner kernel jumps at h = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='8 due to the emergence of  the CDW order parameter as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' 9(c1,c2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' It  is notable that the η-pairing IV with the stripe pattern  shows a difference between x and y directions as shown  in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' 9(c1,c2), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' This characteristic behav-  ior can be intuitively understood from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' 7(a), where  the current along the x-axis flows with experiencing a  staggered pair potential, whereas the current in the y-  direction feels an uniform pair potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' In the Meissner  response, Kxx shows a characteristic behavior of the η-  pairing, while Kyy is qualitatively the same as the kernel  of BCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Thus, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' 9(c1), the diamagnetic  response in the x-axis direction is related to to the odd-  frequency pair, whereas the diamagnetic response in the  y-axis direction, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' 9(c2), is related to even-  frequency pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  SUMMARY AND OUTLOOK  We have studied the square and the triangular lattice  of the attractive Hubbard model by using the mean-field  theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  Several types of η-pairing have been found in  the triangular lattice where a simple bipartite pattern  is not allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  Using the formulation of the Meissner  kernel for a general tight-binding lattice, we have inves-  tigated the electromagnetic stability of η-pairings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' We  have confirmed that the electromagnetic stability of the  η-pairing correlates with the internal energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' In a narrow  parameter range, we also find that the η-pairing state can  show an unphysical paramagnetic response if we assume  the two or three sublattice structure in the mean-field  calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  In this case, another solution with longer  periodicity needs to be sought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  When the current path experiences the staggered  phase of the superconducting order parameter, the odd-  frequency component of the pair amplitude contributes  to the diamagnetic response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' This is in contrast to the  conventional BCS case in which the even-frequency com-  ponent of the pair amplitude contributes to the diamag-  netism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  We have further clarified that one of the η-  pairing states on the triangular lattice has a stripe pat-  tern and shows an anisotropic Meissner response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' In this  case, the odd-frequency pair contributes diamagnetically  or paramagnetically depending on the direction of cur-  rent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  We comment on some issues which are not explored in  this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' We expect that the η-pairing without a simple  staggered phase will appear on pyrochlore, kagome and  quasicrystalline lattice, whose phase-alignment could be  qualitatively different from the triangular lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' In ad-  dition, there is another model that shows η-pairing in  equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' A two-channel Kondo lattice (TCKL) is an  example of a model in which η-pairing appears even in  the absence of a Zeeman field [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Our preliminary cal-  culation for the TCKL shows a number of ordered states  which have similar energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  These additional studies  provide more insight into the exotic superconductivity  (a)  (b)  (c)  °1  0  1  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
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+page_content='n) ° F # "(i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='n)]/  p  2  h=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
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+page_content='n)]/  p  2  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' (a) The difference between the DOSs of the η-pairing  and normal states in the cubic lattice model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' The values of  the DOS are shifted by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='2 for each magnetic field, and the  gray dotted lines are the zero axes for each magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  We also show the Matsubara frequency dependence of (b) the  imaginary part of [F↓↓(iωn) − F↑↑(iωn)] /  √  2 and (c) the real  part of [F↑↓(iωn) − F↓↑(iωn)] /  √  2 for each magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' The  values of the pair amplitudes are shifted by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  characteristic for the η-pairing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  ACKNOWLEDGEMENT  This work was supported by KAKENHI Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  18H01176, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' 19H01842, and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' 21K03459.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  Appendix A: Self-consistent equations in mean-field  theory  We derive self-consistent equations for the general in-  teracting Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Let us begin with the Hamilto-  nian  H =  �  12  ε12c†  1c2 +  �  1234  U1234c†  1c†  2c4c3  (A1)  where site-spin indices are written as 1 = (i1, σ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' The  mean-field Hamiltonian is introduced as  HMF =  �  12  �  E12c†  1c2 + ∆12c†  1c†  2 + ∆∗  12c2c1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  (A2)  We assume ⟨H ⟩ = ⟨HMF⟩ where the statistical average  is taken with HMF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Then the self-consistent equation is  obtained as  E12 = ∂⟨H ⟩  ∂⟨c†  1c2⟩  = ε12 +  �  34  (U1324 + U3142 − U1342 − U3124)⟨c†  3c4⟩  (A3)  ∆12 = ∂⟨H ⟩  ∂⟨c†  1c†  2⟩  =  �  34  U1234⟨c4c3⟩  (A4)  11  where the Wick’s theorem is used for the derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Al-  though the variational principle for the free energy also  gives the same equation, the above formalism gives a sim-  ple procedure to derive the self-consistent equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  Appendix B: Attractive Hubbard model on Cubic  lattice  We analyze the η-pairing on the cubic lattice, whose  DOS does not have a van Hove singularity near zero en-  ergy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Here we choose the parameter U = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='375 and  the electron concentration is half-filled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' As a result, the  DOS for the η-pairing around zero energy for each mag-  netic filed on the cubic lattice is smaller than the DOS of  the normal state as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' 10(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' For reference,  we also show in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  10(b) and (c) the pair amplitude  similar to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' 4(b) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' In addition, the  odd-frequency pair amplitude increases when DOS near  zero energy is enhanced as seen from Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' 10(a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  [1] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Fulde and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Ferrell, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' 135, A550 (1964).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  [2] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Larkin and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Ovchinnikov, Zh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Eksp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Teor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Fiz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
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+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Morr, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  Nayak, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
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+page_content=' Morr, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
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+page_content=' Wang, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
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+page_content=' Tamura, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Ogata, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  Jpn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' 85, 124707 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
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+page_content=' Yokoyama, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Toga, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' :  Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
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+page_content=' Yamazaki, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Tsuchiura, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Ogata,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' : Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
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+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Jpn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
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+page_content=' Tanaka and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Golubov, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
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+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Golubov, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
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+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
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+page_content=' Hoshino, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Yada, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Tanaka, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' B 93,  224511 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  [50] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Tsuruta, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Hyodo, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Miyake, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Jpn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  83, 063706 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  [51] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Chakraborty and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Black-Schaffer, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' B  106, 024511 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content='  [52] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Pearl, App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}
+page_content=' 5, 65 (1964).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf'}