# Continuous transition from a Landau quasiparticle to a neutral spinon

Jing-Yu Zhao  Institute for Advanced Study, Tsinghua University, Beijing
100084, China  Shuai A. Chen  Department of Physics, Hong Kong University of
Science and Technology, Clear Water Bay, Hong Kong, China  Rong-Yang Sun
Computational Materials Science Research Team, RIKEN Center for Computational
Science (R-CCS), Kobe, Hyogo 650-0047, Japan  Quantum Computational Science
Research Team, RIKEN Center for Quantum Computing (RQC), Wako, Saitama
351-0198, Japan  Zheng-Yu Weng  Institute for Advanced Study, Tsinghua
University, Beijing 100084, China

###### Abstract

We examine a wavefunction ansatz in which a doped hole can experience a
quantum transition from a charge $+e$ Landau quasiparticle to a neutral spinon
as a function of the underlying spin-spin correlation. As shown variationally,
such a wavefunction accurately captures all the essential features revealed by
exact diagonalization and density matrix renormalization group simulations in
a two-leg $t$-$J$ ladder. Hence its analytic form can provide an explicit
understanding of the mechanism for the unconventional ground state. The
transition in the phase diagram is accompanied by a change of the hole
composite from a tight charge-spin binding to a loosely-bound hole-spin pair.
In the latter, the hole carries a _finite_ spin current but with vanishing
charge current in the degenerate ground states. We show that the charge of the
hole composite here is dynamically diminished due to an internal relative
hole-spin motion, which is fundamentally distinct from a simple charge-spin
separation in a one-dimensional case. We further show that the same novel
effect is also responsible for a strong pairing between two doped holes in
such a non-Landau quasiparticle regime.

###### Contents

  1. I Introduction
  2. II Benchmarking Wavefunction Ansatz with ED and DMRG via VMC calculation
    1. II.1 The two-leg anisotropic $t$-$J$ model
    2. II.2 Single-hole wavefunction ansatz
    3. II.3 Variational ground-state energy
    4. II.4 Phase diagram
  3. III Further characterization: Novel charge renormalization at $\alpha>\alpha_{c}$
    1. III.1 Emergent spin-current around the hole at $\alpha>\alpha_{c}$: Incoherent charge component
    2. III.2 Response to external electromagnetic flux
    3. III.3 Vanishing charge
  4. IV Discussion
    1. IV.1 QCP at $\alpha_{c}$
    2. IV.2 The two-component structure at $\alpha>\alpha_{c}$
    3. IV.3 Disappearance of charge response to external magnetic flux at $\alpha>\alpha_{c}$
    4. IV.4 Binding force between two doped holes
  5. V Conclusion
  6. A Phase-shift operator on a two-leg ladder with PBC
  7. B Longitudinal spin-polaron effect on the varational ansatz
  8. C Variational Monte Carlo procedure

##  I Introduction

How to properly characterize a single-particle excitation is one of the most
essential challenges in the study of the strongly correlated Mott insulators
Mott (1949); Anderson (1987); Imada  _et al._ (1998); Lee  _et al._ (2006). In
particular, a single chargon (hole) may serve as a building block for
constructing a doped Mott insulator. The central issue under debate is whether
such a chargon moving a quantum spin background will still behave like a
conventional Landau-type quasiparticle or be fundamentally renormalized into a
non-Landau quasiparticle via “twisting” the surrounding many-body spins in the
background.

In analogy to the electronic “cloud” associated with a quasiparticle, it was
widely believed earlier on that such a hole in the Mott insulator may still be
the Landau-type after considering the longitudinal spin-polaron effect
Schmitt-Rink  _et al._ (1988); Kane  _et al._ (1989), which involves a
distortion in the amplitude of the local spin magnetization around the hole.
On the other hand, it was conjectured Anderson (1990) that a nontrivial many-
body response from the Mott insulator background may lead to an
“unrenormalizable Fermi-surface phase shift”, which can result in an
“orthogonality catastrophe” to turn the doped hole into a non-Landau
quasiparticle.

Such an “unrenormalizable phase shift” has been explicitly identified with the
phase-string effect in the $t$-$J$ Sheng  _et al._ (1996); Wu  _et al._ (2008)
and Hubbard Zhang and Weng (2014) models as a singular nonintegrable Berry
phase acquired by the doped hole(s) moving in the quantum spin background of
the Mott insulator. Physically, it implies that the hopping of the doped hole
should generate a spin-current backflow. Exact diagonalization (ED) and
density matrix renormalization group (DMRG) simulations have recently
confirmed Zheng  _et al._ (2018) such an unconventional behavior of the doped
hole in the two-dimensional (2D) square lattice. The hidden spin-current
backflow overlooked by previous numerical works has been revealed Zheng  _et
al._ (2018) to accompany and facilitate the hopping of the hole, leading to a
nontrivial quantum number as a direct manifestation of a non-Landau-like
quasiparticle. Making use of the variational Monte Carlo (VMC) method, a
single-hole wavefunction ansatz has been constructed Chen  _et al._ (2019),
which well interpret the numerical results including the nontrivial angular
momenta $L_{z}=\pm 1$ under a $C_{4}$ rotational symmetry for a finite-size 2D
sample up to $8\times 8$ Zheng  _et al._ (2018).

The phase-string effect or the spin-current backflow here leads to a
_transverse_ spin twist to renormalize the doped hole. But previously a Landau
quasiparticle behavior had been still inferred in a semiclassical field-theory
approach Shraiman and Siggia (1988, 1989) even with incorporating a long-range
transverse dipolar spin twist beyond the longitudinal spin-polaron effect. It
is because a singular coupling between the doped hole and the spin currents at
the _short distance_ was omitted Weng (1991). The latter effect has been
carefully and consistently implemented in the wavefunction approach Chen  _et
al._ (2019) to reproduce the correct behavior. Nevertheless, in spite of the
good agreement of the VMC result Chen  _et al._ (2019) and exact numerics
Zheng  _et al._ (2018) at finite (small) sizes, the 2D single-hole problem can
be further complicated by the antiferromagnetic (AF) long-range order which
sets in as the thermodynamic limit is taken, which may further lead to a self-
localization of the hole. Thus, a thorough understanding of the 2D single-hole
problem has to handle both singular effects in short-range and long-range
physics together, which is beyond the finite-size exact numerical methods.

In this regard, a single-hole-doped $t$-$J$ model on a two-leg ladder may be a
more ideal system where the spin-spin correlation length remains finite even
in the thermodynamic limit. The previous DMRG studies Zhu  _et al._ (2013);
Zhu and Weng (2015); Zhu  _et al._ (2015) have established an exotic ground
state for the single hole, which behaves like a non-Landau-quasiparticle with
breaking charge translational symmetry Zhu  _et al._ (2018). Note that the
above non-Landau-quasiparticle picture was previously contested by another
DMRG study White  _et al._ (2015), in which a finite quasiparticle spectral
weight $Z_{k}$ has been explicitly measured. To reconcile both DMRG results,
it was pointed out Zhu  _et al._ (2018) that the Landau’s one-to-one
correspondence principle is actually broken down even though $Z_{k}$ still
remains finite in the non-Landau-quasiparticle regime, where the hole object
is of a two-component structure, composed of a translation-invariant Bloch
wave component and a charge-incoherent component. In other words, the DMRG
results indicate the existence of a new type of non-Landau quasiparticle.
Especially a quantum transition for the doped hole to become a true Landau
quasiparticle has been also found by reducing the spin-spin correlation length
along the ladder direction Zhu and Weng (2015); Zhu  _et al._ (2015, 2018);
White  _et al._ (2015), with the translation symmetry and charge coherence
being eventually restored. Therefore, a wavefunction description of such a
single-hole problem in a two-leg ladder can offer a valuable proof-of-
principle on how a bare doped hole can be specifically “twisted” into a non-
Landau-quasiparticle due to the intrinsic Mott physics.

In this paper, we shall present a simple variational wavefunction ansatz by
incorporating the aforementioned phase-string or transverse spin-current
effect. It can accurately capture all the essential ground-state features of a
single-hole-doped two-leg $t$-$J$ ladder under a periodic boundary condition
(PBC) via the VMC method. In particular, it will describe a doped hole as a
Landau quasiparticle with charge $q=+e$ in the strong anisotropic limit of
$\alpha\ll 1$ ($\alpha$ is to be defined below). In the isotropic regime of
$\alpha\simeq 1$, the doped hole will acquire nonzero total momenta with
finite spin currents, but its charge current will vanish in the large sample
limit. The results clearly indicate that the doped hole is a charge-neutral
“spinon” in the regime of $\alpha>\alpha_{c}$ with $\alpha_{c}$ denoting a
quantum critical point (QCP), as schematically illustrated in Fig. 1(a), which
are in excellent agreement with the DRMG calculation [cf. Fig. 1(b)]. In other
words, we shall demonstrate by an analytic wavefunction that an exotic
transition resembling a metal-insulator transition beyond a Landau paradigm
Zhou and Ng (2013) can happen in a doped Mott insulator.

Figure 1: The phase diagram of the single-hole variational ground state as a
function of the underlying spin-spin correlation controlled by an anisotropic
parameter $\alpha$ in a two-leg $t$-$J$ ladder (see text). (a) A schematic
illustration of a quantum transition point at $\alpha=\alpha_{c}$ for a Landau
quasiparticle of charge $q=+e$ state to become a “twisted” quasiparticle with
diminished charge $q=0$ to be shown in this work; (b) $Q_{0}$ [cf. Eq. (3)] as
a function of $\alpha$, which measures the momentum splitting in the ground
state (the insets) as calculated by both DMRG and VMC methods on a $48\times
2$ ladder under PBC. The insets show the quasiparticle weight $Z_{k}$ at two
typical $\alpha=0.4<\alpha_{c}$ and $\alpha=1.0>\alpha_{c}$, respectively,
determined by VMC with $\alpha_{c}\approx 0.68$ at $t/J=3$.

Such a single-hole wavefunction ansatz is given by

$|\Psi_{\mathrm{G}}\rangle_{\mathrm{1h}}\propto\sum_{i}e^{i\left(k_{0}x_{i}-\hat{\Omega}_{i}\right)}c_{i\downarrow}|\phi_{0}\rangle,$
(1)

where $|\phi_{0}\rangle$ denotes an undoped spin-singlet background of the
two-leg ladder, and the electron annihilation operator $c_{i\downarrow}$
removes an electron of spin $\downarrow$ (without loss of generality) to
create a _bare_ hole at site $i$. The ground state momentum $k_{0}$ is along
the quasi-1D ladder direction, with each rung labeled by $x_{i}$. Note that if
one turns off the phase-shift operator $\hat{\Omega}_{i}$ in Eq. (1), the
wavefunction is simply reduced to a conventional Bloch-wave state:

$|\Psi_{\mathrm{B}}(k_{0})\rangle_{\mathrm{1h}}\propto\sum_{i}e^{ik_{0}x_{i}}c_{i\downarrow}|\phi_{0}\rangle,$
(2)

which is explicitly translational invariant (with the translationally
invariant spin bakground $|\phi_{0}\rangle$) to describe a Landau-like
quasiparticle of momentum $k_{0}$. The phase-shift field $\hat{\Omega}_{i}$ as
a nonlocal spin operator is thus the only unconventional quantity in the
ground state of Eq. (1), which is to incorporate the phase-string effect as
stated above. An excellent agreement of the VMC calculation based on Eq. (1)
and its variant (i.e., a further incorporation of the longitudinal spin-
polaron effect) with the DMRG and ED results will be demonstrated in this
work.

As a matter of fact, a QCP is found in the ground state as a function of the
anisotropic parameter $\alpha$, by which the AF correlation in
$|\phi_{0}\rangle$ can be continuously tuned. At strong rung limit ($\alpha\ll
1$), $k_{0}$ is found at $\pi\mod 2\pi$ (taking the lattice constant as the
unit), which corresponds to a non-degenerate state essentially the same as the
Bloch-wave state in Eq. (2). A double-degeneracy in the ground state arises on
the other side of the QCP at larger $\alpha$’s, which is characterized by
nontrivial momentum splitting $k_{0}^{\pm}=\pi\pm\kappa\mod 2\pi$. The split
as characterized by the wavevector:

$Q_{0}\equiv 2\kappa$ (3)

is shown in Fig. 1(b) as a function of $\alpha$ calculated by VMC and DMRG,
respectively, which both indicate a QCP at $\alpha_{c}\simeq 0.68$ where
$Q_{0}$ vanishes as determined by DMRG Zhu and Weng (2015); Zhu  _et al._
(2015, 2018); White  _et al._ (2015).

In the insets of Fig. 1(b), the sharp peak(s) of the quasiparticle spectral
weight $Z_{k}$ specifies $k_{0}$ in the ground state. Here $Z_{k}$ is defined
as the absolute-value squared of the overlap between the wavefunctions in Eqs.
(1) and (2) (after normalization) as follows

$Z_{k}\equiv\frac{1}{2}\left|\langle\Psi_{\mathrm{B}}(k)|\Psi_{\mathrm{G}}\rangle_{\mathrm{1h}}\right|^{2}=\left|\langle\phi_{0}|c^{\dagger}_{k\downarrow}|\Psi_{\mathrm{G}}\rangle_{\mathrm{1h}}\right|^{2}~{},$
(4)

where
$c_{k\downarrow}^{\dagger}=1/N\sum_{i}c_{i\downarrow}^{\dagger}e^{ikx_{i}}$ is
the $k$ space electron with $k_{y}=0$. On both sides of the QCP in Fig. 1(b),
$Z_{k}$ is always finite, indeed consistent with the DMRG result first shown
in Ref. White  _et al._ , 2015. In particular, the non-degenerate ground state
at $\alpha<\alpha_{c}$ is a Landau quasiparticle, which can be smoothly
connected to the Bloch-wave state in Eq. (2). However, we shall show that the
charge of the doped hole will actually _disappear_ at $\alpha>\alpha_{c}$,
whereas its spin-1/2 remains unrenormalized. In other words, the QCP
represents a fundamental transition of the doped hole from a Landau-like
quasiparticle to a pure charge-neutral spinon, which is schematically
illustrated in Fig. 1(a). Such a non-Landau-like quasiparticle with a finite
$Z_{k_{0}^{\pm}}$ indicates a two-component structure in the wavefunction
where the Landau’s one-to-one correspondence hypothesis fails at
$\alpha>\alpha_{c}$. Indeed, besides a finite amplitude of the Bloch-wave
component (with $Z_{k^{\pm}_{0}}\neq 0$), another many-body component is also
explicitly identified in the ground state, in which a spin current pattern
associated with the doped charge is always present. The latter is found to be
charge _incoherent_ as the total momentum $k_{0}^{\pm}$ is now continuously
shared between the hole and spin degrees of freedom.

Finally, it is briefly discussed that the pairing between two doped holes also
becomes substantially enhanced at $\alpha>\alpha_{c}$ as previously revealed
by the DMRG calculation Zhu and Weng (2015); Zhu  _et al._ (2014). An explicit
pairing-mediated spin current pattern is shown based on the present
wavefunction ansatz, which illustrates how a strong binding can be indeed
realized by eliminating the phase-string effect through the pairing of two
holes.

The rest of the paper is organized as follows. In Sec. II, we introduce the
two-leg anisotropic $t$-$J$ model and construct a single-hole-doped
wavefunction ansatz under the PBC. A systematical comparison between the DMRG
and VMC methods are shown on both sides of the QCP at $\alpha_{c}$. In Sec.
III, the properties of the wavefunction at $\alpha>\alpha_{c}$ are further
analyzed to show that, different from a Landau quasiparticle, here the
“twisted” quasiparticle carries a finite spin current in the degenerate ground
state but vanishing charge current in the thermodynamic limit. In Sec. IV, a
further discussion of the underlying physics of the non-Landau quasiparticle
behavior is made. In particular, how the incoherent charge component is
crucial to the pairing between doped holes is pointed out. Finally, the
conclusion and perspectives are given in Sec. V.

Figure 2: Illustration of a two-leg ladder with anisotropic coupling
parameters $t_{ij}$ and $J_{ij}$ of the $t$-$J$ model under the PBC (see
text). Here the total number of the ladder sites is $N=N_{x}\times 2$ with
$N_{x}$ denoting the total number along each of the two legs, which are
embedded in 2D with a spatial ring configuration. Note that the two legs of
the ladder as rings are with different radii: $r_{\mathrm{in}}=2-\lambda$ and
$r_{\mathrm{out}}=2+\lambda$, respectively, in which $\lambda$ is a
variational parameter to specify the phase-string operator $\hat{\Omega}_{i}$
of Eq. (9). Here the angle field $\theta_{i}(l)$, satisfying Eq. (10), is
defined accordingly in the 2D configuration.

##  II Benchmarking Wavefunction Ansatz with ED and DMRG via VMC calculation

###  II.1 The two-leg anisotropic $t$-$J$ model

In this work, we shall study the single-hole-doped ground state of the $t$-$J$
model on an anisotropic two-leg ladder with system size $N=N_{x}\times 2$.
Here the $t$-$J$ Hamiltonian is given by
$H=\mathcal{P}_{s}(H_{t}+H_{J})\mathcal{P}_{s}$, where

$\displaystyle H_{t}$ $\displaystyle=-\sum_{\langle
ij\rangle,\sigma}t_{ij}(c_{i\sigma}^{\dagger}c_{j\sigma}+h.c.)~{},$ (5)
$\displaystyle H_{J}$ $\displaystyle=\sum_{\langle
ij\rangle}J_{ij}\left(\mathbf{S}_{i}\cdot\mathbf{S}_{j}-\frac{1}{4}n_{i}n_{j}\right)~{},$
(6)

with $\langle ij\rangle$ denoting a nearest-neighbor (NN) bond. Here
$\mathbf{S}_{i}$ and $n_{i}$ are spin and electron number operators on site
$i$, respectively. The strong correlation nature of the $t$-$J$ model
originates from the no double occupancy constraint
$\sum_{\sigma}c^{\dagger}_{i\sigma}c_{i\sigma}\leq 1$ on each site, which is
imposed via the projection operator $\mathcal{P}_{s}$. Generally, a two-leg
ladder is anisotropic along the chain direction (denoted as ${x}$ direction)
and the rung direction (denoted as $y$ direction) as illustrated in Fig. 2
under PBC, where we choose the rung-direction couplings as $t_{ij}=t$ and
$J_{ij}=J$, and the chain-direction couplings as $t_{ij}=\alpha t$ and
$J_{ij}=\alpha J$, with $\alpha>0$ as the anisotropic parameter. The
superexchange coupling constant $J$ is taken as the unit and the hopping term
$t/J=3$ is used throughout the paper. The DMRG calculation of this work is
done with 2500 saved states to fit a truncation error up to $10^{-10}$ with
200 sweeps for convergence.

###  II.2 Single-hole wavefunction ansatz

At half-filling, where the $t$-$J$ model is reduced to the Heisenberg spin
model on a bipartite square lattice, the ground state is a spin singlet state,
with a finite spin-gap opened up for the two-leg ladder case Zhu  _et al._
(2013). In the following, we shall denote it as $|\phi_{0}\rangle$.

Then, based on a bare hole state created at site $i$ by removing an electron
of spin $\downarrow$ from the spin-singlet background, i.e.,
$c_{i\downarrow}|\phi_{0}\rangle$, a Bloch-wave-like single-hole state may be
constructed as

$|\Psi_{\mathrm{B}}\rangle_{\mathrm{1h}}=\sum_{i}\varphi^{\mathrm{B}}_{h}(i)c_{i\downarrow}|\phi_{0}\rangle~{},$
(7)

where $\varphi^{\mathrm{B}}_{h}(i)\propto e^{ikx_{i}}$ is a Bloch-wave with a
momentum $k$ along the quasi-1D ladder direction under the translation
symmetry. In general, the doped hole will induce a many-body response from the
spin background, known as the phase-string effect Sheng  _et al._ (1996); Wu
_et al._ (2008), such that the single-hole state can be significantly
renormalized beyond the Bloch-wave-like one in Eq. (7). How to treat such an
effect is therefore the central issue in the study of the doped Mott physics.

An ansatz ground state has been previously proposed for the $t$-$J$ model,
which is generally given in the one-hole case as follows Weng (2011); Wang
_et al._ (2015); Chen  _et al._ (2019)

$|\Psi_{\mathrm{G}}\rangle_{\mathrm{1h}}=\sum_{i}\varphi_{h}(i)e^{-i\hat{\Omega}_{i}}c_{i\downarrow}|\phi_{0}\rangle~{},$
(8)

where a new phase factor $e^{-i\hat{\Omega}_{i}}$ is explicitly introduced to
represent the many-body phase shift or the phase-string effect from the spin
background when a hole is created at site $i$. In other words, the
corresponding spin background is modified from $|\phi_{0}\rangle$ to
$e^{-i\hat{\Omega}_{i}}|\phi_{0}\rangle$. Here $\hat{\Omega}_{i}$ is
explicitly given by Weng (2011); Wang  _et al._ (2015); Chen  _et al._ (2019)

$\hat{\Omega}_{i}=\sum_{l(\neq i)}\theta_{i}(l)n_{l\downarrow}~{},$ (9)

where $n_{l\downarrow}$ is the number operator of the down spin at site $l$.
The statistical angle $\theta_{i}(l)$ must satisfy the condition

$\theta_{i}(l)-\theta_{l}(i)=\pm\pi~{},$ (10)

for two NN sites $i$ and $l$ such that a sign change can be instantly produced
by the phase factor $e^{-i\hat{\Omega}_{i}}$ when the hole exchanges with a
spin of $\sigma=-1$ during an NN hopping. As the result, the singular part of
the phase-string effect of the $t$-$J$ model can be precisely compensated via
$e^{-i\hat{\Omega}_{i}}$ due to Eq. (10) Weng (2011); Wang  _et al._ (2015);
Chen  _et al._ (2019). It is important to point out that the phase-string
cannot be truly “gauged away” by $e^{-i\hat{\Omega}_{i}}$, which only serves
as a unitary/duality transformation to turn the singular phase-string into a
smooth nonlocal/topological effect such that $\varphi_{h}(i)$ may be still
treated as a conventional (Bloch-type) wavefunction (see below).

Note that Eq. (10) alone does not completely specify $\theta_{i}(l)$. A simple
choice of $\theta_{i}(l)$ satisfying Eq. (10) may be given by
$\theta_{i}(l)=\pm\mathrm{Im}\ln(z_{i}-z_{l})$ in an isotropic 2D plane Chen
_et al._ (2019), with $z_{i}=x_{i}+iy_{i}$ being the complex coordinate of
site $i$. For the two-leg ladder case, previously an anisotropic definition of
$\theta_{i}(l)$ was introduced Wang  _et al._ (2015) with an extra variational
parameter. But it can only apply to a finite ladder with an open boundary
condition (OBC). In the present work, in order to study the one-hole ground
state properties under the PBC, a distinct choice of $\hat{\Omega}_{i}$ will
be needed. Here we shall still use the same isotropic definition of the
statistical phase $\theta_{i}(l)$ given above for the isotropic 2D Weng
(2011); Chen  _et al._ (2019) but put the two-leg ladder in a 2D plane with a
spatial configuration shown in Fig. 2. With the two legs of the ladder being
bent into two rings, the PBC is realized. By making two rings with distinct
radii of $r_{\mathrm{in}}=2-\lambda$ and $r_{\mathrm{out}}=2+\lambda$,
respectively, $\lambda$ can be taken as a variational parameter to tune the
anisotropy (due to the two-leg instead of 2D) in the phase-string operator
$\hat{\Omega}_{i}$ in replace of an anisotropic $\theta_{i}(l)$ originally
defined in Ref. Wang  _et al._ , 2015.

It is emphasized that such a detailed spatial configuration in Fig. 2 is
useful for implementing the phase-string operator $\hat{\Omega}_{i}$ without
changing the model Hamiltonian, which is apparently translational invariant
along the ladder direction due to PBC. We point out that some basic important
features of the single-hole-doped two-leg ladder have been already well
captured by the wavefunction approach in Ref. Wang  _et al._ , 2015, including
the existence of the QCP and the charge modulation pattern in the non-Landau
quasiparticle regime. But in contrast to OBC used in Ref. Wang  _et al._ ,
2015, the two-leg ladder under PBC will enable us to study the whole problem
under the translational symmetry, where the spin and charge currents, total
momentum, and the response under inserting magnetic flux, etc., can be well
defined to characterize the ground state. In other words, the present PBC
approach will consistently cover all the main results obtained under the OBC
in Ref. Wang  _et al._ , 2015 and at the same time provide a more complete
understanding of the single-hole problem as to be presented below.

Besides $\lambda$ in the definition of $\hat{\Omega}_{i}$ [via $\theta_{i}(l)$
defined in the geometry of Fig. 2], another variational parameter is the
single-hole wavefunction $\varphi_{h}(i)$ in Eq. (8), which may be determined
by optimizing the total ground energy variationally. A further simplification
is that $\varphi_{h}(i)$, up to a $U(1)$ phase, is also Bloch-wave-like:
$\varphi_{h}(i)\propto e^{ik_{0}x_{i}}$, since generally the spin-spin
correlation is short-ranged in $|\phi_{0}\rangle$ such that the new “twisted”
hole created by $\tilde{c}_{i\downarrow}\equiv
e^{-i\hat{\Omega}_{i}}c_{i\downarrow}$ should be of a finite scale which is
translationally invariant moving on $|\phi_{0}\rangle$. Namely, the
variational parameter of $\varphi_{h}(i)$ reduces to determining $k_{0}$,
which will become nontrivial in general. The detailed VMC procedure is a
generalization of Refs. Wang  _et al._ (2015); Chen  _et al._ (2019), which is
outlined in Appendix C.

Finally, the wavefunction ansatz in Eq. (8) can be further improved by
introducing a “longitudinal spin-polaron” correction to make
$|\phi_{0}\rangle\rightarrow\hat{\Pi}_{i}|{\phi}_{0}\rangle$ around the hole
site $i$, such that the spin background becomes
$e^{-i\hat{\Omega}_{i}}\hat{\Pi}_{i}|{\phi}_{0}\rangle$. Such a procedure will
be straightforward as discussed in Appendix B. But as we shall see below, the
main body of the physical properties of the single-hole problem is already
excellently captured by the simplest form in Eq. (8), even though some
quantitative results can be slightly improved by the extra variational
parameters, such as the total energy and the position of the QCP point
$\alpha_{c}$ (cf. Appendix B). Physically such a longitudinal spin-polaron
effect can serve as a key improvement for the Bloch-wave state in Eq. (7)
without changing its nature as a Landau’s quasiparticle. Similarly, its
correction to the present one-hole state in Eq. (8) will not change its non-
Landau features either, with only a quantitative improvement in the
variational energy as to be shown in the following.

Table 1: The energies and quantum number of the single-hole variational ground
states in comparison with the ED results on a $8\times 2$ ladder with
$\alpha=1.0$ and $\alpha=0.4$: $E_{\mathrm{G}}$ is the total energy; $E_{t}$
and $E_{J}$ are the kinetic and the superexchange energies, respectively;
$k_{0}$ denotes the corresponding momentum along the $x$-direction. Ground
states $|\Psi_{\mathrm{B}}\rangle_{\mathrm{1h}}$ and
$|\Psi_{\mathrm{G}}\rangle_{\mathrm{1h}}$ are given in Eqs. (7) and (8),
respectively; $|\tilde{\Psi}_{\mathrm{B}}\rangle_{\mathrm{1h}}$ and
$|\tilde{\Psi}_{\mathrm{G}}\rangle_{\mathrm{1h}}$ denote the corresponding
variational states further improved by incorporating the longitudinal spin-
polaron effect (see the main text and Appendix B).

| $\alpha$ | $E_{\mathrm{G}}$ | $E_{t}$ | $E_{J}$ | $k_{0}$
---|---|---|---|---|---
$|\Psi_{\mathrm{B}}\rangle_{\mathrm{1h}}$ | $1.0$ | $-15.84$ | $-2.46$ | $-13.37$ | $0$
$|\tilde{\Psi}_{\mathrm{B}}\rangle_{\mathrm{1h}}$ | $1.0$ | $-18.03$ | $-6.26$ | $-11.77$ | $0$
$|\Psi_{\mathrm{G}}\rangle_{\mathrm{1h}}$ | $1.0$ | $-18.15$ | $-5.55$ | $-12.60$ | $\pm\pi/2$
$|\tilde{\Psi}_{\mathrm{G}}\rangle_{\mathrm{1h}}$ | $1.0$ | $-19.43$ | $-6.96$ | $-12.47$ | $\pm\pi/2$
ED | $1.0$ | $-19.77$ | $-7.30$ | $-12.47$ | $\pm\pi/2$

| $\alpha$ | $E_{\mathrm{G}}$ | $E_{t}$ | $E_{J}$ | $k_{0}$
---|---|---|---|---|---
$|\Psi_{\mathrm{B}}\rangle_{\mathrm{1h}}$ | $0.4$ | $-11.78$ | $-2.90$ | $-8.88$ | $\pi$
$|\tilde{\Psi}_{\mathrm{B}}\rangle_{\mathrm{1h}}$ | $0.4$ | $-13.03$ | $-4.44$ | $-8.60$ | $\pi$
$|\Psi_{\mathrm{G}}\rangle_{\mathrm{1h}}$ | $0.4$ | $-12.67$ | $-3.89$ | $-8.78$ | $\pi$
$|\tilde{\Psi}_{\mathrm{G}}\rangle_{\mathrm{1h}}$ | $0.4$ | $-13.10$ | $-4.42$ | $-8.69$ | $\pi$
ED | $0.4$ | $-13.14$ | $-4.42$ | $-8.71$ | $\pi$

###  II.3 Variational ground-state energy

Based on the wavefunction ansatz $|\Psi_{\mathrm{G}}\rangle_{\mathrm{1h}}$ in
Eq. (8), the ground state energy, $E_{\mathrm{G}}$, can be determined
variationally by the VMC method outlined above. Table 1 presents the VMC
results of $E_{\mathrm{G}}$ and the hopping and superexchange energies,
$E_{t}$ and $E_{J}$, at two typical values of the anisotropic parameter:
$\alpha=1.0$ and $\alpha=0.4$, respectively, in comparison with the ED results
for a $8\times 2$ ladder. In the same Table, the corresponding energies of the
Bloch-wave state $|\Psi_{\mathrm{B}}\rangle_{\mathrm{1h}}$ [Eq. (7)] are also
given for comparison.

Table 1 shows that the ansatz state $|\Psi_{\mathrm{G}}\rangle_{\mathrm{1h}}$
gives a much better ground energy than the Bloch-like state
$|\Psi_{\mathrm{B}}\rangle_{\mathrm{1h}}$ at $\alpha=1.0$. Most importantly,
the ansatz state $|\Psi_{\mathrm{G}}\rangle_{\mathrm{1h}}$ captures the
correct ground state momenta $k_{0}^{\pm}=\pm\pi/2\mod 2\pi$ at $\alpha=1.0$,
which is totally missed by $|\Psi_{\mathrm{B}}\rangle_{\mathrm{1h}}$. By
contrast, at $\alpha=0.4$, both $|\Psi_{\mathrm{G}}\rangle_{\mathrm{1h}}$ and
$|\Psi_{\mathrm{B}}\rangle_{\mathrm{1h}}$ have the same momentum $k_{0}=\pi$
and the ground state energies are also relatively closer even though $E_{t}$
in the former is still much improved.

According to the phase diagram of Fig. 1(a), which is to be elaborated below,
the single-hole ground states at $\alpha=1.0$ and $\alpha=0.4$ are on the two
sides of the critical $\alpha_{c}\simeq 0.68$, e.g., $\alpha=0.4<\alpha_{c}$
is in the Landau quasiparticle regime where
$|\Psi_{\mathrm{G}}\rangle_{\mathrm{1h}}$ and
$|\Psi_{\mathrm{B}}\rangle_{\mathrm{1h}}$ may be adiabatically connected, but
$\alpha=1.0>\alpha_{c}$ is a distinct regime where
$|\Psi_{\mathrm{G}}\rangle_{\mathrm{1h}}$ cannot be reduced to
$|\Psi_{\mathrm{B}}\rangle_{\mathrm{1h}}$ because of a nontrivial “transverse
spin-polaron effect” due to the phase-shift factor $e^{-i\hat{\Omega}_{i}}$.

Before we explore such distinction between the two phases and the quantum
transition at $\alpha_{c}$ below, we further examine an improvement of the
ground-state energy by incorporating the aforementioned “longitudinal spin-
polaron effect”, which will turn $|\Psi_{\mathrm{G}}\rangle_{\mathrm{1h}}$ and
$|\Psi_{\mathrm{B}}\rangle_{\mathrm{1h}}$ into
$|\tilde{\Psi}_{\mathrm{G}}\rangle_{\mathrm{1h}}$ and
$|\tilde{\Psi}_{\mathrm{B}}\rangle_{\mathrm{1h}}$, respectively (cf. Appendix.
B for the details). The corresponding variational energies on a $8\times 2$
ladder are also shown in Table 1. At $\alpha=1.0$, the ground-state energy of
$|\tilde{\Psi}_{\mathrm{G}}\rangle_{\mathrm{1h}}$ can be optimized to be
within the precision of $1.7\%$ as compared to the ED result, with
$k_{0}^{\pm}$ unchanged. But for the Landau quasiparticle state
$|\tilde{\Psi}_{\mathrm{B}}\rangle_{\mathrm{1h}}$, the energy is still
relatively much worse than that of the ansatz wavefunction, with the wrong
momentum of $k_{0}=0$ unchanged. However, at $\alpha=0.4$, both
$|\tilde{\Psi}_{\mathrm{G}}\rangle_{\mathrm{1h}}$ and
$|\tilde{\Psi}_{\mathrm{B}}\rangle_{\mathrm{1h}}$ get optimized to be within
$0.3\%$ and $0.8\%$ of the exact ED value, consistent with the fact that the
longitudinal spin-polaron effect is solely responsible for the renormalization
of the doped hole in the Landau quasiparticle regime.

Figure 3:  Ground state hole density distributions $n_{x_{i}}^{h}$ and its
Fourier transformation $N_{q}=\sum_{x_{i}=1}^{N_{x}}n_{x_{i}}^{h}e^{iqx_{i}}$
calculated by DMRG method on a $48\times 2$ PBC ladder for
$\alpha=1.0>\alpha_{c}$ [(a) and (b)] and $\alpha=0.4<\alpha_{c}$ [(c) and
(d)], respectively. The results for a $\Phi=\pi$ flux inserting through the
center of the PBC rings are also shown. Note that at $\alpha=1.0$ the ground
states are double degenerate, and a real wavefunction state is used for the
measurement.  Figure 4:  Ground state hole density distributions
$n_{x_{i}}^{h}$ and its Fourier transformation
$N_{q}=\sum_{x_{i}=1}^{N_{x}}n_{x_{i}}^{h}e^{iqx_{i}}$ calculated by VMC based
on the ground-state ansatz in Eq. (8). The parameters are the same as in Fig.
3 for a $48\times 2$ PBC ladder at $\alpha=1.0>\alpha_{c}$ [(a) and (b)] and
$\alpha=0.4<\alpha_{c}$ [(c) and (d)], respectively. The results for a
$\Phi=\pi$ flux inserting through the center of the PBC rings are also shown.

###  II.4 Phase diagram

The ground-state phase diagram of the single hole doped anisotropic two-leg
$t$-$J$ ladder has already been carefully studied by DMRG calculations Zhu and
Weng (2015); Zhu  _et al._ (2015, 2018); White  _et al._ (2015). A key finding
is that the analyticity of the ground state energy has a singularity as a
function of the anisotropic parameter $\alpha$ at $\alpha=\alpha_{c}\approx
0.68$ ($t/J=3$), which resembles a second-order phase transition Zhu and Weng
(2015). Across the critical point $\alpha_{c}$, the physical properties are
also qualitatively changed. For example, the non-degenerate ground state at
$\alpha<\alpha_{c}$ becomes double-degenerate at $\alpha>\alpha_{c}$ (for a
given total spin-1/2). The corresponding hole density distribution changes
from flat to a charge modulation characterized by an incommensurate wavevector
$Q_{0}$ for each of the degenerate (real wavefunction) states. In Fig. 1(b),
the critical point $\alpha_{c}$ is well indicated by the emergence of a finite
$Q_{0}$ as a function of $\alpha$ as calculated by DMRG under PBC for an
$N=48\times 2$ ladder.

Figure 5: (a) The quasiparticle weight $Z_{k}$ calculated by VMC at
$\alpha=0.4$ and $\alpha=1.0$, respectively, on a $48\times 2$ ladder with
PBC; (b) Momentum distribution of the hole substracted by the spectral weight,
$n_{k}^{h}-Z_{k}$ . Here the dashed lines mark the positions of the peaks of
the quasiparticle weight. For $\alpha>\alpha_{c}$ a real wavefunction state is
used.

Such a QCP at $\alpha_{c}$ can be well quantitatively reproduced based on the
ansatz wavefunction given in Eq. (8). The VMC result is also presented in Fig.
1(b) obtained by optimization with regard to the variational parameters of
$\lambda$ and $\varphi_{h}(i)$, which is then further improved by
incorporating the longitudinal spin-polaron effect in $|\phi_{0}\rangle$ as
discussed in the previous subsection (cf. Appendix B). One can see an
excellent overall agreement between the VMC and the DMRG result. We mention
that a similar QCP has been previously determined under the OBC Wang  _et al._
(2015) based on the singularity in second-derivative of the ground state
energy. In the following, to illustrate the quantum transition and distinct
behaviors manifested in Fig. 1(b), we present two benchmarking calculations,
at $\alpha=1.0>\alpha_{c}$ and $\alpha=0.4<\alpha_{c}$, with the DMRG results
given in Fig. 3 and the corresponding VMC results in Fig. 4, respectively.

As comparatively shown in Figs. 3(a) and 4(a), respectively, the hole average
density $n^{h}_{x_{i}}$ on a rung $x_{i}$ of the ladder at $\alpha=1.0$
($>\alpha_{c}$) is presented. Here the density profile shows a two-component
structure: a flat background plus a modulation with the wavevector $Q_{0}\neq
0$, with the Fourier transformation along the ladder direction further
illustrated in Figs. 3(b) and 4(b), respectively. In particular, if a flux
$\Phi=\pi$ is inserted into the center of the rings formed by the ladder (cf.
Fig. 2), a shift of $Q_{0}$ by $2\pi/N_{x}$ is shown for the charge modulation
component, while the flat component at the momentum $q=0$ remains unchanged as
Figs. 3(b) and 4(b) indicate.

By contrast, at $\alpha=0.4$ ($<\alpha_{c}$), a distinct hole profile is given
in Figs. 3(c) and 4(c), whose Fourier transformation is shown in Figs. 3(d)
and 4(d), respectively. They show that the hole profile is uniform with
$Q_{0}=0$. Upon inserting a $\pi$ flux into the rings, however, a node is
exhibited in Figs. 3(c) and 4(c), which corresponds to a momentum shift by
$\pm 2\pi/N_{x}$ in Figs. 3(d) and 4(d). It is consistent with the change of
the PBC to an anti-periodic boundary condition (APBC) for a free Bloch wave,
indicating that the single-hole state is a Landau quasiparticle which carries
a spin-1/2 and charge $+e$ to satisfy both the translation and $U(1)$
symmetries Zhu and Weng (2015); Zhu  _et al._ (2018).

Such a Landau quasiparticle picture at $\alpha=0.4$ is consistent with a
finite overlap between Eq. (8) and Eq. (7). As a matter of fact, for
$\alpha=0.4<\alpha_{c}$, a single sharp peak of $Z_{k_{0}}$ [cf. Eq. (4)] at
momentum at $k_{0}=\pi$ is always seen [cf. Fig. 5(a) or the left inset of
Fig. 1(b)], which is in agreement with the DMRG White  _et al._ (2015); Zhu
_et al._ (2018). However, as first indicated White  _et al._ (2015) by DMRG,
at $\alpha>\alpha_{c}$, each of the double-degenerate ground states still has
a finite overlap with the Bloch states in Eq. (7), such that
$Z_{k^{\pm}_{0}}\neq 0$ at momenta $k_{0}^{\pm}=\pi\pm\kappa\mod 2\pi$ with
$Q_{0}=2\kappa$, as indicated by Fig. 5(a) or the right inset of Fig. 1(b),
which is confirmed by the present VMC. Two typical $Z_{k}$’s at
$\alpha=0.4<\alpha_{c}$ and $\alpha=1.0>\alpha_{c}$ as determined by VMC are
shown in Fig. 5(a), respectively.

However, in contrast to the speculation in Ref. White  _et al._ , 2015, the
present ansatz wavefunction will directly show that the single-hole state is
no longer a Landau quasiparticle state at $\alpha>\alpha_{c}$ even though
$Z_{k^{\pm}_{0}}\neq 0$. Based on the ansatz wavefunction in Eq. (8), one can
calculate the momentum distribution of the hole by
$n_{k}^{h}=1-\sum_{\sigma}n_{k\sigma}$, where

$n_{k\sigma}=\frac{1}{N}\sum_{ij}e^{i{k}(x_{i}-x_{j})}\tensor{[}_{\mathrm{1h}}]{\langle\Psi_{G}|c^{\dagger}_{i\sigma}c_{j\sigma}|\Psi_{G}\rangle}{{}_{\mathrm{1h}}}~{},$
(11)

with $k$ the momentum along the chain direction ($k_{y}=0$). Then, besides
$Z_{k}$ shown in Fig. 5(a), the VMC results of $n_{k}^{h}-Z_{k}$ are presented
in Fig. 5(b) at $\alpha=0.4$ and $\alpha=1.0$, respectively, which clearly
shows a residual broad peak at $\alpha=1.0$, which will be related to an
incoherent charge component. In fact, Figs. 3(a) and (b) and Figs. 4(a) and
(b) have already indicated the two-component structure of the spatial hole
density: the charge modulation with a finite wavevector $Q_{0}$ and the
uniform background. The former responds to inserting a $\pi$ flux into the
ring center of the ladder by $2\pi/N_{x}$ shift in $Q_{0}$, which is
consistent with two quasiparticle components with $Z_{k^{\pm}_{0}}\neq 0$ at
$k_{0}^{\pm}=\pi\pm\kappa$, whereas the latter has no response as it relates
to the incoherent background of Fig. 5(b). In the next section we shall see
that such an incoherent part is charactered by a hidden spin current which
continuously carries away momentum from the hole. Note that a rather weak
broad peak is also seen $n_{k}^{h}-Z_{k}$ for $\alpha=0.4$ in Fig. 5(b), which
has been argued based on DMRG Zhu  _et al._ (2018) as completely contributed
by the conventional many-body “cloud” effect of a Landau quasiparticle. In
fact, it does not lead to an incoherent component as shown in Figs. 3(c) and
(d) and Figs. 4(c) and (d). Furthermore, the hidden spin current will be
completely absent at $\alpha<\alpha_{c}$ as to be discussed later.

##  III Further characterization: Novel charge renormalization at
$\alpha>\alpha_{c}$

In the last section, the wavefunction ansatz based on Eq. (8) has been shown
to very accurately capture the essential physics of the single-hole ground
state at both $\alpha>\alpha_{c}$ and $\alpha<\alpha_{c}$, including the
critical point $\alpha_{c}$, in comparison with both DMRG and ED results. In
the following, we shall focus on the unconventional case at
$\alpha>\alpha_{c}$ and demonstrate that the charge carried by the doped hole
is renormalized to zero. In other words, the QCP at $\alpha_{c}$ marks the
transition of the doped hole from a Laudau quasiparticle to a charge-neutral
spinon.

###  III.1 Emergent spin-current around the hole at $\alpha>\alpha_{c}$:
Incoherent charge component

Let us start by emphasizing that the double-degenerate ground states at
$\alpha>\alpha_{c}$ have a two-component structure. In the real wavefunction
description as shown by DMRG and VMC in Figs. 3 and 4, respectively, the
spatial distribution of the doped hole has a charge modulation component
composed of two Bloch-like waves with momenta $k_{0}^{\pm}=\pi\pm\kappa$ and a
uniform background. The latter is not sensitive to the change of the boundary
condition via inserting a flux into the ring of the ladder and has been argued
to be an incoherent charge component, which we shall examine more carefully
below.

Figure 6: Neutral spin current $J_{ij}^{s}$ [cf. Eq. (12)] (blue arrows)
surrounding the hole, which is projected onto a fixed position (read circle)
at momentum $k^{-}_{0}=\pi-\kappa$ for $\alpha=1.0$: (a) DMRG on a $24\times
2$ ladder and (b) VMC on a $48\times 2$ ladder. The thickness of the blue
lines represents the strength of the spin current, which conserves $S^{z}$.

One may also focus on a translational invariant ground state with a total
momentum, say, $k_{0}^{-}=\pi-\kappa$, which has a finite overlap with the
Bloch-wave state in Eq. (2) at the given $k=k_{0}^{-}$. Then the incoherent
charge component in the ground state can be distinguished from the Landau
quasiparticle component by the presence of a spin current pattern around the
hole, which is shown in Figs. 6 as calculated by (a) DMRG and (b) VMC at
$\alpha=1.0$, respectively. Here the neutral spin-current operator
$J^{s}_{ij}$, which conserves $S^{z}$, is defined on the NN link of $i$ and
$j$ by

$J_{ij}^{s}=i\frac{J}{2}(S^{+}_{i}S^{-}_{j}-S^{-}_{i}S^{+}_{j})~{},$ (12)

while the backflow spin current of the hopping term is zero as the hole is
projected onto a given site (marked by open red circle) as shown in Fig. 6.
Since the Landau component (Bloch-wave) does not contribute to $J_{ij}^{s}$,
the spin current pattern in Fig. 6 should entirely come from the non-Landau-
component in the ground state. The above results are also consistent with the
chiral spin currents observed previously for the single-hole problem in the 2D
$t$-$J$ systems by ED and DMRG Zheng  _et al._ (2018) and verified by VMC Chen
_et al._ (2019); Zhao  _et al._ (2022).

Figure 7: Spin-charge correlator $\langle n_{i}^{h}S^{z}_{i+r}\rangle$ (solid
cricle) calculated by VMC on a $48\times 2$ ladder. It shows a qualitative
change from a tight hole-spin binding to a loosely-bound hole-spin composite
from (a) $\alpha=0.4<\alpha_{c}$ to (b) $\alpha=1.0>\alpha_{c}$ in agreement
with the DMRG result Zhu and Weng (2015); White  _et al._ (2015). The label
$r$ is defined in the inset of (a). The red line and the dashed line in (b)
are the fitted curve and envelope function.

So the spin currents around the hole indicate that the spin partner
$S^{z}=1/2$ is only loosely bound to the hole, which forms a spin current
vortex in the hole composite. On the other hand, no spin currents are seen by
both VMC and DMRG methods at $\alpha<\alpha_{c}$, where the non-Landau-
component disappears completely. The “longitudinal spin-polaron” correction
here does not create a transverse spin current surrounding the hole. The
absence of the spin currents can be understood as that the spin partner is
tightly bound to the doped hole in the strong rung limit to form a
conventional Landau quasiparticle with charge $+e$ and spin-1/2.

This picture can be also confirmed by measuring the spin-charge correlator
$\langle n_{i}^{h}S_{j}^{z}\rangle$, which characterizes the relative distance
between the doped hole and an unpaired spin-1/2. The results calculated by VMC
are shown in Fig. 7 for both $\alpha=0.4$ and $\alpha=1.0$, respectively. At
$\alpha=0.4$, the spin-1/2 and the hole are tightly bound at the same rung as
$\langle n_{i}^{h}S_{j}^{z}\rangle$ decays quickly at $r>1$ [cf. Fig. 7 (a)].
On the other hand, the unpaired spin is only loosely bound with the doped hole
at $\alpha=1.0$ as shown in Fig. 7 (b). Both behaviors are again in good
agreement with the DMRG results Zhu and Weng (2015); White  _et al._ (2015).

Figure 8: The ground-state energy change, $\Delta E_{\mathrm{G}}$, under a
$\pi$ flux insertion ladder as a function of lattice size $N_{x}$, which
determined by VMC at $\alpha=1.0$. The red solid line is a fitted curve whose
equation is given in Eq. (13) and the blue dashed lines are the envelope
function of an exponential decay with a correlation length $\xi\approx 14.6$
in agreement with the DMRG result Zhu  _et al._ (2013).

###  III.2 Response to external electromagnetic flux

Even though the single-hole ground state at $\alpha>\alpha_{c}$ still has a
finite overlap with the Bloch-wave state, i.e., $Z_{k_{0}}\neq 0$, the
presence of an intrinsic _incoherent_ component makes the Landau’s one-to-one
hypothesis invalid as the latter cannot be completely specified by a total
momentum $k_{0}$ alone. In the above, one has seen that the hopping of the
hole will always generate a spin current via the backflow, which means that
the momentum $k_{0}$ is now shared between the (bare) hole and its of its
$S=1/2$ partner, which result in a broad momentum distribution of the bare
hole as has been carefully examined by DMRG Zhu  _et al._ (2018). In other
words, the origin of the incoherent component comes from the internal relative
motion inside the loosely-bound hole-spin composite at $\alpha>\alpha_{c}$.

The non-Landau behavior may be understood as follows. Because of the finite
off-diagonal transition between the coherent and incoherent components, the
hole composite as a single entity cannot exhibit a definite charge in response
to an external electromagnetic field. Imaging that such a hole circles through
the ladder once with a flux $\Phi$ inserting in the hole of the rings formed
by the ladder. A charge $q$ will pick up a Berry phase $q\Phi$, which has been
well demonstrated by DMRG Zhu and Weng (2015) at $\alpha<\alpha_{c}$, but an
exponentially diminished effect of the flux with the increase of the ladder
length $N_{x}$ has been seen at $\alpha>\alpha_{c}$ Zhu  _et al._ (2013); Zhu
and Weng (2015). In the following, we repeat the calculation of the same
effect based on the wavefunction ansatz in Eq. (8).

In Sec. II, the response of the charge density distribution to an external
$\Phi=\pi$ flux insertion has already been shown. The vanishing charge
response for the incoherent component is clearly demonstrated there. Here we
calculate the change of the ground state energy $E_{\mathrm{G}}$ under the
flux insertion: $\Delta E_{\mathrm{G}}\equiv
E_{\mathrm{G}}(\Phi=\pi)-E_{\mathrm{G}}(\Phi=0)$. Figure 8 shows $\Delta
E_{\mathrm{G}}$ as a function of the ladder length $N_{x}$ calculated by VMC
at $\alpha=1.0$. The black dots are the calculated data, which is fitted by
the red solid curve with the following expression

$\Delta E_{\mathrm{G}}=\Delta E_{0}e^{-N_{x}/\xi}\cos(k_{0}N_{x}+\phi)~{},$
(13)

where $k_{0}$ is the ground-state momentum at $\alpha=1.0$, $\Delta E_{0}$,
$\xi$ and, $\phi$ are fitting parameters. The blue dashed curves are the
envelope functions of Eq. (13), which show an exponential decay behaviors
$\pm\Delta E_{0}e^{-N_{x}/\xi}$ with a length scale $\xi\approx 14.6$. It
agrees with a previous calculation by DMRG Zhu  _et al._ (2013) with an
exponential decay $\xi\approx 14.5$.

Figure 9: The doped hole becomes a novel spinon at $\alpha=1.0>\alpha_{c}$:
the scaling behaviors of the neutral spin current $N_{x}J^{s}_{\parallel}$
(open triangular) and the charge current $N_{x}J^{h}_{\parallel}$ (open
square) along the ladder direction for a doped hole with given momentum
$k^{\pm}_{0}$. The charge current oscillates and decays to zero, while the
neutral spin current saturates to a fixed value (dashed blue) in the long
ladder limit. The solid black line is a fitted curve given in Eq. (17) with
the red dashed line as the envelope function.

###  III.3 Vanishing charge

The vanishing response to an external electromagnetic field can be seen in a
more explicit way by directly measuring the charge currents of the ground
state

$J^{c}_{ij}=iqt\sum_{\sigma}(c^{\dagger}_{i\sigma}c_{j\sigma}-c^{\dagger}_{j\sigma}c_{i\sigma})~{},$
(14)

where $i,j$ are NN bonds of the ladder direction and $q=+e$ is the charge of a
bare hole. For a given momentum $k_{0}$, a Landau quasiparticle with charge
$q$ and is expected to carry a finite charge current $J^{c}_{\parallel}$ along
the ladder direction

$J^{c}_{\parallel}\equiv 2J_{i,i+\mathbf{e}_{x}}^{c}=\rho\frac{q}{m^{*}}\sin
k_{0}~{},$ (15)

where $\rho=1/N_{x}$ is the density of a single doped hole on every rung of
the ladder, $m^{*}$ is the effective mass of the quasiparticle, which is found
finite on both sides of the QCP Zhu and Weng (2015), and $\mathbf{e}_{x}$ is a
unit vector along the chain direction. On the other hand, the neutral spin
current of the spin background is expected to vanish

$J^{s}_{\parallel}\equiv 2J^{s}_{i,i+\mathbf{e}_{x}}=0~{}.$ (16)

The above results can be easily verified at $\alpha<\alpha_{c}$ (for a low-
lying excited state with $k_{0}$ deviates from $\pi$). However, at
$\alpha=1.0>\alpha_{c}$, one finds that the currents associated with the doped
hole of momentum $k_{0}^{-}=\pi-\kappa$ are dramatically changed as shown in
Fig. 9. Instead of the finite charge current
$J^{c}_{\parallel}/\rho=N_{x}J^{c}_{\parallel}$ as predicted by Eq. (15), an
oscillating decay behavior of the charge currents is seen, satisfying a fitted
curve

$N_{x}J^{c}_{\parallel}=\left.9.6\left(\left\\{\frac{k_{0}}{2\pi}(N_{x}-\phi)+0.5\right\\}-0.5\right)\middle/N_{x}^{0.94}\right.~{},$
(17)

where the curly bracket in the expression takes the decimal part of the number
inside it, $k_{0}$ is the momentum of the ground state, and $\phi$ is a phase
parameter to be optimized. The overall $N_{x}^{0.94}$ in the denominator
approximate a $1/r$ decay of the envelope function.

On the other hand, the neutral spin current $N_{x}J^{s}_{\parallel}$ in the
spin background (not the spin current associated with the hopping of the
hole), which is expected to vanish in the quasiparticle picture as Eq. (16),
saturates to a finite value $N_{x}J^{s}_{\parallel}\sim 1.6J$ instead. These
results confirm that the doped hole is no longer a charged object. Instead,
the finite neutral spin current $J^{s}_{\parallel}/\rho$ indicate that the
true quasiparticle is now a spinon.

##  IV Discussion

As described in the previous sections, the systematic agreements between the
VMC and exact numerical calculations indicate that the wavefunction ansatz of
Eq. (1) has reasonably captured all the fundamental physics of the single-hole
ground state in the two-leg ladder. It therefore enables one to further
examine the underlying mechanism based on the analytic structure of the
wavefunction.

The sole distinction between the wavefunction in Eq. (1) and a conventional
bare-hole state in Eq. (2) lies in the phase-string factor
$e^{-i\hat{\Omega}_{i}}$, which contributes to a many-body phase shift to the
quasiparticle wavefunction $\varphi_{h}\propto e^{ik_{0}x_{i}}$. Here
$\hat{\Omega}_{i}$ involves the background spins nonlocally according to the
definition given in Eq. (9), which is apparently non-perturbative in nature.
Since the undoped spin system $|\phi_{0}\rangle$ is short-range-AF-correlated
in the two-leg ladder, the exotic quantum entanglement between the doped hole
with the surrounding spins are also short-ranged. In particular, the AF
correlation length along the ladder direction can be continuously tuned by the
anisotropic parameter $\alpha$, which results in a quantum transition at
$\alpha_{c}$ as shown in Fig. 1.

###  IV.1 QCP at $\alpha_{c}$

At the QCP, the ground-state momentum $k_{0}$ splits from $\pi$ to
$\pi\pm\kappa$. The incommensurate wavevector $Q_{0}\equiv 2\kappa$ shown in
Fig. 1 is directly related to the phase-shift operator $\hat{\Omega}_{i}$ in
Eq. (1). As a matter of fact, based on Eq. (2) without $\hat{\Omega}_{i}$, one
always finds $k_{0}=\pi$ with absence of $\alpha_{c}$.

In the strong-rung limit $\alpha\ll 1$, the singlet-pairing of spins in
$|\phi_{0}\rangle$ is mainly concentrated along rungs. In this limit
$e^{-i\hat{\Omega}_{i}}$ is ineffective along the ladder such that the ansatz
state (1) reduces to the Bloch-wave one in Eq. (2) at a single momentum
$k_{0}=\pi$. It has been shown that a longitudinal spin-polaron correction may
further improve the ground state energy, but the hole as a rigid entity of
spin-1/2 and charge $+e$ with the same $k_{0}$ remains robust, which thus
satisfies the one-to-one correspondence principle for a Landau quasiparticle,
so long as the spin-spin correlation is sufficiently short (than a lattice
constant) along the quasi-1D direction.

In the opposite limit of $\alpha\gg 1$, the longer-range spin singlet pairing
along the chain direction become more and more important. As previously shown,
in the 1D case, $e^{-i\hat{\Omega}_{i}}$ will play a crucial role to result in
a momentum at $k_{0}=\pm\pi/2$ Zhu  _et al._ (2016). Indeed
$Q_{0}\rightarrow\pi$ at $\alpha\gg\alpha_{c}$ as shown in Fig. 1(a).

Thus, $\alpha=\alpha_{c}$ is the point where $e^{-i\hat{\Omega}_{i}}$ starts
to play a nontrivial role. Note that because of the spin-singlet pairing,
$|\phi_{0}\rangle$ itself does not significantly contribute to a phase shift
in $e^{-i\hat{\Omega}_{i}}$. But in the bare hole state
$c_{i\downarrow}|\phi_{0}\rangle$ with removing a spin $\downarrow$ at site
$i$, a spin from the original singlet pair in $|\phi_{0}\rangle$ will be left
unpaired, which in general can make a nontrivial contribution to the momentum
shift via $e^{-i\hat{\Omega}_{i}}$ acting on
$c_{i\downarrow}|\phi_{0}\rangle$. In the limit of $\alpha\ll 1$, this
unscreened spin mainly stays at the rung direction across the hole as already
shown in Fig. 7(a), which does not contribute to a momentum shift along the
ladder direction. At $\alpha>\alpha_{c}$, the unpaired spin (spinon) is
loosely separated from the hole along the ladder direction, as previously
shown in Fig. 7(b). Due to the presence of $e^{-i\hat{\Omega}_{i}}$, such a
composite effect is clearly manifested by a chiral spin current surrounding
the hole in Fig. 6, which contributes to the finite momentum splitting, i.e.,
$k^{\pm}_{0}=\pi\pm\kappa$ with $Q_{0}\neq 0$. The VMC calculation shows a
self-consistent procedure, which minimizes the total energy, giving rise to a
QCP at $\alpha_{c}\neq 0$ and the total momentum $k_{0}$ as a function of
$\alpha$. One may find some more detailed account of the underlying mechanism
in Appendix. A.

###  IV.2 The two-component structure at $\alpha>\alpha_{c}$

We have seen that the double-degenerate ground state at $\alpha>\alpha_{c}$
must be characterized by a two-component structure, i.e., a Laudau-like
quasiparticle with $Z_{k_{0}^{\pm}}\neq 0$, charge $+e$, and spin $1/2$, and
an incoherent component with $Z_{k_{0}^{\pm}}=0$ and charge $0$. The latter
exhibits a chiral spin current pattern around the hole, which cannot be
produced by the quasiparticle component as shown in Sec. III.1. Generally for
the left-moving and right-moving hole states at momenta $k_{0}^{\pm}$, the
chiralities of the spin currents are opposite. In a real wavefunction
representation discussed in Sec. II D by a superposition of the
$k_{0}^{+}=\pi+\kappa$ and $k_{0}^{-}=\pi-\kappa$ states, the incoherent
component corresponds to a charge uniform component with no response to an
external inserting flux.

Let us first consider a conventional Landau quasiparticle with momenta
$k_{0}^{\pm}=\pi\pm\kappa$. Mathematically, a real-wavefunction state as a
superposition of the $\pi+\kappa$ and $\pi-\kappa$ states may take the form

$|\psi_{\mathrm{B}}\rangle_{\mathrm{1h}}\propto\sum_{i}(-1)^{x_{i}}\left(e^{-i{\kappa}x_{i}}+c.c.\right)c_{i\downarrow}\hat{\Pi}_{i}|\phi_{0}\rangle~{},$
(18)

where $\hat{\Pi}_{i}$ is a local spin-polaron operator around the doped hole,
which is translationally invariant. Equation (18) will result in a hole
density modulation by

$n_{x_{i}}^{h}\propto\sum_{y_{i}}\langle\phi_{0}|\hat{\Pi}^{\dagger}_{i}n_{i\downarrow}\hat{\Pi}_{i}|\phi_{0}\rangle(1+\cos(2\kappa
x_{i}))~{},$ (19)

which has nodes at $1+\cos(2\kappa x_{i})=0$ with $\kappa\neq 0$.

In contrast, for the wavefunction ansatz in Eq. (1) with a nontrivial phase
factor $e^{-i\hat{\Omega}_{i}}$. By similarly constructing a real-wavefunction
state via a superposition of $k_{0}^{\pm}=\pi\pm\kappa$ states as follows

$|\psi_{\mathrm{G}}\rangle_{\mathrm{1h}}\propto\sum_{i}(-1)^{x_{i}}\left(e^{-i{\kappa}{x}_{i}-i\hat{\Omega}_{i}}+c.c.\right)c_{i\downarrow}|\phi_{0}\rangle~{},$
(20)

one finds a hole density modulation given by

$n_{x_{i}}^{h}\propto\sum_{y_{i}}(\langle\phi_{0}|n_{i\downarrow}|\phi_{0}\rangle+\langle\phi_{0}|n_{i\downarrow}\cos(2\kappa+2\hat{\Omega}_{i})|\phi_{0}\rangle)~{}.$
(21)

The first term is just the uniform density background shown Figs. 3 (a) and 4
(a), while the second term corresponds to the modulation part. (For simplicity
the longitudinal spin-polaron effect has been set as $\hat{\Pi}_{i}=1$ here.)
Different from the modulation of a quasiparticle in Eq. (19), the phase factor
of modulation part in the cosine function is scrambled by the phase operator
$\hat{\Omega}_{i}$. As a result, it is smaller than the first term because of
the inequality

$|\langle\phi_{0}|n_{i\sigma}e^{-2i\hat{\Omega}_{i}}|\phi_{0}\rangle|\leq|\langle\phi_{0}|n_{i\sigma}|\phi_{0}\rangle|~{},$
(22)

which leads to a nodeless charge modulation at $\alpha>\alpha_{c}$ (cf. Fig. 4
in Sec. II D).

###  IV.3 Disappearance of charge response to external magnetic flux at
$\alpha>\alpha_{c}$

A simple Landau-like quasiparticle state at small $\alpha$ experiences a
quantum phase transition to a novel quasiparticle as the AF correlation length
is increased, leading to the nontrivial (incommensurate) momenta in the ground
state at $\alpha>\alpha_{c}$. In a sharp contrast to a conventional Landau-
like quasiparticle with a finite momentum, the new particle no longer carries
a finite charge current like the former, even though it still carries a spin
current contributed by spin-1/2 as a quasiparticle, i.e., a spinon, as show in
Sec. III.

To understand how the charge current of twisted quasi-particle
$\tilde{c}_{i\sigma}$ is renormalized to zero, one may inspect the averaged
charge current $J^{c}_{\parallel}$ in terms of the ground state of Eq. (8),

$\displaystyle J^{c}_{\parallel}=$
$\displaystyle\sum_{y_{i}}\tensor{[}_{\mathrm{1h}}]{\langle\Psi_{\mathrm{G}}|J^{c}_{i,i+1}|\Psi_{\mathrm{G}}\rangle}{{}_{\mathrm{1h}}}$
(23) $\displaystyle\sim$ $\displaystyle\frac{\sigma
t}{N_{x}}\langle\phi_{0}|\sin(\tilde{k}_{0}^{\pm}\mp(\hat{A}_{ij}^{s}-\phi^{0}_{ij}))(n_{i\bar{\sigma}}n_{j\bar{\sigma}}-S_{i}^{\sigma}S_{j}^{\bar{\sigma}})|\phi_{0}\rangle~{},$

where we have used
$\varphi_{h}(i)\sim\frac{1}{\sqrt{2N_{x}}}e^{ik_{0}^{\pm}x_{i}}$. In addition
to the phase factor $e^{ik_{0}^{\pm}}$ contributed by the variational
parameter, a gauge field induced by the spins $\hat{A}^{s}_{ij}=\sum_{l(\neq
i,j)}(\theta_{i}(l)-\theta_{j}(l))S_{l}^{z}$, and a uniform $\pi$ flux
$\phi^{0}_{ij}=\frac{1}{2}\sum_{l(\neq i,j)}(\theta_{i}(l)-\theta_{j}(l))$
emerges from the phase factor $e^{-i\hat{\Omega}_{i}}$. For the present
single-hole doped case where a single unpaired spin $S=1/2$ is present in the
spin background, the local flux fluctuation of $\hat{A}^{s}_{ij}$ is large
($\sim\pi$) as the unpaired single spin is not tightly bound to the doped hole
along the ladder direction at $\alpha>\alpha_{c}$. As the result, the charge
current contributed by the finite momentum wave function
$\mathrm{Im}(e^{ik_{0}^{\pm}})$ is well compensated by the _internal_ gauge
field generated by the background spins, i.e., $\hat{A}^{s}_{ij}$.

In the same spirit, the strong fluctuating gauge field $\hat{A}^{s}_{ij}$
(with strength $\sim\pi$) can effectively screen out a weak effect from the
external electromagnetic field $A^{e}_{ij}$. Indeed, an exponential decaying
behavior in the response to inserting an external $\Phi=\pi$ flux has been
shown in Fig. 8, which is in contrast to the power law ($1/N_{x}^{2}$)
behavior of a charged Landau quasiparticle state at $\alpha<\alpha_{c}$.

Figure 10: Neutral spin currents with on a $40\times 2$ OBC ladder with
$\alpha=1.0$. The two holes are projected at the red circle positions. While
the spin currents is indicated by the blue arrows. The strength of the spin
currents is represented by the thickness of the arrows.

###  IV.4 Binding force between two doped holes

Finally, we briefly discuss the ground state of two doped holes in the two-leg
ladder based on the present single-hole wavefunction. Previously the DMRG
calculation has also indicated Zhu and Weng (2015); Zhu  _et al._ (2014) that
the pairing between two holes can be significantly enhanced at
$\alpha>\alpha_{c}$ beyond a simple RVB picture. In the following, an
underlying mechanism is discussed based on the incoherent behavior of a single
doped hole found in this work at $\alpha>\alpha_{c}$.

According to the single-hole wavefunction in Eq. (8), the two-hole ground
state may be constructed accordingly in the following form Zhao  _et al._
(2022)

$|\Psi\rangle_{\mathrm{2h}}=\sum_{ij}g(i,j)c_{i\uparrow}c_{j\downarrow}e^{-i(\hat{\Omega}_{i}-\hat{\Omega}_{j})}|\phi_{0}\rangle+\cdots~{},$
(24)

where the two-hole wavefunction $g(i,j)$ will be taken as the variational
parameter and the $\cdots$ term denotes the opposite chirality (the complex
conjugate) of the phase-shift operator
$e^{-i(\hat{\Omega}_{i}-\hat{\Omega}_{j})}$. Such a two-hole state has been
already variationally studied in Ref. Zhao  _et al._ , 2022 for both the 2D
square lattice and the isotropic two-leg ladder and an extremely anisotropic
two-leg ladder in Ref. Chen  _et al._ , 2018. The results are in good and
systematic agreement with the exact numerics.

Our variational study of the wavefunction in Eq. (24) confirms the strong
binding between the two holes at $\alpha>\alpha_{c}$. In particular, for each
term in Eq. (24), the distribution of the spin currents around the holes is
shown in Fig. 10 at different hole configurations (an opposite chirality is
not shown) at $\alpha=1.0$. It shows that as two holes are separated
spatially, opposite chiral spin currents emerge around the individual holes
similar to the pattern in Fig. 6. Note that since the opposite chirality of
the spin current in Fig. 6 corresponds to the opposite momentum $k^{\pm}_{0}$,
such a pairing state is Cooper-pair-like with a zero total momentum and zero
total spin. Once two holes are tightly paired in Fig. 10, the spin currents
are completely cancelled out.

Such a spin current pattern for two holes is quite similar to the 2D case Zhao
_et al._ (2022) where a roton-like (vortex-antivortex) pattern of the spin
current has been identified as mediating the important pairing force between
the holes. On the other hand, at $\alpha<\alpha_{c}$, the spin current is
absent in the single-hole-doped case, where the hole simply behaves like a
Landau quasiparticle with diminishing pairing strength Zhu and Weng (2015).
Therefore, the novel spin current associated with a non-Landau quasiparticle
at $\alpha>\alpha_{c}$ is important to the pairing between the holes. In other
words, the incoherent motion of the doped hole becomes the critical source for
pairing by which the dynamic frustration as an intrinsic component of the
single hole’s motion can get eliminated. It is pointed out that by forming a
tight pair, a charge $2e$ of two holes can be recovered Zhu and Weng (2015);
Zhu  _et al._ (2014), which otherwise is diminished to zero in the unpaired
single-hole case as studied in this work.

##  V Conclusion

The single-hole problem of a $t$-$J$ ladder is one of the simplest of doped
Mott insulators, given the gapped (undoped) spin background for an even-leg
ladder. Nevertheless, a novel phenomenon emerges in such a quasi-1D system,
which can be entirely attributed to a quantum entanglement between the doped
hole with the spin background via the nonlocal phase-string factor
$e^{-i\hat{\Omega}_{i}}$ in the ground state ansatz of Eq. (1). It is non-
perturbative in nature, whose effect is thoroughly explored by the VMC
calculation in comparison with the exact numerics in this work.

Generally speaking, the phase-string effect incorporated by
$e^{-i\hat{\Omega}_{i}}$ in the single-hole wavefunction of Eq. (1) is
important to facilitate the hopping of the doped hole on a (short-range) AF
spin background as illustrated by the ground-state energies in Table 1. Since
each hole is always accompanied by a spin-1/2 from a broken spin singlet pair,
the effect of $e^{-i\hat{\Omega}_{i}}$ can further get explicitly exhibited
via the behavior of such a hole-spin composite, whole size depends on the
spin-spin correlation in the background. The detailed consequences as
determined by the present VMC study are as follows.

Such a single-hole wavefunction ansatz can produce an excellent description of
the phase diagram for the anisotropic two-leg $t$-$J$ ladder. It covers two
distinct regions at $\alpha<\alpha_{c}$ and $\alpha>\alpha_{c}$, respectively.
At $\alpha<\alpha_{c}$, it is a conventional Landau-like quasiparticle which
can be adiabatically connected to a Bloch-wave of the bare hole created in the
spin background $|\phi_{0}\rangle$:

$c_{k_{0}\downarrow}|\phi_{0}\rangle$ (25)

at a non-degenerate $k_{0}=\pi\mod 2\pi$ with a total spin $S^{z}=1/2$. On the
other hand, at $\alpha>\alpha_{c}$, a “twisted” quasiparticle emerges as

$\tilde{c}_{k_{0}^{\pm}\downarrow}|\phi_{0}\rangle$ (26)

with double-degenerate ground state at $k_{0}^{\pm}=\pi\pm\kappa\mod 2\pi$
($\kappa\neq 0$). Here $\tilde{c}_{k_{0}\downarrow}$ may be regarded as a 1D
Bloch-wave (the Fourier transformation along the chain direction) of a twisted
particle created by
$\tilde{c}_{i\downarrow}\equiv{c}_{i\downarrow}e^{-i\hat{\Omega}_{i}}$. Since
the half-filling ground state $|\phi_{0}\rangle$ is gapped, the usual
longitudinal spin-polaron correction to either Eq. (25) or Eq. (26) is
perturbatively weak, which has been incorporated by the Lanczos method via
$|\phi_{0}\rangle\rightarrow\hat{\Pi}_{i}|\phi_{0}\rangle$ with only
quantitative improvements of the variational results.

It is found that at $\alpha>\alpha_{c}$, Eq. (26) still has a finite overlap
with Eq. (25) with the quasiparticle spectral weight $Z_{k_{0}^{\pm}}\neq 0$.
But it does not mean that the two states can be smoothly connected to each
other at the same $k_{0}^{\pm}$. The Landau’s one-to-one correspondence
principle is broken down here as there are two components in Eq. (26) for a
loosely bound hole-spin composite. Namely, besides a finite amplitude White
_et al._ (2015) for a Landau quasiparticle, an incoherent component is also
present in which a relative motion between the hole and spin-1/2 emerges
inside the composite. In the latter, the relative spin current around the hole
can carry away a continuum spectrum of momentum such that the hole becomes
incoherent, which violates the _charge_ translational invariance Zhu  _et al._
(2018) and leads to a null response to an external magnetic flux. In fact, it
is explicitly shown that the doped hole created by
$\tilde{c}_{k_{0}^{\pm}\downarrow}$ becomes a charge-neutral spinon at
$\alpha>\alpha_{c}$. The charge current carried by the doped hole disappears
in the long ladder limit, while the spin current still remains unchanged as
contributed by a spin $S^{z}=1/2$ at a finite momentum $k_{0}^{\pm}$. In
contrast to the spin-charge separation in the 1D $t$-$J$ model chain, the
charge of the doped hole simply “vanishes” in the two-leg ladder as the novel
consequence of the _transverse_ spin-polaron effect introduced by
$e^{-i\hat{\Omega}_{i}}$.

In short, the earlier DMRG discovery Zhu  _et al._ (2013) that the charge of
the doped hole is self-localized while a neutral object (spinon) still behaves
like a _free_ particle at $\alpha=1$ may be well reconciled with the latter
DMRG result White  _et al._ (2015) that there is no 1D-like spin-charge
separation and the quasiparticle $Z_{k}\neq 0$. Based on the present
wavefunction description, the doped hole actually becomes a loosely bound
hole-spin composite here such that it still carries an well-defined spin-1/2
but its charge degree of freedom becomes incoherent due to the unscreened
phase-string effect. As the essence of the doped Mott physics, the latter will
generally break the charge translational symmetry, which may only be recovered
either at $\alpha<\alpha_{c}$ or by pairing up of two holes Zhu  _et al._
(2014); Zhao  _et al._ (2022). Lastly it is noted that the two-component
structure in the present single-hole wavefunction may be further considered as
a precursor of the “Fermi arc” physics at finite doping as recently explored
in Ref. Zhang and Weng, 2022.

Finally, we have examined the pairing between between two doped holes. As
shown by DMRG Zhu and Weng (2015), the pairing strength is rather weak at
$\alpha<\alpha_{c}$, but gets substantially enhanced at $\alpha>\alpha_{c}$.
It is due to the spin-current-carrying component that is strongly compensated
once two holes form a tightly-bound pair to gain a substantial binding energy
at $\alpha>\alpha_{c}$. It means that a novel pairing mechanism is also
crucially related to the non-Landau behavior of the single hole at
$\alpha>\alpha_{c}$ via the nonlocal phase-shift factor originated from the
hidden phase-string sign structure in the $t$-$J$ model.

###### Acknowledgements.

Very helpful discussions with Qingrui Wang, Yang Qi, Zheng Zhu, Donna Sheng,
Xiaoliang Qi, and Jan Zaanen are acknowledged. This work is partially
supported by MOST of China (Grant No. 2021YFA1402101).

##  Appendix A Phase-shift operator on a two-leg ladder with PBC

Despite that the essential physics should not be affected by boundary
conditions in the long ladder limit, using PBC makes it more convenient to
detect translationally invariant properties, such as momentum and any non-zero
current configurations at finite-size calculations. It also makes it simpler
to measure the response to external electromagnetic fields by inserting flux
into the ring formed by the ladder. However, the original phase-shift operator
$\hat{\Omega}_{i}$ is defined on a lattice under OBC Wang  _et al._ (2015). To
capture these properties explicitly in the variational wavefunction Eq. (8),
we need to generalize the original single-hole wavefunction Wang  _et al._
(2015) to the case under PBC:

$|\Psi_{\mathrm{G}}\rangle_{\mathrm{1h}}=\sum_{i,a}\varphi_{h}^{(a)}(i)e^{-i\hat{\Omega}_{i}^{a}}c_{i\bar{\sigma}}|\phi_{0}\rangle~{},$
(27)

where the phase-shift operator is defined by

$\hat{\Omega}^{a}_{i}=\sum_{l(\neq i)}\theta^{a}_{i}(l)n_{l\downarrow}~{},$
(28)

and the definition of the statistical angles $\theta^{a}_{i}(l)$ on a PBC
ladder is shown in Fig. 2. The two legs of the ladder are bent into two rings
with different radii on the same plane, i.e., $r_{\mathrm{in}}=2-\lambda$ and
$r_{\mathrm{out}}=2+\lambda$, with $\lambda$ a variational parameter to be
determined by minimizing the variational energy. The phase $\theta^{a}_{i}(l)$
can be defined as a conventional 2D angle:
$\theta^{a}_{i}(l)=\pm\mathrm{Im}\ln(z_{i}-z_{l})$ where $z_{i}=x_{i}+iy_{i}$
is the complex coordinate of site $i$. The superscript $a=1,2$ here labels two
choices of the radii for the two chains of the ladder, i.e. for $a=1$ the
first chain is bent into the inner ring and the second chain into the outer
ring; while for $a=2$ the second chain becomes the inner ring and the first
chain the outer one.

Under the PBC with explicit translational invariant symmetry, the variational
parameter $\varphi_{h}^{(a)}$ is constraint to be a plane wave

$\varphi_{h}^{(a)}(i)\propto
e^{i\tilde{\mathbf{k}}^{\pm}\cdot\mathbf{r}_{i}}~{},$ (29)

where $\tilde{\mathbf{k}}^{\pm}=(\tilde{k}_{x}^{\pm},\tilde{k}_{y}^{\pm})$ is
a variational parameter to be determined by minimizing the ground state
energy. The relation between $\tilde{{k}}^{\pm}_{x}$ and the total momentum of
the ground state ${k}^{\pm}_{x}$ along the chain direction can be derived by a
one-step translational transformation $\hat{T}$ along the chain direction
acting on the translational invariant ground state
$|\Psi_{\mathrm{G}}\rangle_{\mathrm{1h}}$

$\displaystyle\hat{T}|\Psi_{\mathrm{G}}\rangle_{\mathrm{1}h}\equiv$
$\displaystyle e^{-ik_{x}^{\pm}}|\Psi_{\mathrm{G}}\rangle_{\mathrm{1}h}$ (30)
$\displaystyle=$ $\displaystyle\sum_{i}\varphi_{h}^{(a)}(i)e^{-i\sum_{l(\neq
i)}\theta^{a}_{i}(l)n_{l+\mathbf{e}_{x},\downarrow}}c_{{i+\mathbf{e}_{x}},\bar{\sigma}}|\phi_{0}\rangle$
$\displaystyle=$
$\displaystyle\sum_{i}\varphi_{h}^{(a)}(i-\mathbf{e}_{x})e^{-i\sum_{l(\neq
i)}\theta_{i-\mathbf{e}_{x}}^{a}(l-\mathbf{e}_{x})n_{l\downarrow}}c_{i\bar{\sigma}}|\phi_{0}\rangle$
$\displaystyle=$ $\displaystyle e^{-i\tilde{k}_{x}^{\pm}}e^{-i\sum_{l(\neq
i)}(\theta_{i}^{a}(l)\mp
2\pi/N_{x})n_{l\downarrow}}c_{i\bar{\sigma}}|\phi_{0}\rangle$ $\displaystyle=$
$\displaystyle e^{-i\tilde{k}_{x}^{\pm}\mp
i(\sigma+1)\pi/N_{x}}|\Psi_{\mathrm{G}}\rangle_{\mathrm{1}h}~{},$

where $\mathbf{e}_{x}$ denotes the unit vector along the chain direction, and
the relation
$\theta^{a}_{i-\mathbf{e}_{x}}(l-\mathbf{e}_{x})=\theta^{a}_{i}(l)\mp
2\pi/N_{x}$ is used. The above expression shows that
$k_{x}^{\pm}=\tilde{k}_{x}^{\pm}+K_{x}^{\pm}(\hat{\Omega})$, Where
$K_{x}^{\pm}(\hat{\Omega})=\pm(\sigma+1)\pi/N_{x}$ denotes the momentum
contributed by the phase factor $e^{\mp i\hat{\Omega}_{i}}$. As $K_{x}$
vanishes for $\sigma=-1$, and can be cancelled for $\sigma=+1$ by a
redefinition of the phase factor $\hat{\Omega}_{i}\rightarrow\sum_{l(\neq
i)}\theta^{a}_{i}(l)n_{l\uparrow}$, we no longer distinguish $k_{x}$ and
$\tilde{k}_{x}$ by only considering the $\sigma=-1$ case without loss of
generality.

For the $y$ component momentum ${k}_{y}$, we can get a similar expression
$k_{y}^{\pm}=\tilde{k}_{y}^{\pm}+K_{y}^{\pm}(\hat{\Omega})$. But for a given
index $a$, the $Z_{2}$ symmetry by exchanging the two chains is broken in a
single configuration shown in Fig. 2. This symmetry can be restored after the
summation over the index $a$, and the total momentum $k_{y}^{\pm}$ remains a
good quantum number. For all the parameter regions we are interested in this
article, we always find $k_{y}^{\pm}=0$ for the ground state. We therefore
keep only the $x$ component momentum $k_{x}$ in the main text and denote it
simply as $k$, keeping in mind that it represents a momentum along the chain
direction with $k_{y}=0$.

The advantage of the circular configuration shown in Fig. 2 can be explicitly
seen by taking the single-hole doped 1D $t$-$J$ chain as an example, which is
detailed studied in Ref. Zhu  _et al._ , 2016. By constructing a similar
circular configuration to define $\theta_{i}(l)$ in a 1D ring, one sees that
every time when the hole hops from site $i$ to its neighboring site $i\pm 1$,
it will pick up a phase shift $\theta_{i}(l)-\theta_{i\pm 1}(l)=\pm\pi/N_{x}$
from each down spin at site $l$ (which is just the circular angle and thus
independent to $i$ and $l$). When the phase shifts from all the down spins are
summed over, one gets an effective phase $\pm
N_{\downarrow}\pi/N_{x}\approx\pm\pi/2$ every time when the doped hole hops.
Such $\pm\pi/2$ phase as contributed by all the background spins just
constitutes the ground state momentum of a single hole doped $t$-$J$ chain,
giving rise to a simple example of Anderson’s phase shift idea.

Figure 11: The phase diagrams based on $Q_{0}$ calculated by VMC before
(green) and after (red) a power 1 Lanczos procedure is applied to incorporate
a longitudinal spin-polaron effect. Note that the two phase diagrams are in
qualitative agreement with only a global shift along the $\alpha$ axis with
the correction.

##  Appendix B Longitudinal spin-polaron effect on the varational ansatz

In the main text, we have shown that the phase-shift operator
$e^{-i\hat{\Omega}_{i}}$ plays the essential role in the non-Landau-type
single-hole state of Eq. (8). It represents the singular phase-string effect
which results in the transverse spin current surrounding the doped hole to
facilitate its hopping. The novel physical properties of the non-Landau
quasiparticle at $\alpha>\alpha_{c}$ can be entirely attributed to such a
transverse spin distortion as a many-body “cloud” associated with the hole.

On the other hand, the conventional spin distortion or the “longitudinal spin-
polaron effect” should be generally present in both regimes of
$\alpha<\alpha_{c}$ and $\alpha>\alpha_{c}$. Since the spin background is
gapped at half-filling, such a local spin distortion is expected to be
perturbatively weak such that one may use an approximation similar to the
power 1 Lanczos method to incorporate it to further improve the variational
ground state $|\Psi_{\mathrm{G}}\rangle$. In the following we outline the
corresponding procedure.

For a trial state $|\Psi_{\mathrm{G}}\rangle$, the power 1 Lanczos method is
to construct a new trial state $|\Psi_{\mathrm{G}}\rangle+\gamma
H|\Psi_{\mathrm{G}}\rangle$. Here $H$ is the system Hamiltonian, and $\gamma$
is a new variational parameter to be optimized. Physically, such a one-step
evolution of the Hamiltonian $H$ on $|\Psi_{\mathrm{G}}\rangle$ takes a local
distortion into consideration, and the exact ground state can be achieved if
the above procedure is iterated continuously until convergence.

In the spirit of power 1 Lanczos method, then we consider
$c_{i\downarrow}|\phi_{0}\rangle\rightarrow
c_{i\downarrow}\hat{\Pi}_{i}|\phi_{0}\rangle$ with $\hat{\Pi}_{i}$
incorporating the local hopping effect produced by $H_{t}$ to the first order
of correction:

$c_{i\downarrow}\hat{\Pi}_{i}=c_{i\downarrow}+\sum_{j=\in\mathrm{NN}(i)}\left[a^{1}(i,j)c_{i\downarrow}n_{j\downarrow}+a^{2}(i,j)c_{i\uparrow}S_{j}^{+}\right]$
(31)

where $a^{1}(i,j)$ and $a^{2}(i,j)$ are two new variational parameters. Then a
new variational wavefunction is constructed

$|\tilde{\Psi}_{\mathrm{G}}\rangle_{\mathrm{1h}}=\sum_{i}\varphi^{(a)}_{h}(i)e^{-i\hat{\Omega}^{a}_{i}}c_{i\downarrow}\hat{\Pi}_{i}|\phi_{0}\rangle\\\
$ (32)

Figure. 11 shows the phase diagram determined by VMC before and after the
above power 1 Lanczos method is applied. It shows that the overall line shape
of $Q_{0}$ vs. $\alpha$ is unchanged, but there is a global shift along the
$\alpha$ axis, which makes the VMC result in excellent agreement with the DMGR
result as shown in Fig. 1(b). We therefore conclude that although the
qualitative physics are unchanged by the longitudinal local spin-polaron
effect, which is solely decided by the transverse phase-shift field due to the
phase-string, the local spin distortion can still effectively improve the
variational energy, momentum $k_{0}^{\pm}$, and even the value of $\alpha_{c}$
to be quantitative agreement with the DMRG results.

Finally, we mention that the standard power Lanczos method up to power 2 is
used in the benchmarking calculation of the ground state energy in a small
lattice size as shown in Table. 1, with the optimized wavefunction also
denoted as $|\tilde{\Psi}_{\mathrm{G}}\rangle$ and
$|\tilde{\Psi}_{\mathrm{B}}\rangle$ All the other VMC results presented in the
main text are optimized based on Eq. (32).

##  Appendix C Variational Monte Carlo procedure

The procedures of the variational Monte Carlo calculations were already
explained in great detail in previous works Wang  _et al._ (2015); Chen  _et
al._ (2019, 2018); Zhao  _et al._ (2022). For completeness of the article, in
this appendix, we briefly introduce the variational Monte Carlo procedures
used to optimize the variational ground state and to measure other
observables.

Before turning to the hole doped problem, we first need a wavefunction
$|\phi_{0}\rangle$ of the half filling state, where the $t$-$J$ model reduces
to the Heisenberg spin model on a bipartite square lattice. It was shown that
the ground state $|\phi_{0}\rangle$ can be well simulated by the Liang-Doucot-
Anderson type wave function Liang  _et al._ (1988).

$|\phi_{0}\rangle=\sum_{v}\omega_{v}|v\rangle~{},$ (33)

where the valence bond (VB) state

$|v\rangle=|(a_{1},b_{1})\cdots(a_{n},b_{n})\rangle~{}$ (34)

consist of singlet pairs
$|(a,b)\rangle=|\uparrow_{a}\downarrow_{b}\rangle-|\downarrow_{a}\uparrow_{b}\rangle$.
Here $a$ and $b$ comes from the two different sublattices A and B of the
bipartite lattice, respectively. According to the Marshall sign rule on a
bipartite lattice, the variational parameter $w_{v}$ of each valence bond
state $|v\rangle$ is always positive, which can therefore be used to construct
a probability distribution for the Monte Carlo procedure

$P(v^{\prime},v)=\frac{w_{v^{\prime}}w_{v}\langle
v^{\prime}|v\rangle}{\langle\phi_{0}|\phi_{0}\rangle}~{},$ (35)

which satisfies $\sum_{v^{\prime},v}P(v^{\prime},v)=1$ and
$P(v^{\prime},v)>0$. The expectation value of any observable $\hat{O}$ can
then be evaluated as

$\frac{\langle\phi_{0}|\hat{O}|\phi_{0}\rangle}{\langle\phi_{0}|\phi_{0}\rangle}=\sum_{v^{\prime},v}P(v^{\prime},v)O(v^{\prime},v)~{},$
(36)

as long as one gets the expression of $O(v^{\prime},v)$.

After a hole is doped into the spin background $|\phi_{0}\rangle$, we follow
the same idea to express all the observables on $|\Psi_{\mathrm{G}}\rangle$ in
terms of expectation values of the spin background $|\phi_{0}\rangle$. By
fixing the normalization condition of
$|\Psi_{\mathrm{G}}\rangle_{\mathrm{1h}}$ as

$\frac{\tensor{[}_{1\mathrm{h}}]{\langle\Psi_{\mathrm{G}}|\hat{\mathds{1}}|\Psi_{\mathrm{G}}\rangle}{{}_{1\mathrm{h}}}}{\langle\phi_{0}|\phi_{0}\rangle}=1~{},$
(37)

the expectation value of an arbitrary observable $\hat{O}$ is then a quadratic
form of the variational parameters $\varphi^{a}_{h}(i)$

$\langle\hat{O}\rangle\equiv\frac{\tensor{[}_{1\mathrm{h}}]{\langle\Psi_{\mathrm{G}}|\hat{O}|\Psi_{\mathrm{G}}\rangle}{{}_{1\mathrm{h}}}}{\langle\phi_{0}|\phi_{0}\rangle}=\bm{\varphi}^{\dagger}\mathbf{O}\bm{\varphi}~{},$
(38)

where the variational parameters $\varphi_{h}^{a}(i)$ are written as a vector
$\bm{\varphi}$, and $\mathbf{O}$ is a Hermitian matrix with its matrix
elements as

$\mathbf{O}_{ia}^{i^{\prime}a^{\prime}}=\sum_{v^{\prime},v}P(v^{\prime},v)\frac{\langle
v^{\prime}|e^{+i\hat{\Omega}^{a^{\prime}}_{i^{\prime}}}c^{\dagger}_{i^{\prime}\downarrow}\hat{O}c_{i\downarrow}e^{-i\hat{\Omega}_{i}^{a}}|v\rangle}{\langle
v^{\prime}|v\rangle}~{}.$ (39)

The Monte Carlo sampling procedure can then be used here to measure the matrix
elements in Eq. (39) by sampling different transposition-graph covers
$(v^{\prime},v)$. To improve the Monte Carlo sampling efficiency, the loop
update procedure introduced in Ref. Sandvik and Evertz (2010) is used here in
the updating progress of $(v^{\prime},v)$.

We note that Eq. (37) actually also falls into the form of Eq. (38), with
$\hat{O}$ taken as the identity operator $\hat{\mathds{1}}$,

$1=\langle\hat{\mathds{1}}\rangle=\bm{\varphi}^{\dagger}\mathbf{A}\bm{\varphi}~{},$
(40)

which is also a quadratic form with the corresponding matrix denoted as
$\mathbf{A}$ here. To get the variational energies, we take the operator
$\hat{O}$ in Eq. (38) as the Hamiltonian $\hat{H}$

$E_{\mathrm{G}}\equiv\langle\hat{H}\rangle=\bm{\varphi}^{\dagger}\mathbf{H}\bm{\varphi}~{}.$
(41)

Combined with the normalization condition Eq. (40), the procedure to optimize
$E_{\mathrm{G}}$ then turns out to be a generalized eigenvalue problem

$\mathbf{H}\bm{\varphi}=E_{\mathrm{G}}\mathbf{A}\bm{\varphi}~{}.$ (42)

To explicitly calculate the matrix elements Eq. (39), we follow Ref. Zhao  _et
al._ , 2022 to transform all the operators to the following form:

$\frac{\langle
v^{\prime}|e^{+i\hat{\Omega}_{i^{\prime}}^{a^{\prime}}}n_{k_{1}\sigma_{1}}n_{k_{2}\sigma_{2}}\cdots
n_{k_{n}\sigma_{n}}S_{l_{1}}^{\sigma_{1}}S_{l_{2}}^{\sigma_{2}}\cdots
S_{l_{s}}^{\sigma_{s}}e^{-i\hat{\Omega}_{i}^{a}}|v\rangle}{\langle
v^{\prime}|v\rangle}~{}.$ (43)

Then for every pair of $v^{\prime}$ and $v$, one can check the compatibility
of the $n_{k_{i}\sigma_{i}}$ s and $S_{l_{j}}^{\sigma_{j}}$ s with the loop
configurations in the transposition-graph covers $(v^{\prime},v)$, as well as
calculate the phase factors $e^{i\hat{\Omega}_{i}^{a}}$ for every loop. In the
following we give the explicit formulas of Eq. (39) for some important
operators measured in the main text.

For the normalization condition where $\hat{O}=\hat{\mathds{1}}$, the matrix
elements of $\mathbf{A}$ are given by

$\mathbf{A}^{i^{\prime}a^{\prime}}_{ia}=\sum_{v^{\prime},v}P(v^{\prime},v)\frac{1}{\langle
v^{\prime}|v\rangle}\delta_{ii^{\prime}}\langle
v^{\prime}|e^{+i\hat{\Omega}_{i^{\prime}}^{a^{\prime}}}n_{i\downarrow}e^{-i\hat{\Omega}_{i}^{a}}|v\rangle~{}.$
(44)

The hole density at every site $n_{i}^{h}$ can be directed evaluated from the
$\mathbf{A}$ matrix with

$n^{h}_{i}=\sum_{a,a^{\prime}}\mathbf{A}_{ia}^{ia^{\prime}}~{}.$ (45)

The hopping Hamiltonian $\hat{H}_{t}$ Eq. (5) connectes two NN sites $i$ and
$i^{\prime}\pm\mathbf{e}_{\alpha}$

$\displaystyle(\mathbf{H}_{t})_{ia}^{i^{\prime}a^{\prime}}=$ $\displaystyle
t\sum_{v^{\prime},v,\alpha=x,y}P(v^{\prime},v)\frac{1}{\langle
v^{\prime}|v\rangle}\delta_{ii^{\prime}\pm\mathbf{e}_{\alpha}}$ (46)
$\displaystyle\times\langle
v^{\prime}|e^{+i\hat{\Omega}_{i^{\prime}}^{a^{\prime}}}(n_{i\downarrow}n_{i^{\prime}\downarrow}+S_{i}^{+}S_{i^{\prime}}^{-})e^{-i\hat{\Omega}_{i}^{a}}|v\rangle~{},$

where $\mathbf{e}_{x,y}$ are the $x$ and $y$ direction unit vectors. The
charge current $J^{c}_{ij}$ can also be derived from the hopping matrix by
just taking the imaginary part of it

$J^{c}_{ij}=\sum_{a,a^{\prime}}\mathrm{Im}\left((\varphi^{a^{\prime}}_{h}(j))^{*}(\mathbf{H}_{t})_{ia}^{ja^{\prime}}\varphi_{h}^{a}(i)\right)~{}.$
(47)

The superexchange Hamiltonian $\hat{H}_{J}$ Eq. (6) does not change the
position of the hole

$\displaystyle(\mathbf{H}_{J})_{ia}^{i^{\prime}a^{\prime}}=$
$\displaystyle\frac{J}{2}\sum_{v^{\prime},v,\langle kl\rangle(\neq
i)}P(v^{\prime},v)\frac{1}{\langle
v^{\prime}|v\rangle}\delta_{ii^{\prime}}\langle
v^{\prime}|e^{+i\hat{\Omega}_{i^{\prime}}^{a^{\prime}}}n_{i\downarrow}$ (48)
$\displaystyle\times(S_{k}^{+}S_{l}^{-}+S_{k}^{-}S_{l}^{+}-n_{k\uparrow}n_{l\downarrow}-n_{k\downarrow}n_{l\uparrow})e^{-i\hat{\Omega}_{i}^{a}}|v\rangle$

The neutral spin current $\hat{J}_{kl}^{s}$ defined in Eq. (12) has similar
matrix elements with the superexchange term:

$\displaystyle(\mathbf{J}_{kl}^{s})^{i^{\prime}a^{\prime}}_{ia}=$
$\displaystyle-i\frac{J}{2}(1-\delta_{ik})(1-\delta_{il})\sum_{v^{\prime},v}P(v^{\prime},v)\frac{1}{\langle
v^{\prime}|v\rangle}\delta_{ii^{\prime}}$ (49) $\displaystyle\times\langle
v^{\prime}|e^{+i\hat{\Omega}_{i^{\prime}}^{a^{\prime}}}n_{i\downarrow}(S_{k}^{+}S_{l}^{-}-S_{k}^{-}S_{l}^{+})e^{-i\hat{\Omega}_{i}^{a}}|v\rangle$

Finally, for the quasiparticle weight, $Z_{k}$, we need to normalize Eq. (4)
as

$\displaystyle Z_{k}=$
$\displaystyle\left|\frac{\langle\phi_{0}|c_{k\downarrow}^{\dagger}|\Psi_{\mathrm{G}}\rangle_{\mathrm{1}h}}{\sqrt{\langle\phi_{0}|\phi_{0}\rangle\tensor{[}_{\mathrm{1}h}]{\langle\Psi_{\mathrm{G}}|\Psi_{\mathrm{G}}\rangle}{{}_{\mathrm{1}h}}}}\right|^{2}$
(50) $\displaystyle=$
$\displaystyle\left|\sum_{i,a}\varphi^{a}_{h}(i)\frac{\langle\phi_{0}|c^{\dagger}_{k\downarrow}c_{i\downarrow}e^{-i\hat{\Omega}^{a}_{i}}|\phi_{0}\rangle}{\langle\phi_{0}|\phi_{0}\rangle}\right|^{2}$
$\displaystyle=$
$\displaystyle\left|\sum_{i^{\prime}a^{\prime},ia}\frac{1}{N}e^{ikx_{i^{\prime}}}\varphi^{a}_{h}(i)\sum_{v^{\prime},v}P(v^{\prime},v)\frac{\langle
v^{\prime}|n_{i\downarrow}e^{-i\hat{\Omega}_{i}^{a}}|v\rangle}{\langle
v^{\prime}|v\rangle}\right|^{2},$

which is a similar quadratic formula with Eq. (38) if we treat
$\frac{1}{N}e^{-ikx_{i^{\prime}}}$ as a new wavefunction with $a=0$ as
$\varphi^{0}_{h}(i^{\prime})$.

When the Lanczos procedure in Appendix B is taken into considered, the above
expressions will become too complicated to be calculated by hands. Actually,
we write a program to generate the expressions automatically in realistic
calculations, and the explicit expressions will be not given here.

## References

  * Mott (1949) N. F. Mott, “The Basis of the Electron Theory of Metals, with Special Reference to the Transition Metals,” Proc. Phys. Soc. A 62, 416–422 (1949).
  * Anderson (1987) P. W. Anderson, “The Resonating Valence Bond State in $\mathrm{La}_{2}\mathrm{CuO}_{4}$ and Superconductivity,” Science 235, 1196–1198 (1987).
  * Imada  _et al._ (1998) Masatoshi Imada, Atsushi Fujimori, and Yoshinori Tokura, “Metal-Insulator Transitions,” Rev. Mod. Phys. 70, 1039–1263 (1998).
  * Lee  _et al._ (2006) Patrick A. Lee, Naoto Nagaosa, and Xiao Gang Wen, “Doping a Mott Insulator: Physics of High-Temperature Superconductivity,” Rev. Mod. Phys. 78, 17–85 (2006), arXiv:cond-mat/0410445 .
  * Schmitt-Rink  _et al._ (1988) S. Schmitt-Rink, C. M. Varma, and A. E. Ruckenstein, “Spectral Function of Holes in a Quantum Antiferromagnet,” Phys. Rev. Lett. 60, 2793–2796 (1988).
  * Kane  _et al._ (1989) C. L. Kane, P. A. Lee, and N. Read, “Motion of a Single Hole in a Quantum Antiferromagnet,” Phys. Rev. B 39, 6880–6897 (1989).
  * Anderson (1990) P. W. Anderson, ““Luttinger-Liquid” Behavior of the Normal Metallic State of the 2D Hubbard Model,” Phys. Rev. Lett. 64, 1839–1841 (1990).
  * Sheng  _et al._ (1996) D. N. Sheng, Y. C. Chen, and Z. Y. Weng, “Phase String Effect in a Doped Antiferromagnet,” Phys. Rev. Lett. 77, 5102–5105 (1996).
  * Wu  _et al._ (2008) K. Wu, Z. Y. Weng, and J. Zaanen, “Sign Structure of the $t$-$J$ model,” Phys. Rev. B 77, 155102 (2008), arXiv:0802.0273 .
  * Zhang and Weng (2014) Long Zhang and Zheng Yu Weng, “Sign structure, electron fractionalization, and emergent gauge description of the Hubbard model,” Phys. Rev. B 90, 165120 (2014), arXiv:1406.6867 .
  * Zheng  _et al._ (2018) Wayne Zheng, Zheng Zhu, D. N. Sheng, and Zheng-Yu Weng, “Hidden Spin Current in Doped Mott Antiferromagnets,” Phys. Rev. B 98, 165102 (2018), arXiv:1802.05977 .
  * Chen  _et al._ (2019) Shuai Chen, Qing-Rui Wang, Yang Qi, D. N. Sheng, and Zheng-Yu Weng, “Single-Hole Wave Function in Two Dimensions: A Case Study of the Doped Mott Insulator,” Phys. Rev. B 99, 205128 (2019), arXiv:1812.05627 .
  * Shraiman and Siggia (1988) Boris I. Shraiman and Eric D. Siggia, “Mobile Vacancies in a Quantum Heisenberg Antiferromagnet,” Phys. Rev. Lett. 61, 467–470 (1988).
  * Shraiman and Siggia (1989) Boris I. Shraiman and Eric D. Siggia, “Spiral Phase of a Doped Quantum Antiferromagnet,” Phys. Rev. Lett. 62, 1564–1567 (1989).
  * Weng (1991) Z. Y. Weng, “Dynamical Spiral State at Finite Doping,” Phys. Rev. Lett. 66, 2156–2159 (1991).
  * Zhu  _et al._ (2013) Zheng Zhu, Hong-Chen Jiang, Yang Qi, Chushun Tian, and Zheng-Yu Weng, “Strong Correlation Induced Charge Localization in Antiferromagnets,” Solar Phys. 3, 2586 (2013), arXiv:1212.6634 .
  * Zhu and Weng (2015) Zheng Zhu and Zheng-Yu Weng, “Quasiparticle Collapsing in an Anisotropic $t$-$J$ Ladder,” Phys. Rev. B 92, 235156 (2015), arXiv:1409.3241 .
  * Zhu  _et al._ (2015) Zheng Zhu, Chushun Tian, Hong-Chen Jiang, Yang Qi, Zheng-Yu Weng, and Jan Zaanen, “Charge modulation as fingerprints of phase-string triggered interference,” Phys. Rev. B 92, 035113 (2015).
  * Zhu  _et al._ (2018) Zheng Zhu, D. N. Sheng, and Zheng-Yu Weng, “Intrinsic Translational Symmetry Breaking in a Doped Mott Insulator,” Phys. Rev. B 98, 035129 (2018), arXiv:1707.00068 .
  * White  _et al._ (2015) S. R. White, D. J. Scalapino, and S. A. Kivelson, “One Hole in the Two-Leg $t$-$J$ Ladder and Adiabatic Continuity to the Noninteracting Limit,” Phys. Rev. Lett. 115, 056401 (2015), arXiv:1502.04403 .
  * Zhou and Ng (2013) Yi Zhou and Tai-Kai Ng, “Spin Liquid States in the Vicinity of a Metal-insulator Transition,” Phys. Rev. B 88, 165130 (2013), arXiv:1302.0157 .
  * Zhu  _et al._ (2014) Zheng Zhu, Hong Chen Jiang, D. N. Sheng, and Zheng Yu Weng, “Nature of Strong Hole Pairing in Doped Mott Antiferromagnets,” Sci. Rep. 4, 5419 (2014), arXiv:1312.6893 .
  * Weng (2011) Zheng-Yu Weng, “Superconducting Ground State of a Doped Mott Insulator,” New J. Phys. 13, 103039 (2011), arXiv:1105.3027 .
  * Wang  _et al._ (2015) Qing-Rui Wang, Zheng Zhu, Yang Qi, and Zheng-Yu Weng, “Variational Wave Function for an Anisotropic Single-hole-doped $t$-$J$ Ladder,” (2015), arXiv:1509.01260 .
  * Zhao  _et al._ (2022) Jing-Yu Zhao, Shuai A. Chen, Hao-Kai Zhang, and Zheng-Yu Weng, “Two-Hole Ground State: Dichotomy in Pairing Symmetry,” Phys. Rev. X 12, 011062 (2022), arXiv:2106.14898 .
  * Zhu  _et al._ (2016) Zheng Zhu, Qing-Rui Wang, D. N. Sheng, and Zheng-Yu Weng, “Exact Sign Structure of the $t$-$J$ Chain and the Single Hole Ground State,” Nucl. Phys. B 903, 51–77 (2016), arXiv:1510.07634 .
  * Chen  _et al._ (2018) Shuai Chen, Zheng Zhu, and Zheng-Yu Weng, “Two-hole Ground State Wavefunction: Non-BCS Pairing in a $t$-$J$ Two-leg Ladder,” Phys. Rev. B 98, 245138 (2018), arXiv:1808.06173 .
  * Zhang and Weng (2022) Jia-Xin Zhang and Zheng-Yu Weng, “Crossover from Fermi Arc to Full Fermi Surface,” , 1–20 (2022), arXiv:2208.10519 .
  * Liang  _et al._ (1988) S. Liang, B. Doucot, and P. W. Anderson, “Some New Variational Resonating-Valence-Bond-Type Wave Functions for the Spin-$\frac{1}{2}$ Antiferromagnetic Heisenberg Model on a Square Lattice,” Phys. Rev. Lett. 61, 365–368 (1988).
  * Sandvik and Evertz (2010) Anders W. Sandvik and Hans Gerd Evertz, “Loop Updates for Variational and Projector Quantum Monte Carlo Simulations in the Valence-Bond Basis,” Phys. Rev. B 82, 024407 (2010), arXiv:0807.0682 .