# Nonlocal Fermions with Local Interactions and the SYK Model

G. C. Levine  Department of Physics and Astronomy, Hofstra University,
Hempstead, NY 11549

###### Abstract

One of the most promising routes to non-fermi liquids and strange metals has
been through SYK models Sachdev:2010um , which necessarily involve large
flavor degrees of freedom and interactions with imposed disorder. We introduce
an interacting model of nonlocal spinless fermions in which large flavor and
effective disorder emerge spontaneously in the extreme nonlocal limit. This
model may be thought of as an expansion in a dimensionless ”locality scale,”
$\alpha$, with the limit $\alpha\rightarrow 0$ recovering a conventional local
action. For finite $\alpha$, the resulting interacting nonlocal action
exhibits a large number of low-energy degrees of freedom—proportional to
$\alpha$—and the structure factor mediating the local interactions between
these fermions is effectively random for large $\alpha$. In one dimension,
with finite $\alpha$, we show that interactions are marginal to the one loop
level (as they are in the conventional local case), preserving a gapless
phase. At large $\alpha$ the interaction strength becomes comparable to the
bandwidth and we analyze this large flavor limit in a manner similar to the
diagrammatic approach to SYK. As $\alpha$ is increased we argue that the
gapless phase established by conventional RG possibly crosses over to a
gapless SYK phase, although the ”melon” diagrams are not exclusively dominant.
We speculate on how this model might arise physically from an S-matrix
connecting pure excited states—rather than the vacuum—to access finite
temperature interacting fermions.

††preprint: cond-mat

##  I Introduction

Nonlocal quantum lattice models and field theories have appeared in many
subfields of physics, spanning quantum information, condensed matter physics
and quantum gravity. Random models such as Sachdev:2010um  have been
introduced in connection with the phenomenology of strange metals, but also
appear to be important in constructing exactly solvable quantum models with
gravitational duals Maldacena:2016hyu . Nonlocal models have also appeared in
the study of the thermalization hypothesis providing the rigorous basis of the
canonical ensemble of statistical mechanics Magan:2015yoa . Attempts to
understand the internal degrees of freedom of black holes, consistent with the
no-cloning theorem, have led to the notion of ”fast-scrambling” Hayden:2007cs
; Sekino:2008he  which also can be realized with certain nonlocal models
Magan:2016ojb ; Swingle:2016var . Entanglement entropy of nonlocal bosons was
first studied analytically in Li:2010dr  and numerically in Shiba:2013jja ,
with nonlocality extended to fermions in Levine (2019). There have been
several analytical works on the role that nonlocal QFT plays in flat space
holography Bagchi:2014iea ; Kachru:2018 ; Pang:2014tpa .

The motivation for the present work is threefold. First, we introduce a model
to study interactions of nonlocal fermions. This model becomes—in effect—a
large flavor model and we argue that interactions may be treated by
conventional renormalization group techniques. Second, we explore the
possibility of local interactions producing an effective random interaction,
transforming the large flavor model into an effective SYK model. The need for
a mechanism to produce disorder in applications of SYK physics to condensed
matter is self-evident, but the need for such a mechanism in applications of
SYK to quantum gravity (see for instance, Witten (2019)) has also been
emphasized. Finally, there is the established correspondence between the
ground state of fermionic nonlocal models and an excited state (with a
specific energy) of the corresponding local, free fermion models Jafarizadeh &
Rajabpour (2019). As such free excited states have been shown to thermalize on
a sufficiently short length scale Lai & Yang (2015), nonlocal physics with
interactions may provide new theoretical machinery to apply to finite
temperature transport and relaxation phenomenon such as in the phenomenology
of strange metals.

To introduce this topic generally, consider the following 1-d hamiltonian for
free, nonrelativistic, spinless fermions in the continuum Levine (2019):

$H_{\rm
c}=\frac{\epsilon}{\alpha^{2}}\int{dx\psi^{\dagger}(x)\cos{(i\alpha\partial_{x})}\psi(x)}$
(1)

where $\partial_{x}$ is a spatial derivative, $\alpha$ is a dimensionful
locality scale and $\epsilon$ is an energy parameter with units of energy
$\times$ length. Noting the generator of finite translations,

$\psi(x+\alpha)=e^{\alpha\partial_{x}}\psi(x)$ (2)

the nonlocal kinetic operator simply expresses finite translations to adjacent
points rather than a diffusion operator. The limit $\alpha\rightarrow 0$
recovers the conventional non-relativistic kinetic energy operator. Since the
kinetic operator is a harmonic function, we refer to this model as ”compact”
nonlocality.

Figure 1:  Depiction of entanglement entropy spanning local and nonlocal
regimes based upon numerical calculations in Levine (2019). The 2d entropy has
been rescaled by $L$. Entropy satisfies a volume law for small $L$ and a
logarithmic law characteristic of gapless fermions for large $L$. The locality
scale, $\alpha$, dividing nonlocal $(L<\alpha)$ and local $(L>\alpha)$ regimes
is also proportional to the number of fermion flavors (the central charge) in
the microscopic calculation. Thus the large $L$ (local) behavior for the
entropy is proportional to $\alpha\log{L}$. The prefactor $\alpha$ in the
latter expression is also suggested ”naturally” by the continuity of the
graph.

The kinetic energy operator now involves derivatives of all orders and
nonlocality but leads to a well-defined theory. Placing this theory on a
lattice, so that $\alpha$ is dimensionless and the fundamental degrees of
freedom are finite, this hamiltonian simply leads to an energy momentum
dispersion relation that is periodic in $\alpha/2\pi$. Considering an average
occupancy of 1/2, it is readily seen that rather than two fermi flavors (two
fermi points in 1-d), there are $\cal{O}(\alpha)$ fermi flavors.

If $\alpha/\pi$ is large and irrational, nearby energy states below the Fermi
level will be drawn from widely disparate wavenumbers, maintaining the energy
volume of the Fermi sea but scrambling the energy order of the translation
invariant eigenfunctions. Thus the nonlocal hamiltonian has a ground state
that corresponds to an excited state of the underlying local model. Quantum
many body states exhibiting an entropy volume law were first realized as
excited states of integrable hamiltonians Alba:2009th  and a remarkable
momentum/spatial duality feature introduced in Lai & Yang (2015) (see also
Vidmar, Hackl, Bianchi, & Rigol (2017); Vidmar:2018rqk ). Thus there is a
close connection between free nonlocal models and excited states in that the
ground state of our model for a given locality parameter $\alpha$ must
correspond to a particular excited state of the corresponding local model.

In the limit $\alpha\rightarrow\infty$, the irrational values of $\alpha/\pi$
lead to random correlation functions resembling those of models with quenched
disorder, although the present hamiltonian is translation invariance. This is
the specific feature leading to the volume entanglement entropy law when
entropy is computed for a spatial subregion smaller than $\alpha$. Fig 1
depicts the behavior of the entanglement entropy computed in Levine (2019) for
regimes smaller and large than $\alpha$. To summarize, $\alpha$ appears in two
roles: (1) the number of fermion flavors (specifically, int($\alpha/\pi$) =
central charge c in 1-d ) (2) $\alpha$ is the locality scale designating the
boundary between extensive and logarithmic entanglement entropy.

##  II 1-d nonlocal fermion model

Consider a conventional one-dimensional noninteracting kinetic hamiltonian for
spinless fermions:

$H=\sum^{N-1}_{x,y=0}{\psi^{\dagger}_{j}S_{jk}\psi_{k}}$ (3)

where $j$ and $k$ are one-dimensional site indices and the operators
$\psi_{j}$ ($\psi^{\dagger}_{j},$) destroys (creates) fermions at site $j$ of
a periodic $N$ site lattice and obey the conventional Dirac fermion algebra.
$S_{jk}$ and $T_{jk}$ are the symmetric and antisymmetric lattice operators:

$\displaystyle S_{jk}$ $\displaystyle=$
$\displaystyle\delta_{j,k+1}+\delta_{j,k-1}$ (4) $\displaystyle T_{jk}$
$\displaystyle=$ $\displaystyle\delta_{j,k+1}-\delta_{j,k-1}$

We now aim to write $H$ in a chiral lattice form, suited for making a nonlocal
extension along the lines of equation (1). With the transformation to a new
Dirac fermion, $d_{j}$,

$\psi_{j}=e^{i\frac{\pi}{2}j}d_{j}$ (5)

the hamiltonian may be rewritten where odd and even sublattices are coupled

$H=i\sum_{j\,{\rm(even)}}{[d^{\dagger}_{j}(d_{j+1}-d_{j-1})+d^{\dagger}_{j+1}(d_{j+2}-d_{j})]}$
(6)

and in matrix form,

$H=\sum_{j\in{\rm e};k\in{\rm o}}\left(\begin{array}[]{c}d_{j+1}\\\
d_{j}\end{array}\right)^{\dagger}\left(\begin{array}[]{cc}0&iT_{jk}\\\
iT_{jk}&0\end{array}\right)\left(\begin{array}[]{c}d_{k}\\\
d_{k+1}\end{array}\right)$ (7)

where ”e” and ”o” refer to even and odd sublattices. $H$ may be diagonalized
by introducing chiral fermions $L_{j}$ and $R_{j}$ defined as follows:

$\displaystyle L_{j}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(d_{j+1}+d_{j})\,\,\,\,\,\,R_{j}=\frac{1}{\sqrt{2}}(d_{j+1}-d_{j})$
(8) $\displaystyle\bar{L}_{j}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(d_{j+1}^{\dagger}+d_{j}^{\dagger})\,\,\,\,\,\,\bar{R}_{j}=\frac{1}{\sqrt{2}}(d_{j}^{\dagger}-d_{j+1}^{\dagger})$
(9)

Note that $\bar{R}_{j}$ may be alternatively defined as $R^{\dagger}$.
Finally, $H$ may be written compactly in terms of the Pauli matrix
$\sigma_{z}$,

$H=i\sum_{j\in{\rm e};k\in{\rm o}}{\bar{\phi}_{j}T_{jk}\sigma_{z}\phi_{k}}$
(10)

with the definitions:

$\phi_{j}=\left(\begin{array}[]{c}L_{j}\\\
R_{j}\end{array}\right)\,\,\,\,\bar{\phi}_{j}=\left(\begin{array}[]{c}\bar{L}_{j}\\\
\bar{R}_{j}\end{array}\right)$ (11)

The extension to a ”compact” nonlocal model as previously developed in Levine
(2019) is now straightforward:

$H_{\alpha}=\frac{1}{\alpha}\sum_{j\in{\rm e};k\in{\rm
o}}{\bar{\phi}_{j}[\sin{(i\alpha\sigma_{z}\bf{T}})]_{jk}\phi_{k}}$ (12)

In the continuum limit, where $\alpha$ is dimensionful, $H$ may be written:

$H^{\rm cont}_{\alpha}=\frac{1}{\alpha}\int
dx\bar{\phi}(x)\sin{(i\alpha\sigma_{z}\partial_{x})}\phi(x)$ (13)

Introducing momentum space operators for the original Dirac fermions and
comparable expressions for the $L_{j}$ and $R_{j}$ fermions

$\psi_{j}=\frac{1}{N}\sum_{n=-\frac{N}{2}+1}^{\frac{N}{2}}{\psi_{n}e^{ik_{n}j}}\rightarrow\psi_{j}=\frac{1}{2\pi}\int_{-\pi}^{\pi}{dk\psi(k)e^{ikj}}$
(14) $L_{j}=\frac{1}{2\pi}\int_{-\pi}^{\pi}{dkL(k)e^{ijk}}$ (15)

where $k_{n}=2n\pi/N$. With $N\rightarrow\infty$, $H_{\alpha}$ may be written

$H_{\alpha}=\frac{1}{2\pi\alpha}\int_{-\pi}^{\pi}{dk(\bar{R}(k)R(k)-\bar{L}(k)L(k))\sin{[\alpha}\sin{k}]}$
(16)

As the locality parameter $\alpha$ grows large, the energy as a function of
momentum in the Brillouin zone crosses zero $O(\alpha)$ times leading to a
series of pairs of $R,L$ moving branches (depicted as in figure (2)). Using
the nonlocal dispersion relation, the total number of branches $B$ is related
to $\alpha$ by $B=4\lfloor\alpha/\pi\rfloor$.

Our goal is to study spinless, nonlocal fermions interacting through a local,
nearest neighbor interaction. In terms of the original Dirac fermions, this
interaction is taken to be:

$V=\frac{1}{2!2!}\frac{U}{B}\sum^{N-1}_{j=0}{\psi^{\dagger}_{j}\psi_{j}\psi^{\dagger}_{j+1}\psi_{j+1}}$
(17)

where the bare interaction strength has been rescaled by $B$. Noting the
$1/\alpha$ in the hamiltonian (12) necessary to recover the local limit, it is
seen that both kinetic and interacting energy scales are rescaled by
$B\sim\cal{O}(\alpha)$. Such a scaling gives strong coupling in the local
limit, and coupling comparable to the bandwidth in the nonlocal limit. But it
should be emphasized that there is no rationale for this particular scaling of
the interaction other than that it yields sensible results in both the RG
analysis and SYK analysis of the model.

In momentum space,

$V=\frac{1}{2!2!}\frac{U}{B}\int{\frac{dkdk^{\prime}dq}{(2\pi)^{3}}F(k,k^{\prime},q)\psi^{\dagger}(k)\psi^{\dagger}(k^{\prime})\psi(k^{\prime}-q)\psi(k+q)}$
(18)

where the form factor,
$F(k,k^{\prime},q)={\frac{1}{2}}(\cos{q}-\cos{(k-k^{\prime}+q}))$. The form
factor may also be written in a more symmetric form:

$F(\Delta_{i},\Delta_{f})=-\cos{\frac{\Delta_{i}}{2}}\cos{\frac{\Delta_{f}}{2}}$
(19)

where $\Delta_{i}$ is the difference in the ingoing momenta to the vertex and
$\Delta_{f}$ is the difference in the outgoing momenta from the vertex.

##  III RG analysis

The low energy physics of the noninteracting nonlocal hamiltonian $H_{\alpha}$
is described by a collection of $B\sim\cal{O}(\alpha)$ pairs of linearly
dispersing chiral modes $R(k_{b}),L(k_{b})$ that we will label by a ”branch”
index $b\in(-B/2,B/2)$ index and a momentum index, $k_{b}$, referring to the
specific branch. This series of Fermi ”points” is depicted in figure 2.
Although they are not evenly spaced as depicted, owing to the
$\epsilon_{k}=\sin{(\alpha\sin{k})}$ dispersion, we will treat them as evenly
spaced in much of what follows and argue that the spacing variations are
irrelevant within RG. The distinct quantum fields on different branches are
labeled by the momentum variables so that the notation is less cluttered. It
should be stressed that, for instance, $R(k_{b})$ and $R(k_{b^{\prime}})$ are
distinct quantum fields. This feature is supported by the inference of a
central charge $c\propto\alpha$ from the asymptotic logarithmic behavior of
entanglement entropy in numerical treatments of $H_{\alpha}$ Levine (2019). A
full treatment (for instance, within nonabelian bosonization) would include
zero-mode anomalies so that these fields necessarily anticommute. We will
concentrate on the renormalization of the zero momentum vertices where
$(L_{i},R_{i})\rightarrow(\bar{L}_{f},\bar{R}_{f})$ are chosen to be
$(L(k_{-b}=0),R(k_{b}=0))\rightarrow(\bar{L}(k_{-b^{\prime}}=0),\bar{R}(k_{b^{\prime}}=0))$;
that is, a $\pm b$ pair of fields scattering to another $\pm b^{\prime}$ pair
of fields. Thus, there is a large ${\cal{O}}(\alpha^{2})$ set of coupling
constants, one for each $(b,b^{\prime})$ choice, but they are all renormalized
identically.

Figure 2:  Depiction of multi-branch energy dispersion and processes
corresponding to Feynman diagrams in figure (3) : ZS′ (top) and BCS (bottom).
Because of the specific form of the energy-momentum relationship in the
hamiltonian $H_{\alpha}$ (equation 16 ) the branches are not evenly spaced as
shown here… This figure illustrates processes contributing to the
renormalization of one particular coupling constant (for the LRLR vertex
shown) of O($\alpha^{2}$) possible coupling constants. For the ZS′ process
(top) the nesting vector $Q$ is set by the particular choice of vertex fields
$(\bar{L}_{f}\bar{R}_{f}L_{i}R_{i})$, but the internal loop momentum $p_{b}$
is free to range over $\cal{O}(\alpha)$ branches, generating a singular
contribution in the loop integration at each one. For the BCS process
(bottom), the nesting vector $Q_{b}$ is itself free to range over
$\cal{O}(\alpha)$ branches generating a singular contribution at each one.

Interactions create particle hole pairs within one branch that interact with
modes of the same branch—these are Zero Sound (ZS) diagrams figure 3(a). But
there are also ZS′ diagrams that correspond to excitations in one branch
interacting with modes on another branch, as well as BCS diagrams that
exchange Cooper pairs on different branches. These latter two processes are
specifically depicted in figures 3 (a,b). Similar to the conventional (local)
1-d case, we will argue that ZS′ and BCS diagrams exactly cancel—although the
reason has subtleties distinct from the 1-d case—and that weak interactions
are marginal to the one loop level. (As in the conventional case, ZS diagrams
are nonsingular). We closely follow the formalism and notation of Shankar
Shankar  in his exceptionally lucid treatment of the renormalization group for
nonrelativistic fermions.

In the RG that follows, we adopt a euclidean path integral formalism for the
partition function:

$Z=\int{D\bar{\psi}D\psi
e^{+S}}\,\,\,\,\,\,\,\,S=-\int{(\bar{\psi}\partial_{\tau}\psi+H_{0}(\bar{\psi}\psi)+V(\bar{\psi}\psi))}$
(20)

Defining fourier expansions as follows. The noninteracting action implicitly
defines the greens function $G_{0}$ as follows:

$S_{0}=\int\frac{dk}{2\pi}\int\frac{d\omega}{2\pi}\bar{\psi}(k\omega)\psi(k\omega)(i\omega-\epsilon_{k}+\mu)=\int\frac{dk}{2\pi}\int\frac{d\omega}{2\pi}\bar{\psi}(k\omega)G_{0}^{-1}\psi(k\omega)$
(21)

where $\epsilon_{k}$ and $\mu$ are the energy dispersion and chemical
potential for $H_{0}$.

Figure 3:  (a) Feynman diagram for Zero Sound (ZS), (b) another Zero Sound
channel (ZS’) and (c) Pairing (BCS). As in the local case, the ZS diagram is
finite and does not contribute to the RG analysis.

ZS′ diagrams. Even though the RG analysis is completely standard, it is useful
to show sample contractions for one diagram in detail because of the inherent
complication of multiple branches. For simplicity, frequency variables have
been suppressed. Expanding the interaction in equation (20) to second order,
we make assignments of $\psi$ and $\bar{\psi}$ to four external lines
following figure 3(b). The form factors $F(k,k^{\prime},q)$ are taken to be
unity at this order.

$\displaystyle\int{dk_{1}dk_{2}dq_{1}}\int{dk_{3}dk_{4}dq_{2}}F(k_{1}k_{2}q_{1})F(k_{3}k_{4}q_{2})\times{\frac{1}{2}}(\frac{U}{2!2!})^{2}\times$
$\displaystyle\underbrace{\bar{\psi}(k_{1})}\underbrace{\bar{\psi}(k_{2})}\underbrace{\psi(k_{2}-q_{1})}\underbrace{\psi(k_{1}+q_{1})}\,\,\,\,\bar{\psi}(k_{3})\bar{\psi}(k_{4})\psi(k_{4}-q_{2})\psi(k_{3}+q_{2})$
(22)
$\bar{L}_{f}\,\,\,\,\bar{R}(p_{b})\,\,L(p_{b}-Q)\,\,\,\,\,\,\,\,R_{i}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\bar{R}_{f}\,\,\,\,\bar{L}(p_{b}-Q)\,\,\,\,R(p_{b})\,\,\,\,L_{i}.\,\,\,\,\,\,\,\,$
(23)

As usual, the multiplicity in the choice of external lines cancels the
$(1/2!2!)^{2}$ factor, and the two possible assignments of in/out modes to
vertices cancels the $1/2$ factor in the exponential expansion. Note that $Q$,
the analog of a ”nesting” vector in a conventional Brillouin zone, is fixed by
the choice of external fields whereas $b$, the index of the internal loop
momentum, is free to range over all branches. Adopting a canonical order for
the external lines of the vertex based upon equation 18, the ZS′ diagram may
be expressed symbolically, ZS′ = $\bar{L}\bar{R}LR(-GG)$. The correction to
the coupling constant $U$ from one loop integration is then:

$\Delta U_{\rm
ZS^{\prime}}=-(\frac{U}{B})^{2}\int\frac{dp_{b}}{2\pi}\int\frac{d\omega}{2\pi}\frac{1}{i\omega-\epsilon_{p_{b}}}\frac{1}{i\omega-\epsilon_{p_{b}-Q}}=+(\frac{U}{B})^{2}\int\frac{dp_{b}}{2\pi}\int\frac{d\omega}{2\pi}\frac{1}{\epsilon_{p_{b}}-\epsilon_{p_{b}-Q}}$
(24)

For a given branch index, $b$, and nesting vector, $Q$, the modes to be
integrated out may lie within any one of four cutoff regions as depicted in
figure 5. The flow of the coupling constant corresponding to integrating out
the designated modes over an infinitesimal region $\log{s}=dl$ may be written:

$\Delta U_{\rm ZS^{\prime}}=4\times
BU^{2}\frac{1}{2\pi}\int_{\Lambda/s}^{\Lambda}{\frac{dk}{2k}}=B\frac{U^{2}}{\pi}\Delta
l$ (25)

where we have included the range of $p_{b}$ over $B$ branches. Since the bare
vertex is proportional to $U/B$, the flow of the coupling constant from ZS′
diagrams is

$\frac{dU}{dl}=\frac{U^{2}}{\pi}$ (26)

Figure 4:  Depiction of ZS′ loop momenta. The nesting vector, $Q$ is fixed by
the momenta of external lines. $p_{b}\in[-\pi,\pi]$ is free to range over
branches $b$; $k$ is a momentum variable relative to the particular branch
$p_{b}$ lies on. As in the conventional (local) spinless fermion case, the
energy of a mode $\epsilon_{k}=k$. Figure 5:  Depiction of ZS′ loop momenta
indicating four cutoff regions. $p_{b}$ is assigned to one of these four
regions and modes between $\Lambda$ and $\Lambda/s$ are integrated out. In
this instance $p_{b}>0$ is located on $L$ mode and $p_{b}-Q$ on an $R$ mode.
Figure 6:  Depiction of BCS diagram momenta. In the BCS case, the nesting
vector $Q_{b}$ is free to range over branches $b$. In contrast to the ZS′
diagrams, there are eight cutoff regions. $Q_{b}-p$ is assigned to one of
these eight regions and modes between $\Lambda$ and $\Lambda/s$ are integrated
out.

BCS diagrams. Examining the contractions as in equation (III) there are eight
ways of choosing external lines and two ways of choosing internal contractions
for each leaving a net factor of $1/2$ (the diagrammatic loop ”symmetry”
factor) from the ${\frac{1}{2}}(\frac{1}{2!2!})^{2}$ factor in the expansion
to second order.

$\Delta U_{\rm
BCS}=-{\frac{1}{2}}(\frac{U}{B})^{2}\int\frac{dp}{2\pi}\int\frac{d\omega}{2\pi}\frac{1}{-i\omega+p}\frac{1}{i\omega+p}$
(27)

Refer to figure showing nesting, momentum integration regions. Now including
the range of the nesting vector $Q_{b}$ over $B$ branches, the flow of the
coupling constant from BCS diagrams corresponding to integrating out the
designated modes over an infinitesimal region $\log{s}=dl$ may be written:

$\Delta U_{\rm
BCS}=-8\times{\frac{1}{2}}BU^{2}\frac{1}{2\pi}\int_{\Lambda/s}^{\Lambda}{\frac{dp}{2p}}=-B\frac{U^{2}}{\pi}\Delta
l$ (28)

Again, since the bare vertex is proportional to $U/B$, the flow of the
coupling constant from BCS diagrams is

$\frac{dU}{dl}=-\frac{U^{2}}{\pi}$ (29)

The two contributions cancel and thus, to the one loop level, interactions are
exactly marginal.

##  IV Relationship to SYK4

One might consider whether this gapless critical state persists as the
locality scale $\alpha$ is increased arbitrarily. At some point, performing
momentum shell integrations about each fermi point and ignoring the scattering
between fermi points (analogous to Umklapp) becomes unphysical. Since the
number of branches, $B$, has becomes large—in some sense, of measure
equivalent to the momentum integrations in a single branch—we argue that the
sensible model in this limit is a collection of $B$ independent fermions,
lying at a sequence of fermi points, that interact through the local
interaction (17). Considering the form factor (19), summing the momenta of a
high order Feynman diagram is expected to produce rapidly oscillating phases
in that the momenta of the fermi points are not rationally related (for
arbitrary large $\alpha$). Thus the momentum sums produce an effect resembling
disorder averaging Sachdev:2010um .

To this end, we rewrite the four-fermion interaction (17) truncating the
degrees of freedom to the fermions lying at the sequence of $B$ fermi points,
as described above. The effective model for the interactions of $B$ fermions
flavors follows (where we have dropped the combinatoric factor in (17) for
simplicity):

$H=\frac{U}{B}\sum_{lkq}^{B}{F(l,k,q)\psi^{\dagger}_{k}\psi_{k-q}\psi^{\dagger}_{l}\psi_{l+q}}$
(30)

To be clear, the $B$ flavors represent fermions with a specific (conserved)
momentum, thus the sum is intended represent $B$ points in a $[-\pi,\pi]$
Brillouin zone corresponding to the zeroes of the nonlocal dispersion
relation. Compared with SYK4 this interaction has the added constraint of
momentum conservation and the form factor $F(l,k,q)$ replaces the interaction
weight in SYK that is chosen from a random ensemble. Going over to Grassman
variables the action is now:

$S=-\int{(\sum_{j}^{B}\bar{\psi}_{j}(\partial_{\tau}-\mu)\psi_{j}+H(\bar{\psi}_{j}\psi_{j}))d\tau}$
(31)

and the free propagator follows:

$G_{0}^{j}(\tau)=-\langle\psi_{j}(\tau)\bar{\psi}_{j}(0)\rangle$ (32)

Following the general scheme of SYK, we compute the second order correction to
the propagator corresponding to the melon diagram shown in figure 7(a):

$\displaystyle\langle\psi_{i}(\tau)\bar{\psi}_{i}(0)\rangle$ $\displaystyle=$
$\displaystyle G_{0}^{i}(\tau)+$ (33) $\displaystyle+$
$\displaystyle{\frac{1}{2}}\frac{U^{2}}{B^{2}}\sum_{p,q}F_{1}^{2}(p,q)\int_{0}^{\tau_{2}}{d\tau_{1}}\int_{\tau_{1}}^{\tau_{2}}{d\tau_{2}}G_{0}^{i}(\tau-\tau_{2})G_{0}^{i}(\tau_{1})\times$
(34) $\displaystyle\times$ $\displaystyle
G_{0}^{i-q}(\tau_{2}-\tau_{1})G_{0}^{i-p}(\tau_{2}-\tau_{1})G_{0}^{i-q-p}(\tau_{1}-\tau_{2})$
(35)

The propagators $G_{0}^{i}(\tau)$ are all dispersion-less and therefore
identical; that is, the only importance of the momentum superscript is to
identify the reduction in phase space of the diagram due to momentum
conservation. The form factor for both vertices evaluates to:

$F_{1}(p,q)={\frac{1}{2}}(\cos{p}-\cos{q})=-F_{2}(p,q)$ (36)

To perform the momentum sums, we approximate the distribution of fermi points
in the BZ to be $B$ evenly spaced points, $p_{j}\approx 2\pi j/B$. Thus, for
instance,

$\sum_{j=-B/2}^{B/2}{\sin^{2}{p_{j}}}\approx\frac{B}{2\pi}\int_{-\pi}^{\pi}{dp\sin^{2}p}=B\overline{\sin^{2}p}=\frac{B}{2}$
(37)

Noting that $\overline{\sin{p}\cos{p}}=0$, the double momentum sums produce a
$B^{2}$ factor canceling reciprocal $B^{2}$ factor from the vertex.

$\displaystyle G(\tau)$ $\displaystyle=$ $\displaystyle G_{0}^{i}(\tau)+$ (38)
$\displaystyle+$
$\displaystyle\frac{1}{8}U^{2}\int_{0}^{\tau_{2}}{d\tau_{1}}\int_{\tau_{1}}^{\tau_{2}}{d\tau_{2}}G_{0}(\tau-\tau_{2})G_{0}(\tau_{1})G_{0}^{2}(\tau_{2}-\tau_{1})G_{0}(\tau_{1}-\tau_{2})$
(39)

The success of SYK relies critically upon the suppression of higher order loop
diagrams (such as figure 7(b)) through disorder averaging. In the melon
diagram computed above, loop momenta appear in even powers of harmonic
functions and thus have a nonzero average. To assess the suppression arising
from the averaging over harmonic functions in the structure factor, we
examined the higher loop diagrams in figure 7(b). In the expression below, we
have only reproduced the sums over structure factors for the diagram given
with external momentum $i$ and expressed the internal propagators
symbolically.

$\displaystyle\frac{1}{B^{4}}\sum_{pqrk}{G\ldots
G\times{\frac{1}{2}}(\cos{p}-\cos{q}){\frac{1}{2}}(\cos{(q-r)}-\cos{p+r})}\times$
(40)
$\displaystyle{\frac{1}{2}}(\cos{(i-p-k)}+\cos{r}){\frac{1}{2}}(\cos{(i-q-k+r)}+\cos{r})$
(41)

Of the 16 terms, only 2 survive the momentum averaging. For a term to survive,
all momentum labels must appear an even number of times in the product of
harmonic functions. for instance, the first term in expression (40)
$\cos{p}\cos{(q-r)}\cos{(i-p-k)}\cos{(i-q-k+r)}$, has a nonzero average, but
$\cos{p}\cos{(p-r)}\cos{(i-p-k)}\cos{(i-q-k+r)}$ vanishes summed over free
internal momenta. As the number of internal loops grow, the relative number of
factors containing only even pairings is expected to vanish if the internal
momenta appear randomly within the harmonic functions.

This assertion leads to an estimate of the contribution of diagrams containing
internal loops. The melon diagram nominally contains two free momenta, and
each additional internal loop contains two additional momenta. Such a diagram
containing $N$ vertices will contribute the product of $N$ harmonic function
factors coming from the form factor, each containing some combination of $N$
free internal momenta. If we assume that in a high order diagram, the
combination of free momenta within a given harmonic function is essentially
random, the probability of any given momentum variable appearing an even
number of times is 1/2. Thus an ${\cal{O}}(U^{N})$ diagram that is should be
suppressed by a factor of $(1/2)^{N}$.

In summary, internal bubble diagrams which include corrections to the two
particle propagator do not vanish, as they do under disorder averaging in the
SYK model. This incomplete disorder averaging may be traced to the additional
constraint of momentum conservation in the action (31). High order in these
internal bubble diagrams are exponentially suppressed and we thus regard the
sum of melon diagrams as an uncontrolled approximation.

Figure 7:  SYK melon diagram (a) and melon diagram with internal loop (b).

##  V Discussion and Further Directions

In this manuscript we study local density-density interactions in a 1-d model
of nonlocal fermions based upon a hamiltonian (12) and (13). The free version
of this model exhibit a crossover between volume and anomalous (logarithmic)
area law behavior at scales larger than the locality scale, $\alpha$. Owing to
the periodic nature of the hamiltonian $\alpha$ is also a measure of the
number of fundamental degrees of freedom in the theory—the effective central
charge, $c_{\rm{eff}}$. At weak coupling, we find that interactions are
exactly marginal and the system remains gapless. As $\alpha\rightarrow\infty$
we argue that the interaction becomes effectively random due to the rapidly
fluctuating phase of the interaction structure factor written in momentum
space. Due to momentum conservation, this quasi-randomness is not as complete
as the quenched disorder in the SYK model and the melon diagrams are not
exclusively dominant. However, we argue that higher order non-melon diagrams
are suppressed and that an SYK phase may be stabilized.

We now speculate about the possible physical origin of a nonlocal model such
as (12). In effect, it might follow from reinterpreting the path integral
constructed for (12) as representing amplitudes connecting a pure excited
state—rather than the vacuum. It has been shown that typical excited
eigenstates in a model of free fermions manifest thermal behavior on a
sufficiently short spatial scale (call this scale $\alpha$). Specifically, the
reduced density matrix of a single typical excited state matches the Fermi-
Dirac distribution with a temperature reflecting the average single particle
energy of the excited state. Informally, this result suggests that a
calculation of finite temperature interacting fermions might be approached
from the S-matrix of a pure excited state rather than a thermal ensemble.

To explore this approach, one might regard the density matrix composed from
hamiltonian (12) as a projection operator onto an excited state of the
underlying local model.

$\rho_{0}(\alpha)=\lim\limits_{\beta\to\infty}e^{-\beta H_{\alpha}}$ (42)

and consider time evolution given by the local hamiltonian $H_{0}+V$ (defined
in equations (12) and (17)). Also note that we take the fictitious temperature
$\beta\rightarrow\infty$ to select a single excited state. There are various
ways of including interactions of a system prepared in an initial
nonequilibrium density matrix (e.g. Schwinger-Keldysh closed time-loop Kamenev
, Kadanoff-Baym KadanoffBaym , and, most recently, Chakraborty, Gorantla, &
Sensarma (2019)) In the closed time loop approach, the Keldysh component of
the Greens function will carry the single particle distribution function
formed from the initial density matrix, $\rho_{0}(\alpha)$; thus, this
distribution function describes multiple 1-d Fermi surfaces such as depicted
in figure 2. Treating the fermion fields at these fermi points as independent
fields and taking the interaction energy scale as much larger than the kinetic
scale of the local model ($V>>H_{0}$) approximately realizes the large flavor
model studied in section IV. In principle, one might be able to study
transport properties—for instance, the decay of a current—generated ”on top”
of a zero current excited state. Specifically, the Keldysh Dyson equation (or
quantum kinetic equation) governs the relaxation of the distribution function
to determine transport properties. Unfortunately, the procedure sketched here
may not work for the ”typical” excited states that produce thermal RDMs as
these excited (pure) states may not necessarily be written in terms of
exponentials of single particle operators and thus obey Wick’s theorem.
However, the novel approach to time evolution of athermal inital density
matrices, recently proposed in Chakraborty, Gorantla, & Sensarma (2019), may
be useful.

The nonlocal model may be a route to studying finite temperature fermion
systems with strong interactions. The potential benefit of such an approach is
that physical temperature—appearing for instance in a transport calculation—is
encoded (through the energy of the excited state) in a parameter that
represents the number of fundamental degrees of freedom, the effective central
charge. Just as central charge appears in thermodynamic and transport
quantities conventionally, the physical temperature would appear through the
parameter $\alpha$ in a zero temperature nonlocal model.

We should point out that extensions to higher dimensional lattices might be
accomplished by considering a generalization of (1) in which a scalar
diffusion operator is constructed that respects the underlying lattice
orthogonality. One possible extension in 2-d might be found by following the
scheme of the Dirac equation,

$H_{\rm
2d}=\frac{\epsilon}{\alpha^{2}}\int{d^{2}x\psi^{\dagger}(x)\cos{(i\alpha\sigma^{\mu}\partial_{\mu})}\psi(x)}$
(43)

where $\\{\sigma^{\mu}\\}$ are two anticommuting Pauli matrices BB .
Similarly, this model becomes a large flavor model (with additional flavors
resulting from the $\sigma$ matrices) and interactions may also be studied by
conventional RG techniques. Perturbation theory and RG in both 1-d and higher
dimensional models would benefit from support by numerical investigation.
Unfortunately, it is likely that the nonzero locality length (and
corresponding volume entropy law) would lead to a rapid increase in the
required hilbert space dimension ($O(\exp{L^{d}})$) even in a 1-d DMRG
numerical scheme and, therefore, direct diagonalization may be the only
possible approach.

Lastly, we remark that this large flavor model resembles the relativistic
Gross-Neveu (GN) model. The essential difference is that the scalar
interaction term in GN leads to the ZS’ diagrams without the BCS diagrams. In
condensed matter parlance, the charge density wave symmetry breaking dominates
superconductivity (when the ZS’ channel dominate the BCS channel). The charge
density wave phase is essentially equivalent to the UV asymptotic freedom
associated with the GN model.

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