# Green’s Function Zeros in Fermi Surface Symmetric Mass Generation

Da-Chuan Lu  Department of Physics, University of California, San Diego, CA
92093, USA  Meng Zeng  Department of Physics, University of California, San
Diego, CA 92093, USA  Yi-Zhuang You  Department of Physics, University of
California, San Diego, CA 92093, USA

###### Abstract

The Fermi surface symmetric mass generation (SMG) is an intrinsically
interaction-driven mechanism that opens an excitation gap on the Fermi surface
without invoking symmetry-breaking or topological order. We explore this
phenomenon within a bilayer square lattice model of spin-1/2 fermions, where
the system can be tuned from a metallic Fermi liquid phase to a strongly-
interacting SMG insulator phase by an inter-layer spin-spin interaction. The
SMG insulator preserves all symmetries and has no mean-field interpretation at
the single-particle level. It is characterized by zeros in the fermion Green’s
function, which encapsulate the same Fermi volume in momentum space as the
original Fermi surface, a feature mandated by the Luttinger theorem. Utilizing
both numerical and field-theoretical methods, we provide compelling evidence
for these Green’s function zeros across both strong and weak coupling regimes
of the SMG phase. Our findings highlight the robustness of the zero Fermi
surface, which offers promising avenues for experimental identification of SMG
insulators through spectroscopy experiments despite potential spectral
broadening from noise or dissipation.

##  I Introduction

Symmetric mass generation (SMG) [1, 2, 3, 4, 5, 6, 7, 8] is an interaction-
driven mechanism that creates many-body excitation gaps in anomaly-free
fermion systems _without_ condensing any fermion bilinear operator or
developing topological orders. It has emerged as a alternative symmetry-
preserving approach for mass generation in relativistic fermion systems, which
is distinct from the traditional symmetry-breaking Higgs mechanism [9, 10, 11,
12, 13, 14]. The prospect of SMG offering a potential solution to the long-
standing fermion doubling problem [15, 16, 17, 18, 19, 20, 21] has sparked
significant interest in the lattice gauge theory community [22, 23, 24, 25,
26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44,
45, 46]. In condensed matter physics, SMG was initially explored within the
framework of the interaction-reduced classification of fermionic symmetry
protected topological (SPT) states [1, 2, 47, 48, 49, 50, 51, 52, 53, 54, 55,
56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68], and has been recently
extended to systems with Fermi surfaces [69, 70, 71, 72, 73, 74], given the
growing understanding that Fermi liquids can be perceived as fermionic SPT
states within the phase space [75, 76].

One important feature of the SMG gapped state lies in the zeros of fermion
Green’s function [77, 78, 79, 80, 81, 82] at low-energy. Investigations reveal
that the poles of the fermion Green’s function in the pristine gapless fermion
state will be replaced by zeros in the gapped SMG state as the fermion system
goes across the SMG transition upon increasing the interaction strength. This
pole-to-zero transition was postulated [78] as a direct indicator of the SMG
transition [80, 83] that can be probed by spectroscopy experiments. However,
the presence of similar zeros in the Green’s function within Fermi surface SMG
states has not been investigated yet, and it is the focus of our present
research.

Fermi surface SMG [74] refers to the occurrence of SMG phenomena on Fermi
surfaces with non-zero Fermi volumes. It describes scenarios where the fermion
interaction transforms a gapless Fermi liquid state (metal) into a non-
degenerate, gapped, direct product state (trivial insulator), without breaking
any symmetry (for example, without invoking Cooper pairing or density wave
orders). Such a metal-insulator transition is viable when Fermi surfaces
collaboratively cancel the Fermi surface anomaly [84, 85, 74]. This anomaly
can be perceived as a mixed anomaly between the translation symmetry and the
charge conservation $\mathrm{U}(1)$ symmetry on the lattice [86, 87, 88, 89,
85, 84, 90], or as an anomaly of an emergent loop $\mathrm{LU}(1)$ symmetry
[91, 92, 93] in the infrared theory.

In this work, we present evidence of robust Green’s function zeros in Fermi
surface SMG states. Let $t$ be the energy scale of band dispersion and $J$ be
the energy scale of SMG gapping interaction, we investigate the problem from
two parameter regimes:

  * •

Deep in the SMG phase ($J/t\gg 1$), we start with an exact-solvable SMG
product state in a lattice model and calculate the fermion Green’s function by
treating the fermion hopping as perturbation [94]. We find that the Green’s
function $G_{\text{SMG}}(\omega,{{\bm{k}}})$ deep in the SMG phase takes the
following form

$G_{\text{SMG}}(\omega,{{\bm{k}}})=\frac{\omega+\alpha\epsilon_{{{\bm{k}}}}/J^{2}}{(\omega-\epsilon_{{\bm{k}}}/2)^{2}-J^{2}},$
(1)

where $(\omega,{{\bm{k}}})$ labels the frequency-momentum of the the fermion.
$\epsilon_{{\bm{k}}}$ is the energy dispersion of the original band structure
in the free-fermion limit, and $\alpha$ is an order-one number depending on
other details of the system. One salient feature of $G_{\text{SMG}}$ is that
it has a series of zeros at $\omega=-\alpha\epsilon_{{\bm{k}}}/J^{2}$ in the
frequency-momentum space. At $\omega=0$, the Green’s function zeros form a
zero Fermi surface that replaces the original Fermi surface.

  * •

If the SMG phase is adjacent to a spontaneous symmetry breaking (SSB) phase,
we use perturbative field theory to argue that the Green’s function in the SMG
phase near the symmetry-breaking transition ($J/t\gtrsim 1$) should take the
form of

$G^{\prime}_{\text{SMG}}(\omega,{{\bm{k}}})=\frac{\omega+\epsilon_{{\bm{k}}}}{\omega^{2}-\epsilon_{{\bm{k}}}^{2}-\Delta_{0}^{2}}$
(2)

where we assume that the SSB order parameter retains a finite amplitude
$\Delta_{0}$ in the SMG phase, but its phase is randomly fluctuating [95].
Again, $G^{\prime}_{\text{SMG}}$ features a series of zeros at
$\omega=-\epsilon_{{\bm{k}}}$, with the same zero Fermi surface.

Many previous works [96, 97, 98, 99] suggest that the Luttinger theorem [100]
will not be violated in the presence of the interaction that preserves the
translation and charge conservation symmetry. However, quasi-particles (poles
of Green’s function) may not exist in the strongly correlated systems, the
Fermi surface is instead defined by the surface of Green’s function zeros at
zero frequency, i.e., $G(0,{{\bm{k}}})=0$, and the Green’s function changes
sign on the two sides of the _zero Fermi surface_ , or the so-called Luttinger
surface [101, 98, 91, 102, 103]. This can be regarded as the remnant of the
conventional Fermi surface in the strongly interacting gapped phase. Our
analysis shows that the volume enclosed by the zeros of the Green’s function
in the SMG phase is the same as the Fermi volume in the Fermi liquid phase,
which agrees with the Luttinger theorem.

The paper will be structured as follows. We start by introducing a concrete
lattice model for Fermi surface SMG in Sec. II.1 and briefly discussing its
phase diagram. We give theoretical arguments for Green’s function zeros in the
SMG phase from the Luttinger theorem in Sec. II.2 (general), and the particle-
hole symmetry in Sec. II.3 (specific). We provide numerical and field
theoretical evidence of Green’s function zeros from both the strong coupling
Sec. III.1 and the weak coupling Sec. III.2 perspectives. We comment on the
robustness of probing the zero structure in spectroscopy experiments in Sec.
IV. We conclude in Sec. V with a discussion of the relevance of our model to
the nickelate superconductor La3Ni2O7.

##  II Argument For Green’s Function Zeros

###  II.1 Lattice Model and Phase Diagram

As a specific example of Fermi surface SMG, we consider a bilayer square
lattice [104, 105, 106] model of spin-1/2 fermions, as illustrated in Fig.
1(a). Let $c_{il\sigma}$ be the fermion annihilation operator on site-$i$
layer-$l$ ($l=1,2$) and spin-$\sigma$ ($\sigma=\uparrow,\downarrow$). The
model is described by the following Hamiltonian

$H=-t\sum_{\langle
ij\rangle,l,\sigma}(c_{il\sigma}^{\dagger}c_{jl\sigma}+\text{h.c.})+J\sum_{i}{\bm{S}}_{i1}\cdot{\bm{S}}_{i2},$
(3)

where
${\bm{S}}_{il}:=\tfrac{1}{2}c_{il\sigma}^{\dagger}{\bm{\sigma}}_{\sigma\sigma^{\prime}}c_{il\sigma^{\prime}}$
denotes the spin operator with
${\bm{\sigma}}:=(\sigma^{1},\sigma^{2},\sigma^{3})$ being the Pauli matrices.
The Hamiltonian $H$ contains a nearest-neighbor hopping $t$ of the fermions
within each layer and an inter-layer Heisenberg spin-spin interaction with
antiferromagnetic coupling $J>0$. The Heisenberg interaction should be
understood as a four-fermion interaction, that there is no explicitly formed
local moment degrees of freedom. Unlike the standard $t$-$J$ model [107], we
do _not_ impose any on-site single-occupancy constraint [108] here. We assume
that the fermions are half-filled in each layer.

Figure 1: (a) Bilayer square lattice model with intra-layer hopping and inter-
layer spin interaction. (b) Fermi sea and Fermi surface at $J=0$ in the
Brillouin zone. A high-symmetry path is traced out in gray. (c) A conjectured
phase diagram consist of a Fermi liquid (FL) fixed point, a spontaneous
symmetry breaking (SSB) phase, a XY transition, and a SMG insulating phase.

In the non-interacting limit ($J/t\to 0$), the ground state of the tight-
binding Hamiltonian in Eq. (3) is a Fermi liquid with a four-fold degenerated
(two layers and two spins) square-shaped Fermi surface in the Brillouin zone,
as shown in Fig. 1(b). The fermion system is gapless in this limit. However,
given that the fermion carries one unit charge under the $\mathrm{U}(1)$
symmetry, the Fermi surface anomaly vanishes due to [87, 76]

$\sum_{a=1}^{4}q_{a}\nu_{a}=4\times 1\times\frac{1}{2}=0\mod 1,$ (4)

where $a$ indexes the four-fold degenerated Fermi surface with $q_{a}=1$ being
the $\mathrm{U}(1)$ charge carried by the fermion and $\nu_{a}=1/2$ being the
filling fraction. This implies there must be a way to gap out the Fermi
surface into a trivial insulator while preserving both the translation and the
$\mathrm{U}(1)$ charge conservation symmetries. Nevertheless, these symmetry
requirements are restrictive enough to rule out all possible fermion bilinear
gapping mechanisms, leaving Fermi surface SMG the only available option.

One possible SMG gapping interaction is the interlayer Heisenberg spin-spin
interaction $J$ in Eq. (3). In the strong interaction limit ($J/t\to\infty$),
the system has a unique ground state, given by

$|0\rangle=\bigotimes_{i}(c_{i1\uparrow}^{\dagger}c_{i2\downarrow}^{\dagger}-c_{i1\downarrow}^{\dagger}c_{i2\uparrow}^{\dagger})|\text{vac}\rangle,$
(5)

which is a direct product of the inter-layer spin-singlet state on every site.
$|\text{vac}\rangle$ stands for the vacuum state of fermions (i.e.
$c_{il\sigma}|\text{vac}\rangle=0$). The SMG ground state $|0\rangle$ does not
break any symmetry and does not have topological order. All excitations are
gapped by an energy of the order $J$ from the ground state. Any local
perturbation far below the energy scale $J$ can not close this excitation gap,
so the SMG phase is expected to be stable in a large parameter regime as long
as $J\gg t$.

Given the distinct ground states in the two limits of $J/t$, we anticipate at
least one quantum phase transition separating the Fermi liquid and the SMG
insulator. However, due to the perfect nesting of the Fermi surface, the Fermi
liquid state is unstable towards spontaneous symmetry breaking (SSB) upon
infinitesimal interaction, so a more plausible phase diagram should look like
Fig. 1(c), where an intermediate SSB phase sets in. A mean-field analysis
based on the Fermi liquid fixed point shows that there are two degenerated
leading instabilities: (i) the inter-layer exciton condensation (EC) and (ii)
the inter-layer superconductivity (SC). They are respectively described by the
following order parameters

$\phi_{\text{EC}}=\sum_{i,\sigma}(-)^{i}c_{i1\sigma}^{\dagger}c_{i2\sigma},\quad\phi_{\text{SC}}=\sum_{i,\sigma}(-)^{\sigma}c_{i1\sigma}^{\dagger}c_{i2\bar{\sigma}}^{\dagger}.$
(6)

Here, $(-)^{i}$ denotes the stagger sign on the square lattice of lattice
momentum $(\pi,\pi)$. $(-)^{\sigma}=+1$ for $\sigma=\uparrow$ and $-1$ for
$\sigma=\downarrow$. $\bar{\sigma}$ stands for the opposite spin of $\sigma$.

The energetic degeneracy of these two SSB orders can be explained by the fact
that their order parameters $\phi_{\text{EC}}$ and $\phi_{\text{SC}}$ are
related by a particle-hole transformation
$c_{i2\sigma}\to(-)^{i}(-)^{\sigma}c_{i2\bar{\sigma}}^{\dagger}$ in the second
layer only, which is a symmetry of the model Hamiltonian in Eq. (3). The EC
$\langle\phi_{\text{EC}}\rangle\neq 0$ spontaneously breaks the translation
and interlayer $\mathrm{U}(1)$ symmetry, and the SC
$\langle\phi_{\text{SC}}\rangle\neq 0$ spontaneously breaks the total
$\mathrm{U}(1)$ symmetry. Both of them gap out the Fermi surfaces fully,
leading to an SSB insulator (or superconductor). The SSB and SMG phases are
likely separated by an XY transition, at which the symmetry gets restored. We
will leave the numerical verification of the proposed phase diagram Fig. 1(c)
for future study, as the main focus of this research is to investigate the
structure of fermion Green’s function in the SMG insulating phase.

We note that the model Eq. (3) was also introduced as the “coupled ancilla
qubit” model to describe the pseudo-gap physics in the recent literature [70,
72, 73]. Its honeycomb lattice version has been investigated in recent
numerical simulations [109], where a direct quantum phase transition between
semimetal and insulator phases was observed.

###  II.2 Luttinger Theorem and Green’s Function Zeros

The Luttinger theorem [110, 100] asserts that in a fermion many-body system
with lattice translation and charge $\mathrm{U}(1)$ symmetries, the ground
state charge density $\langle N\rangle/V$ (i.e., the $\mathrm{U}(1)$ charge
per unit cell) is tied to the momentum space volume in which the real part of
the zero-frequency fermion Green’s function is positive
$\operatorname{Re}G(0,{{\bm{k}}})>0$. This can be formally expressed as

$\frac{\langle
N\rangle}{V}=N_{f}\int_{\operatorname{Re}G(0,{{\bm{k}}})>0}\frac{\mathrm{d}^{2}{{\bm{k}}}}{(2\pi)^{2}}.$
(7)

Here, the $\mathrm{U}(1)$ symmetry generator
$N=\sum_{i,l,\sigma}c_{il\sigma}^{\dagger}c_{il\sigma}$ measures the total
charge, and the volume $V=\sum_{i}1$ is defined as the number of unit cells in
the lattice system. $N_{f}=4$ counts the fermion flavor number (or the Fermi
surface degeneracy), including two layers and two spins. The Green’s function
$G(\omega,{{\bm{k}}})$ in Eq. (7) is defined by the fermion two-point
correlation as

$\langle
c_{l\sigma}(\omega,{{\bm{k}}})c_{l^{\prime}\sigma^{\prime}}(\omega,{{\bm{k}}})^{\dagger}\rangle=G(\omega,{{\bm{k}}})\delta_{ll^{\prime}}\delta_{\sigma\sigma^{\prime}}.$
(8)

The correlation function is proportional to an identity matrix in the flavor
(layer-spin) space because of the layer
$\mathrm{U}(1):c_{l\sigma}\to\mathrm{e}^{(-)^{l}\mathrm{i}\theta}c_{l\sigma}$,
the layer interchange $\mathbb{Z}_{2}:c_{1\sigma}\leftrightarrow c_{2\sigma}$,
and the spin
$\mathrm{SU}(2):c_{l\sigma}\to(\mathrm{e}^{\mathrm{i}{\bm{\theta}}\cdot{\bm{\sigma}}/2})_{\sigma\sigma^{\prime}}c_{l\sigma^{\prime}}$
symmetries.

The Luttinger theorem applies to the Fermi liquid and SMG states in the
bilayer square lattice model Eq. (3), as both states preserve the translation
and charge $\mathrm{U}(1)$ symmetries. Given that the fermions are half-filled
($\nu=1/2$) in the system, the Fermi volume should be

$\int_{\operatorname{Re}G(0,{{\bm{k}}})>0}\frac{\mathrm{d}^{2}{{\bm{k}}}}{(2\pi)^{2}}=\frac{\langle
N\rangle}{VN_{f}}=\nu=\frac{1}{2}.$ (9)

The Fermi volume is enclosed by the Fermi surface, across which
$\operatorname{Re}G(0,{{\bm{k}}})$ changes sign. The sign change can be
achieved either by poles or zeros in the Green’s function.

In the Fermi liquid state, the required Fermi volume is satisfied via Green’s
function poles along the Fermi surface, as pictured in Fig. 1(b). However, the
SMG insulator is a fully gapped state of fermions that has no low-energy
quasi-particles (below the energy scale $J$). Consequently, the Green’s
function $G(\omega,{{\bm{k}}})$ cannot develop poles at $\omega=0$, meaning
the required Fermi volume can only be satisfied by Green’s function zeros.
Therefore, the Lutinger theorem implies that there must be robust Green’s
function zeros at low energy in the SMG phase, and the zero Fermi surface must
enclose half of the Brillouin zone volume in place of the original pole Fermi
surface.

It is known that the Luttinger theorem can be violated in the presence of
topological order [111, 112, 113, 114, 115, 86, 102, 116, 88, 117]. However,
this concern does not affect our discussion in the SMG phase, because the SMG
insulator is a trivial insulator without topological order.

###  II.3 Particle-Hole Symmetry and Zero Fermi Surface

The Luttinger theorem only constrains the Fermi volume but does not impose
requirements on the shape of the Fermi surface. However, in this particular
example of the bilayer square lattice model Eq. (3), the system has sufficient
symmetries to determine even the shape of the Fermi surface.

The key symmetry here is a particle-hole symmetry $\mathbb{Z}_{2}^{C}$, which
acts as

$c_{il\sigma}\to(-)^{i}(-)^{\sigma}c_{il\bar{\sigma}}^{\dagger}.$ (10)

The Hamiltonian $H$ in Eq. (3) is invariant under this transformation. Since
the Green’s function is an identity matrix in the flavor space Eq. (8) which
is invariant under any flavor basis transformation, we can omit the flavor
indices and focus on the frequency-momentum dependence of the Green’s
function, written as

$G(\omega,{\bm{k}})=\sum_{t,{\bm{x}},t^{\prime},{\bm{x}}^{\prime}}\langle
c(t,{\bm{x}})c(t^{\prime},{\bm{x}}^{\prime})^{\dagger}\rangle\mathrm{e}^{\mathrm{i}(\omega(t-t^{\prime})-{\bm{k}}\cdot({\bm{x}}-{\bm{x}}^{\prime}))}.$
(11)

Given Eq. (10), the fermion field $c(t,{\bm{x}})$ transforms under the
$\mathbb{Z}_{2}^{C}$ symmetry as

$c(t,{\bm{x}})\to
c(t,{\bm{x}})^{\dagger}\mathrm{e}^{\mathrm{i}{\bm{Q}}\cdot{\bm{x}}},\quad
c(t,{\bm{x}})^{\dagger}\to
c(t,{\bm{x}})\mathrm{e}^{-\mathrm{i}{\bm{Q}}\cdot{\bm{x}}},$ (12)

where ${\bm{Q}}=(\pi,\pi)$ is the momentum associated with the stagger sign
factor $(-)^{i}$ on the square lattice. As a consequence, the Green’s function
transforms as

$G(\omega,{\bm{k}})\to-G(-\omega,{\bm{Q}}-{\bm{k}}).$ (13)

Furthermore, there are also two diagonal reflection symmetries on the square
lattice, which maps ${{\bm{k}}}=(k_{x},k_{y})$ to $(k_{y},k_{x})$ or
$(-k_{y},-k_{x})$ in the momentum space.

Both the Fermi liquid and the SMG states preserve the particle-hole symmetry
$\mathbb{Z}_{2}^{C}$ and the lattice reflection symmetry, which requires the
Green’s function to be invariant under the combined symmetry transformations.
So the zero-frequency Green’s function must satisfy

$G(0,k_{x},k_{y})=-G(0,\pi\pm k_{y},\pi\pm k_{x}),$ (14)

meaning that the sign change of $G(0,{{\bm{k}}})$ should happen along
$k_{x}\pm k_{y}=\pi\mod 2\pi$, which precisely describes the shape of the
Fermi surface. The Fermi surface is pole-like in the Fermi liquid state and
becomes zero-like in the SMG state, but its shape and volume remain the same.

However, it should be noted that the precise overlap of the zero Fermi surface
in the SMG insulator and the pole Fermi surface in the Fermi liquid is a fine-
tuned feature of the bilayer square lattice model Eq. (3). In more general
cases, such as including further neighbor hopping in the model, the particle-
hole symmetry would cease to exist, thus the invariance in the shape of the
Fermi surface is no longer guaranteed. Nevertheless, the Luttinger theorem can
still ensure the invariance in the Fermi volume, thereby providing the SMG
insulator with robust Green’s function zeros.

To verify this proposition, we will analyze the behavior of the Green’s
function in the SMG phase from both strong and weak coupling perspectives in
Sec. III. Our calculations suggest that, for this specific model, the SMG
state indeed possesses a Fermi surface (of Green’s function zeros) that is
identical in shape to that in the Fermi liquid state.

##  III Evidence of Green’s Function Zeros

###  III.1 Strong Coupling Analysis

We will first focus on the strong interaction limit $(J/t\to\infty)$, where
the system is deep in the SMG phase and the exact ground state is known (see
Eq. (5)). We start from this limit and turn on the hopping term as a
perturbation. We employ exact diagonalization and cluster perturbation theory
(CPT) [94, 118] to compute the Green’s function in the SMG phase. The details
of our method are described in Appendix A. It is valid to use a small cluster
to reconstruct the Green’s function in the SMG phase since the ground state is
close to a product state that does not have long-range correlation or long-
range quantum entanglement. This is quite different from the Hubbard model,
where the Fermi surface anomaly is non-vanishing, and the infrared phase must
be either SSB order or topological order [111, 114, 115, 86]. In either case,
the ground state wave functions cannot be reconstructed from the small
clusters due to the long-range correlation/entanglement. This argument has
been noted in the original paper on the CPT method [94].

Figure 2: Partition the square lattice into $2\times 2$ clusters. The many-
body Hamiltonian is exactly diagonalized within each cluster. The effect of
inter-cluster hopping is included in an RPA-like approach.

To be specific, we first partition the square lattice (including both layers)
into $2\times 2$ square clusters as shown in Fig. 2. Let us first ignore the
inter-cluster hopping. Within each cluster, we represent the Hamiltonian in
the many-body Hilbert space and use the Lanczos method to obtain the lowest
$\sim 2000$ eigenvalues and eigenvectors. The Green’s function in the cluster
can then be obtained by the Källén–Lehmann representation

$G_{0}(\omega)_{ij}=\sum_{m>0}\frac{\langle 0|c_{i}|m\rangle\langle
m|c_{j}^{\dagger}|0\rangle}{\omega-(E_{m}-E_{0})}+\frac{\langle
m|c_{i}|0\rangle\langle 0|c_{j}^{\dagger}|m\rangle}{\omega+(E_{m}-E_{0})},$
(15)

where $|m\rangle$ is the $m$th excited state with energy $E_{m}$, and
$|0\rangle$ is the ground state with energy $E_{0}$, whose wave function was
previously given in Eq. (5). Since the four fermion flavors (two spins and two
layers) are identical under the internal flavor symmetry, we can drop the
flavor index in the Green’s function and only focus on one particular flavor
with the site indices $i,j$, where $i,j=1,2,3,4$ as indicated in Fig. 2. The
convergence of the Green’s function can be verified by including more
eigenstates from the Lanczos method. We checked that increasing the number of
eigenpairs to $\sim 8000$ will not change the result significantly, indicating
that the result with $\sim 2000$ eigenpairs has already converged.

Now we restore the inter-cluster hoping to extend the Green’s function from
small clusters to the infinite lattice. The Green’s function of super-lattice
momentum ${{\bm{k}}}$ can be obtained from the random phase approximation
(RPA) approach [94],

$G(\omega,{{\bm{k}}})_{ij}=\left(\frac{G_{0}(\omega)}{1-T({{\bm{k}}})G_{0}(\omega)}\right)_{ij},$
(16)

where the $T({{\bm{k}}})$ matrix

$T({{\bm{k}}})=-t\begin{pmatrix}0&\mathrm{e}^{-\mathrm{i}2k_{x}}&0&\mathrm{e}^{\mathrm{i}2k_{y}}\\\
\mathrm{e}^{\mathrm{i}2k_{x}}&0&\mathrm{e}^{\mathrm{i}2k_{y}}&0\\\
0&\mathrm{e}^{-\mathrm{i}2k_{y}}&0&\mathrm{e}^{\mathrm{i}2k_{x}}\\\
\mathrm{e}^{-\mathrm{i}2k_{y}}&0&\mathrm{e}^{-\mathrm{i}2k_{x}}&0\end{pmatrix}$
(17)

describes the inter-cluster fermion hopping. The resulting Green’s function
$G(\omega,{{\bm{k}}})_{ij}$ is defined in the folded Brillouin zone
${{\bm{k}}}\in(-\pi/2,\pi/2]^{\times 2}$ with sub-lattice indices $i,j$. To
unfold the Green’s function to the original Brillouin zone
${{\bm{k}}}\in(-\pi,\pi]^{\times 2}$, we perform the following (partial)
Fourier transform

$G(\omega,{{\bm{k}}})=\frac{1}{L}\sum_{i,j}\mathrm{e}^{-\mathrm{i}{{\bm{k}}}\cdot({{\bm{r}}}_{i}-{{\bm{r}}}_{j})}G(\omega,{{\bm{k}}})_{ij}.$
(18)

Figure 3: Fermion Green’s function Eq. (18) deep in the SMG insulator phase,
at $J=8t$. (a) The imaginary part (spectral function)
$-2\operatorname{Im}G(\omega+\mathrm{i}0_{+},{\bm{k}})$ shows the pole
(spectral peak) structure. (b) The real part
$\operatorname{Re}G(\omega,{\bm{k}})$ shows the pole (divergence) and zero
(purple contour) structures. (c) Same as (b) but zoomed in near $\omega=0$ to
show the dispersion of Green’s function zeros.

We numerically calculated the unfolded Green’s function $G(\omega,{{\bm{k}}})$
using the above-mentioned cluster perturbation method. We take a large
interaction strength $J/t=8$ deep in the SMG phase and present the resulting
Green’s function in Fig. 3. From Fig. 3(a), the poles of the Green’s function
form two dispersing bands around $\omega=\pm J$, which resembles the upper and
lower Hubbard bands in the Hubbard model. This indicates the quasi-particles
are fully gapped in the SMG phase. Meanwhile, from Fig. 3(b,c), the zeros of
the Green’s function appear around $\omega=-\alpha\epsilon_{{\bm{k}}}/J^{2}$
with some non-universal but positive coefficient $\alpha>0$. We find that the
“dispersion” of zeros is reversed compared to the original band dispersion
$\epsilon_{{\bm{k}}}$. In Fig. 4, we also numerically confirmed that the
“bandwidth” $w_{\text{zero}}$ of zeros is suppressed by the interaction $J$ as
$w_{\text{zero}}\sim J^{-2}$ as $J\to\infty$.

Figure 4: Scaling of the Green’s function zero “bandwidth” $w_{\text{zero}}$
with the interaction strength $J$. Circles represent the numerically
calculated $w_{\text{text}}$ at different $J$, and the line is a fit to the
data.

Building upon the above observation of the poles and zeros of the Green’s
function, we put forth the following empirical formula:

$G_{\text{SMG}}(\omega,{{\bm{k}}})=\frac{\omega+\alpha\epsilon_{{{\bm{k}}}}/J^{2}}{(\omega-\epsilon_{{\bm{k}}}/2)^{2}-J^{2}},$
(19)

as an approximate description of our numerical result Eq. (18). An important
aspect of this formula is the positioning of the Green’s function zeros
precisely around the initial Fermi surface (where $\epsilon_{{\bm{k}}}=0$) at
$\omega=0$. This is indicated by the small arrows in Fig. 3(c).

Assuming $\operatorname{Re}G_{\text{SMG}}(0,{{\bm{k}}})=0$ as the definition
of the zero Fermi surface in the SMG phase, it would encompass the same Fermi
volume as the pole Fermi surface in the Fermi liquid phase. As both
translation and charge conservation symmetries remain unbroken in the SMG
phase, the Luttinger theorem mandates the preservation of the Fermi volume.
Given that the SMG state is a fully gapped trivial insulator, there is no pole
(no quasi-particle) at low energy, thus the Green’s function can only rely on
zeros to fulfill the Fermi volume required by the Luttinger theorem, which is
explicitly demonstrated by Eq. (19).

###  III.2 Weak Coupling Analysis

Nevertheless, SMG is not the sole mechanism for gapping out the Fermi surface.
SSB might also open a full gap on the Fermi surface, which corresponds to the
Higgs mechanism for fermion mass generation. Specifically, in the bilayer
square lattice model Eq. (3), due to the perfect nesting of the Fermi surface,
the Fermi liquid exhibits strong instability toward SSB orders. Without loss
of generality, we will focus on the inter-layer exciton condensation in the
weak coupling limit. The corresponding order parameter $\phi_{\text{EC}}$ was
introduced in Eq. (6), which carries momentum ${\bm{Q}}=(\pi,\pi)$. The
exciton condensation leads to an SSB insulating phase, as noted in the phase
diagram Fig. 1(c). However, there are significant differences between the SMG
insulator and the SSB insulator, especially in terms of the structure of
Green’s function zeros.

Figure 5: Fermion Green’s function Eq. (20) $G_{\text{SSB}}$ in the SSB
insulator phase, assuming a gap size of $|\Delta|=t$. (a) The imaginary part
$-2\operatorname{Im}G(\omega+\mathrm{i}0_{+},{{\bm{k}}})_{11}$ in the $\langle
c_{{\bm{k}}}^{\dagger}c_{{\bm{k}}}\rangle$ channel, showing the pole (quasi-
particle peak) along gapped bands. (b) The real part of the determinant
$\operatorname{Re}\det G(\omega,{{\bm{k}}})$. No zero within the gap. In both
plots, the frequency is shifted by a small imaginary part
$\omega\to\omega+0.01\mathrm{i}t$ for better visualization of spectral
features.

In the SSB insulator phase, the Brillouin zone folds by the nesting vector
${\bm{Q}}=(\pi,\pi)$. The fermion Green’s function can be written in the
$(c_{{{\bm{k}}}},c_{{{\bm{k}}}+{\bm{Q}}})^{\intercal}$ basis (omitting layers
and spins freedom) as

$G_{\text{SSB}}(\omega,{{\bm{k}}})=\frac{\omega\sigma^{0}+\epsilon_{{\bm{k}}}\sigma^{3}+\operatorname{Re}\Delta\sigma^{1}+\operatorname{Im}\Delta\sigma^{2}}{\omega^{2}-\epsilon_{{\bm{k}}}^{2}-|\Delta|^{2}},$
(20)

where $\Delta=J\langle\phi_{\text{EC}}\rangle$ denotes the exciton gap induced
by the exciton condensation $\langle\phi_{\text{EC}}\rangle\neq 0$. The
properties of $G_{\text{SSB}}$ are illustrated in Fig. 5. The spectral
function in Fig. 5(a) depicts the quasi-particle peak along the band
dispersion, reflecting a gapped (insulating) band structure.

Since $G_{\text{SSB}}$ is a matrix, its zero structure should be defined by
its determinant being zero, i.e., $\det G_{\text{SSB}}(\omega,{{\bm{k}}})=0$,
which is the only way to define the zero structure in a basis independent
manner. Fig. 5(b) indicates the determinant of $G_{\text{SSB}}$ remains the
same sign within the band gap induced by the exciton condensation. Since
$G_{\text{SSB}}$ does not preserve the translation symmetry (as
$\Delta\to-\Delta$ is translation-odd), and $\Delta$ is non-zero, $\det
G_{\text{SSB}}$ does not have zeros crossing $\omega=0$ at the original Fermi
surface. These two observations are linked: the absence of translation
symmetry makes the Luttinger theorem ineffective, hence there is no
expectation for the zero Fermi surface in the SSB insulator.

As the interaction $J$ intensifies, the SSB insulator ultimately transitions
into the SMG insulator, as depicted in the phase diagram Fig. 1(c). During
this transition, the broken symmetry is restored, yet the fermion excitation
gap remains intact, similar to the pseudo-gap phenomenon seen in correlated
materials [119, 120]. In the context of modeling fermion spectral functions,
the pseudo-gap phenomenon can be interpreted as a consequence of the phase (or
orientation) fluctuations of fermion bilinear order parameters [121, 122, 123,
124, 125, 126, 127, 128]. In this picture, the order parameter
$\Delta=\Delta_{0}\mathrm{e}^{\mathrm{i}\theta}$ maintains a finite amplitude
$\Delta_{0}$ as we enter the SMG phase from the adjacent SSB phase, but its
phase $\theta$ is disordered by long-wavelength random fluctuations.
Consequently, on the large scale, $\Delta$ cannot condense to form long-range
order; but on a smaller scale, $\Delta_{0}$ still provides a local excitation
gap everywhere for fermions.

Based on this picture of the SMG state, the simplest treatment is to focus on
the long wavelength fluctuation of $\Delta$ and estimate its self-energy
correction for the fermion by

$\Sigma(\omega,{{\bm{k}}})=\raisebox{-3.0pt}{\includegraphics[scale={0.65}]{diag_Sigma}}=\mathop{\mathbb{E}}_{\Delta}\hat{\Delta}^{\dagger}G_{0}(\omega,{{\bm{k}}})\hat{\Delta}=\frac{\Delta_{0}^{2}}{\omega\sigma^{0}+\epsilon_{{\bm{k}}}\sigma^{3}},$
(21)

where the vertex operator is
$\hat{\Delta}:=\operatorname{Re}\Delta\sigma^{1}+\operatorname{Im}\Delta\sigma^{2}$
and the bare Green’s function is
$G_{0}(\omega,{{\bm{k}}})=(\omega\sigma^{0}-\epsilon_{{\bm{k}}}\sigma^{3})^{-1}$.
Here we have assumed that the correlation length $\xi$ of the bosonic field
$\Delta$ is long enough that its momentum is negligible for fermions. This
assumption is valid near the transition to the SSB phase, as the correlation
length diverges ($\xi\to\infty$) at the transition.

Using this self-energy to correct the bare Green’s function, we obtain

$\begin{split}G(\omega,{{\bm{k}}})&=(G_{0}(\omega,{{\bm{k}}})^{-1}-\Sigma(\omega,{{\bm{k}}}))^{-1}\\\
&=\frac{\omega\sigma^{0}+\epsilon_{{\bm{k}}}\sigma^{3}}{\omega^{2}-\epsilon_{{\bm{k}}}^{2}-\Delta_{0}^{2}}.\end{split}$
(22)

Since the translation symmetry has been restored in the SMG phase, we can
unfold the Green’s function back to the original Brillouin zone (by taking the
$G(\omega,{{\bm{k}}})_{11}$ component), which leads to a weak coupling
description of the Green’s function in the shallow SMG phase near the
transition to the SSB phase

$G^{\prime}_{\text{SMG}}(\omega,{{\bm{k}}})=\frac{\omega+\epsilon_{{\bm{k}}}}{\omega^{2}-\epsilon_{{\bm{k}}}^{2}-\Delta_{0}^{2}}.$
(23)

A more rigorous treatment of a similar problem can be found in Ref. 95, which
includes finite momentum fluctuations of $\Delta$. The major effect of these
fluctuations is to introduce a spectral broadening for the fermion Green’s
function as if replacing $\omega\to\omega+\mathrm{i}\delta$ in Eq. (23). It
was also found that the broadening $\delta\sim\xi^{-1}$ scales inversely with
the correlation length $\xi$ of the order parameter, which justifies our
simple treatment in the large-$\xi$ regime. Similar Green’s functions as Eq.
(23) was previously constructed to describe non-Fermi liquid [98] statisfying
the Luttinger theorem. However, its physical meaning is now clarified as
Green’s function in the SMG phase.

Figure 6: Fermion Green’s function Eq. (23) $G^{\prime}_{\text{SMG}}$ in the
SMG insulator phase near the phase transition to an adjacent SSB phase,
assuming a local gap size of $\Delta_{0}=t$. (a) The imaginary part (spectral
function) $-2\operatorname{Im}G(\omega+\mathrm{i}0_{+},{{\bm{k}}})$ shows the
pole (quasi-particle peak) along gapped bands. (b) The real part
$\operatorname{Re}G(\omega,{{\bm{k}}})$ exhibits the zero (purple contour)
crossing $\omega=0$ at the original Fermi surface. In both plots, the
frequency is shifted by a small imaginary part
$\omega\to\omega+0.01\mathrm{i}t$ for better visualization of spectral
features.

The features of $G^{\prime}_{\text{SMG}}$ in Eq. (23) are presented in Fig. 6.
When comparing Fig. 6(a) and Fig. 5(a), we can observe that the pole structure
of $G^{\prime}_{\text{SMG}}$ is identical to that of $G_{\text{SSB}}$ (in the
diagonal component), both showcasing a gapped spectrum. However, they
significantly differ in their zero structures, as seen by comparing Fig. 6(b)
and Fig. 5(b). Due to the restoration of symmetry, the low-energy zeros
reemerge in the Green’s function in the SMG phase. Additionally, its zero
Fermi surface perfectly aligns with the original pole Fermi surface,
fulfilling the Luttinger theorem’s requirement for the Fermi volume.

Comparing the Green’s function in the SMG phase derived from the strong
coupling analysis Eq. (19) and the weak coupling analysis Eq. (23) (see also
Fig. 3 and Fig. 6), we find that despite the apparent difference in high-
energy spectral features, the zero Fermi surface defined by
$G(0,{{\bm{k}}})=0$ remains a resilient low-energy feature. The persistent
zero Fermi surface in the SMG phase is a consequence of the Luttinger theorem.

Nonetheless, besides the low-energy zero structure, it is also intriguing to
understand how the high-energy spectral feature deforms from the weak coupling
case to the strong coupling case. However, this problem requires non-
perturbative numerical simulations. Fortunately, the bilayer square lattice
model Eq. (3) admits a sign-problem-free [129] quantum Monte Carlo [130, 131,
132, 133, 134] simulation. We will leave this interesting direction for future
research.

##  IV Probing Green’s Function Zeros

While Green’s function zeros are an important feature of the SMG insulator,
they are not directly observable in experiments. Spectroscopy experiments,
such as angle-resolved photoemission spectroscopy (ARPES), can directly probe
the fermion’s spectral function
$A(\omega,{{\bm{k}}})=-2\operatorname{Im}G(\omega+\mathrm{i}0_{+},{{\bm{k}}})$,
which is the imaginary part of Green’s function. By employing the Kramers-
Kronig (KK) relation to recover the real part of Green’s function from the
spectral function,

$\operatorname{Re}G(\omega,{{\bm{k}}})=\frac{1}{2\pi}\mathcal{P}\int
d\omega^{\prime}\frac{A(\omega^{\prime},{{\bm{k}}})}{\omega^{\prime}-\omega},$
(24)

we can indirectly study the zero structure of the Green’s function.

Figure 7: (a) Broadened spectral function from the one in Fig. 3. (b)
Reconstructed Green’s function real part by the KK relation, showing robust
Green’s function zeros (purple contour) crossing $\omega=0$.

However, the spectral function might be broadened in experimental data due to
noise or dissipation. We are interested in studying how sensitive the
reconstructed Green’s function zero is to these disturbances, in order to
understand the stability of the method. Following Sec. III.1, we start from
the strong coupling limit and use the CPT approach to calculate Green’s
function. To account for the spectral broadening effect, we replace $\omega$
with $\omega+\mathrm{i}\delta$, where $\delta$ is relatively large, say, about
the order of the hopping $t$. Based on the broadened spectral function in Fig.
7(a), we use the KK relation to reconstruct the real part, as shown in Fig.
7(b). We find that the zero Fermi surface maintains the same shape, but the
zero “dispersion” bandwidth gets larger.

The increase in bandwidth can be understood by taking the SMG Green’s function
$G_{\text{SMG}}(\omega,{{\bm{k}}})$ in Eq. (19), and solving for its zeros
$\operatorname{Re}G(\omega+\mathrm{i}\delta,{{\bm{k}}})=0$. To the leading
order of $1/J$ and $\delta$, the solution is given by

$\omega({{\bm{k}}})=-\Big{(}1+\frac{\delta^{2}}{\alpha}\Big{)}\frac{\alpha\epsilon_{{\bm{k}}}}{J^{2}}+\cdots,$
(25)

meaning that the bandwidth of Green’s function zero dispersion will increase
by $\delta^{2}/\alpha$, but the corresponding Luttinger surface remains
unchanged. Therefore, the Green’s function zero in the SMG phase is a robust
feature that can be potentially identified from spectroscopy measurements,
even in the presence of noises or dissipations.

##  V Summary and Discussions

In this paper, we investigated the Fermi surface SMG in a bilayer square
lattice model. A crucial finding of this study lies in the robust Green’s
function zero in the SMG phase. Traditionally, a Fermi liquid state is
characterized by poles in the Green’s function along the Fermi surface.
However, as the fermion system is driven into the SMG state by interaction
effects, these poles are replaced by zeros. This is a robust phenomenon
underlined by the constraints of the Luttinger theorem.

Our exploration is not limited to theoretical assertions. We also offer a
tangible demonstration of this occurrence in the bilayer square lattice model.
By applying both strong and weak coupling analyses, we provide a comprehensive
portrayal of the fermion Green’s function across different interaction
regimes. We highlight that the emergence of the zero Fermi surface is not an
ephemeral or fine-tuned phenomenon, but rather a robust and enduring feature
of the SMG phase. We show that even when the system is subjected to spectral
broadening, the zero Fermi surface persists, retaining the Fermi volume.

The results of this study confirm the robustness of the zero Fermi surface and
underscore the possibility of observing it in experimental setups, such as
through ARPES. Despite not being directly observable, the zero structure of
the Green’s function could be inferred indirectly via the KK relation.

The bilayer square lattice model may be relevant to the nickelate
superconductor recently discovered in pressurized La3Ni2O7 [135, 136], which
is a layered two-dimensional material where each layer consists of nickel
atoms arranged in a bilayer square lattice. The Fermi surface is dominated by
$d_{z^{2}}$ and $d_{x^{2}-y^{2}}$ electrons of Ni. The $d_{z^{2}}$ electron
has a relatively small intra-layer hopping $t$ due to the rather localized
$d_{z^{2}}$ orbital wave function in the $xy$-plane but enjoys a large
interlayer antiferromagnetic Heisenberg interaction $J$ due to the super-
exchange mechanism mediated by the apical oxygen. This likely puts the
$d_{z^{2}}$ electrons in an SMG insulator phase in the bilayer square lattice
model and opens up opportunities to investigate the proposed Green’s function
zeros in real materials. The potential implication of SMG physics on the
nickelate high-$T_{c}$ superconductor still requires further theoretical
research in the future.

###### Acknowledgements.

We acknowledge the helpful discussions with Liujun Zou, Zi-Xiang Li, Fan Yang,
Yang Qi, Subir Sachdev, and Ya-Hui Zhang. All authors are supported by the NSF
Grant DMR-2238360.

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##  Appendix A Cluster Perturbation Theory

Here we review the details of cluster perturbation theory (CPT) originally
developed in [94]. Denote the superlattice lattice points by ${{\bm{R}}}$,
then the position of any original lattice point would be given by
${{\bm{R}}}+{{\bm{r}}}$, where ${{\bm{r}}}$ is the relative position of the
lattice point to the location ${{\bm{R}}}$ of the cluster containing that
particular lattice point. For clusters of size $L$, the generic Green’s
function in real space can be denoted by
$G_{i,j}^{{{\bm{R}}},{{\bm{R}}}^{\prime}}$, with $i,j=1,...,L$, where the
time-dependence is implicitly assumed and same goes for the frequency-
dependence in Fourier space. Due to the translation invariance of the clusters
on the superlattice, the real space Green’s function can be firstly partially
Fourier-transformed to give,

$G_{i,j}^{{{\bm{R}}},{{\bm{R}}}^{\prime}}=\frac{1}{N}\sum_{{\bm{q}}}G({\bm{q}})_{ij}\mathrm{e}^{i{\bm{q}}\cdot({{\bm{R}}}-{{\bm{R}}}^{\prime})},$
(26)

where the ${\bm{q}}$-summation is over the Brillouin zone (BZ) of the
superlattice and $N$ is the number of clusters on the superlattice, which goes
to infinity in the thermodynamic limit. In contrast to the translation
invariance of the $({{\bm{R}}},{{\bm{R}}}^{\prime})$-part of
$G_{i,j}^{{{\bm{R}}},{{\bm{R}}}^{\prime}}$, or equivalently it only depends on
the difference ${{\bm{R}}}-{{\bm{R}}}^{\prime}$ as can be seen in Eq. (26),
the $(i,j)$-part of the Green’s function loses translation invariance due to
the introduction of clusters. This is so because correlation between two
points within the same cluster is not manifestly the same with the correlation
between another pair of equally separated points across clusters. Therefore,
it takes two lattice momenta to fully characterize
$G_{i,j}^{{{\bm{R}}},{{\bm{R}}}^{\prime}}$ in Fourier space. More precisely,
we have,

$G({{\bm{k}}},{{\bm{k}}}^{\prime})=\frac{1}{NL}\sum_{{{\bm{R}}},{{\bm{R}}}^{\prime}}\sum_{i,j}G_{i,j}^{{{\bm{R}}},{{\bm{R}}}^{\prime}}\mathrm{e}^{i{{\bm{k}}}\cdot({{\bm{R}}}+{{\bm{r}}}_{i})-i{{\bm{k}}}^{\prime}\cdot({{\bm{R}}}^{\prime}+{{\bm{r}}}_{j})}.$
(27)

Then we can plug Eq. (26) into Eq. (27) and integrate out the superlattice
lattice vectors ${{\bm{R}}},{{\bm{R}}}^{\prime}$ to obtain the following,

$G({{\bm{k}}},{{\bm{k}}}^{\prime})=\frac{1}{L}\sum_{i,j}\sum_{{\bm{q}}}G({\bm{q}})_{ij}\tilde{\delta}_{{{\bm{k}}},{\bm{q}}}\tilde{\delta}_{{{\bm{k}}}^{\prime},{\bm{q}}}\mathrm{e}^{i({{\bm{k}}}\cdot{{\bm{r}}}_{i}-{{\bm{k}}}^{\prime}\cdot{{\bm{r}}}_{j})},$
(28)

where the $\tilde{\delta}$-function denotes the fact that the two wavevectors
are equivalent only up to a superlattice reciprocal lattice vector ${\bm{Q}}$
because ${\bm{Q}}\cdot{{\bm{R}}}=2\pi\mathbb{Z}$ in the phase factor. More
precisely, we have

$\tilde{\delta}_{{{\bm{k}}},{\bm{q}}}=\sum_{s=1}^{L}\delta_{{{\bm{k}}},{\bm{q}}+{\bm{Q}}_{s}},$
(29)

where ${\bm{Q}}_{s}$ with $s=1,...,L$ are the $L$ inequivalent wave vectors in
the reciprocal lattice of the original lattice (see the 1d case shown in Fig.
8). Then we can perform the ${\bm{q}}$-summation in Eq. (28) to have,

$\begin{split}G({{\bm{k}}},{{\bm{k}}}^{\prime})&=\frac{1}{L}\sum_{i,j}\sum_{s,s^{\prime}}G({{\bm{k}}}-{\bm{Q}}_{s})_{ij}\delta_{{{\bm{k}}}^{\prime}-{{\bm{k}}},{\bm{Q}}_{s}-{\bm{Q}}_{s^{\prime}}}\mathrm{e}^{i({{\bm{k}}}\cdot{{\bm{r}}}_{i}-{{\bm{k}}}^{\prime}\cdot{{\bm{r}}}_{j})}\\\
&=\sum_{i,j}\sum_{\Delta{\bm{Q}}}G({{\bm{k}}})_{ij}\delta_{{{\bm{k}}}^{\prime}-{{\bm{k}}},\Delta{\bm{Q}}}\mathrm{e}^{i({{\bm{k}}}\cdot{{\bm{r}}}_{i}-{{\bm{k}}}^{\prime}\cdot{{\bm{r}}}_{j})},\end{split}$
(30)

where we have used the fact that $G({\bm{q}})_{ij}$ is invariant under the
shift by a superlattice reciprocal lattice vector ${\bm{Q}}_{s}$.

Figure 8: Reciprocal lattice in 1d for a 4-site cluster. $K$ labels the
reciprocal lattice vector for the original lattice and $Q$ labels the
reciprocal lattice vector for the superlattice. More precisely,
$K_{s}=\frac{2\pi}{a}s$ and $Q_{s}=\frac{2\pi}{La}s$, where $a$ is the lattice
constant of the original lattice, $L=4$ here and $s\in\mathbb{Z}$ .

The translation invariant approximation for the Green’s function on the
original lattice is obtained when $\Delta{\bm{Q}}=0$, i.e.
${{\bm{k}}}={{\bm{k}}}^{\prime}$. Therefore, the Green’s function becomes,

$G({{\bm{k}}})=\sum_{i,j}G({{\bm{k}}})_{ij}\mathrm{e}^{i{{\bm{k}}}\cdot({{\bm{r}}}_{i}-{{\bm{r}}}_{j})}.$
(31)

Figure 9: Cluster diagram showing the hopping between neighboring clusters
(dashed line). The four sites inside each cluster are numbered as shown.

Now we just need to calculate $G_{i,j}({{\bm{k}}})$ using cluster
perturbation. The idea is to treat hopping between clusters as perturbation
when consider strong on-site interactions. In particular,

$\hat{H}=\hat{H}_{0}+\hat{V},$ (32)

where $\hat{H}_{0}$ contains intra-cluster terms and $\hat{V}$ contains inter-
cluster hopping. Considering nearest-neighbor hopping between the square
clusters used in the main text. The cluster construction is reproduced in Fig.
9 with the 4 sites in each cluster labeled by $1,2,3,4$. The hopping matrix is
given by (setting lattice constant $a=1$)

$\begin{split}V_{i,j}^{{{\bm{R}}},{{\bm{R}}}^{\prime}}=&-t\delta_{{{\bm{R}}},{{\bm{R}}}^{\prime}-2\hat{x}}(\delta_{i,2}\delta_{j,1}+\delta_{i,3}\delta_{j,4})\\\
&-t\delta_{{{\bm{R}}},{{\bm{R}}}^{\prime}+2\hat{x}}(\delta_{i,1}\delta_{j,2}+\delta_{i,4}\delta_{j,3})\\\
&-t\delta_{{{\bm{R}}},{{\bm{R}}}^{\prime}-2\hat{y}}(\delta_{i,1}\delta_{j,4}+\delta_{i,2}\delta_{j,3})\\\
&-t\delta_{{{\bm{R}}},{{\bm{R}}}^{\prime}+2\hat{y}}(\delta_{i,3}\delta_{j,2}+\delta_{i,4}\delta_{j,1})\end{split}$
(33)

Fourier transforming $V_{i,j}^{{{\bm{R}}},{{\bm{R}}}^{\prime}}$ into the
superlattice reciprocal space, we have

$\begin{split}V_{i,j}({\bm{q}})&=-t\mathrm{e}^{i2q_{x}}(\delta_{i,2}\delta_{j,1}+\delta_{i,3}\delta_{j,4})\\\
&\quad-t\mathrm{e}^{-i2q_{x}}(\delta_{i,1}\delta_{j,2}+\delta_{i,4}\delta_{j,3})\\\
&\quad-t\mathrm{e}^{i2q_{y}}(\delta_{i,1}\delta_{j,4}+\delta_{i,2}\delta_{j,3})\\\
&\quad-t\mathrm{e}^{-i2q_{y}}(\delta_{i,3}\delta_{j,2}+\delta_{i,4}\delta_{j,1})\\\
&=-t\begin{pmatrix}0&e^{-i2{\bm{q}}_{x}}&0&e^{i2{\bm{q}}_{y}}\\\
e^{i2{\bm{q}}_{x}}&0&e^{i2{\bm{q}}_{y}}&0\\\
0&e^{-i2{\bm{q}}_{y}}&0&e^{i2{\bm{q}}_{x}}\\\
e^{-i2{\bm{q}}_{y}}&0&e^{-i2{\bm{q}}_{x}}&0\end{pmatrix}_{i,j},\end{split}$
(34)

which is the form presented in Eq. (17) in the main text. Then the interacting
Green’s function is given by

$\begin{split}\hat{G}({\bm{q}})&=\frac{1}{\omega-\hat{H}}=\frac{1}{\omega-\hat{H}_{0}-\hat{V}({\bm{q}})}\\\
&=\frac{\hat{G}_{0}}{1-\hat{V}({\bm{q}})\hat{G}_{0}},\end{split}$ (35)

where $\hat{G}_{0}\equiv(\omega-\hat{H}_{0})^{-1}$ is the intra-cluster
Green’s function that can be easily obtained by exact diagonalization as long
as the cluster size is not too big. The obtained $G({\bm{q}})_{ij}$ can now be
plugged into Eq. (31) to calculate the CPT Green’s function for the
interacting system.