# Optical Conductivity in Symmetric Mass Generation Insulators

Meng Zeng  Department of Physics, University of California at San Diego, La
Jolla, CA 92093, USA  Fu Xu  Department of Physics, Nanjing University,
Nanjing, Jiangsu 210093, China  Da-Chuan Lu  Department of Physics, University
of California at San Diego, La Jolla, CA 92093, USA  Yi-Zhuang You
<EMAIL_ADDRESS>Department of Physics, University of California at San
Diego, La Jolla, CA 92093, USA

###### Abstract

Symmetric mass generation (SMG) insulators are interaction-driven, featureless
Mott insulating states in quantum many-body fermionic systems. Recent
advancements suggest that zeros in the fermion Green’s function could lead to
non-vanishing negative optical conductivity in SMG insulators, even below the
charge excitation gap. This study explores the origin of this unusual behavior
through the lens of pole-zero duality, highlighting a critical issue where the
current operator becomes unbounded, rendering the response function
unphysical. By employing a lattice model, we derive a well-behaved lattice
regularization of the current operator, enabling a detailed study of optical
conductivity in SMG insulators. Utilizing both analytical and numerical
methods, including strong-coupling expansions, we confirm that SMG insulators
exhibit no optical conductivity at low energies below the charge gap,
effectively resolving the paradox. This work not only deepens our
understanding of quantum many-body phenomena but also lays a robust
theoretical groundwork for future experimental explorations of SMG materials.

_Introduction._ — Symmetric mass generation (SMG) insulators [1, 2, 3, 4, 5,
6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25,
26, 27, 28, 29, 30] represent a novel class of interaction-driven
_featureless_ Mott insulating states in quantum many-body systems of fermions.
These systems feature the cancellation of all quantum anomalies [31, 32, 33],
such that a symmetric gapped state without spontaneous symmetry breaking (SSB)
or topological order is allowed, which we refer to as “featureless”. SMG
insulators are characterized by a full energy gap to all fermionic and bosonic
excitations, including collective charge and current excitations. The
excitation gap arises from non-perturbative interaction effects and eludes
mean-field theoretical explanations.

Central to the theoretical understanding of SMG insulators is the behavior of
the fermion Green’s function, defined as:
$G(k):=-\langle\psi_{k}\psi_{k}^{\dagger}\rangle$, where $\psi_{k}$ is the
fermion operator at the energy-momentum $k=(\omega,{\bm{k}})$. A key aspect of
the SMG insulator is that its fermion Green’s function determinant approaches
zero as energy-momentum tends to zero [34, 35, 36, 37, 38, 39, 40, 41, 42, 43,
44, 45, 46, 47, 48, 49], i.e.

$\det G(k)=0\text{ as }k_{\mu}k^{\mu}\to 0.$ (1)

This property, the zero of the Green’s function, raises an intriguing question
regarding its experimental implications [46, 29, 50, 45, 47, 48]. Recently,
Golterman and Shamir [49] proposed that these zeros might significantly
influence the electromagnetic response of the SMG insulator when the fermions
are coupled to a background $\mathrm{U}(1)$ gauge field, particularly
suggesting a scenario where the SMG insulator exhibits non-vanishing charge
conductivity at low energy, even below the energy gap of all excitations.

This apparent puzzle of how an _insulator_ might exhibit _conductive_ behavior
without charge excitations below the insulating gap presents a fascinating
paradox. This research aims to postulate a potential resolution of this
paradox, offering new insights into the behavior of the optical conductivity
in SMG insulators and expanding our understanding of quantum many-body
phenomena.

_Pole-Zero Duality._ — Let us first revisit the Golterman-Shamir construction
[49], and reproduce their results through the broader lens of pole-zero
duality in the fermion Green’s function.

The analysis starts with the fermion two-point correlation function
$G(k)=-\langle\psi_{k}\psi_{k}^{\dagger}\rangle$ in the energy-momentum space.
At this point, Golterman and Shamir introduced a pivotal _assumption_ that the
fermion system can be approximated by a _free_ effective action,

$S[\psi]=-\sum_{k}\psi_{k}^{\dagger}G(k)^{-1}\psi_{k}=-\int\mathrm{d}^{d}x\;\psi^{\dagger}G(\mathrm{i}\partial)^{-1}\psi,$
(2)

such that the Green’s function will be consistently reproduced by the fermion
path integral:

$G(k)=\frac{1}{Z}\int\mathcal{D}[\psi]\,(-\psi_{k}\psi_{k}^{\dagger})e^{-S[\psi]},$
(3)

with $Z=\int\mathcal{D}[\psi]e^{-S[\psi]}$ being the partition function. It is
crucial to acknowledge that this approach to reconstructing the effective
action from the two-point correlation is only valid under the premise that the
fermions behave as free or generalized free fields [51, 52, 53, 54, 55]. In
such cases, higher-point correlations decompose into two-point correlations
via Wick’s theorem. Should the fermions deviate from generalized free field
behavior, it becomes necessary to incorporate higher-order terms in the
effective action to model higher-point correlation functions.

If we accept the effective action $S[\psi]$ in Eq. (2), we can proceed to
gauge the $\mathrm{U}(1)$ symmetry of the fermion field $\psi$, under which
$\psi\to\mathrm{e}^{\mathrm{i}\theta}\psi$. By introducing the $\mathrm{U}(1)$
gauge field $A$ through minimal coupling, the effective action becomes:

$S[\psi,A]=-\int\mathrm{d}^{d}x\;\psi^{\dagger}G(\mathrm{i}\partial-A)^{-1}\psi.$
(4)

Integrating out the fermion field $\psi$, the fermion path integral
$\mathrm{e}^{-S[A]}=\int\mathcal{D}[\psi]\mathrm{e}^{-S[\psi,A]}$ defines an
effective action $S[A]$ for the gauge field $A$:

$\begin{split}S[A]&=\operatorname{Tr}\log G(\mathrm{i}\partial-A)\\\
&=\sum_{n=0}^{\infty}\frac{1}{n!}\Pi_{n}^{\mu_{1}\mu_{2}\cdots\mu_{n}}A_{\mu_{1}}A_{\mu_{2}}\cdots
A_{\mu_{n}},\end{split}$ (5)

where
$\Pi_{n}^{\mu_{1}\mu_{2}\cdots\mu_{n}}:=\delta_{A_{\mu_{1}}}\delta_{A_{\mu_{2}}}\cdots\delta_{A_{\mu_{n}}}S[A]$
corresponds to the $n$th order current correlation, or loosely denoted as
$\Pi_{n}=\delta_{A}^{n}S[A]$. These correlations $\Pi_{n}$ encode the
responses of the fermion system to the external electromagnetic field at
different orders. The goal is to understand their behaviors across the SMG
transition.

The SMG transition refers to the fermion gap-opening transition driven by the
fermion interaction [33]. On the weakly interacting side, the fermion system
is metallic, characterized by gapless single-particle excitations at low
energy, manifested as poles in the fermion Green’s function. In contrast, on
the strongly interacting side, the system transitions into the SMG insulating
phase, where the original poles in the Green’s function are replaced by zeros.
For instance, in the case of the relativistic fermions discussed in Ref. 49,
the propagator poles and zeros are respectively modeled by $G_{\text{Dirac}}$
and $G_{\text{SMG}}$ as

$G_{\text{Dirac}}(k)=\frac{1}{\gamma^{0}\gamma^{\mu}k_{\mu}},\quad
G_{\text{SMG}}(k)=-\frac{\gamma^{0}\gamma^{\mu}k_{\mu}}{m^{2}}.$ (6)

The concept of pole-zero duality [35, 42, 43, 46] offers a compelling
framework for relating the low-energy behaviors of Green’s functions across
the SMG transition. This duality is articulated through a transformation of
the fermion Green’s function,

$G(k)\to\tilde{G}(k)\propto G(-k)^{-1},$ (7)

under which poles and zeros replace each other. For example,
$G_{\text{Dirac}}$ and $G_{\text{SMG}}$ in Eq. (6) are related by the pole-
zero duality.

Under the pole-zero duality defined in Eq. (7), the effective gauge action
$S[A]=\operatorname{Tr}\log G(\mathrm{i}\partial-A)$ in Eq. (5) transforms as

$S[A]\to\operatorname{Tr}\log G(-\mathrm{i}\partial+A)^{-1}=-S[-A],$ (8)

then the $n$th order current correlation $\Pi_{n}=\delta_{A}^{n}S[A]$
transforms as

$\Pi_{n}\to-\delta_{A}^{n}S[-A]=-(-1)^{n}\Pi_{n}.$ (9)

This implies that if the fermion Green’s functions across the SMG transition
are related by pole-zero duality, as exemplified by $G_{\text{Dirac}}$ and
$G_{\text{SMG}}$ in Eq. (6), then their corresponding electromagnetic response
functions will also be related by (for $n=2,3$)

$\Pi_{2,\text{SMG}}=-\Pi_{2,\text{Dirac}},\quad\Pi_{3,\text{SMG}}=\Pi_{3,\text{Dirac}},$
(10)

which reproduce the main conclusions in Ref. 49 that, compared to free Dirac
fermions, the vacuum polarization $\Pi_{2}$ changes sign in the SMG insulator
while the triangle diagram $\Pi_{3}$ remains the same.

These results are remarkably general and do not depend on the specific form of
the Green’s function $G(k)$. Provided we accept the effective action
$S[\psi,A]$ in Eq. (4) as our starting point, the conclusions outlined above
are inevitable under the principle of pole-zero duality.

The implications of these results are significant. Since vacuum polarization
is connected to optical conductivity by
$\operatorname{Re}\sigma(\omega,{\bm{k}})=-\frac{1}{\omega}\operatorname{Im}\Pi_{2}(\omega,{\bm{k}})$
[56], the relationship $\Pi_{2,\text{SMG}}=-\Pi_{2,\text{Dirac}}$ would imply
that $\sigma_{\text{SMG}}=-\sigma_{\text{Dirac}}$, indicating that the gapped
SMG insulator would exhibit a conductivity that is finite and opposite to that
of the gapless Dirac semimetal. However, we should not anticipate finite
conductivity in an insulator below the charge excitation gap. Additionally,
the notion of negative conductivity raises concerns about the potential
violation of the fluctuation-dissipation theorem and the loss of unitarity in
the theory.

_Unbounded Current Operator_ — Given the perplexing behavior of conductivity,
we are motivated to examine the foundational assumptions of the Golterman-
Shamir construction. Specifically, it is presumed that the effective action
$S[\psi,A]$ described in Eq. (4) models the physics of the SMG insulator in a
background electromagnetic field. Starting from this premise, the current
operator in the system should be given by

$J^{\mu}=-\delta_{A_{\mu}}S[\psi,A\to
0]=\sum_{k}\psi_{k}^{\dagger}\partial_{k_{\mu}}G(k)^{-1}\psi_{k}.$ (11)

For Dirac fermions described by $G_{\text{Dirac}}(k)$ in Eq. (6), Eq. (11)
gives the current operator in the conventional form
$J^{\mu}=\sum_{k}\bar{\psi}_{k}\gamma^{\mu}\psi_{k}$ (where
$\bar{\psi}:=\psi^{\dagger}\gamma^{0}$), which has a bounded spectrum.
However, for the SMG insulator, if we naively substitute the Green’s function
$G_{\text{SMG}}(k)$ from Eq. (6) into Eq. (11),

$J^{\mu}=\sum_{k}\frac{m^{2}}{k^{4}}(2k_{\mu}k_{\nu}-k^{2}\delta_{\mu\nu})\bar{\psi}_{k}\gamma^{\nu}\psi_{k},$
(12)

we find that the resulting current operator $J^{\mu}$ would diverge as
$k^{2}:=k_{\mu}k^{\mu}\rightarrow 0$. This divergence is a direct consequence
of the Green’s function zeros along the light cone ($k^{2}=0$) below the SMG
insulating gap. Such a current operator has an unbounded spectrum, which is
unphysical because this would imply the existence of quantum states in which
the velocity of charge movement could potentially exceed the speed of light.

Starting from such an unbounded current operator to define the current-current
correlation $\Pi_{2}^{\mu\nu}=-\langle J^{\mu}J^{\nu}\rangle$ could
potentially lead to unphysical results. This perspective makes the unusual
behavior of $\Pi_{2}$ less surprising. It implies that $S[\psi,A]$ in Eq. (4)
might not be a complete theory for describing the SMG insulator. Given that
the SMG insulator is intrinsically a strongly interacting system, it is
reasonable to suspect that the action should include various higher-order
terms to cancle the divergence of the current operator, thereby ensuring a
well-defined bounded current operator.

_Lattice Modeling_ — Having recognized the critical issue with the unbounded
current operator, our goal is to move beyond the effective action $S[\psi,A]$
and explore the electromagnetic response of the SMG insulator from a more
fundamental perspective.

To address this challenge, we turn to a concrete lattice model for the SMG
insulator. Consider a system comprising four flavors of fermions $c_{ia}$
(where $a=1,2,3,4$) on each lattice site $i$, governed by the following
Hamiltonian

$H=-\sum_{ij}t_{ij}\mathrm{e}^{\mathrm{i}A_{ij}}c_{ia}^{\dagger}c_{ja}-g\sum_{i}c_{i1}^{\dagger}c_{i2}^{\dagger}c_{i3}c_{i4}+\text{h.c.},$
(13)

where the repeated flavor index $a$ in $c_{ia}^{\dagger}c_{ja}$ will be
implicitly summed over, and “h.c.” represents the Hermitian conjugate terms.
In this model, a background $\mathrm{U}(1)$ gauge connection $A_{ij}$ is
introduced between every pair of sites to gauge the global $\mathrm{U}(1)$
symmetry of the fermions (acting as
$c_{ia}\to\mathrm{e}^{\mathrm{i}\theta}c_{ia}$).

We further specify that the fermions in the system are _half-filled_ , a
crucial condition for cancelling the Fermi-surface anomaly [57, 58, 59, 60,
61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73] and enabling an SMG
insulating state [26, 27]. To enforce the half-filling condition without fine-
tuning the chemical potential, a simple approach is to first assume a
bipartite lattice structure (e.g., a square or honeycomb lattice that can be
partitioned into $A$ and $B$ sublattices), and then impose an anti-unitary
sublattice particle-hole symmetry $\mathbb{Z}_{2}^{\mathcal{S}}$ (also known
as the chiral symmetry [74, 75]), under which
$c_{ia}\to\mathcal{K}(-)^{i}c_{ia}^{\dagger}$, with an alternating sign
$(-)^{i}=\pm 1$ for the site $i$ in $A$ and $B$ sublattices respectively. Here
$\mathcal{K}$ represents the complex conjugation operator that
$\mathcal{K}^{2}=1$ and $\mathcal{K}\mathrm{i}\mathcal{K}=-\mathrm{i}$.

Let us first turn off the background gauge field by setting $A_{ij}=0$. In the
free fermion limit ($g=0$), the Hamiltonian $H$ describes a fermion hopping
model on a bipartite lattice with a chiral symmetry
$\mathbb{Z}_{2}^{\mathcal{S}}$. In the momentum space, the
$\mathbb{Z}_{2}^{\mathcal{S}}$ symmetry transforms the fermions as
$c_{{\bm{k}}a}\to\mathcal{K}\sigma^{3}c_{{\bm{k}}a}^{\dagger}$, enforcing the
Hamiltonian to take the form
$H=\sum_{{\bm{k}}}c_{{\bm{k}}a}^{\dagger}\xi_{{\bm{k}}}\sigma^{1}c_{{\bm{k}}a}$,
where $\sigma^{\alpha}$ ($\alpha=0,1,2,3$) denote the Pauli matrices acting
within the sublattice Hilbert space. The specific details of the band
dispersion $\xi_{{\bm{k}}}$ are not crucial to our discussion. Without fine-
tuning the band structure, $\xi_{\bm{k}}$ typically exhibits a Fermi surface,
rendering the fermion system as a gapless Fermi liquid in general.

Conversely, in the strong interaction limit ($g\to\infty$), the hopping term
$t_{ij}$ can be omitted relative to $g$, and the model is decoupled to
individual sites, permitting an independent solution for each site. In this
limit, the exact many-body ground state of $H$ is a product state:

$|\Psi_{\text{SMG}}\rangle=\prod_{i}\frac{1}{\sqrt{2}}(c_{i1}^{\dagger}c_{i2}^{\dagger}-c_{i3}^{\dagger}c_{i4}^{\dagger})|0\rangle,$
(14)

where $|0\rangle$ represents the vacuum state of the fermions. This solution
arises because the four-fermion interaction $g$ directly hybridizes the two-
fermion states $c_{i1}^{\dagger}c_{i2}^{\dagger}|0\rangle$ and
$c_{i3}^{\dagger}c_{i4}^{\dagger}|0\rangle$ on each site, thereby lowering the
energy of the particular superposition state of them in Eq. (14). The
resulting product state $|\Psi_{\text{SMG}}\rangle$ maintains the full
symmetry of the Hamiltonian $H$ and exhibits a gap of order $g$ to all
excitations, therefore realizing an SMG insulator (in its ideal limit). It is
noteworthy that the fermions are automatically half-filled on every site,
which is precisely why we emphasize the half-filling condition from the
outset. Otherwise, we would have to violate the fermion number conservation
when driving the system into the SMG insulating state as we increase the
interaction strength $g$.

_Ideal SMG Limit._ — As long as at half-filling, regardless of the band
structure of $\xi_{{\bm{k}}}$, strong enough interaction $g$ in this model
will always drive the fermion system into the SMG insulating phase.
Understanding how the SMG transition happens, as a metal-insulator transition,
is a fascinating yet challenging problem. However, this inquiry is beyond the
scope of our current analysis. Instead, our focus will be on the strongly
interacting regime where $g\gg t_{ij}$, and we aim to study the current
correlation in the SMG insulating phase.

First, to properly define the current operator in the lattice model Eq. (13),
we start by introducing the background gauge field $A_{ij}$, differentiating
$H$ with respect to $A_{ij}$ and subsequently taking the limit of $A_{ij}\to
0$,

$J_{ij}=\frac{\delta H}{\delta
A_{ij}}=-(\mathrm{i}t_{ij}c_{ia}^{\dagger}c_{ja}+\text{h.c.}).$ (15)

This current operator $J_{ij}$, well-defined on the lattice, has a bounded
spectrum and does not suffer from the previous problem of unbounded current in
the effective action approach. We can observe that the on-site interaction $g$
has no influence on the definition of the current operator in Eq. (15), as the
interaction term was not modified by the $\mathrm{U}(1)$ background field
$A_{ij}$ in the Hamiltonian Eq. (13) to begin with. The lattice current
operator $J_{ij}$ is entirely determined by the hopping term. Therefore, in
momentum space, the current operator
${\bm{J}}=\sum_{\bm{k}}c_{{\bm{k}}a}^{\dagger}\partial_{\bm{k}}\xi_{\bm{k}}\sigma^{1}c_{{\bm{k}}a}$
can also be expressed as solely dependent on the band dispersion
$\xi_{\bm{k}}$, the same as in the free fermion limit.

Then, we can compute the current-current correlation on the lattice as
$\Pi(t)=-\mathrm{i}\langle[J_{ij}(t),J_{kl}(0)]\rangle\Theta(t)$, where
$J_{ij}(t)=\mathrm{e}^{-\mathrm{i}Ht}J_{ij}\mathrm{e}^{\mathrm{i}Ht}$.
Evaluating the operator expectation values on the ideal ground state
$|\Psi_{\text{SMG}}\rangle$ in Eq. (14) of SMG insulator in the $g\gg t_{ij}$
limit, we find $\Pi(t)=4|t_{ij}|^{2}\sin(2gt)\Theta(t)\delta_{il}\delta_{jk}$.
Fourier transform to the frequency domain and averaging over all sites, the
optical conductivity
$\operatorname{Re}\sigma(\omega)=-\frac{1}{\omega}\operatorname{Im}\Pi(\omega)$
reads

$\operatorname{Re}\sigma(\omega)=\frac{2\pi|t|^{2}}{2g}\big{(}\delta(\omega-2g)+\delta(\omega+2g)\big{)},$
(16)

where $|t|^{2}:=\sum_{j}|t_{ij}|^{2}$ characterizes the overall hopping
strength. This result illustrates the expected reasonable behavior of the
conductivity in an ideal SMG insulator: $\operatorname{Re}\sigma(\omega)$
should vanish at low frequencies $|\omega|<2g$ within the charge gap $2g$. The
presence of sharp peaks at $\omega=\pm 2g$ is attributed to the fact that all
excitations exhibit flat dispersion in the ideal SMG state, as the system
decouples among independent sites. Deviating from this ideal limit, for a
finite $t_{ij}/g$, we should anticipate the peaks to broaden into continua
above the charge gap.

_Beyond Ideal Limit._ — To elucidate the general behavior of optical
conductivity in the SMG insulator beyond the strong interaction limit, we
engage a perturbative expansion in $t_{ij}/g$ around the ideal state
$|\Psi_{\text{SMG}}\rangle$ and calculate the correlation functions in the
perturbed state. The perturbation theory has been thoroughly analyzed in Ref.
46. The results indicate that the fermion Green’s function in the SMG
insulator of a $\mathbb{Z}_{2}^{\mathcal{S}}$-symmetric two-band system can be
approximated by:

$G_{\text{SMG}}(k)=\frac{\omega\sigma^{0}+\xi_{\bm{k}}\sigma^{1}}{\omega^{2}-\xi_{\bm{k}}^{2}-\Delta^{2}},$
(17)

where $\Delta\sim g$ represents the single-particle gap (i.e., the fermion
mass) renormalized from its bare value $g$ set by the interaction strength. It
is worth emphasizing that the gap $\Delta$ here is not a symmetry-breaking
order parameter and has no mean-field interpretation. Notably,
$G_{\text{SMG}}(k)$ exhibits zeros along $\omega=\pm\xi_{{\bm{k}}}$, a
distinguishing feature of SMG insulators. By contrast, in systems where the
gap opens due to spontaneous symmetry breaking (SSB), the fermion Green’s
function would take the following form:

$G_{\text{SSB}}(k)=\frac{\omega\sigma^{0}+\xi_{\bm{k}}\sigma^{1}+\Delta\sigma^{3}}{\omega^{2}-\xi_{\bm{k}}^{2}-\Delta^{2}},$
(18)

where the extra term $\Delta\sigma^{3}$ on the numerator breaks the
$\mathbb{Z}_{2}^{\mathcal{S}}$ symmetry, which is particularly absent in
$G_{\text{SMG}}$.

In any scenario, the charge $\Pi^{00}$ and current $\Pi^{ij}$ correlation
functions can be characterized by the following equations:

$\begin{split}\Pi^{00}(q)&=\operatorname{Tr}(G(k+q)G(k)),\\\
\Pi^{ij}(q)&=\operatorname{Tr}(v_{i}G(k+q)v_{j}G(k)).\end{split}$ (19)

In these expressions, the charge vertex operator is always $\sigma^{0}$, and
the current vertex operator is
${\bm{v}}=\partial_{{\bm{k}}}\xi_{{\bm{k}}}\sigma^{1}$. These vertex operators
align with the lattice current operators derived in Eq. (15), ensuring a
bounded operator spectrum.

Figure 1: The dynamic charge susceptibility
$\chi^{\prime\prime}(\nu,{\bm{q}})$ (imaginary part) and optical conductivity
$\sigma(\nu,{\bm{q}})$ as a function of frequency $\nu$ along a cut through
$\Gamma(0,0)$, X$(\pi,0)$, M$(\pi,\pi)$ points in the momentum space, for 2D
square lattice fermions in the Fermi liquid (FL), spontaneous symmetry
breaking (SSB), and symmetric mass generation (SMG) phases respectively.

For example, consider a square lattice fermion model characterized by the
dispersion relation $\xi_{\bm{k}}=-2t(\cos k_{x}+\cos k_{y})$ given the
hopping parameter $t$ (set as the energy unit). We can numerically evaluate
the spectral weight of the dynamic charge susceptibility
$\chi^{\prime\prime}(\nu,{\bm{q}})=-2\operatorname{Im}\Pi^{00}(\nu+\mathrm{i}0_{+},{\bm{q}})$
and the optical conductivity
$\sigma(\nu,{\bm{q}})=-\frac{1}{\nu}\operatorname{Im}\Pi^{ii}(\nu+\mathrm{i}0_{+},{\bm{q}})$
as outlined in Eq. (19). Plugging in the fermion Green’s function in different
phases: $G_{\text{FL}}(k)=(\omega\sigma^{0}-\xi_{\bm{k}}\sigma^{1})^{-1}$,
$G_{\text{SSB}}(k)$ in Eq. (18), and $G_{\text{SMG}}(k)$ in Eq. (17) (assuming
$\Delta=t$ in the SSB and SMG cases), their resulting electromagnetic response
functions are compared in Fig. 1. Much like that in the SSB insulator, the SMG
insulator’s electromagnetic response is gapped, displaying only subtle
differences in detail. Contrarily, it does not resemble the gapless response
typical of a Fermi liquid.

This analysis can be extended to SMG in non-chiral Dirac/Weyl fermions, where
on-site local gapping interactions generally exist, ensuring that the current
operator remains unaffected by interactions. This is a crucial element for our
argument concerning the lattice regularization of the current operator.
However, for chiral fermions, such as in the 3-4-5-0 model [76, 19, 24, 77],
where the SMG interaction is not on-site, regularizing the current operator is
an open problem for future research. Thus, while the paradox regarding how the
SMG insulator can exhibit finite optical conductivity despite its gap is
effectively resolved for non-chiral fermions, further work is needed for
chiral systems. This resolution hinges on a nuanced understanding of the
current operator in the SMG phase using lattice regularization. These insights
reinforce the SMG state’s insulating nature while clarifying its distinctive
low-energy electromagnetic properties, laying a theoretical foundation for
future experimental exploration of featureless Mott insulators.

###### Acknowledgements.

We acknowledge the discussions with Simon Catterall, Cenke Xu, Srimoyee Sen,
David Tong, Yigal Shamir, Maarten Golterman, and Lei Su. We thank Lei Su for
sharing their related upcoming work [78] with us. MZ, DCL, and YZY are
supported by a startup fund by UCSD and the National Science Foundation (NSF)
Grant No. DMR-2238360. FX is supported by the Zhenggang Scholarship Program at
Nanjing University. This research was supported in part by grant NSF
PHY-2309135 to the Kavli Institute for Theoretical Physics (KITP), through the
KITP Program “Correlated Gapless Quantum Matter” (2024). We acknowledge the
OpenAI GPT4 for providing language and editing suggestions throughout the
process of writing this paper.

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