# Spectral properties of critical 1+1D Abelian-Higgs model

Titas Chanda<EMAIL_ADDRESS>Department of Physics, Indian Institute of
Technology Indore, Khandwa Road, Simrol, Indore 453552, India  The Abdus Salam
International Centre for Theoretical Physics (ICTP), Strada Costiera 11, 34151
Trieste, Italy  Marcello Dalmonte  The Abdus Salam International Centre for
Theoretical Physics (ICTP), Strada Costiera 11, 34151 Trieste, Italy
International School for Advanced Studies (SISSA), via Bonomea 265, 34136
Trieste, Italy  Maciej Lewenstein  ICFO-Institut de Ciències Fotòniques, The
Barcelona Institute of Science and Technology, Av. Carl Friedrich Gauss 3,
08860 Barcelona, Spain  ICREA, Passeig Lluis Companys 23, 08010 Barcelona,
Spain  Jakub Zakrzewski  Instytut Fizyki Teoretycznej, Uniwersytet
Jagielloński, Łojasiewicza 11, 30-348 Kraków, Poland  Mark Kac Complex Systems
Research Center, Jagiellonian University in Krakow, Łojasiewicza 11, 30-348
Kraków, Poland.  Luca Tagliacozzo<EMAIL_ADDRESS>Instituto de
Física Fundamental IFF-CSIC, Calle Serrano 113b, Madrid 28006, Spain

###### Abstract

The presence of gauge symmetry in 1+1D is known to be redundant, since it does
not imply the existence of dynamical gauge bosons. As a consequence, in the
continuum, the Abelian-Higgs model, the theory of bosonic matter interacting
with photons, just possesses a single phase, as the higher dimensional Higgs
and Coulomb phases are connected via non-perturbative effects. However, recent
research published in [Phys. Rev. Lett. 128, 090601 (2022)] has revealed an
unexpected phase transition when the system is discretized on the lattice.
This transition is described by a conformal field theory with a central charge
of $c=3/2$. In this paper, we aim to characterize the two components of this
$c=3/2$ theory – namely the free Majorana fermionic and bosonic parts –
through equilibrium and out-of-equilibrium spectral analyses.

##  I Introduction

A physical theory is most often characterized by fixing its symmetries, the
dimensions of space-time it operates on, and its field content Coleman (1985);
Francesco  _et al._ (1997); Blumenhagen and Plauschinn (2009); Shankar (2017);
Fradkin (2013). This concept forms the basis of the renormalization group
construction and our efforts to categorize its fixed points Wilson (1971a, b,
1975); Shankar (1994); Gell-Mann and Low (1954). For instance, in a four-
dimensional space-time, in a system featuring $U(1)$ gauge-invariance and
fermionic matter, we expect the physics to be described by quantum
electrodynamics, where the fermions correspond to electrons and the gauge
bosons correspond to photons Peskin and Schroeder (1995); Schwartz (2013).

Here we analyze a scenario that shows that this simple picture does not
necessarily hold. The lattice discretization of the Abelian-Higgs model
Anderson (1963); Englert and Brout (1964); Higgs (1964); Guralnik  _et al._
(1964); Peskin and Schroeder (1995); Schwartz (2013) (see Refs. Endres  _et
al._ (2012); Kasamatsu  _et al._ (2013); Pekker and Varma (2015); Kuno  _et
al._ (2017); González-Cuadra  _et al._ (2017); Zhang  _et al._ (2018); Unmuth-
Yockey  _et al._ (2018); Park  _et al._ (2019); Meurice (2021) for recent
discussions on realizing Abelian-Higgs model in quantum simulators), defined
on one spatial and one time dimensions (1+1D), shares the same dimensionality
and field content with the same in the continuum. However, as shown recently
in Ref. Chanda  _et al._ (2022), strong correlations induce an emergent
physical theory on the lattice that can be described, at low energies, as one
massless relativistic fermion together with a massless relativistic boson.

The Abelian-Higgs model is among the simplest $U(1)$ gauge theory in 1+1D. It
contains bosonic matter (the Higgs part of the model) and constitutes a well-
known textbook example of gauge theory Tong ; Peskin and Schroeder (1995);
Schwartz (2013). The model is described by the action (with the metric
convention $g_{\mu\nu}=\text{diag}(-1,1)$):

$\displaystyle S=\int d^{2}x\Big{(}$
$\displaystyle-\frac{1}{2}F^{01}F_{01}-D^{\nu}\phi\left(D_{\nu}\phi\right)^{*}$
$\displaystyle-m^{2}|\phi|^{2}-\frac{\lambda}{2}|\phi|^{4}+\frac{\theta}{2\pi}qF^{01}\Big{)},$
(1)

where $D_{\nu}=\partial_{\nu}+iqA_{\nu}$ is the covariant derivative, with $q$
being the charge of the scalar matter field described by the complex-valued
field $\phi$. $m$ and $\lambda$ are the bare mass and self-interaction of the
scalar field, respectively, and $\theta$ is a periodic parameter,
$-\pi\leq\theta\leq\pi$, that controls the background field.

Despite the similarity of the above action with higher-dimensional Abelian-
Higgs models, the physics in 1+1D is very different Coleman (1985);
Komargodski  _et al._ (2019); Tong . In higher dimensions, the model displays
two well-known phases, a Coulomb phase described by massless photons
interacting with the matter field, and a Higgs phase where the photons are
screened and acquire mass, and the Higgs field condenses Fradkin and Shenker
(1979); Callaway and Carson (1982). In 1+1D, for positive $m^{2}$ the scalar
matter is confined. Differently from the higher-dimensional models, instanton
effects lead to confinement also for large and negative $m^{2}$, destroying
the expected screening caused by the condensation of the Higgs field Coleman
(1985); Komargodski  _et al._ (2019); Tong . The only exception to this simple
scenario involving the occurrence of a single gapped phase in the continuum is
at $\theta=\pi$ (see e.g., Tong ; Gattringer  _et al._ (2018); Göschl  _et
al._ (2018); Sulejmanpasic  _et al._ (2020); Sulejmanpasic (2021)), where
there is a line of first-order transition that culminates into a second-order
critical point in the Ising universality class (akin to what happens in the
Schwinger model Coleman  _et al._ (1975); Coleman (1976)).

In the recent works Chanda  _et al._ (2020, 2022), the same model has been
discretized on the lattice in the Hamiltonian formalism. Having $L$ sites
(with lattice spacing $a$) and $L-1$ bonds the corresponding Hamiltonian, at
$\theta=0$, reads

$\displaystyle\hat{H}=\sum_{j}\Big{[}\hat{L}^{2}_{j}$ $\displaystyle+2x\
\hat{\Pi}^{\dagger}_{j}\hat{\Pi}_{j}+(4x-\frac{2\mu^{2}}{q^{2}})\hat{\phi}^{\dagger}_{j}\hat{\phi}_{j}$
$\displaystyle+\frac{\lambda}{q^{2}}(\hat{\phi}_{j}^{\dagger})^{2}\hat{\phi}_{j}^{2}-2x(\hat{\phi}^{\dagger}_{j+1}\hat{U}_{j}\hat{\phi}_{j}+\text{h.c.})\Big{]},$
(2)

with $x=1/a^{2}q^{2}$ and $\mu^{2}=-m^{2}$. The matter field operators,
$\\{\hat{\phi}_{j},\hat{\phi}_{j}^{\dagger},\hat{\Pi}_{j},\hat{\Pi}_{j}^{\dagger}\\}$,
act on the Hilbert space at site $j$, while the gauge-field operators,
$\\{\hat{L}_{j},\hat{U}_{j},\hat{U}^{\dagger}_{j}\\}$, act on Hilbert space
defined on the bond linking sites $j$ and $j+1$. The operators fulfil the
standard canonical commutation relations:

$\displaystyle\
[\hat{\phi}_{j},\hat{\Pi}_{k}]=[\hat{\phi}^{\dagger}_{j},\hat{\Pi}^{\dagger}_{k}]=i\delta_{jk};$
$\displaystyle[\hat{L}_{j},\hat{U}_{j}]=-\hat{U}_{j},\ \
[\hat{L}_{j},\hat{U}^{\dagger}_{j}]=\hat{U}^{\dagger}_{j}.$ (3)

The usual continuum limit is taken as $x\rightarrow\infty$.

Rather than considering the usual continuum limit, $x=2$ was fixed in Chanda
_et al._ (2022), and the phase diagram as a function of $\lambda$ and $\mu$
was characterized numerically with matrix product state (MPS) techniques
Schollwöck (2011); Orús (2014) (see Refs. Bañuls  _et al._ (2013); Buyens  _et
al._ (2014); Kühn  _et al._ (2014); Bañuls  _et al._ (2015); Buyens  _et al._
(2016a); Pichler  _et al._ (2016); Bañuls  _et al._ (2017); Buyens  _et al._
(2016b); Magnifico  _et al._ (2020); Bañuls  _et al._ (2020); Aidelsburger
_et al._ (2021) for recent applications of MPS techniques in models of lattice
gauge theory). On the lattice, the model presents a rich phase diagram. Even
at $\theta=0$, there is first-order phase transition between two distinct
regimes, a confined regime characterized by the suppression of the kinetic
term of the Hamiltonian
$(\hat{\phi}^{\dagger}_{j+1}\hat{U}_{j}\hat{\phi}_{j}+\text{h.c.})$ and low
entanglement entropy in the ground state, and a gapped “Higgs” regime,
characterized by a larger (but finite) kinetic energy and entanglement
entropy. The first-order line ends in a second-order critical point. This
critical point is described by a conformal field theory (CFT) with central
charge of $c=3/2$. This implies that the interplay of lattice effects and non-
perturbative physics can give rise to new emerging phenomena, whose field
theory description is completely different from the original continuum field
theory discretized to obtain the lattice model.

As a result, in the continuum, we have two completely different models
emerging from the same microscopic model. A gapped model of confined bosonic
charges obtained by taking $x\to\infty$ in any region of the phase diagram on
the lattice model whose physics is well known and described in textbooks
Peskin and Schroeder (1995); Tong ; Schwartz (2013). At the critical point on
the lattice, however, we can take a different continuum limit at fixed $x$ and
obtain a gapless conformal field theory with $c=3/2$ that originates from the
same microscopic model – relativistic bosons interacting with a $U(1)$ gauge
field. This continuum theory is less understood.

In this work, we proceed to better characterize this lattice microscopic
model. We start solving some of the puzzles presented in Ref. Chanda  _et al._
(2022). In particular, we show that all Renyi entropies, differently from what
was claimed originally, scale as expected from the conformal field theory
predictions, once the sub-leading corrections are taken into account.

We then extract the sound velocity and the dispersion relation of the system
via real-time dynamics, and combine those with an accurate finite size-scaling
analysis of the Hamiltonian spectrum. From those, we identify the presence of
a charge gap, and identify the portion of the low energy spectrum that is
responsible for critical behavior. Since the model is gauge invariant, we can
target different background charge sectors. In the zero background charge
sector, we only find the Majorana fermionic or Ising part of the spectrum. On
the other hand, using known relation about the fluctuations of the charges, we
also identify the value of the Luttinger liquid parameter of the bosonic
sector of the theory.

The resulting picture is thus much clearer than before but still incomplete.
As far as we know, the critical point of the discretized Abelian Higgs model
in 1+1D provides the first example of a relativistic CFT where a discrepancy
between the scaling of entanglement and the low-energy spectrum is observed.

We thus extend our analysis to the other sectors of background charges, where
we find many low-energy excitations that should encode the bosonic part of the
spectrum. However, with our current tools, we have limited precision in
extracting the spectra in the other background charge sectors to properly
identify all of them. Thus we are still not able to completely identify the
full low-energy spectrum, something we plan to do in the future.

##  II A Brief Recap: The system

The Abelian-Higgs model in 1+1D is described by the Hamiltonian (2). We can
then define creation and annihilation operators for particles ‘$a$’ and anti-
particles ‘$b$’ as $\hat{a}_{j}$ and $\hat{b}_{j}$ fulfilling
$[\hat{a}_{j},\hat{a}^{\dagger}_{k}]=[\hat{b}_{j},\hat{b}^{\dagger}_{k}]=\delta_{jk}$
The operators are defined as

$\displaystyle\hat{\phi}_{j}=\frac{1}{\sqrt{2}}\left(\hat{a}_{j}+\hat{b}_{j}^{\dagger}\right),\
\hat{\Pi}_{j}=\frac{i}{\sqrt{2}}\left(\hat{a}^{\dagger}_{j}-\hat{b}_{j}\right),$
$\displaystyle\hat{\phi}^{\dagger}_{j}=\frac{1}{\sqrt{2}}\left(\hat{a}^{\dagger}_{j}+\hat{b}_{j}\right),\
\hat{\Pi}^{\dagger}_{j}=\frac{i}{\sqrt{2}}\left(\hat{b}^{\dagger}_{j}-\hat{a}_{j}\right),$

as discussed in e.g., Chanda  _et al._ (2020).

###  II.1 Gauge invariance and Gauss law generators

In the absence of external charges, the local $U(1)$ symmetry implies that all
physical states $|\Psi\rangle$ satisfy Gauss law $\hat{G}_{j}|\Psi\rangle=0,\
\forall j$, where the generators are Chanda  _et al._ (2020):

$\hat{G}_{j}=\hat{L}_{j}-\hat{L}_{j-1}-{\hat{Q}_{j}},$ (4)

with
$\hat{Q}_{j}=\hat{a}_{j}^{\dagger}\hat{a}_{j}-\hat{b}_{j}^{\dagger}\hat{b}_{j}$
describing the dynamical charge. Using Gauss law, we can integrate-out the
gauge fields in a chain with open boundary conditions by the non-local
transformations:

$\displaystyle\left(\prod_{l\leq
j}\hat{U}_{j}\right)\hat{\phi}_{j}\rightarrow\hat{\phi}_{j},\
\left(\prod_{l\leq
j}\hat{U}^{\dagger}_{j}\right)\hat{\Pi}_{j}\rightarrow\hat{\Pi}_{j},\
\hat{L}_{j}=\sum_{l\leq j}\hat{Q}_{l}.$ (5)

This introduces a long-range potential for the matter fields Schwinger (1951)
as follows:

$\displaystyle\hat{H}=\sum_{j}\Big{[}\left(\sum_{l\leq
j}\hat{Q}_{l}\right)^{2}+2x\
\hat{\Pi}^{\dagger}_{j}\hat{\Pi}_{j}+(4x-\frac{2\mu^{2}}{q^{2}})\hat{\phi}^{\dagger}_{j}\hat{\phi}_{j}$
$\displaystyle+\frac{\lambda}{q^{2}}(\hat{\phi}_{j}^{\dagger})^{2}\hat{\phi}_{j}^{2}-2x(\hat{\phi}^{\dagger}_{j+1}\hat{\phi}_{j}+\text{h.c.})\Big{]}.$
(6)

In the rest of the paper, unless explicitly stated otherwise, we shall
consider Eq. (6) as the system Hamiltonian111It is to be noted that since the
mapping in Eq. (5) is exact, the long-range Hamiltonian (6) describes the same
physics as that of the short-range matter-gauge Hamiltonian (2). We verify
this numerically in Appendix B. that respects global $U(1)$ symmetry
corresponding to the conservation of the total dynamical charge
$\hat{Q}=\sum_{j}\hat{Q}_{j}$. We use the density matrix renormalization group
(DMRG) algorithm Schollwöck (2011); Orús (2014); White (1992, 1993, 2005);
Schollwöck (2005) in the framework of the matrix product states (MPS) to find
the ground state or low-lying excited states of the Hamiltonian (6). The
diagrammatic depictions of the matrix product operator (MPO) for the
Hamiltonian (6) and the corresponding MPS are given in Fig. 1(a) and (b)
respectively. In Fig. 1(c) and (d) we also present the same for the short-
range matter-gauge description of Eq. (2), where we also impose global $U(1)$
symmetry to conserve of the total dynamical charge $\hat{Q}$. For time-
evolution, we use MPS based time-dependent variational principle (TDVP)
Haegeman  _et al._ (2011, 2016); Koffel  _et al._ (2012); Paeckel  _et al._
(2019) concentraing only on the long-range description.

Figure 1: (Color online.) Diagrammatic depictions of the MPO (panels (a) and
(c) and MPS (panels (b) and (d) representations used in this paper. Panels (a)
and (b) correspond to the long-range matter-only description of Eq. (6), while
(c) and (d) presents the same for the matter-gauge picture of Eq. (2).

###  II.2 The phase diagram

The phase diagram of the Abelian-Higgs model in 1+1D on a discrete lattice has
been studied recently in Chanda  _et al._ (2022). Here we present a different
characterization leading to the same phase-diagram. In continuum, the phase
diagram of the Abelian-Higgs model is somewhat uninteresting as there exists
only one gapped confined phase without any phase transition in absence of
background field. However, the lattice discretization unveils a rich landscape
of phase transitions between the confined and the Higgs phase. At weak
coupling (i.e., at small $\lambda/q^{2}$) these two phases are separated by a
line of first order quantum phase transition (FOQPT) that ends at a critical
second order quantum phase transition (SOQPT). Above this SOQPT point these
two phases are smoothly connected by a crossover. In Fig. 2, we depict the
phase diagram of system in the $(\mu^{2}/q^{2},\lambda/q^{2})$-plane.

Figure 2: Phase diagram of the Abelian-Higgs model in 1+1D (6) in the
$(\mu^{2}/q^{2},\lambda/q^{2})$-plane for a system of size $L=60$. The left
panel shows the behavior of the expectation value
$\braket{\hat{\phi}^{\dagger}\hat{\phi}}$ while the right panel is for the
variance $\sigma^{2}(\hat{L})$. Both the quantities are averaged over lattice
sites/bonds.

In the Higgs phase, since the gauge field becomes massive, its quantum
fluctuations as measured by the variance
$\sigma^{2}(\hat{L}_{j})=\braket{\hat{L}^{2}_{j}}-\braket{\hat{L}_{j}}^{2}$
becomes non-zero, while it remains vanishingly small in the confined region.
Oppositely, since the matter field attains finite expectation value in the
Higgs phase, $\braket{\hat{\phi}_{j}^{\dagger}\hat{\phi}_{j}}$ becomes large
as seen in Fig. 2.

The SOQPT point has been located precisely to be
$(\lambda_{c}/q^{2}=0.0565(1),\ \mu_{c}^{2}/q^{2}=0.447(1))$ in Chanda  _et
al._ (2022) using predictions about scaling of entanglement entropy for a CFT.
There it was also shown that the entanglement entropy and the kinetic term in
the Hamiltonian could be used to illustrate the phase diagram as we have done
here with the fluctuation of the electric field and the onsite occupation of
the bosons.

The analysis of Ref. Chanda  _et al._ (2022) using the well-established
machinery of CFT predicted the existence of a direct sum of two critical
theories, free Majorana fermion and free boson, at the critical point. In this
work, we complement the previous work with a proper characterization of these
two gapless modes.

###  II.3 Sound velocity at criticality

In order to verify that the critical point
$(\mu_{c}^{2}/q^{2}=0.447,\lambda_{c}/q^{2}=0.0565)$ displays Lorentz
invariance in its low-energy description, we calculate the sound velocity
$v_{s}$ at the critical point from the scaling of ground state energy –
recapitulating the result presented in the supplementary material of Ref.
Chanda  _et al._ (2022). For a Lorentz invariant system exhibiting a linear
dispersion at low energies, the ground state energy $E_{0}$, under open
boundary conditions, scales with the system size $L$ according to Blöte  _et
al._ (1986); Affleck (1986); Affleck  _et al._ (1989):

$E_{0}(L)=\epsilon_{0}^{\infty}L+\epsilon_{1}^{\infty}-\frac{\pi
cv_{s}}{24L},$ (7)

where $\epsilon_{0}^{\infty}$ is the ground state energy density in the bulk
and $\epsilon_{1}^{\infty}$ is the surface free energy in the thermodynamic
limit, $c$ is the central charge of the corresponding CFT, and $v_{s}$ is the
sound velocity. It should be noted that $v_{s}$ vanishes for Lorentz non-
invariant critical points characterized by a quadratic dispersion.

Figure 3:  Finite-size scaling of the ground state energy $E_{0}$ at the
critical point according to Eq. (7) that gives $cv_{s}\simeq 6$, where $c$ is
the central charge and $v_{s}$ is the sound velocity.

To estimate $v_{s}$, we perform the finite-size scaling of the ground state
energy according to Eq. (7) at the critical point (for similar scaling
analysis, see Hallberg  _et al._ (1996); Xavier (2010); Dalmonte  _et al._
(2012); Chepiga and Mila (2017)). The finite-size scaling (Fig. 3) reveals

$cv_{s}=6.02\pm 0.08,$ (8)

confirming the Lorentz invariance of the critical point
$(\mu_{c}^{2}/q^{2}=0.447,\lambda_{c}/q^{2}=0.0565)$ and the applicability of
CFT framework.

##  III Numerical results

Here we describe our numerical results that contribute to clarifying the
nature of the field theory emerging at the critical point of the lattice
model. For this, we set $\lambda/q^{2}=0.0565$ and $\mu^{2}/q^{2}=0.447$ in
the rest of the paper.

To reiterate, we employ DMRG simulations to extract the low-lying spectrum of
the Hamiltonian (6) with $U(1)$ symmetric MPS Singh  _et al._ (2010, 2011).
The MPS bond dimension used in our study is up to $\chi=600$, sufficient for
convergence up to system sizes $120$. Each of the bosonic species, ‘$a$’ and
‘$b$’, has been truncated to $11$ levels, leading to the physical dimension of
$121$ for the MPS tensors. Furthermore, in Sec. III.2.1, we also simulate the
full matter-gauge Hamiltonian (2). In that case, the MPS ansatz contains
alternating tensors for matter and gauge degrees of freedom (see Fig. 1). The
gauge fields are then truncated to up to 15 levels in the electric basis.
Details about the MPS based numerical calculations are presented in Appendix
A.

For the out-of-equilibrium analysis presented in Sec. III.4, we use TDVP
algorithms as described in Chanda  _et al._ (2020). To reduce the
computational complexity of TDVP runs, we truncate the bosonic species upto
$8$ levels and the maximum MPS bond dimension is restricted upto $\chi=400$.

###  III.1 The scaling of Renyi entanglement entropy

The Renyi entanglement entropy of order $n$ for a block of $l$ sites is
defined as

$S_{n}(l)=\frac{1}{1-n}\log\left(\text{Tr}\rho_{l}^{n}\right),$ (9)

where $\rho_{l}=\text{Tr}_{l+1,l+2,...,L}\ket{\psi}\bra{\psi}$ is the reduced
density matrix after tracing out the rest of the system. As limiting cases of
the Renyi entropy, we get the von Neumann entropy as $S=\lim_{n\rightarrow
1}S_{n}$ and the entanglement ground state energy
$\varepsilon_{0}=\lim_{n\rightarrow\infty}S_{n}$, where $\varepsilon_{0}$ is
the lowest eigenvalue of the entanglement Hamiltonian $H_{l}=-\log(\rho_{l})$,
i.e., $\varepsilon_{0}=-\log\lambda_{0}$ where $\lambda_{0}$ is the smallest
eigenvalue of the density matrix $\rho_{l}$. It is to be noted that since we
are integrating out the gauge fields, the resulting Hilbert space of matter
has a well-defined tensor-product structure and there are no ambiguities in
defining the bipartitions associated with the gauge invariance (c.f., Casini
_et al._ (2014); Ghosh  _et al._ (2015); Soni and Trivedi (2016)).

In a CFT, the finite-size scaling of the Renyi entanglement entropy of the
bipartition of size $l$ in a chain with open boundary conditions and length
$L$ is Callan and Wilczek (1994); Vidal  _et al._ (2003); Calabrese and Cardy
(2004)

$S_{n}(l,L)=\frac{c}{12}(1+\frac{1}{n})W+b_{n}^{\prime},$ (10)

where $c$ is the central charge of the corresponding CFT, $b_{n}^{\prime}$ is
a non-universal constant and $W$, the chord length, is a function of both $L$
and $l$:

$W(l,L)=\log\left[\frac{2L}{\pi}\sin(\pi l/L)\right].$ (11)

Figure 4: (Color online.) (a) The variation of the von Neumann entanglement
entropy $S=S_{n\rightarrow 1}$ and the entanglement ground state energy
$\varepsilon=S_{n\rightarrow\infty}$ as a function of the chord length $W$
according to Eq. (10) at the critical SOQPT point. (b) The extracted values of
the central charge after fitting the Renyi entropy data to Eq. (10). The
bluish shade describes the error-bars in fitted values of $c_{\text{eff}}(n)$.
For these plots, we have analyzed the data for system-sizes
$L\in[40,60,80,100,120]$ at the critical point
$(\lambda_{c}/q^{2}=0.0565,\mu_{c}^{2}/q^{2}=0.447)$.

In Ref. Chanda  _et al._ (2022), using the above scaling of von Neumann
entropy at the critical point, a central charge of $c=3/2$ ($c=1.49(1)$ to be
precise) has been extracted (see Fig. 4(a)). It has been argued in Chanda  _et
al._ (2022) that this value of the central charge can be explained as the sum
of two different minimal models, each contributing to a piece of the total
central charge, a $c_{f}=1/2$ for a free Majorana fermion (the Ising sector)
and a $c_{b}=1$ for a free boson.

However, performing the same scaling for the entanglement ground state energy
$\varepsilon_{0}$, the effective central charge comes out to be
$c_{\text{eff}}(n\rightarrow\infty)=1.20(1)$ (see Fig. 4(a)), which
contradicts the value $c=3/2$. Moreover, on a closer inspection, we find that
the values of central charge obtained by fitting the entropy data to Eq. (10)
is heavily dependent on the order $n$ as seen in Fig. 4(b)).

Figure 5: (Color online.) (a) The exponent $\beta_{n}$ in the finite-size
correction to the entanglement entropy scaling (see Eq. (12)). as a function
of the Renyi order $n$. Here we fix $c=3/2$ (see text). The bluish shade
describes the error bars in fitted values of $\beta_{n}$. Note that the error
in the fitting is large when $c_{\text{eff}}\sim 3/2$ as the curve-fitting
becomes ill-formed with redundant parameters within that regime. (b) The
fitting of von Neumann entanglement entropy $S=S_{1}$ and the entanglement
ground state energy $\varepsilon=S_{n\rightarrow\infty}$ including the
correction term as in Eq. (12). Note that the entropic data (symbols) are
exactly same as in Fig. 4(b) – the difference lies in the use of Eq. (12) in
fitting these data. All other descriptions are the same as those in Fig. 4.

Nevertheless, the analysis in Sec. II.3 and in the following Sec. III.4
suggest that the central charge of the system is indeed compatible with
$c=3/2$. In Sec. II.3, by utilizing the finite-size scaling of ground state
energy we have shown that $cv_{s}\simeq 6$, whereas using the out-of-
equilibrium local quench spectrocopy in the following Sec. III.4 we verify
that the sound velocity is consistent with $v_{s}=4$ – thereby leaving $c=3/2$
as the only possibility. These results are obviously independent of the
entropy scaling.

Here, we try to explain the discrepancies in the scaling of different Renyi
entropies by finite-size corrections in the scaling law of Eq. (10).
Specifically, it has been shown in Cardy and Calabrese (2010) that due to
irrelevant and marginally irrelevant bulk operators in the Hamiltonian, the
Renyi entropies can attain corrections in finite systems, where the correction
terms can depend on $n$ and on the scaling dimensions of such bulk operators
(see also Eriksson and Johannesson (2011); Laflorencie  _et al._ (2006)).
Since the numerical determination of the scaling dimensions of such irrelevant
(or marginally irrelevant) bulk operators can be very non-trivial, here we
consider a generic correction term, having the same form as in Cardy and
Calabrese (2010), to the scaling formula of Eq. (10) as follows:

$S_{n}(l,L)=\frac{c}{12}(1+\frac{1}{n})W+b_{n}^{\prime}+d_{n}\left(\exp(W)\right)^{-\beta_{n}},$
(12)

where $d_{n}$ and $\beta_{n}$ are $n$-dependent quantities to be determined.
It is to be noted that since Eq. (12) has four independent parameters, the
curve fitting process is underconstraint for the given entropic data leading
to unfaithful extraction of these parameters. Because of this, we fix the
central charge to $c=3/2$, verified independently from the ground-state energy
scaling in Sec. II.3 (see Eq. (8)) and the local quench spectroscopy in Sec.
III.4, when fitting fitting Eq. (12) to the entropic data.

The variation of the exponent $\beta_{n}$ as a function of the order $n$ is
shown in Fig. 5(a) along with its errors for fixed $c=3/2$. For small $n$,
corrections to the original formula are small (the corresponding error is not
informative as such term does not contribute to the fit). Oppositely, for
large $n$, where we observe $c_{\text{eff}}\not\approx 3/2$ (Fig. 4(b)),
corrections are indeed strong and the coefficient $\beta_{n}$ is essentially
$n$-independent. This strongly confirms the fact that extracting the central
charge from such large-$n$ Renyi entropies is not reliable using the original
scaling relation when finite-size effects are strong. Fig. 5(b) shows the
actual entropy scalings for the von Neumann entanglement entropy $S=S_{1}$ and
the entanglement ground state energy $\varepsilon=S_{n\rightarrow\infty}$
according to Eq. (12) with fixed $c=3/2$.

###  III.2 Spectral analysis at equilibrium

To detect and characterize the Majorana fermionic and the bosonic part of the
theory, we first analyze the energies of the ground state and low-lying
excited states and their finite-size scaling at the critical point
$(\lambda_{c}/q^{2}=0.0565,\mu_{c}^{2}/q^{2}=0.447)$ obtained from DMRG
simulations.

####  III.2.1 The neutral gap and the charge gap

We calculate the neutral gap ($\Delta E_{N}$) and the charge gap ($\Delta
E_{C}$) at the critical point. The gaps are defined as

$\displaystyle\Delta E_{N}=E_{1}(Q=0)-E_{0}(Q=0),$ $\displaystyle\Delta
E_{C}=E_{0}(Q=1)+E_{0}(Q=-1)-2E_{0}(Q=0).$ (13)

where $E_{n}(Q)$ represents the $n^{th}$ energy eigenvalue at the
$Q=\braket{\hat{Q}}$ quantum sector extracted by means of excited state DMRG.

Figure 6: (Color online.) The scalings of (a) the neutral gap $\Delta E_{N}$
and (b) the charge gap $\Delta E_{C}$ with the system-size $L$ at the critical
SOQPT point. The neutral gap decays to zero in the thermodynamic limit as
$\sim 1/L$, while the charge gap remains finite and its thermodynamic value
($5.104(4)$) is extracted by fitting a quadratic function in $1/L$.

As expected, the neutral gap $\Delta E_{N}$ decays to zero in the
thermodynamic limit as $\sim 1/L$ (see Fig. 6(a)). On the other hand, quite
interestingly, the charge gap $\Delta E_{C}$ saturates to a finite value
$\Delta E_{C}=5.104(4)$ in the thermodynamic limit (see Fig. 6(b)). This
happens due to the gauge invariance of the system, as a single charge
excitation must be accompanied by a string of electric fields (semi-infinite
in the thermodynamic limit) whose energy cannot vanish.

Therefore, by imposing the Gauss law, we are throwing away a large chunk of
the low-energy spectrum. This is illustrated in Fig. 7, where we show that
there are indeed many low-lying states that come from other gauge sectors
(i.e., non-zero background fields) by performing the excited state DMRG
calculations for the matter-gauge Hamiltonian (2). It is to be noted that
since gauge invariance cannot be broken spontaneously, excited state DMRG
cannot scan all possible low-lying gauge sectors, and its reliability in
iteratively finding excited states one-after-another degrades very quickly
when we do not impose the Gauss law.

Figure 7: (Color online.) Energy gaps of few excited states obtained from
excited-state DMRG when we do not enforce Gauss law by integrating out the
gauge fields, and instead treat the Hamiltonian (2) as it is with conserved
total charge $Q=0$. Here, we truncate the gauge fields to 15 electric levels.
The x-axis, $\sum_{j=1}^{L}|\braket{\hat{G}_{j}}|$, denotes the deviation from
the Gauss law $\braket{\hat{G}_{j}}=0$ for all $j=1,\cdots,L$. The red circle
demarcates the first neutral excitations that obey the Gauss law. Note here
that the excited state DMRG is unable to find all the low-lying states coming
from different gauge sectors, specifically when
$\sum_{j=1}^{L}|\braket{\hat{G}_{j}}|\neq 0$. Nevertheless, we can see the
existence of many low-lying states coming from different gauge sectors in the
low-energy spectrum.

####  III.2.2 Conformal towers

Figure 8: (Color online.) First few boundary scaling dimensions
$x_{n}=(E_{n}-E_{0})L/\pi v_{s}$ in the conformal tower of the critical point
using excited-state DMRG.

Here, we analyze the excitation spectra of the system at a critical point
using the predictions of CFT by using the excited-state DMRG. The excited
state energies vary with the system-size as

$E_{n}(L)=E_{0}(L)+x_{n}\frac{\pi v_{s}}{L},$ (14)

where $x_{n}$ are boundary scaling dimensions, organized in conformal towers
Cardy (1986a, b). In Fig. 8, we plot $x_{n}=(E_{n}-E_{0})L/\pi v_{s}$ for
first $9$ excited states along with their degeneracies obtained from the
excited-state DMRG calculations. The boundary exponents that we get from Fig.
8 are $x_{n}=[2,3,4,5,6,...]$ with degeneracies $[1,1,2,2,3,...]$. These
values of the boundary exponents and corresponding degeneracies match with the
CFT prediction for the critical Ising model with fixed boundary condition
Rocha-Caridi (1985); Evenbly and Vidal (2014); Chepiga and Mila (2017). This
structure of the conformal tower confirms the Ising or free Majorana fermion
part of the critical point. On the other hand, we completely miss the bosonic
part of the spectrum at the critical point. However, we can make an educated
guess based on the results of Sec. III.2.1 (specifically, of Fig. 7) that the
bosonic excitations, that carry charges, might be hidden in other gauge
sectors.

Figure 9: (Color online.) (a) The graphical representation of the mid-chain
effective Hamiltonian constructed from the MPO and MPS tensors to target the
ground state of the Hamiltonian (6). (b) The excitation spectra of the
effective Hamiltonian depicted in (a) at the critical point.

To validate this result further, we analyze the spectra of the effective
Hamiltonian constructed to target the ground states of the Hamiltonian (6)
during DMRG sweeps (see Fig. 9(a)). In Ref. Chepiga and Mila (2017), it has
been shown that at the critical point, the low-lying eigenstates of the
effective Hamiltonian, within the framework of the standard ground state
algorithm, gives correct excitation spectra of the original Hamiltonian. In
Fig. 9(b) we show the first few eigenstates of the effective Hamiltonian at
the critical point. Once again, the extracted boundary exponents and
corresponding degeneracies match with the Isinf CFT.

###  III.3 Determination of the Luttinger parameter

Since we could not characterize the bosonic part of the critical theory from
the low-energy excitation spectrum, we take the route of determining the
Luttinger parameter $K$ associated with the free bosonic part that is
described by the Tomonaga-Luttinger liquid theory Giamarchi (2003); Gogolin
_et al._ (2004). Here, we extract $K$ from the scaling of the bipartite
fluctuations as described in Refs. Song  _et al._ (2010, 2012); Rachel  _et
al._ (2012), under the assumption that charge excitations are indeed those
responsible for the emergence of the $c_{b}=1$ sector of theory. It has been
shown that in a Tomonaga-Luttinger liquid, for a global $U(1)$ conserved
quantity $\mathcal{O}$, the local fluctuations

$\mathcal{F}_{l}(\mathcal{O})=\Braket{\left(\sum_{j\leq
l}\mathcal{O}_{j}\right)^{2}}-\Braket{\sum_{j\leq l}\mathcal{O}_{j}}^{2}$ (15)

obey the scaling of the form

$\mathcal{F}_{l}(\mathcal{O})=\frac{K}{2\pi^{2}}W(l,L)+\text{const.}$ (16)

In the present scenario, the global conserved quantity is the total dynamical
charge $\hat{Q}$. Using the local gauge invariance, we can also rewrite the
local fluctuations of the dynamical charge as

$\mathcal{F}_{l}(\hat{Q})=\Braket{\hat{L}_{l}^{2}}-\Braket{\hat{L}_{l}}^{2},$
(17)

where we have used the fact that $\hat{L}_{l}=\sum_{j\leq l}\hat{Q}_{j}$. In
Fig. 10, we present the variations of local fluctuations as a function of both
bipartition and system size. This figure provides evidence that the Luttinger
parameter is $K=2.10(1)$, which characterizes the existence of a strongly
interacting bosonic component in the theory.

Figure 10: (Color online.) The scaling of local fluctuations
$\mathcal{F}_{l}(\hat{Q})$ in relation to the chord length $W(l,L)$. (a) The
pattern of local fluctuations with respect to the bipartition size for
different system sizes is plotted and indicated in the figure. (b) The fit
performed with the data for $L\in[40,60,80,100,120]$, based on Eq. (16),
suggests that Luttinger parameter is $K=2.11(2)$. The inset shows the
negligible dependence of $K$ on the system size. To estimate Luttinger
parameter $K$ at the thermodynamic limit, we extrapolate the extracted values
of $K$ for different system-sizes $L\in[60,120]$ using a linear fit in $1/L$,
yielding $K=2.10(1)$.

###  III.4 Spectral analysis from out-of-equilibrium dynamics

We now analyze the critical point by considering the response of the system to
a local quench as suggested in Villa  _et al._ (2019, 2020). For that, we
create a local excitation at the middle of the chain using the operator

$\hat{\mathcal{M}}^{Q}=\hat{\phi}^{\dagger}_{L/2}\hat{\phi}_{L/2+1}$ (18)

that will create $Q=\pm 1$ dynamical charges at sites $L/2$ and $L/2+1$
respectively on top of the ground state $\ket{\Omega}$, such that our initial
state becomes

$\ket{\psi^{Q}}(t=0)=\mathcal{N}\hat{\mathcal{M}}\ket{\Omega},$ (19)

where $\mathcal{N}$ is a normalization constant. Since, the operator
$\hat{\mathcal{M}}^{Q}$ is a local element of the Hamiltonian (6), we can
expect that the overlaps of $\ket{\psi^{Q}}$ with low-lying excited states are
non-vanishing.

For any local observable $\hat{\mathcal{O}}_{j}$, we analyze the spectral
properties by means of the (discrete) Fourier transform:

$\mathcal{F}_{\mathcal{O}}(k,\omega)=\frac{2\pi}{LT}\delta
t\sum_{j=1}^{L}e^{-ik(j-\frac{L}{2})}\sum_{n=0}^{t_{N}}e^{-i\omega
t_{n}}\left(\braket{\mathcal{O}_{j}}(t_{n})-\braket{\mathcal{O}_{j}}_{\Omega}\right),$
(20)

where $t_{n}=n\delta t$ is the discrete time-steps, $T=t_{N}\delta t$ is the
total time of the evolution, and $\braket{\mathcal{O}_{j}}_{\Omega}$ defines
the expectation value of $\hat{\mathcal{O}}_{j}$ with respect to the the
ground state $\ket{\Omega}$ (see Paeckel  _et al._ (2019) for details).

Since the neutral gap vanishes while the charge gap remains finite at the
critical point, we expect that for neutral operators such as
$\hat{\phi}^{\dagger}_{j}\hat{\phi}_{j}$,
$\hat{\Pi}^{\dagger}_{j}\hat{\Pi}_{j}$, or $\hat{L}^{2}_{j}=(\sum_{l\leq
j}\hat{Q}_{l})^{2}$ the Fourier transform will provide a gapless dispersion
with $\omega(k)\approx v_{s}|k|$ for $k\ll 1$, and for any charge carrying
operators like $\hat{Q}_{j}$ or $\hat{L}_{j}=\sum_{l\leq j}\hat{Q}_{l}$, the
$\mathcal{F}_{\mathcal{O}}(k,\omega)$ will give a gapped spectrum Villa  _et
al._ (2019, 2020).

Figure 11: (Color online.) The dispersion relations from the Fourier transform
(Eq. (20)) of the local operators (a) $\hat{\Pi}_{j}^{\dagger}\hat{\Pi}_{j}$
and (b) $\hat{L}_{j}$ at the critical point for the initial state (19) for a
system of size $L=60$. Here we apply the operator $\hat{\mathcal{M}}^{Q}$ (see
Eq. (18)) to excite the ground state.

From the Fourier transform of the neutral operators, e.g.,
$\mathcal{F}_{\braket{\hat{\Pi}^{\dagger}_{j}\hat{\Pi}_{j}}}(k,\omega)$ in
Fig. 11(a), we are getting the gapless spectrum as expected. The signal in the
Fourier transform matches with $v_{s}\approx 4$ (red dashed line in Fig.
11(a)) extracted from the scaling of ground state energy (see Sec. II.3). On a
careful inspection, we find another faint signal corresponding to
$v_{s}\approx 2.4$. However, this value of the sound velocity is not
consistent with earlier equilibrium analysis coming from the scaling of ground
state energy (Eq. (8)). This faint signal might be a spurious effect coming
from numerical errors, as also seen in Paeckel  _et al._ (2019). There is also
a possibility that this might come from the dispersion in $Q=\pm 1$ sectors as
the excitation performed by the operator $\hat{M}^{Q}$ couples these sectors
to the present $Q=0$ one. On the other hand, in Fig. 11(b), we see that the
dispersion relation from $\mathcal{F}_{\braket{\hat{L}_{j}}}(k,\omega)$ is
gapped and the gap is exactly the same as the charge gap $\Delta E_{C}$.

To be sure that there is only one sound velocity at $v_{s}=4$, we perform two
other forms of time-evolution in such a way that the local excitation now does
not create any dynamical charges so that we can solely examine the neutral
excitations. Specifically, we excite the ground state $\ket{\Omega}$ with the
neutral operators

$\displaystyle\hat{\mathcal{M}}^{1}=\hat{\phi}^{\dagger}_{L/2}\hat{\phi}_{L/2}\hat{\phi}^{\dagger}_{L/2+1}\hat{\phi}_{L/2+1}$
(21)
$\displaystyle\hat{\mathcal{M}}^{2}=\hat{\phi}^{\dagger}_{L/2}\hat{\phi}_{L/2}+\hat{\phi}^{\dagger}_{L/2+1}\hat{\phi}_{L/2+1},$
(22)

and then perform a similar Fourier analysis as before.

Figure 12: (Color online.) Same as in Fig. 11(a) but for the initial states
excited with the operators given in (a) Eq. (21) and (b) Eq. (22).

In Fig. 12, we show the Fourier transforms of
$\hat{\Pi}_{j}^{\dagger}\hat{\Pi}_{j}$ for such initial states. Clearly, there
is only one sound velocity of $v_{s}=4$ that we can observe.

##  IV Conclusions

In this article, we presented an in-depth analysis of the critical properties
of the 1+1D Abelian-Higgs model on a discretized lattice. Our approach
provides a comprehensive understanding of the model’s behavior, including its
bosonic and fermionic components.

First, we addressed the ambiguity surrounding the value of the central charge
at the critical point. We confirmed that the central charge is $c=3/2$, which
can be interpreted as a direct sum of gapless compactified bosons with
$c_{b}=1$ and critical Majorana fermions with $c_{f}=1/2$. We independently
corroborated this value from both equilibrium and out-of-equilibrium analysis.
Our results also reveal that the discrepancy in extracting the central charge
from different Renyi entropies can be attributed to finite size effects coming
from irrelevant or marginally irrelevant bulk operators.

To identify the bosonic and fermionic components of the model, we have used a
two-fold approach. Firstly, we find that the excitation spectrum of the model
is identical to the one of the critical Ising chain with fixed boundary
conditions, i.e., the $c=1/2$ free fermionic part. Quite interestingly, we
find no sign of the bosonic excitations in the gauge-invariant spectrum. Our
examination of the excitation spectrum in different superselection sectors
leads us to make an informed guess that the bosonic excitation spectrum may be
‘hidden’ in other gauge sectors. We have then determined the Luttinger
parameter, which confirms the existence of a gapless Tomonaga-Luttinger liquid
with a value of Luttinger parameter $K\approx 2.1$.

The fact that the bosonic spectrum that contributes to the entanglement
entropy at the critical point could be hidden in different superselection
sectors suggests that a better understanding of the boundary CFT involved with
the computation of the entanglement entropy here could be necessary. Also,
some further hints could be obtained from the analysis of the symmetry-
resolved entanglement spectrum that will be performed elsewhere.

Finally, we analyzed the local quenched dynamics of the critical system. By
performing spectroscopy via discrete Fourier transform, we extract the
dispersion relations for the interacting system. Our analysis reveals that
there exists only one speed of sound in this gauge invariant system, which is
consistent with our equilibrium analysis.

##  Appendix A Details about the matrix product states (MPS) simulations

In this paper, we use matrix product states (MPS)Schollwöck (2011); Orús
(2014) based tensor network simulations to obtain the results. Specifically,
to extract the ground state and low-lying excited states, we use a strictly
single-site variant of density matrix renormalization group (DMRG) algorithm
White (1992, 1993, 2005); Schollwöck (2005) with subspace expansion Hubig  _et
al._ (2015). For the out-of-equilibrium real-time dynamics, we employ the
time-dependent variational principle (TDVP) method Haegeman  _et al._ (2011,
2016); Koffel  _et al._ (2012); Paeckel  _et al._ (2019).

Figure 13: (Color online.) (a) The convergence of ground-state energy density
with respect to the maximum bosonic occupancy $n_{0}$ for a system of size
$L=120$ at the critical point. Here we plot the relative errors in the energy
density $\Delta E_{0}/L$, where $\Delta
E_{0}=|E_{0}({n_{0}})-E_{0}({n_{0}+1})|$. (b) The profile of the von Neumann
entanglement entropy $S=S_{1}$ with respect to the cord length
$W=\ln\left(\frac{2L}{\pi}\sin(\pi l/L)\right)$ at the critical point for
$L\in[40,120]$ and for different values of $n_{0}$.

In our MPS representation, we truncate the Hilbert spaces of both the bosonic
species, ‘$a$’ and ‘$b$’, upto the maximum occupancy of $n_{0}=10$ (i.e., 11
levels of each type of bosonic matter), that results into the physical
dimension of 121 for the MPS tensors on the matter sites. In order to verify
the convergence with respect to this cutoff, we vary the maximum bosonic
occupancy $n_{0}$ in the range $[4,15]$, and check for convergence of
different observables (see Fig. 13). Specifically, we show in Fig. 13(a) that
for $n_{0}=10$ the error in the ground-state energy density for system-size
$L=120$ falls below $10^{-7}$ at the critical point. Moreover, the entropy
profile for $n_{0}=10$ is essentially identical to the one with $n_{0}=15$,
capturing the proper entropy scaling with respect to the cord length (Fig.
13(b)). For simulating short-range matter-gauge Hamiltonian (2), we truncate
the gauge field Hilbert spaces to 15 levels electric basis (see Appendix B).

Figure 14: (Color online.) The convergence of (a) the energy density ($\Delta
E_{0}/L$) and (b) the half-chain von Neumann entanglement entropy ($\Delta
S_{L/2}$) and the half-chain entanglement ground state energy
($\Delta\varepsilon_{L/2}$) with respect to the maximum bond dimension used in
the simulation, respectively $\chi\in\\{100,200,300,400,500,600\\}$. Here we
plot relative errors in the quantities as
$\Delta\mathcal{O}=|\mathcal{O}_{\chi}-\mathcal{O}_{\chi+100}|$. In (a) we
choose three points in the phase diagram, namely (1) a point in the confined
phase ($\mu^{2}/q^{2}=0.2$), (2) the critical point ($\mu^{2}/q^{2}=0.447$),
and (3) a point in the Higgs phase ($\mu^{2}/q^{2}=0.6$) for the system-size
$L=120$. In (b) we show the convergence of the entropic quantities as a
function of the bond dimension at the critical point.

For the DMRG simulations, the MPS bond dimension is truncated upto $\chi=600$.
Fig. 14(a) shows the convergence of ground-state energy density with respect
to the bond dimension $\chi\leq 600$ for systems of sizes $L\leq 120$. The
energy density converges close to the machine precision within $\chi\leq 500$
in the gapped regions – confined ($\mu^{2}/q^{2}=0.2$) and Higgs
($\mu^{2}/q^{2}$). On the other hand, as expected, the convergence is slower
at the critical point due to the diverging correlation length. However, as
shown in Fig. 14(b), the precision attained at the critical point for
$\chi=500,600$ is sufficient to perform the precise scaling analysis reported
here. To confirm the convergence of the DMRG sweeps, we continue the DMRG
iterations until the energy difference in subsequent sweeps falls below
$10^{-13}$.

In case of out-of-equilibrium simulations with TDVP, we truncate the bosonic
species upto the maximum occupancy of $n_{0}=7$ (i.e, 8 levels) levels and
restrict the maximum MPS bond upto $\chi=400$.

##  Appendix B Numerical comparisons between the local matter-gauge and the
matter-only long-range descriptions of the system

Figure 15:  (a) In presence of gauge fields at the links between two nearest-
neighbor sites, there are two options to take bipartition at a given link,
i.e., cut-1 and cut-2 depicted in the figure, for the calculation of
entanglement entropies. The entropies, however, remain same between these two
options since gauge fields in 1+1D are not dynamical variables. (b) The
comparison between von Neumann entropy $S=S_{1}$ and the entanglement ground
state energy $\varepsilon_{0}=S_{n\rightarrow\infty}$ calculated from the
long-range Hamiltonian after integrating-out the gauge fields and from the
original local matter-gauge Hamiltonian for system-size $L=100$ at the
critical point. For the local gauge theory Hamiltonian, we truncate the gauge-
field Hilbert spaces upto $d_{\text{truc}}=21$ electric states. (c) We plot
the absolute differences in ground state energy densities and in the mid-chain
von Neumann entanglement entropies between the truncated local matter-gauge
Hamiltonian with respect to the long-range system as a function of the
truncated gauge-field Hilbert-space dimension $d_{\text{trunc}}$. Here we
consider the system-size $L=60$. The differences reduce to the numerical
accuracy attained with the MPS bond dimension $\chi=600$ at the truncation
level $d_{\text{trunc}}=17$.

In the main text, we have favored the long-range Hamiltonian (6) after
integrating-out the gauge fields by an exact mapping utilizing the Gauss law.
This is because of the advantage in the long-range formulation, where we avoid
dealing with extra degrees of freedom corresponding to the gauge fields.
Moreover, since the Hilbert spaces of the gauge fields are formally infinite
dimensional, they need to be truncated to a finite dimensional subspace in the
short-range formulation of Eq. (2), thereby introducing an extra source of
approximation that needs to be accurately controlled (see e.g., Kühn  _et al._
(2014)). Here, we show that the short-range Hamiltonian, with sufficiently
large truncated gauge-field Hilbert spaces, gives the exactly same results as
that of the long-range Hamiltonian discussed in the main text.

To be noted that, in presence of the gauge fields in the short-range
formulation, we have two options to take bipartitions at each link – one
before the gauge field (cut-1) and another after the gauge field (cut-2) – for
the calculation of entanglement entropies as seen in Fig. 15(a). In Figs.
15(b)-(c), we compare the results between the truncated short-range matter-
gauge Hamiltonian (2) and the long-range Hamiltonian (6). First, we can
observe that in case of local gauge theory, entropies calculated at cut-1 or
cut-2 are the same, which is expected since the gauge fields are not dynamical
variables in 1+1D, see Fig. 15(b). Moreover, with sufficiently large gauge-
field Hilbert space dimension ($d_{\text{trunc}}$), local and long-range
Hamiltonians give exactly same result (upto a numerical accuracy, see attached
Fig. 15(c)). This is understandable as in 1+1D, integrating-out the gauge
fields using Gauss law is exact. Specifically, Fig. 15(c) shows that the
differences between the results coming from truncated local gauge theory
Hamiltonian and the long-range Hamiltonian diminute to the numerical
accuracies (attained with the MPS bond dimension $\chi=600$) for
$d_{\text{trunc}}\geq 17$.

###### Acknowledgements.

We acknowledge the PL-GRID infrastructure for providing us the high-
performance computing facility under the grant PLG/2022/015613 for the
numerical results reported here. The MPS algorithms have been implemented
using ITensor library Fishman  _et al._ (2022a, b). The work of M.D. was
partly supported by the ERC under grant number 758329 (AGEnTh), by the MIUR
Programme FARE (MEPH), and by the European Union’s Horizon 2020 research and
innovation programme under grant agreement No 817482 (Pasquans). M.L.
acknowledges support from: ERC AdG NOQIA; Ministerio de Ciencia y Innovation
Agencia Estatal de Investigaciones
(PGC2018-097027-B-I00/10.13039/501100011033,
CEX2019-000910-S/10.13039/501100011033, Plan National FIDEUA
PID2019-106901GB-I00, FPI, QUANTERA MAQS PCI2019-111828-2, QUANTERA DYNAMITE
PCI2022-132919, Proyectos de I+D+I “Retos Colaboración” QUSPIN
RTC2019-007196-7); MICIIN with funding from European Union
NextGenerationEU(PRTR-C17.I1) and by Generalitat de Catalunya; Fundació
Cellex; Fundació Mir-Puig; Generalitat de Catalunya (European Social Fund
FEDER and CERCA program, AGAUR Grant No. 2021 SGR 01452, QuantumCAT
U16-011424, co-funded by ERDF Operational Program of Catalonia 2014-2020);
Barcelona Supercomputing Center MareNostrum (FI-2022-1-0042); EU Horizon 2020
FET-OPEN OPTOlogic (Grant No 899794); EU Horizon Europe Program (Grant
Agreement 101080086 — NeQST), ICFO Internal “QuantumGaudi” project; European
Union’s Horizon 2020 research and innovation program under the Marie-
Skłodowska-Curie grant agreement No 101029393 (STREDCH) and No 847648 (“La
Caixa” Junior Leaders fellowships ID100010434: LCF/BQ/PI19/11690013,
LCF/BQ/PI20/11760031, LCF/BQ/PR20/11770012, LCF/BQ/PR21/11840013). This
research was also funded by National Science Centre (Poland) by grant
2021/03/Y/ST2/00186 within the QuantEra II Programme that has received funding
from the European Union’s Horizon 2020 research and innovation programme under
Grant Agreement No 101017733 DYNAMITE (J.Z.). The partial support by the
Strategic Programme Excellence Initiative at Jagiellonian University is also
acknowledged. LT acknowledges the support from the Spanish project
PGC2018-095862-B-C22, the CSIC Interdisciplinary Thematic Platform (PTI)
Quantum Technologies (PTI-QTEP+) and from the Spanish projects
PID2021-127968NB-I00 and TED2021-130552B-C22 funded by r
MCIN/AEI/10.13039/501100011033/FEDER, UE and MCIN/AEI/10.13039/501100011033,
respectively. Views and opinions expressed in this work are, however, those of
the authors only and do not necessarily reflect those of the European Union,
European Climate, Infrastructure and Environment Executive Agency (CINEA), nor
any other granting authority. Neither the European Union nor any granting
authority can be held responsible for them.

## References

  * Coleman (1985) S. Coleman,  _Aspects of Symmetry_ (Cambridge University Press, 1985).
  * Francesco  _et al._ (1997) P. D. Francesco, P. Mathieu, and D. Sénéchal,  _Conformal Field Theory_ (Springer New York, 1997).
  * Blumenhagen and Plauschinn (2009) R. Blumenhagen and E. Plauschinn,  _Introduction to Conformal Field Theory_ (Springer Berlin Heidelberg, 2009).
  * Shankar (2017) R. Shankar,  _Quantum Field Theory and Condensed Matter_ (Cambridge University Press, 2017).
  * Fradkin (2013) E. Fradkin,  _Field Theories of Condensed Matter Physics_ (Cambridge University Press, 2013).
  * Wilson (1971a) K. G. Wilson, Phys. Rev. B 4, 3174 (1971a).
  * Wilson (1971b) K. G. Wilson, Phys. Rev. B 4, 3184 (1971b).
  * Wilson (1975) K. G. Wilson, Rev. Mod. Phys. 47, 773 (1975).
  * Shankar (1994) R. Shankar, Rev. Mod. Phys. 66, 129 (1994).
  * Gell-Mann and Low (1954) M. Gell-Mann and F. E. Low, Phys. Rev. 95, 1300 (1954).
  * Peskin and Schroeder (1995) M. E. Peskin and D. V. Schroeder,  _An introduction to quantum field theory_ (Addison-Wesley Pub. Co, Reading, Mass, 1995).
  * Schwartz (2013) M. D. Schwartz,  _Quantum Field Theory and the Standard Model_ (Cambridge University Press, 2013).
  * Anderson (1963) P. W. Anderson, Phys. Rev. 130, 439 (1963).
  * Englert and Brout (1964) F. Englert and R. Brout, Phys. Rev. Lett. 13, 321 (1964).
  * Higgs (1964) P. W. Higgs, Phys. Rev. Lett. 13, 508 (1964).
  * Guralnik  _et al._ (1964) G. S. Guralnik, C. R. Hagen, and T. W. B. Kibble, Phys. Rev. Lett. 13, 585 (1964).
  * Endres  _et al._ (2012) M. Endres, T. Fukuhara, D. Pekker, M. Cheneau, P. Schau$\upbeta$, C. Gross, E. Demler, S. Kuhr, and I. Bloch, Nature 487, 454 (2012).
  * Kasamatsu  _et al._ (2013) K. Kasamatsu, I. Ichinose, and T. Matsui, Phys. Rev. Lett. 111, 115303 (2013).
  * Pekker and Varma (2015) D. Pekker and C. Varma, Annual Review of Condensed Matter Physics 6, 269 (2015).
  * Kuno  _et al._ (2017) Y. Kuno, S. Sakane, K. Kasamatsu, I. Ichinose, and T. Matsui, Phys. Rev. D 95, 094507 (2017).
  * González-Cuadra  _et al._ (2017) D. González-Cuadra, E. Zohar, and J. I. Cirac, New Journal of Physics 19, 063038 (2017).
  * Zhang  _et al._ (2018) J. Zhang, J. Unmuth-Yockey, J. Zeiher, A. Bazavov, S.-W. Tsai, and Y. Meurice, Phys. Rev. Lett. 121, 223201 (2018).
  * Unmuth-Yockey  _et al._ (2018) J. Unmuth-Yockey, J. Zhang, A. Bazavov, Y. Meurice, and S.-W. Tsai, Phys. Rev. D 98, 094511 (2018).
  * Park  _et al._ (2019) J. Park, Y. Kuno, and I. Ichinose, Phys. Rev. A 100, 013629 (2019).
  * Meurice (2021) Y. Meurice, Phys. Rev. D 104, 094513 (2021).
  * Chanda  _et al._ (2022) T. Chanda, M. Lewenstein, J. Zakrzewski, and L. Tagliacozzo, Phys. Rev. Lett. 128, 090601 (2022).
  * (27) D. Tong, “Lectures on Gauge Theory,” https://www.damtp.cam.ac.uk/user/tong/gaugetheory.html.
  * Komargodski  _et al._ (2019) Z. Komargodski, A. Sharon, R. Thorngren, and X. Zhou, SciPost Phys. 6, 3 (2019).
  * Fradkin and Shenker (1979) E. Fradkin and S. H. Shenker, Phys. Rev. D 19, 3682 (1979).
  * Callaway and Carson (1982) D. J. E. Callaway and L. J. Carson, Phys. Rev. D 25, 531 (1982).
  * Gattringer  _et al._ (2018) C. Gattringer, D. Göschl, and T. Sulejmanpašić, Nuclear Physics B 935, 344 (2018).
  * Göschl  _et al._ (2018) D. Göschl, C. Gattringer, and T. Sulejmanpasic, “The critical endpoint in the 2d U(1) gauge-Higgs model at topological angle $\theta=\pi$,” (2018), arXiv:1810.09671 .
  * Sulejmanpasic  _et al._ (2020) T. Sulejmanpasic, D. Göschl, and C. Gattringer, Phys. Rev. Lett. 125, 201602 (2020).
  * Sulejmanpasic (2021) T. Sulejmanpasic, Phys. Rev. D 103, 034512 (2021).
  * Coleman  _et al._ (1975) S. Coleman, R. Jackiw, and L. Susskind, Annals of Physics 93, 267 (1975).
  * Coleman (1976) S. Coleman, Annals of Physics 101, 239 (1976).
  * Chanda  _et al._ (2020) T. Chanda, J. Zakrzewski, M. Lewenstein, and L. Tagliacozzo, Phys. Rev. Lett. 124, 180602 (2020).
  * Schollwöck (2011) U. Schollwöck, Annals of Physics 326, 96 (2011).
  * Orús (2014) R. Orús, Annals of Physics 349, 117 (2014).
  * Bañuls  _et al._ (2013) M. Bañuls, K. Cichy, J. Cirac, and K. Jansen, Journal of High Energy Physics 2013, 158 (2013).
  * Buyens  _et al._ (2014) B. Buyens, J. Haegeman, K. Van Acoleyen, H. Verschelde, and F. Verstraete, Phys. Rev. Lett. 113, 091601 (2014).
  * Kühn  _et al._ (2014) S. Kühn, J. I. Cirac, and M.-C. Bañuls, Phys. Rev. A 90, 042305 (2014).
  * Bañuls  _et al._ (2015) M. C. Bañuls, K. Cichy, J. I. Cirac, K. Jansen, and H. Saito, Phys. Rev. D 92, 034519 (2015).
  * Buyens  _et al._ (2016a) B. Buyens, F. Verstraete, and K. Van Acoleyen, Phys. Rev. D 94, 085018 (2016a).
  * Pichler  _et al._ (2016) T. Pichler, M. Dalmonte, E. Rico, P. Zoller, and S. Montangero, Phys. Rev. X 6, 011023 (2016).
  * Bañuls  _et al._ (2017) M. C. Bañuls, K. Cichy, J. I. Cirac, K. Jansen, and S. Kühn, Phys. Rev. Lett. 118, 071601 (2017).
  * Buyens  _et al._ (2016b) B. Buyens, J. Haegeman, H. Verschelde, F. Verstraete, and K. Van Acoleyen, Phys. Rev. X 6, 041040 (2016b).
  * Magnifico  _et al._ (2020) G. Magnifico, M. Dalmonte, P. Facchi, S. Pascazio, F. V. Pepe, and E. Ercolessi, Quantum 4, 281 (2020).
  * Bañuls  _et al._ (2020) M. C. Bañuls, R. Blatt, J. Catani, A. Celi, J. I. Cirac, M. Dalmonte, L. Fallani, K. Jansen, M. Lewenstein, S. Montangero, C. A. Muschik, B. Reznik, E. Rico, L. Tagliacozzo, K. V. Acoleyen, F. Verstraete, U.-J. Wiese, M. Wingate, J. Zakrzewski, and P. Zoller, The European Physical Journal D 74, 165 (2020).
  * Aidelsburger  _et al._ (2021) M. Aidelsburger, L. Barbiero, A. Bermudez, T. Chanda, A. Dauphin, D. González-Cuadra, P. R. Grzybowski, S. Hands, F. Jendrzejewski, J. Jünemann, G. Juzeliūnas, V. Kasper, A. Piga, S.-J. Ran, M. Rizzi, G. Sierra, L. Tagliacozzo, E. Tirrito, T. V. Zache, J. Zakrzewski, E. Zohar, and M. Lewenstein, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2021), 10.1098/rsta.2021.0064.
  * Schwinger (1951) J. Schwinger, Phys. Rev. 82, 664 (1951).
  * White (1992) S. R. White, Phys. Rev. Lett. 69, 2863 (1992).
  * White (1993) S. R. White, Phys. Rev. B 48, 10345 (1993).
  * White (2005) S. R. White, Phys. Rev. B 72, 180403(R) (2005).
  * Schollwöck (2005) U. Schollwöck, Rev. Mod. Phys. 77, 259 (2005).
  * Haegeman  _et al._ (2011) J. Haegeman, J. I. Cirac, T. J. Osborne, I. Pižorn, H. Verschelde, and F. Verstraete, Phys. Rev. Lett. 107, 070601 (2011).
  * Haegeman  _et al._ (2016) J. Haegeman, C. Lubich, I. Oseledets, B. Vandereycken, and F. Verstraete, Phys. Rev. B 94, 165116 (2016).
  * Koffel  _et al._ (2012) T. Koffel, M. Lewenstein, and L. Tagliacozzo, Phys. Rev. Lett. 109, 267203 (2012).
  * Paeckel  _et al._ (2019) S. Paeckel, T. Köhler, A. Swoboda, S. R. Manmana, U. Schollwöck, and C. Hubig, Annals of Physics 411, 167998 (2019).
  * Blöte  _et al._ (1986) H. W. J. Blöte, J. L. Cardy, and M. P. Nightingale, Phys. Rev. Lett. 56, 742 (1986).
  * Affleck (1986) I. Affleck, Phys. Rev. Lett. 56, 746 (1986).
  * Affleck  _et al._ (1989) I. Affleck, D. Gepner, H. J. Schulz, and T. Ziman, Journal of Physics A: Mathematical and General 22, 511 (1989).
  * Hallberg  _et al._ (1996) K. Hallberg, X. Q. G. Wang, P. Horsch, and A. Moreo, Phys. Rev. Lett. 76, 4955 (1996).
  * Xavier (2010) J. C. Xavier, Phys. Rev. B 81, 224404 (2010).
  * Dalmonte  _et al._ (2012) M. Dalmonte, E. Ercolessi, and L. Taddia, Phys. Rev. B 85, 165112 (2012).
  * Chepiga and Mila (2017) N. Chepiga and F. Mila, Phys. Rev. B 96, 054425 (2017).
  * Singh  _et al._ (2010) S. Singh, R. N. C. Pfeifer, and G. Vidal, Phys. Rev. A 82, 050301 (2010).
  * Singh  _et al._ (2011) S. Singh, R. N. C. Pfeifer, and G. Vidal, Phys. Rev. B 83, 115125 (2011).
  * Casini  _et al._ (2014) H. Casini, M. Huerta, and J. A. Rosabal, Phys. Rev. D 89, 085012 (2014).
  * Ghosh  _et al._ (2015) S. Ghosh, R. M. Soni, and S. P. Trivedi, Journal of High Energy Physics 2015, 69 (2015).
  * Soni and Trivedi (2016) R. M. Soni and S. P. Trivedi, Journal of High Energy Physics 2016, 136 (2016).
  * Callan and Wilczek (1994) C. Callan and F. Wilczek, Physics Letters B 333, 55 (1994).
  * Vidal  _et al._ (2003) G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev, Phys. Rev. Lett. 90, 227902 (2003).
  * Calabrese and Cardy (2004) P. Calabrese and J. Cardy, Journal of Statistical Mechanics 2004, P06002 (2004).
  * Cardy and Calabrese (2010) J. Cardy and P. Calabrese, Journal of Statistical Mechanics: Theory and Experiment 2010, P04023 (2010).
  * Eriksson and Johannesson (2011) E. Eriksson and H. Johannesson, Journal of Statistical Mechanics: Theory and Experiment 2011, P02008 (2011).
  * Laflorencie  _et al._ (2006) N. Laflorencie, E. S. Sørensen, M.-S. Chang, and I. Affleck, Phys. Rev. Lett. 96, 100603 (2006).
  * Cardy (1986a) J. L. Cardy, Nuclear Physics B 275, 200 (1986a).
  * Cardy (1986b) J. L. Cardy, Nuclear Physics B 270, 186 (1986b).
  * Rocha-Caridi (1985) A. Rocha-Caridi, in  _Mathematical Sciences Research Institute Publications_ (Springer US, 1985) pp. 451–473.
  * Evenbly and Vidal (2014) G. Evenbly and G. Vidal, Journal of Statistical Physics 157, 931 (2014).
  * Giamarchi (2003) T. Giamarchi,  _Quantum Physics in One Dimension_, International Series of Monographs on Physics (Oxford University Press, 2003).
  * Gogolin  _et al._ (2004) A. O. Gogolin, A. A. Nersesyan, and A. M. Tsvelik,  _Bosonization and strongly correlated systems_ (Cambridge University Press, Cambridge, England, 2004).
  * Song  _et al._ (2010) H. F. Song, S. Rachel, and K. Le Hur, Phys. Rev. B 82, 012405 (2010).
  * Song  _et al._ (2012) H. F. Song, S. Rachel, C. Flindt, I. Klich, N. Laflorencie, and K. Le Hur, Phys. Rev. B 85, 035409 (2012).
  * Rachel  _et al._ (2012) S. Rachel, N. Laflorencie, H. F. Song, and K. Le Hur, Phys. Rev. Lett. 108, 116401 (2012).
  * Villa  _et al._ (2019) L. Villa, J. Despres, and L. Sanchez-Palencia, Phys. Rev. A 100, 063632 (2019).
  * Villa  _et al._ (2020) L. Villa, J. Despres, S. J. Thomson, and L. Sanchez-Palencia, Phys. Rev. A 102, 033337 (2020).
  * Hubig  _et al._ (2015) C. Hubig, I. P. McCulloch, U. Schollwöck, and F. A. Wolf, Phys. Rev. B 91, 155115 (2015).
  * Fishman  _et al._ (2022a) M. Fishman, S. R. White, and E. M. Stoudenmire, SciPost Phys. Codebases , 4 (2022a).
  * Fishman  _et al._ (2022b) M. Fishman, S. R. White, and E. M. Stoudenmire, SciPost Phys. Codebases , 4 (2022b).