# Phase diagram of 1+1D Abelian-Higgs model and its critical point

Titas Chanda<EMAIL_ADDRESS>The Abdus Salam International Centre for
Theoretical Physics (ICTP), Strada Costiera 11, 34151 Trieste, Italy  Instytut
Fizyki Teoretycznej, Uniwersytet Jagielloński, Łojasiewicza 11, 30-348 Kraków,
Poland  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  Instituto de Física Fundamental IFF-CSIC, Calle
Serrano 113b, Madrid 28006, Spain  Departament de Física Quàntica i
Astrofísica and Institut de Ciències del Cosmos (ICCUB), Universitat de
Barcelona, Martí i Franquès 1, 08028 Barcelona, Catalonia, Spain

###### Abstract

We determine the phase diagram of the Abelian-Higgs model in one spatial
dimension and time (1+1D) on a lattice. We identify a line of first order
phase transitions separating the Higgs region from the confined one. This line
terminates in a quantum critical point above which the two regions are
connected by a smooth crossover. We analyze the critical point and find
compelling evidences for its description as the product of two non-interacting
systems, a massless free fermion and a massless free boson. However, we find
also some surprizing results that cannot be explained by our simple picture,
suggesting this newly discovered critical point to be an unusual one.

### Introduction.

Gauge theories in 1+1 dimensions (1D in space and time) are ideal playgrounds
to characterize the effects of strong-coupling between matter and gauge
fields. Many of the non-perturbative aspects of 3+1 dimensional gauge theories
relevant to our understanding of particle physics, such as quark confinement
and chiral symmetry breaking, have a 1+1D analogue. Furthermore, 1+1D field
theories can often be treated analytically Borgs and Nill (1987); Brydges  _et
al._ (1979) providing important insights to the physics of 1+1D systems
relevant also to condensed matter physics.

In this work, we study the lattice version of a relativistic bosonic field
that interacts with a photonic field Chanda  _et al._ (2020), the bosonic
version of the Schwinger model Schwinger (1951, 1962a, 1962b); Coleman (1976).
In contrast to (polarized) fermions, bosons can have contact interactions that
are described by the well known Abelian-Higgs model (AHM) in 1+1D (AHM2)
Anderson (1963); Englert and Brout (1964); Higgs (1964); Guralnik  _et al._
(1964); Peskin and Schroeder (1995).

In AHM2, a weak matter-field coupling limit Coleman (1985) suggests that the
phase diagram is shared by two phases characteristic of the Higgs mechanism
Anderson (1963); Englert and Brout (1964); Higgs (1964); Guralnik  _et al._
(1964), a superfluid phase 1 with the quasi-condensation of bosons (Higgs
phase) and a Mott-insulating phase 2 with strong interactions. However, non-
perturbative calculations show that the phenomenology in the phase 1 is the
same as in the phase 2, and bosons are always tightly confined Coleman (1985)
(for a recent discussion see Komargodski  _et al._ (2019); Tong ).

One can certify the presence of a phase transition in $d+1$ dimension for any
$d>2$ Fradkin and Shenker (1979); Callaway and Carson (1982), but due to the
(boring) expectation of a single phase in the continuum, the phase diagram of
AHM2 has never been computed on the lattice in 1+1D. This work aims at filling
this gap.

Our work is strongly motivated by the current prospects of simulating lattice
gauge theories using cold atomic setups Dutta  _et al._ (2017); Schweizer  _et
al._ (2019); Görg  _et al._ (2019); Mil  _et al._ (2020); Bañuls  _et al._
(2020); Aidelsburger  _et al._ (2021). Since bosons are easier to cool down
than fermions, experiments along the lines proposed in Kasamatsu  _et al._
(2013); Kuno  _et al._ (2017); González-Cuadra  _et al._ (2017); Zhang  _et
al._ (2018); Unmuth-Yockey  _et al._ (2018); Park  _et al._ (2019); Meurice
should soon explore the phase diagram of AHM2.

Figure 1: Phase diagram of the AHM2 (1) in the
$(\mu^{2}/q^{2},\lambda/q^{2})$-plane for a system of size $L=60$. At small
couplings, the system occupies two qualitatively different regions, a confined
and a Higgs region, separated by a line of FOQPT as witnessed by the average
tunneling amplitude $\mathcal{O}_{\text{tunn}}$ (left panel) (effectively zero
in the confined region, and finite in the Higgs region) and the entanglement
entropy $\mathcal{S}_{L/2}$ measured at the center of the chain (right panel)
(small in the confined region and large in the Higgs region). The line of
FOQPT ends at a SOQPT, above which two regions are smoothly connected
representing different aspects of a single phase.

Physicists have been working hard to measure the Higgs mode in experiments
with cold atoms for a long-time, as reviewed in Pekker and Varma (2015). In
2+1D, an explicit particle-hole symmetry protects the decay of the Higgs mode
into Goldstone modes allowing a proper measurement of its mass. These
conditions are only met at the tip of the lobe of the Mott insulator to
superfluid transition in Bose-Hubbard systems Schori  _et al._ (2004);
Stöferle  _et al._ (2004); Endres  _et al._ (2012) which is, unfortunately, in
1+1D of the Berezinskii-Kosterlitz-Thouless type (see e.g., Kühner and Monien
(1998); Danshita and Polkovnikov (2011); Cazalilla  _et al._ (2011); Dutta
_et al._ (2015a)) and is not particle-hole symmetric. This observation seems
to strengthen the picture emerging from the presence of a single phase in AHM2
and seems to suggest that a proper Higgs mode does not exist in 1+1D.

The results we present indicate a different picture, still characterized by a
single phase, but with a reach landscape of transitions.

By performing matrix product states (MPS) Schollwöck (2011); Orús (2014)
simulations of the Hamiltonian version of AHM2, we confirm the presence of a
single phase for all the values of the mass of the bosons $\mu^{2}/q^{2}$ and
their interaction strength $\lambda/q^{2}$ (in unit of the bosonic charge $q$)
in agreement with the field theoretical analysis. However, unexpectedly, for
small $\lambda/q^{2}$ we find a line of first order quantum phase transitions
(FOQPT) between the “Higgs” and the “confined” regions. This line ends, for a
finite value of $\lambda/q^{2}$, in a critical second order quantum phase
transition (SOQPT), above which the two regions are continuously connected
through a smooth crossover. Close to FOQPT line the two regions are well
separated enabling identification of a “Higgs” mode and its analysis in the
continuum limit (for a sufficiently small $\lambda/q^{2}$). One can indeed
take a different continuum limit to the standard one by approaching, from the
Higgs region, the newly discovered SOQPT.

We precisely identify the position and the nature of the new critical point.
By assuming Lorentz invariance at the critical point and then using the
machinery of conformal field theories (CFT), we can understand the critical
point as the direct sum of two non-interacting fields: a free fermionic field
describing the Higgs mode and a free bosonic field, a collective mode of the
Goldstone modes and the gauge field. Still the a complete characterization of
this critical point remains an outstanding challenge as some results do not
fit the above picture.

### The model.

Following Chanda  _et al._ (2020), we discretize AHM2 on a finite 1D lattice
with $L$ sites (with spacing $a$) (see sup  for details) arriving at the
Hamiltonian (with open boundary conditions):

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

with $x=1/a^{2}q^{2}$. The matter fields
$\\{\hat{\phi}_{j},\hat{\phi}_{j}^{\dagger},\hat{\Pi}_{j},\hat{\Pi}_{j}^{\dagger}\\}$
operators act in Hilbert space at sites $j$, while the gauge-field
$\\{\hat{L}_{j},\hat{U}_{j},\hat{U}^{\dagger}_{j}\\}$ objects act in Hilbert
space defined on the bond linking sites $j$ and $j+1$. The operators fulfill
the standard commutation relations $\
[\hat{\phi}_{j},\hat{\Pi}_{k}]=[\hat{\phi}^{\dagger}_{j},\hat{\Pi}^{\dagger}_{k}]=i\delta_{jk}$,
$[\hat{L}_{j},\hat{U}_{j}]=-\hat{U}_{j}$ and $\
[\hat{L}_{j},\hat{U}^{\dagger}_{j}]=\hat{U}^{\dagger}_{j}$.

The usual continuum limit is $x\rightarrow\infty$. Here we fix $x=2$ and
characterize the phase diagram on the lattice.

We can define creation and annihilation operators for particles ‘$a$’ and
anti-particles ‘$b$’ as $\hat{a}_{j}$ and $\hat{b}_{j}$ fulfilling
$[\hat{a}_{j},\hat{a}^{\dagger}_{k}]=[\hat{b}_{j},\hat{b}^{\dagger}_{k}]=\delta_{jk}$
111The 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).. We use the density matrix
renormalization group (DMRG) algorithm Schollwöck (2011); Orús (2014); White
(1992, 1993, 2005); Schollwöck (2005) to find the ground state of the
Hamiltonian (1). Specifically, we employ a strictly single-site variant of
DMRG with subspace expansion Hubig  _et al._ (2015). For numerics we limit the
occupations of bosonic modes to at most $n_{0}^{a}=n_{0}^{b}=10$ 222For
details about the DMRG simulations, see sup ..

In absence of external charges, the local $\mathbb{U}(1)$ symmetry implies the
Gauss law $\hat{G}_{j}=0,\ \forall j$, where the generators are Chanda  _et
al._ (2020) $\hat{G}_{j}=\hat{L}_{j}-\hat{L}_{j-1}-{\hat{Q}_{j}},$ and
$\hat{Q}_{j}=\hat{a}_{j}^{\dagger}\hat{a}_{j}-\hat{b}_{j}^{\dagger}\hat{b}_{j}$
encodes the density of dynamical charges. Using the Gauss law, we can
integrate-out the gauge-fields in a chain with open-boundary conditions in
favor of a long-range potential for the matter fields Schwinger (1951).

### Confined and Higgs, two shades of the same phase.

The long-range interactions among bosons destroy the phases of the standard
1+1D Bose Hubbard model – see the phase diagram in Fig. 1. For
$\lambda/q^{2}\geq 0$, the system is in the confined region as far as
$\mu^{2}/q^{2}\leq 0$. In this region the model has a finite mass gap, and the
elementary excitations are mesons, bound pairs of particle-antiparticle. The
gauge bosons are in the lowest eigenstate of $\hat{L}_{j}$, so that the
variance
$\sigma^{2}(\hat{L}_{j})=\braket{\hat{L}^{2}_{j}}-\braket{\hat{L}_{i}}^{2}\approx
0$.

For $\mu^{2}/q^{2}\gg 0$, the system enters the gapped Higgs region where the
variance $\sigma^{2}(\hat{L}_{j})$ becomes large. The effective gauge-field
mediated tunneling amplitude
$\mathcal{O}_{\text{tunn}}=\frac{1}{2L}\sum_{j}\braket{\hat{\phi}^{\dagger}_{j+1}\hat{U}_{j}\hat{\phi}_{j}+h.c.}$
increases, so that it can distinguish the Higgs region from the confined one.

We can also characterize the two regions by considering the behavior of the
entanglement entropy of a block made of $l$ constituents starting from the
boundary, defined as

$\mathcal{S}_{l}=-\text{Tr}\left[\rho_{l}\ln\rho_{l}\right],$ (2)

where $\rho_{l}=\text{Tr}_{l+1,l+2,...,L}\ket{\psi}\bra{\psi}$ is the reduced
density matrix. In the Higgs region, the entanglement entropy is
systematically larger than in the confined region. However, since both phases
are gapped, the entropy follows area-law scaling with respect to the
bipartition size (see sup ).

For sufficiently small $\lambda/q^{2}$, the two regions are separated by a
FOQPT line characterized by discontinuous jumps both in the tunneling
amplitude $\mathcal{O}_{\text{tunn}}$ and in $\mathcal{S}_{L/2}$ measured
across the central bond sup . This line terminates at a critical SOQPT at a
finite value of $\lambda_{c}/q^{2}$ and $\mu_{c}^{2}/q^{2}$, identified by the
red cross in Fig. 1. We discuss precise location and characterization of this
critical point below. Above the SOQPT, the two regions are smoothly connected
as revealed by smooth changes in all the physical quantities while moving from
one region to the another.

### Nature of the critical point.

We can precisely locate and characterize the critical point assuming its
Lorentz invariance, that implies an applicability of a CFT at low energies. In
a CFT, the finite-size scaling of the entanglement entropy of a block of first
$l$ consecutive sites in a chain with open boundary conditions and length $L$
is

$\mathcal{S}(l,L)=\frac{c}{6}W+b^{\prime},$ (3)

where $c$ is the central charge of the corresponding CFT, $b^{\prime}$ is a
non-universal constant and the chord length $W$ is a function of both $L$ and
$l$: $W(l,L)=\ln\left[\frac{2L}{\pi}\sin(\pi l/L)\right]$ Callan and Wilczek
(1994); Vidal  _et al._ (2003); Calabrese and Cardy (2004).

Figure 2: Entropy scaling of the AHM2. (a) The fitted central charge
$c_{\text{fit}}$ according to (3) for fixed $\mu^{2}/q^{2}=0.447$ and
different system sizes. Curves for different system-sizes crosses each other
at $\lambda/q^{2}\approx 0.0565$ and $c_{\text{fit}}\approx 3/2$. (b) The
scaling of the entanglement entropy at the critical point for different system
sizes yields the central charge of the critical theory as $c=1.49(1)$.

We pin-point the SOQPT by adapting the idea of the phenomenological
renormalization group Nightingale (1975) to the scaling of the entropy in (3)
as explained in Koffel  _et al._ (2012); Buyskikh  _et al._ (2019). At the
critical point, the $c$ value should be independent of the system’s size. For
each $L$, we obtain $c_{\text{fit}}$ by fitting (3) to our numerical data for
$\mathcal{S}(l,L)$. The extracted values in the
$(\mu^{2}/q^{2},\lambda/q^{2})$-plane depend on $L$ and become independent of
the system size only at the critical point. Our data suggest that the $L$
dependent central charges collapse to a single value at
$(\mu^{2}/q^{2}=0.447(1),\lambda/q^{2}=0.0565(1))$ (see Fig. 2). The central
charge at the critical point
$(\mu_{c}^{2}/q^{2}=0.447,\lambda_{c}/q^{2}=0.0565)$ is found to be
$c=1.49(1)$ 333Notice that our analysis is based on the full scaling form of
the entropy in (3) that holds for conformally invariant systems only. The
logarithmic divergence of the half chain entropy on the other hand can be
observed also for systems that only posses scale invariance Refael and Moore
(2004); Latorre  _et al._ (2005); Koffel  _et al._ (2012)..

The value of the central charge mentioned above suggests that we are not
dealing with a minimal model. However, we want to argue here that we are in
the presence of the direct 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 and a $c_{b}=1$ for free boson. This scenario is strongly
motivated by the standard Higgs mechanism. The complex Higgs field separates
into its amplitude and its phase. The amplitude mode is effectively described
by a real $\lambda\phi^{4}$ theory that undergoes the standard Ising phase
transition (the $c=1/2$ part). The phase, on the other hand, provides the
longitudinal degree of freedom to the photon field. The latter becomes
massless at the transition and provides the $c=1$ free bosonic part. The value
of $c=1.5$ furthermore suggests, based on the $c$ theorem, that the two parts
should be non-interacting Zomolodchikov (1986).

In order to confirm this scenario we compute the entanglement spectrum that is
also known to encode the central charge of the theory Cardy and Tonni (2016).
The entanglement spectrum, denoted by $\varepsilon_{s}$, is the spectrum of
the entanglement Hamiltonian $H_{l}=-\log(\rho_{l})$. By assuming a factorized
ground state, we should observe that the smallest eigenvalue of $H_{l}$,
$\varepsilon_{0}$ diverge logarithmically. In particular we should see that
Cardy and Tonni (2016)

$\varepsilon_{0}=\left(\varepsilon_{0}^{Ising}+\varepsilon_{0}^{boson}\right)\propto\frac{(c_{f}+c_{b})}{12}W+O(1/W).$
(4)

By fitting our numerical data to (4), we observe a perfect collapse on the
functional form predicted by CFT, but the numerical result for of
$c_{\text{eff}}=1.20(1)$ is not compatible with $1.5$ (see Fig. 3(a)). This
disagreement between the scaling of the entanglement entropy and that for the
entanglement ground state already suggests that the critical point is unusual
and exotic in nature.

Figure 3:  (a) The scaling of the entanglement ground state $\varepsilon_{0}$
matches perfectly the functional form suggested by the CFT analysis, reported
in the text. However, the numerical value for the central charge deviates by
around the $20\%$ from the value we extract from the scaling of the
entanglement entropy as the best fit suggests $c_{\text{eff}}=1.20(1)$. The
red-dotted line depicts the fit assuming $c=1.5$. (b) The scaling of
$\braket{\hat{\Pi}^{\dagger}_{l}\hat{\Pi}_{l}}$ according to the CFT
prediction (Eq. (5)). It couples both to the identity operator and one primary
with scaling dimension $\Delta$ that comes out to be $\Delta=0.51(2)$ from the
fit, with $a$ being $0.5474(2)$

.

We thus turn to analyze the operator content of the model by studying the
correlation functions of local operators. We should be able to identify a set
of primary operators, by studying the large distance two point correlations
function that should decay algebraically as $\phi(0)\phi(r)\sim
1/r^{{\Delta_{\phi}}}$. The presence of gauge symmetry, however, strongly
reduces the set of operators we can consider. Most of the candidates that
should couple to primary operators are either trivial (due to the low-
dimensionality of the system) or vanishing since they are not gauge invariant.
The only non-vanishing operators are indeed Wilson lines terminating on a
boson-antiboson pair, and electric field correlations. We also have access to
local operators such as $\hat{\phi}^{\dagger}\hat{\phi}$ and
$\hat{\Pi}^{\dagger}\hat{\Pi}$ that couple both to the real part and the phase
of the field. By assuming we are dealing with a CFT, we can use the conformal
map that maps the profile of local operators to two points correlation
functions on the full plane (see e.g. car (1988)).

At first we analyze the behavior of $\braket{\hat{L}^{2}_{l}}$ as function of
the chord coordinate $W$. The numerical results show that
$\braket{\hat{L}^{2}_{l}}$ diverges linearly as a function of $W$, unveiling
that $\hat{L}^{2}$ behaves as a free-bosonic field. Furthermore, the slope of
such linear scaling is found to be $1.20(4)/12$, matching that of the
entanglement ground state energy. Turning to analyzing the profile of
$\braket{\hat{\Pi}^{\dagger}_{l}\hat{\Pi}_{l}}$ as a function of $W$ we find
that

$\braket{\hat{\Pi}^{\dagger}_{l}\hat{\Pi}_{l}}\simeq a+b(\exp(W))^{-\Delta},$
(5)

where $a$ and $b$ encode the overlap of the above expectation value with the
identity operator and one of the primaries. The numerical data (Fig. 3(b))
suggest that $\Delta\simeq 0.5$, the conjugate operator to the one that would
match to the derivative of the electric field.

Unfortunately, we do not find any operator that couples to the primary of the
Ising part of the CFT. Summarizing, our data seem to confirm that we have one
part of the system that behaves as free boson and suggest that
$\partial_{x}\hat{L}^{2}$ should have a large overlap with the primary
operator (the derivative of the free bosonic field), while
$\hat{\Pi}^{\dagger}\hat{\Pi}$ should have a strong overlap with the conjugate
primary operator.

Now moving away from the critical point, we can use the standard scaling
hypothesis to extract the exponent $\nu$ from the collapse of the fitted
central charge as

$\displaystyle
c_{\text{fit}}(L)=f\left((\mu^{2}/\lambda-\mu_{c}^{2}/\lambda_{c})L^{1/\nu}\right),$
(6)

where $f(.)$ is a continuous function and $\nu$ is the corresponding critical
exponent. Performing the data collapse according to (6) in the neighborhood of
the critical point $\mu_{c}^{2}/\lambda_{c}$ (see Fig. 4) we find the critical
exponent to be $\nu=1/2\pm 0.02$ that matches the value observed in the
transition from polarized to critical phase in the XX model in a magnetic
field Latorre  _et al._ (2004); Campostrini and Vicari (2010); Dutta  _et al._
(2015b). The same transition can be understood in terms of free bosons that
pass from their Fock vacuum to the superfluid regime as the chemical potential
exceeds the width of the first band. In our case, the strange thing is that
there is no superfluid regime, but just a single critical point where the
gauge-boson condense, while away from the critical point our system passes
from vacuum to a Mott insulator phase. Now using the standard scaling
hypothesis, once we have figured out that $\nu=1/2$ we can deduce that
$\braket{\hat{\Pi}^{\dagger}\hat{\Pi}}\simeq(\mu^{2}/\lambda-\mu_{c}^{2}/\lambda_{c})$,
meaning that $\beta=1$.

Figure 4:  The collapse of $c_{\text{fit}}$ according to the scaling
hypothesis (6) in the neighborhood of the critical point
$\mu_{c}^{2}/\lambda_{c}=0.447/0.0565$ for fixed $\mu^{2}/q^{2}=0.447$ (left)
and for fixed $\lambda/q^{2}=0.0565$ (right). Here, we vary (a)
$\lambda/q^{2}$ in the range $[0.056,0.058]$ and (b) $\mu^{2}/q^{2}$ in the
range $[0.435,0.45]$ for the data collapses. In both the cases, the critical
exponent is found to be $\nu=0.5\pm 0.02$ from data collapses.

It is worth pointing out that $\nu=1/2$, seems inconsistent with a CFT, where
we would expect, that $d-1/\nu=\Delta$ with $\Delta$ the thermal critical
exponent and $d$ being 2 for the 1+1D quantum system, while we find
$d-1/\nu=0$. However, all our results so far have been obtained by assuming a
full conformal invariance in mapping the correlation function of our finite
system to the ones of an infinite plane by means of a conformal
transformations.

The appearance of $\nu=1/2$, together with the failure to identify a local
operator that couples to the primary field of the Ising part of the CFT,
contrasts with the factorization on the critical point. However, by repeating
a similar analysis in a $\mathbb{Z}_{3}$ gauge theory coupled to bosonic
matter we find a $c\simeq 0.8+0.5$ Chanda  _et al._ . As a result we still
believe that the factorization hypothesis is correct, but it requires a
further analysis to be appropriately confirmed. In particular, it would be
interesting to analyze the system under periodic boundary condition, which,
however, is impractical at current computational capabilities using MPS ansatz
but may become a possibility using next-generation tensor network algorithms.

### Discussion and Conclusions.

We have analyzed the phase diagram of AHM2 on a discrete lattice in 1+1D. We
have found two distinct regions, the confined and the Higgs regions that are
separated by line of FOQPT that terminates at a SOQPT. Beyond the SOQPT the
two regions are connected by a smooth crossover. The presence of a SOQPT
allows one to construct an unorthodox continuum limit of the theory that
should be described by free fermions and free bosons that do not interact.

This would result in a CFT with central charge $c=3/2$, compatible with our
numerical result and would have a compelling interpretation in terms of the
standard Higgs mechanism – the real part of the complex field undergoes an
Ising transition (the $c=1/2$ part), while the phase of it provides the
transverse degree of freedom to the photon that becomes dynamical and massless
(the $c=1$ part).

However, further numerical analyses unveil surprizing pieces of the puzzle
that do not fit our interpretation.We did not find a local operator that
couples to the $c=1/2$ part of the CFT. The scaling of the entanglement ground
state should follow a similar law to the one of the entanglement entropy. The
numerical value of the central charge that we extract from it is $c=1.20(1)$.
We also obtain $\nu=1/2$ analyzing the collapse of the data for the
entanglement entropy close to the critical point, in contrast to the expected
$\nu=1$.

Are we actually observing a Lorentz invariant critical point where the Higgs
and photon mode factorize? We believe this is the case as also supported by
the presence of a linear dispersion relation witnessed by the non-zero “sound
velocity” extracted from a finite-size scaling analysis of the ground state
energy sup . Still our study leaves some questions unanswered. We strongly
believe that our paper will open the debate, and that together with the
broader scientific community we will soon have a final picture of the
mechanism behind this newly observed critical point.

###### Acknowledgements.

We are grateful to Marcello Dalmonte for the valuable suggestion regarding the
analysis of the sound velocity. T.C. thanks Subhroneel Chakrabarti for useful
discussions. L.T. would like to acknowledge the discussions with F. Gliozzi,
B. Fiol, and E. Vicari on the topics presented. The numerical computations
have been possible thanks to PL-Grid Infrastructure. The works of T.C. and J.Z
have been realized within the QuantERA grant QTFLAG, financed by National
Science Centre (Poland) via grant 2017/25/Z/ST2/03029. M.L. acknowledges
support from ERC AdG NOQIA, State Research Agency AEI (“Severo Ochoa” Center
of Excellence CEX2019-000910-S, Plan National FIDEUA
PID2019-106901GB-I00/10.13039 / 501100011033, FPI, QUANTERA MAQS
PCI2019-111828-2 / 10.13039/501100011033), Fundació Privada Cellex, Fundació
Mir-Puig, Generalitat de Catalunya (AGAUR Grant No. 2017 SGR 1341, CERCA
program, QuantumCAT U16-011424, co-funded by ERDF Operational Program of
Catalonia 2014-2020), EU Horizon 2020 FET-OPEN OPTOLogic (Grant No. 899794),
and the National Science Centre, Poland (Symfonia Grant No.
2016/20/W/ST4/00314), Marie Skłodowska-Curie grant STRETCH No. 101029393, “La
Caixa” Junior Leaders fellowships (ID100010434), and EU Horizon 2020 under
Marie Skłodowska-Curie grant agreement No 847648 (LCF/BQ/PI19/11690013,
LCF/BQ/PI20/11760031, LCF/BQ/PR20/11770012). L.T. acknowledges support from
the Ramón y Cajal program RYC-2016-20594, the “Plan Nacional Generación de
Conocimiento” PGC2018-095862-B-C22, the State Agency for Research of the
Spanish Ministry of Science and Innovation through the “Unit of Excellence
María de Maeztu 2020-2023” award to the Institute of Cosmos Sciences
(CEX2019-000918-M), and the European Union Regional Development Fund within
the ERDF Operational Program of Catalunya, Spain (project QUASICAT/QuantumCat,
ref. 001-P-001644).

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Supplementary Material:
Phase diagram of 1+1D Abelian-Higgs model and its critical point

##  .1 From the continuum to the lattice

The Lagrangian density of the Abelian-Higgs model is given by Peskin and
Schroeder (1995); Tong

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle-\left[D_{\mu}\phi\right]^{*}D^{\mu}\phi-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-V(\phi),$
(7)

with

$\displaystyle V(\phi)=-\mu^{2}|\phi|^{2}+\frac{\lambda}{2}|\phi|^{4}.$ (8)

Here $\phi$ is the complex scalar field, $D_{\mu}=(\partial_{\mu}+iqA_{\mu})$
is the covariant derivative with $q$ and $A_{\mu}$ being the gauge coupling
and the electromagnetic vector potential respectively, and
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the
electromagnetic field tensor. Here, we use the metric convection $(-1,1,1,1)$
or $(-1,1)$ (in 1+1 dimension).

For $\mu^{2}<0$, the system describes the bosonic Schwinger model (BSM) Chanda
_et al._ (2020) with added $|\phi|^{4}$-interaction, while $\mu^{2}>0$
describes Abelian-Higgs model (AHM) where the potential attains minimum
($V_{\min}=-\frac{\mu^{4}}{2\lambda}$) at non-zero value of the field-strength
$|\phi_{0}|=\sqrt{\mu^{2}/\lambda}$, leading to the spontaneous symmetry
breaking at semiclassical level.

After fixing the temporal gauge $A_{t}(x,t)=0$, we get the quantum 1+1D
Hamiltonian (AHM2) in the continuum as

$\displaystyle\hat{H}$ $\displaystyle=$ $\displaystyle\int
dx\bigg{[}\hat{\Pi}^{\dagger}(x)\hat{\Pi}(x)-\mu^{2}\hat{\phi}^{\dagger}(x)\hat{\phi}(x)+\frac{\lambda}{2}(\hat{\phi}^{\dagger}(x))^{2}(\hat{\phi}(x))^{2}+$
$\displaystyle+$
$\displaystyle\frac{1}{2}\hat{E}_{x}^{2}(x)+\left(\partial_{x}-iq\hat{A}_{x}(x)\right)\hat{\phi}^{\dagger}(x)\left(\partial_{x}+iq\hat{A}_{x}(x)\right)\hat{\phi}(x)\bigg{]},$

where $\hat{E}_{x}(x)$, $\hat{\Pi}(x)$, and $\hat{\Pi}^{\dagger}(x)$ are the
canonical conjugate operators corresponding to $\hat{A}_{x}(x)$,
$\hat{\phi}(x)$, and $\hat{\phi}^{\dagger}(x)$ respectively, satisfying the
canonical commutation relations:

$\displaystyle[\hat{A}_{x}(x_{1}),\hat{E}_{x}(x_{2})]$
$\displaystyle=[\hat{\phi}(x_{1}),\hat{\Pi}(x_{2})]$
$\displaystyle=[\hat{\phi}^{\dagger}(x_{1}),\hat{\Pi}^{\dagger}(x_{2})]=i\delta(x_{1}-x_{2}).$
(10)

Note: The quantization of the $|\phi|^{4}$ term may be done in various
manners. Here we do not enforce any normal ordering but rather take
$|\phi|^{4}\xrightarrow[]{\text{quantization}}(\phi^{\dagger})^{2}\phi^{2}$.

Following Chanda  _et al._ (2020), we can straightforwardly discretize the
above Hamiltonian on a 1D spatial lattice with the lattice spacing $a$, and we
ultimately arrive at the Hamiltonian (1) in the main text.

Figure 5: The study of the first order quantum phase transition at fixed
$\lambda/q^{2}=0.045$ and varying $\mu^{2}/q^{2}$ for different system-sizes
$L\in[20,120]$. (a)-(c) respectively show the variations of
$\mathcal{O}_{\text{tunn}}$, the energy density and its derivative with
respect to the system parameter $\mu^{2}/q^{2}$. (d) The entanglement scaling
with respect to the cord length $W=\ln\left(\frac{2L}{\pi}\sin(\pi
l/L)\right)$ for $L=120$ at the first order transition that unveils the
expected area-law scaling of the entropy.

##  .2 The first order quantum phase transition

As reported in the main text, below the critical interaction strength
$\lambda_{c}/q^{2}=0.0565$ the confined and the Higgs regions are separated by
a first order quantum phase transition (FOQPT) line. To clearly demonstrate
the existence and features of the FOQPT, in Figs. 5 we present different
observables across the transition as a function of $\mu^{2}/q^{2}$ for fixed
$\lambda/q^{2}<\lambda_{c}/q^{2}$. Specifically, we depict the behaviors of
$\mathcal{O}_{\text{tunn}}=\frac{1}{2L}\sum_{j}\braket{\hat{\phi}^{\dagger}_{j+1}\hat{U}_{j}\hat{\phi}_{j}+h.c.}$,
the energy density and its derivative with respect to the system parameter
$\mu^{2}/q^{2}$ respectively in Figs. 5(a)-(c) for different system-sizes
$L\in[20,120]$. The average tunneling amplitude $\mathcal{O}_{\text{tunn}}$
clearly shows discontinuous jumps as we vary $\mu^{2}/q^{2}$ across the phase
transition for sufficiently large system-sizes. On other hand, the ground
state energy shows non-analytic kinks in its profile at the transition point,
such that its derivative (with respect to the parameter $\mu^{2}/q^{2}$)
manifests a discontinuity. To observe the discontinuous jumps or kinks in the
profile of $\mathcal{O}_{\text{tunn}}$ or of the energy, we need sufficiently
large system-size as FOQPTs are associated with large (but finite) correlation
lengths, as a result smaller system sizes cannot properly resolve the FOQPT.
In Fig. 5(d), we also show the scaling of the entanglement entropy at the
FOQPT that features the expected area-law behavior for bipartitions larger
than the correlation length.

##  .3 Entanglement scaling in the gapped phases

Figure 6: The profile of the entanglement entropy with respect to the cord
length $W$ at (a) the confined region and (b) the Higgs region.

As mentioned in the main text, apart from the critical point, the Hamiltonian
in every point of the phase diagram is gapped, therefore the entanglement
entropy follows asymptotically, for large enough blocks, the standard area-law
scaling. To show this, we plot the the entanglement entropy with respect to
the cord length $W$ in the confined and the Higgs regions in Fig. 6. In both
cases, the entanglement entropy becomes flat with increasing values of $W$,
showing that for large enough blocks it saturates to a value that does not
depend on the block size (area-law).

Figure 7:  Finite-size scaling of the ground state energy $E_{0}$ at the
critical point according to Eq. (11) that gives the sound velocity
$v_{s}\simeq 4$.

##  .4 Evidence for Lorentz invariance of the critical point

In the main text, we have analyzed the critical point
$(\mu_{c}^{2}/q^{2}=0.447,\lambda_{c}/q^{2}=0.0565)$ by means of predictions
from conformal field theory (CFT), assuming Lorentz invariance. In order to
verify this assumption, we calculate the “sound velocity” at the critical
point from the scaling of ground state energy. For a Lorentz invariant system
with open boundary condition (OBC) having a linear dispersion at low energies,
the ground state energy $E_{0}$ scales with the system size $L$ as Blöte  _et
al._ (1986); Affleck (1986); Affleck  _et al._ (1989)

$E_{0}(L)=\epsilon_{0}^{\infty}L+\epsilon_{1}^{\infty}-\frac{\pi
cv_{s}}{24L},$ (11)

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. For a Lorentz invariant system $v_{s}$ must be non-zero, while
it vanishes for Lorentz non-invariant critical points with a quadratic
dispersion.

To estimate $v_{s}$, we perform the finite-size scaling of the ground state
energy according to Eq. (11) at the critical point (for similar analysis, see
Hallberg  _et al._ (1996); Xavier (2010); Dalmonte  _et al._ (2012); Chepiga
and Mila (2017)). Such a finite-size scaling (Fig. 7) yields $cv_{s}=6.02\pm
0.08$ ($v_{s}\simeq 4$ for $c=1.5$), confirming the Lorentz invariance of the
critical point analyzed in the main text.

##  .5 Details about numerical simulations

Figure 8: The convergence of (a) the energy density ($\Delta E_{0}/L$) and (b)
the half-chain entanglement entropy ($\Delta\mathcal{S}_{L/2}$) 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 consider three points in the phase diagram, namely (1) a point in
the confined region ($\mu^{2}/q^{2}=0.2$), (2) the critical point
($\mu^{2}/q^{2}=0.447$), and (3) a point in the Higgs region
($\mu^{2}/q^{2}=0.6$) for a system of size $L=120$. In (b) we show the
convergence of the entropic quantities as a function of the bond dimension at
the critical point.

The results reported in the article have been obtained using the density
matrix renormalization group (DMRG) method White (1992, 1993, 2005);
Schollwöck (2005, 2011); Orús (2014) with the matrix product state (MPS)
ansatz Schollwöck (2011); Orús (2014). Specifically, we use a strictly single-
site variant of DMRG (DMRG3S) with the subspace expansion Hubig  _et al._
(2015). In our calculations the gauge fields are integrated out, and thus we
do not need to use gauge-invariant tensor network Tagliacozzo  _et al._
(2014); Buyens  _et al._ (2014); Silvi  _et al._ (2014); Kull  _et al._ (2017)
but we can use standard globally symmetric $\mathbb{U}(1)$ MPS Singh  _et al._
(2010, 2011). This residual global symmetry corresponds to the conservation of
the total dynamical charge $\sum_{j}\hat{Q}_{j}$.

The results from DMRG simulations, that we report, have been performed with
maximum MPS bond dimension of $\chi=600$. To confirm the convergence of the
DMRG sweeps, we continue the DMRG3S iterations until the energy difference in
subsequent sweeps falls below $10^{-13}$. On the other hand, Fig. 8 shows the
convergence of different quantities with respect to the bond dimension
$\chi\leq 600$ for systems of sizes $L\leq 120$. For example, the energy
density converges close to the machine precision within $\chi\leq 500$ in the
gapped regions – confined and Higgs. 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. 8(b), the precision we attain at the
critical point for $\chi=500,600$ is sufficient to perform the precise scaling
analysis that we report in the main text.

Figure 9: (a) The convergence with respect to the maximum boson occupancy
$n_{0}$ in three different points in the phase diagram, namely (1) a point in
the confined region ($\mu^{2}/q^{2}=0.2$), (2) the critical point
($\mu^{2}/q^{2}=0.447$), and (3) a point in the Higgs region
($\mu^{2}/q^{2}=0.6$) for a system of size $L=120$. 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 entanglement
entropy $\mathcal{S}$ 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 $n_{0}$. From the results presented
it is clear that values of $n_{0}<10$ are not sufficient to capture the proper
entropy scaling at the critical point.

Since, our system Hamiltonian does not conserve particle numbers for the
individual bosonic species ‘$a$’ and ‘$b$’, but conserves their number
difference, we verify the convergence with respect to the maximum bosonic
occupancy $n_{0}$ for both the species. Specifically, we vary the bosonic
cutoff $n_{0}$ in the range $[4,15]$, and check for convergence of different
observables (see Fig. 9). Clearly, as seen in Fig. 9(a), even a small
$n_{0}=4$ is sufficient to faithfully capture the confined phase. However,
this is not the case at the critical point or in the Higgs region. This is due
to the fact that semiclassically the field $\phi$ attains non-zero expectation
values in the Higgs region. It is to be noted that in our calculations
$\braket{\hat{\phi}}$ is trivially zero since it violates the global symmetry,
while $\braket{\hat{\phi}^{\dagger}\hat{\phi}}$, and thereby individual
$\braket{\hat{n}_{a}}$ and $\braket{\hat{n}_{b}}$, attain large expectation
values in the Higgs region. Moreover, Fig. 9(b) shows that the smaller values
of the bosonic cutoff $n_{0}$ are insufficient to capture the critical entropy
scaling with respect to the cord length $W$ at the critical point. On the
other hand, 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. Therefore, Fig. 9 clearly demonstrates that the results obtained
with the bosonic cutoff $n_{0}=10$, provide accurate enough results, and this
is the reason we have used this cutoff in our analysis.

We do not exceed $n_{0}=10$ since the local Hilbert space dimension is
$d=(n_{0}+1)^{2}=121$, which is very large, and the complexity of the DMRG3S
algorithm scales as $\sim Ld\chi^{3}$. For this reasons we are forced to
consider moderate bond dimensions ($\chi\leq 600$) that also limit the range
of system sizes ($L\leq 120$) that we can analyze accurately.

Figure 10: The scaling of the entanglement ground state energy
$\varepsilon_{0}$ with respect to the cord length $W$ for system-sizes (a)
$L\in[40,160]$ and (b) $L\in[100,160]$.

On the other hand, Fig. 8(b) shows that the entanglement ground state energy
$\varepsilon_{0}$ converges faster with respect to the bond dimension than the
convergence in the entanglement entropy $\mathcal{S}$ (that requires the
convergence of the full entanglement spectrum, not only of its ground state).
Therefore, for this specific quantity, and just for a comparison, we extend
our finite size scaling analysis to the range $\varepsilon_{0}$ with
$L\in[40,160]$ keeping the bond dimension fixed at $\chi=600$. In Fig. 10 we
show the same scaling first in the whole range of $L\in[40,160]$ and then for
longer chains $L\in[100,160]$. One could indeed think that the discrepancy
between the central charge extracted from the entanglement entropy and the one
extracted from the scaling of the first eigenvalue is ultimately a finite-size
effect and that this would disappear in the thermodynamics limit.

Unfortunately, in both the cases, the slopes are pretty robust to the choice
of system sizes that we use in our fit, and remain fixed to $1.2/12$. Even
using only the larger systems does not seem to induce a systematic deviation
towards the $1.5/12$ extracted from the scaling of the entanglement entropy
reported in the main text. As a result, the level of accuracy, that we are
able to achieve with our computational resources, is extremely good and under
control. These extra analysis seems to confirm the discrepancy between the
value of the central charge extracted from the scaling of the entanglement
ground state with respect to the one extracted from the scaling of the
entanglement entropy.