# Localization of Dirac modes in the $\mathrm{SU}(2)$-Higgs model at finite
temperature

György Baranka<EMAIL_ADDRESS>ELTE Eötvös Loránd University,
Institute for Theoretical Physics, Pázmány Péter sétány 1/A, H-1117, Budapest,
Hungary  Matteo Giordano<EMAIL_ADDRESS>ELTE Eötvös Loránd
University, Institute for Theoretical Physics, Pázmány Péter sétány 1/A,
H-1117, Budapest, Hungary

###### Abstract

We investigate the connection between localization of low-lying Dirac modes
and Polyakov-loop ordering in the lattice $\mathrm{SU}(2)$-Higgs model at
finite temperature, probed with static external staggered fermions. After
mapping out the phase diagram of the model at a fixed temporal extension in
lattice units, we study the localization properties of the low-lying modes of
the staggered Dirac operator, how these properties change across the various
transitions, and how these modes correlate with the gauge and Higgs fields. We
find localized low modes in the deconfined and in the Higgs phase, where the
Polyakov loop is strongly ordered, but in both cases they disappear as one
crosses over to the confined phase. Our findings confirm the general
expectations of the “sea/islands” picture, and the more detailed expectations
of its refined version concerning the favorable locations of localized modes,
also in the presence of dynamical scalar matter.

##  I Introduction

Although it is well established that the finite-temperature QCD transition is
an analytic crossover [1, 2], the microscopic mechanism that drives it is
still being actively studied. The main goals of this line of research are a
better understanding of the connection between deconfinement and restoration
of chiral symmetry, both taking place in the crossover region; and of the fate
of the anomalous $\mathrm{U}(1)_{A}$ symmetry, especially in the chiral limit.
In this context, the fact that also the nature of the low-lying Dirac
eigenmodes changes radically in the crossover region has aroused some
interest. While delocalized in the low-temperature, confined and chirally
broken phase, these modes become in fact spatially localized in the high-
temperature, deconfined and (approximately) chirally restored phase, up to a
critical point in the spectrum known as “mobility edge” [3, 4, 5, 6, 7, 8, 9]
(see Ref. [10] for a recent review). As the strength of chiral symmetry
breaking is controlled by the density of low-lying Dirac modes [11], while the
change in their localization properties is mainly due to the ordering of the
Polyakov loop in the high-temperature phase [12, 13, 14, 15, 10, 16], low-
lying eigenmodes could provide the link between deconfinement and restoration
of chiral symmetry.

The connection between low-mode localization and Polyakov-loop ordering is
qualitatively explained by the “sea/islands” picture, initially proposed in
Ref. [12], and further developed in Refs. [13, 14, 15, 10, 16]. In the
deconfined phase, typical gauge configurations display a “sea” of ordered
Polyakov loops, which on the one hand provides a spatially (approximately)
uniform region where Dirac modes can easily delocalize, and on the other hand
opens a (pseudo)gap in the near-zero spectrum. Polyakov-loop fluctuations away
from order, and more generally gauge-field fluctuations with reduced
correlation in the temporal direction, allow for eigenvalues below the gap;
since in the deconfined phase these fluctuations typically form well separated
“islands”, they tend to “trap” the low eigenmodes, causing their localization.

The sea/islands mechanism is quite general, and requires essentially only the
ordering of the Polyakov loop for low-mode localization to take place 111A
notable exception is when the (untraced) Polyakov loops order along minus the
identity, in which case low modes remain delocalized also in the deconfined
phase [27].. This leads one to expect localization of low Dirac modes to be a
generic phenomenon in the deconfined phase of a gauge theory, an expectation
so far fully confirmed by numerical results, both for pure gauge theories [18,
19, 20, 21, 22, 12, 23, 24, 25, 26, 27, 16] and in the presence of dynamical
fermionic matter [28, 29]. An interesting aspect of the
deconfinement/localization relation is that while the thermal transition can
be a smooth, analytic crossover, the appearance of a mobility edge can only be
sudden, taking place at a well-defined temperature. If the connection between
deconfinement and localization is indeed general, one can then associate the
(possibly smooth) thermal transition with a (definitely sharp) “geometric”
transition (a similar suggestion, although in connection with deconfinement
and center vortices, was made in Ref. [30], from which we borrowed the
terminology). This point of view is supported by the fact that the geometric
and the thermodynamic transition coincide when the latter is a genuine phase
transition [23, 24, 25, 26, 27, 16, 28, 29].

As a further test of the universality of the sea/islands mechanism, one can
investigate whether a change in the localization properties of low modes takes
place across other thermal transitions where the Polyakov loop gets ordered,
besides the usual deconfinement transition. As an example, Ref. [26] studied
low-mode localization across the “reconfinement” transition in trace-deformed
$\mathrm{SU}(3)$ gauge theory at finite temperature [31, 32, 33, 34, 35].
While localized modes are present in the deconfined phase also at nonzero
deformation parameter, where the Polyakov-loop expectation value is different
from zero, they disappear as the system reconfines and the Polyakov-loop
expectation value vanishes.

Yet another test of universality consists in changing the type of dynamical
matter from fermionic to scalar. As long as a phase with ordered Polyakov
loops exists, this should not affect the expectations of the sea/islands
picture, and localized modes should appear in the spectrum of the Dirac
operator in that phase. In this context, the Dirac operator can be seen simply
as a mathematical probe of certain properties of the gauge fields or, more
physically, as a probe of how these fields couple to external, static (i.e.,
infinitely heavy) fermion fields.

A model allowing one to carry out both these tests at once is the lattice
fixed-length $\mathrm{SU}(2)$-Higgs model [36]. At zero temperature the phase
diagram of this model has been studied in depth both with analytical [36] and
numerical [37, 38, 39, 40, 41, 42] methods. This model has two parameters,
namely the (inverse) gauge coupling $\beta$ and the Higgs-gauge coupling
$\kappa$, and it displays two lines of transitions in the $(\beta,\kappa)$
plane as follows [42]:

  * •

a line of crossovers at $\beta\approx\beta_{\mathrm{bulk}}$, starting from the
bulk transition (crossover) of the pure gauge $\mathrm{SU}(2)$ theory [43] at
$(\beta,\kappa)=(\beta_{\mathrm{bulk}},0)$, and ending at some point
$(\beta_{e},\kappa_{e})$;

  * •

a line of crossovers coming down from large $\kappa$ at small $\beta$, meeting
the first line at $(\beta_{e},\kappa_{e})$, turning into a line of first-order
transitions at $(\beta_{f},\kappa_{f})$, and tending to $\kappa\approx 0.6$ as
$\beta\to\infty$.

These transition lines separate three phases of the system: a confined phase
at low $\beta$ and low $\kappa$; a deconfined phase at high $\beta$ and low
$\kappa$; and a Higgs phase at high $\kappa$. A similar phase diagram was
found at finite temperature, although the transition lines were all identified
as crossovers in that case [42]. The absence of a sharp transition between the
confined and the Higgs phase at any $\kappa$ at sufficiently low $\beta$ was
proved in Ref. [36], where it was also shown that in this region all local
correlation functions, and so the spectrum of the theory, depend analytically
on the couplings.

While fermions are absent in the $\mathrm{SU}(2)$-Higgs model, one can still
probe this system using static external fermions coupled to the
$\mathrm{SU}(2)$ gauge field, as pointed out above. One can then study how the
corresponding Dirac spectrum behaves, and check what happens to the
localization properties of its low modes across the various transitions, in
particular as one crosses over to the Higgs phase starting from either the
confined or the deconfined phase. Since eigenvalues and eigenvectors of the
Dirac operator are nonlocal functions of the gauge fields, they can display
non-analytic behavior even in the strip of the $(\beta,\kappa)$ plane where
all local correlators are analytic functions of the couplings, and so they
could allow one to sharply distinguish the confined and the Higgs phase. (A
different approach to this issue, based on the analogies between gauge-Higgs
theories and spin glasses, is discussed in the review Ref. [44] and references
therein.)

In this paper we study the spectrum and the eigenvectors of the staggered
lattice Dirac operator in the $\mathrm{SU}(2)$-Higgs model at finite
temperature. After briefly describing the model, in section II we introduce
the tools we use to investigate the localization properties of staggered
eigenmodes. In section III we map out the phase diagram of the model at finite
temperature, working at fixed temporal extension in lattice units. In section
IV we analyze the staggered eigenmodes, focussing in particular on how their
localization properties change across the transitions between the confined,
deconfined, and Higgs phases. We then study in detail the correlation between
eigenmodes and the gauge and Higgs fields, to identify the field fluctuations
mostly responsible for localization. Finally, in section V we draw our
conclusions and show some prospects for the future.

##  II $\mathrm{SU}(2)$-Higgs model and localization

In this section we describe the fixed-length $\mathrm{SU}(2)$-Higgs model, and
discuss how to characterize the localization properties of Dirac modes, and
how these correlate with the gauge and Higgs fields.

###  II.1 $\mathrm{SU}(2)$-Higgs model on the lattice

We study the lattice $\mathrm{SU}(2)$-Higgs model in 3+1 dimensions, defined
by the action

$S=-\frac{\beta}{2}\sum_{n}\sum_{1\leq\mu<\nu\leq 4}{\rm
tr}\,U_{\mu\nu}(n)-\frac{\kappa}{2}\sum_{n}\sum_{1\leq\mu\leq 4}{\rm
tr}\,G_{\mu}(n)\,,$ (1)

where we omitted an irrelevant additive constant. Here $n=(\vec{x},t)$,
$n_{\mu}=0,\ldots,N_{\mu}-1$, are the sites of a hypercubic $N_{s}^{3}\times
N_{t}$ lattice, i.e., $N_{1,2,3}=N_{s}$ and $N_{4}=N_{t}$, where
$\mu=1,\ldots,4$ denotes the lattice directions and $\hat{\mu}$ the
corresponding unit vectors. The dynamical variables are the $\mathrm{SU}(2)$
matrices $U_{\mu}(n)$ and $\phi(n)$, representing respectively the gauge
variables associated with the link connecting $n$ and $n+\hat{\mu}$, and the
unit-length Higgs field doublet (recast as a unitary matrix) associated with
site $n$, and

$\displaystyle U_{\mu\nu}(n)$
$\displaystyle=U_{\mu}(n)U_{\nu}(n+\hat{\mu})U_{\mu}(n+\hat{\nu})^{\dagger}U_{\nu}(n)^{\dagger}\,,$
(2) $\displaystyle G_{\mu}(n)$
$\displaystyle=\phi(n)^{\dagger}U_{\mu}(n)\phi(n+\hat{\mu})\,,$

are the plaquette variables associated with the elementary lattice squares,
and the nontrivial part of the discretized covariant derivative of the Higgs
field, which we will refer to as the Higgs-gauge field coupling term. Periodic
boundary conditions are imposed on $U_{\mu}(n)$ and $\phi(n)$ in all
directions. In what follows we will also make use of the Polyakov loop winding
around the temporal direction,

$P(\vec{x})={\rm tr}\,\prod_{t=0}^{N_{t}-1}U_{4}(\vec{x},t)\,.$ (3)

Expectation values are defined as

$\displaystyle\langle O\rangle$ $\displaystyle=\frac{1}{Z}\int DU\int
D\phi\,e^{-S(U,\phi)}O(U,\phi)\,,$ (4) $\displaystyle Z$ $\displaystyle=\int
DU\int D\phi\,e^{-S(U,\phi)}\,,$

where $DU$ and $D\phi$ denote the products of the $\mathrm{SU}(2)$ Haar
measures associated with $U_{\mu}(n)$ and $\phi(n)$.

We study this model at finite temperature $T=1/(aN_{t})$, where $a$ is the
lattice spacing, which can be set by suitably tuning the parameters of the
model, namely the inverse gauge coupling $\beta$ and the Higgs-gauge field
coupling $\kappa$. However, since we are not interested here in taking the
continuum limit, we treat the model simply as a two-parameter anisotropic
statistical mechanics system, keeping $N_{t}$ fixed as we take the
thermodynamic limit $N_{s}\to\infty$, and as we change $\beta$ and $\kappa$
freely. To study the phase diagram in the $(\beta,\kappa)$ plane we use the
average plaquette, Polyakov loop, and Higgs-gauge field coupling term,

$\displaystyle\langle U\rangle$
$\displaystyle=\frac{1}{N_{t}V}\sum_{n}\left\langle U(n)\right\rangle\,,$
$\displaystyle\langle P\rangle$
$\displaystyle=\frac{1}{V}\sum_{\vec{x}}\left\langle
P(\vec{x})\right\rangle\,,$ (5) $\displaystyle\langle G\rangle$
$\displaystyle=\frac{1}{N_{t}V}\sum_{n}\left\langle G(n)\right\rangle\,,$

where $V=N_{s}^{3}$ is the lattice volume, and the corresponding
susceptibilities,

$\displaystyle\chi_{U}$
$\displaystyle=\frac{1}{N_{t}V}\left(\left\langle\left({\textstyle\sum_{n}}U(n)\right)^{2}\right\rangle-\left\langle{\textstyle\sum_{n}}U(n)\rule{0.0pt}{12.0pt}\right\rangle^{2}\right)\,,$
(6) $\displaystyle\chi_{P}$
$\displaystyle=\frac{1}{V}\left(\left\langle\left({\textstyle\sum_{\vec{x}}}\,{\rm
tr}\,P(\vec{x})\right)^{2}\right\rangle-\left\langle{\textstyle\sum_{\vec{x}}}\,{\rm
tr}\,P(\vec{x})\rule{0.0pt}{12.0pt}\right\rangle^{2}\right)\,,$
$\displaystyle\chi_{G}$
$\displaystyle=\frac{1}{N_{t}V}\left(\left\langle\left({\textstyle\sum_{n}}G(n)\right)^{2}\right\rangle-\left\langle{\textstyle\sum_{n}}{\rm
tr}\,G(n)\rule{0.0pt}{12.0pt}\right\rangle^{2}\right)\,.$

In Eqs. (5) and (6) we denoted with $U(n)$ and $G(n)$ the average plaquette
and gauge-Higgs coupling term touching a lattice site $n$,

$\displaystyle U(n)$ $\displaystyle=\frac{1}{24}\sum_{1\leq\mu<\nu\leq 4}{\rm
tr}\,(U_{\mu\nu}(n)+U_{\mu\nu}(n-\hat{\mu})$ (7)
$\displaystyle\phantom{\frac{1}{24}\sum_{1\leq\mu<\nu\leq
4}}+U_{\mu\nu}(n-\hat{\nu})+U_{\mu\nu}(n-\hat{\mu}-\hat{\nu}))\,,$
$\displaystyle G(n)$ $\displaystyle=\frac{1}{8}\sum_{1\leq\mu\leq 4}{\rm
tr}\,(G_{\mu}(n)+G_{\mu}(n-\hat{\mu}))\,.$

###  II.2 Localization of staggered eigenmodes

We are interested in the spectrum of the staggered Dirac operator in the
background of the $\mathrm{SU}(2)$ gauge fields for fermions in the
fundamental representation,

$D^{\mathrm{stag}}=\frac{1}{2}\sum_{\mu}\eta_{\mu}(U_{\mu}{\rm T}_{\mu}-{\rm
T}_{\mu}^{\dagger}U_{\mu}^{\dagger})\,,$ (8)

where $\eta_{\mu}$ are the usual staggered phases and ${\rm T}_{\mu}$ are the
translation operators with periodic (resp. antiperiodic) boundary conditions
in space (resp. time), i.e.,

$\displaystyle\eta_{\mu}(n)$
$\displaystyle=(-1)^{\sum_{\alpha<\mu}n_{\alpha}}\,,$ (9) $\displaystyle({\rm
T}_{\mu})_{n,n^{\prime}}$
$\displaystyle=b_{\mu}(n_{\mu})\delta_{n_{\mu}+1,n_{\mu}^{\prime}}\prod_{\alpha\neq\mu}\delta_{n_{\alpha},n_{\alpha}^{\prime}}\,,$

with $n_{\mu}=N_{\mu}$ identified with $n_{\mu}=0$, and $b_{\mu}(n_{\mu})=1$,
$\forall\mu,n_{\mu}$, except for $b_{4}(N_{t}-1)=-1$. Since the staggered
operator is anti-Hermitian and anticommutes with
$\varepsilon(n)=(-1)^{\sum_{\alpha}n_{\alpha}}$, its spectrum is purely
imaginary and symmetric about the origin. We write

$D^{\mathrm{stag}}\psi_{l}(n)=i\lambda_{l}\psi_{l}(n)\,,\qquad\lambda_{l}\in\mathbb{R}\,,$
(10)

with eigenvectors $\psi_{l}(n)$ carrying an internal “color” index,
$\psi_{l,c}(n)$, $c=1,2$, that has been suppressed for simplicity, and focus
on $\lambda_{l}\geq 0$ only. Notice that since
$\sigma_{2}U_{\mu}(n)\sigma_{2}=U_{\mu}(n)^{*}$, $D^{\mathrm{stag}}$ commutes
with the antiunitary “time-reversal” operator $T=\sigma_{2}K$, where $K$
denotes complex conjugation. Since $T^{2}=-\mathbf{1}$, $D^{\mathrm{stag}}$
displays in this case doubly degenerate eigenvalues, and belongs to the
symplectic class in the symmetry classification of random matrices [45, 46].
In the following it is understood that we work with the reduced spectrum,
including only one eigenvalue from each degenerate pair.

#### Participation ratio

The localization properties of the staggered eigenmodes can be studied
directly by looking at the eigenvectors, or indirectly by looking at the
corresponding eigenvalues. In the first case one can study the volume scaling
of the so-called participation ratio (PR) of the modes,

$\mathrm{PR}_{l}=\frac{1}{N_{t}V}\mathrm{IPR}_{l}^{-1}\,,\qquad\mathrm{IPR}_{l}=\sum_{n}\|\psi_{l}(n)\|^{4}\,,$
(11)

where $\|\psi_{l}(n)\|^{2}=\sum_{c=1}^{2}|\psi_{l,c}(n)|^{2}$, modes are
normalized to 1, $\sum_{n}\|\psi_{l}(n)\|^{2}=1$, and IPR is the inverse
participation ratio. The quantity $\mathrm{PR}_{l}$ measures the fraction of
lattice volume $N_{t}V$ occupied by a given mode, and similarly
$N_{t}V\cdot\mathrm{PR}_{l}=\mathrm{IPR}_{l}^{-1}$ gives the “mode size”.
After averaging over an infinitesimally small spectral bin around a point
$\lambda$ in the spectrum and over gauge configurations, as the spatial size
$N_{s}$ grows the resulting average $\mathrm{PR}(\lambda,N_{s})$ tends to a
constant if modes near $\lambda$ are delocalized on the entire lattice, and
goes to zero as the inverse of the lattice volume if they are localized in a
finite region. Equivalently, the similarly averaged mode size diverges
linearly in the lattice volume for delocalized modes and tends to a constant
for localized modes. In this paper we denote the average of any observable
$O_{l}$ associated with mode $l$, following the procedure described above, as

$O(\lambda,N_{s})=\frac{\left\langle\sum_{l}\delta(\lambda-\lambda_{l})O_{l}\right\rangle}{\left\langle\sum_{l}\delta(\lambda-\lambda_{l})\right\rangle}\,,$
(12)

having made explicit the dependence on the spatial size of the lattice. The
volume scaling of $\mathrm{PR}(\lambda,N_{s})$ defines the fractal dimension
of modes in the neighborhood of $\lambda$,

$\alpha(\lambda)=3+\lim_{N_{s}\to\infty}\frac{\log\mathrm{PR}(\lambda,N_{s})}{\log
N_{s}}\,.$ (13)

The multifractal properties of eigenmodes can be investigated by looking at
the generalized inverse participation ratios,

$(\mathrm{IPR}_{q})_{l}=\sum_{n}\|\psi_{l}(n)\|^{2q}\,,$ (14)

with $(\mathrm{IPR}_{2})_{l}=\mathrm{IPR}_{l}$ 222Since
$\|T\psi_{l}(n)\|^{2}=\|\psi_{l}(n)\|^{2}$, eigenmodes in a degenerate pair
have identical $(\mathrm{IPR}_{q})_{l}$.. Their average according to Eq. (12)
scales with the system size as $\mathrm{IPR}_{q}(\lambda,N_{s})\propto
N_{s}^{-D_{q}(\lambda)(q-1)}$, with generalized fractal dimensions $D_{q}$
(notice $D_{2}=\alpha$). One has $D_{q}=3$ for delocalized modes and $D_{q}=0$
for localized modes, while a nontrivial $D_{q}$ signals eigenmode
multifractality [48].

#### Spectral statistics

The localization properties of the eigenmodes reflect on the statistical
properties of the eigenvalues [49]: for localized modes one expects
independent fluctuations of the eigenvalues, while for delocalized modes one
expects to find the correlations typical of dense random matrix models. It is
convenient in this context to study the probability distribution of the so-
called unfolded level spacings [45, 46],

$s_{l}=\frac{\lambda_{l+1}-\lambda_{l}}{\langle\lambda_{l+1}-\lambda_{l}\rangle_{\lambda}}\,,$
(15)

computed locally in the spectrum, i.e.,

$p(s;\lambda,N_{s})=\frac{\left\langle\sum_{l}\delta(\lambda-\lambda_{l})\delta(s-s_{l})\right\rangle}{\left\langle\sum_{l}\delta(\lambda-\lambda_{l})\right\rangle}\,.$
(16)

In Eq. (15), $\langle\lambda_{l+1}-\lambda_{l}\rangle_{\lambda}$ denotes the
average spacing in the relevant spectral region, which for large volumes
equals
$\langle\lambda_{l+1}-\lambda_{l}\rangle_{\lambda}\to\frac{1}{N_{t}V\rho(\lambda)}$,
where $\rho(\lambda)$ is the spectral density,

$\rho(\lambda)=\lim_{V\to\infty}\frac{1}{N_{t}V}\left\langle{\textstyle\sum_{l}}\delta(\lambda-\lambda_{l})\right\rangle\,.$
(17)

The statistical properties of the unfolded spacings are expected to be
universal [45], i.e., independent of the details of the model, and can be
compared to the theoretical predictions obtained from exactly solvable models.
As the system size increases, for localized modes $p(s;\lambda,N_{s})$ should
approach the exponential distribution, $p_{\mathrm{P}}(s)=e^{-s}$, appropriate
for independent eigenvalues obeying Poisson statistics [45]. For delocalized
modes $p(s;\lambda,N_{s})$ should instead approach the distribution
$p_{\mathrm{RMT}}(s)$ predicted by the appropriate Gaussian Ensemble of Random
Matrix Theory, which is the Gaussian Symplectic Ensemble in the case at hand
[45, 46]. This quantity is known exactly, but is not available in closed form.
An accurate approximation is provided by the symplectic Wigner surmise,

$p_{\mathrm{WS}}(s)=\left(\frac{64}{9\pi}\right)^{3}s^{4}e^{-\frac{64}{9\pi}s^{2}}\,.$
(18)

#### Mobility edge

Localized and delocalized modes are generally found in disjoint spectral
regions separated by critical points known as mobility edges, where the
localization length diverges and the system undergoes a phase transition along
the spectrum, known as Anderson transition [48]. At the mobility edge the
critical eigenmodes display a fractal dimension different from those of
localized or delocalized modes, as well as a rich multifractal structure. This
is reflected in critical spectral statistics different from both Poisson and
RMT statistics. To monitor how the localization properties change along the
spectrum using its statistical properties, it is convenient to use the
integrated unfolded level spacing distribution,

$I_{s_{0}}(\lambda,N_{s})=\int_{0}^{s_{0}}ds\,p(s;\lambda,N_{s})\,,$ (19)

where $s_{0}\simeq 0.563$ is chosen so to maximize the difference between the
expectations for Poisson and RMT distributions, $I_{s_{0},\mathrm{P}}\simeq
0.431$ and $I_{s_{0},\mathrm{RMT}}\simeq 0.0797$, estimated using
$p_{\mathrm{P}}$ and $p_{\mathrm{WS}}$, see Eq. (18) above. This quantity
allows one to determine the mobility edge very accurately by means of a
finite-size-scaling analysis [50]. In fact, as the system size increases
$I_{s_{0}}(\lambda,N_{s})$ tends to $I_{s_{0},\mathrm{P}}$ or
$I_{s_{0},\mathrm{RMT}}$ depending on the localization properties of the modes
in the given spectral region, except at the mobility edge where it is volume-
independent and takes the value $I_{s_{0},c}$ corresponding to the critical
statistics. This, however, requires large-scale simulations to achieve a
sufficient quality of the data, and several large volumes.

One can give up some of the accuracy but save a lot in computing effort by
using the critical value of the spectral statistic, expected to be universal,
to determine the mobility edge simply by looking for the point where the curve
for $I_{s_{0}}$ crosses its critical value, $I_{s_{0},c}$ (see, e.g, Refs.
[28, 23, 25, 29]). This critical value is not known for the symplectic class,
but it can be determined by identifying the scale-invariant point in the
spectrum at some point in the parameter space of the model under study (if one
can find an Anderson transition, of course); the corresponding critical value
can then be used in the rest of the analysis. Notice that one could estimate
the mobility edge in a finite volume as the point where $I_{s_{0}}$ takes any
chosen value intermediate between the RMT and the Poisson predictions, and
this would converge to the correct value in the infinite-volume limit. In this
respect, the choice of $I_{s_{0},c}$ is only the most convenient, as it is
expected to minimize the magnitude of finite-size effects.

#### Correlation with bosonic observables

To investigate the correlation between staggered eigenmodes and gauge and
Higgs fields we considered the following observables,

$\displaystyle\mathcal{U}_{l}$
$\displaystyle=\sum_{n}U(n)\|\psi_{l}(n)\|^{2}\,,$
$\displaystyle\mathcal{P}_{l}$
$\displaystyle=\sum_{t,\vec{x}}P(\vec{x})\|\psi_{l}(\vec{x},t)\|^{2}\,,$ (20)
$\displaystyle\mathcal{G}_{l}$
$\displaystyle=\sum_{n}G(n)\|\psi_{l}(n)\|^{2}\,,$

averaged according to Eq. (12). Recall that $U(n)$ and $G(n)$ are the average
plaquette and gauge-Higgs coupling term touching a lattice site $n$, defined
in Eq. (7). For delocalized modes $\|\psi_{l}\|^{2}\sim\frac{1}{V}$, and the
averages $\mathcal{U}(\lambda,N_{s})$, $\mathcal{P}(\lambda,N_{s})$, and
$\mathcal{G}(\lambda,N_{s})$ of the observables in Eq. (20) are approximately
equal to the average of the corresponding bosonic observable, i.e., $\langle
U\rangle$, $\langle P\rangle$, and $\langle G\rangle$, respectively [see Eq.
(5)]. For localized modes $\|\psi_{l}\|^{2}$ is non-negligible only inside a
region of finite spatial volume, so $\mathcal{P}(\lambda,N_{s})$ measures the
average Polyakov loop inside the localization region, and
$\mathcal{U}(\lambda,N_{s})$ and $\mathcal{G}(\lambda,N_{s})$ measure
respectively the average plaquette and gauge-Higgs coupling term in a
neighborhood of the localization region. One should, however, keep in mind
that there are 24 neighboring squares and 8 neighboring links to each site, so
that a possible correlation of modes with the plaquette and gauge-Higgs
coupling term fluctuations get diluted.

More informative than the averages of the observables in Eq. (20) are the
corresponding centered and rescaled averages,

$\displaystyle\widehat{\mathcal{U}}(\lambda,N_{s})$
$\displaystyle=\frac{\mathcal{U}(\lambda,N_{s})-\langle U\rangle}{\delta
U}\,,$ (21) $\displaystyle(\delta U)^{2}$ $\displaystyle=\langle
U(n)^{2}\rangle-\langle U(n)\rangle^{2}\,,$
$\displaystyle\widehat{\mathcal{P}}(\lambda,N_{s})$
$\displaystyle=\frac{\mathcal{P}(\lambda,N_{s})-\langle P\rangle}{\delta
P}\,,$ $\displaystyle(\delta P)^{2}$ $\displaystyle=\langle
P(\vec{x})^{2}\rangle-\langle P(\vec{x})\rangle^{2}\,,$
$\displaystyle\widehat{\mathcal{G}}(\lambda,N_{s})$
$\displaystyle=\frac{\mathcal{G}(\lambda,N_{s})-\langle G\rangle}{\delta
G}\,,$ $\displaystyle(\delta G)^{2}$ $\displaystyle=\langle
G(n)^{2}\rangle-\langle G(n)\rangle^{2}\,.$

These quantities measure the correlation of the eigenmodes with fluctuations
in the gauge and Higgs fields, normalized by the average size of these
fluctuations. Indeed, writing these quantities out explicitly, one has, e.g.,

$\displaystyle\widehat{\mathcal{U}}(\lambda,N_{s})=\left\langle{\textstyle\sum_{n}}\frac{{\textstyle\sum_{l}}\delta(\lambda-\lambda_{l})\|\psi_{l}(n)\|^{2}}{N_{t}V\rho(\lambda)}\,\frac{U(n)-\langle
U\rangle}{\delta U}\right\rangle\,.$ (22)

As a consequence, the observables in Eq. (21) vanish in the absence of
correlation, and are strongly suppressed for delocalized modes. The
normalization factor takes into account that for observables with a strongly
peaked probability distribution even a correlation with small deviations from
average is significant, indicating that eigenmodes are attracted by the
corresponding type of fluctuations, and favor the locations where they show up
in a field configuration. In particular, for localized modes this allows one
to identify the most favorable type of fluctuations for localization.

#### Sea/islands picture

We also study the correlation between eigenmodes and the “islands” of the
refined “sea/islands” picture of localization discussed in Ref. [16]. These
are defined using the “Dirac-Anderson Hamiltonian” representation of the
staggered Dirac operator [14], obtained by diagonalizing the temporal hopping
term in $D^{\mathrm{stag}}$ [i.e., the term with $\mu=4$ in the sum in Eq.
(8)] by means of a unitary transformation $\Omega$ [16],

$H^{\mathrm{DA}}\equiv\Omega^{\dagger}(-iD^{\mathrm{stag}})\Omega=\mathcal{E}\mathbf{1}_{s}+\frac{1}{2i}\sum_{j=1}^{3}\eta_{j}({\cal
V}_{j}{\rm T}_{j}-{\rm T}_{j}^{\dagger}{\cal V}_{j}^{\dagger})\,,$ (23)

where $\mathbf{1}_{s}$ is the $V\times V$ identity matrix
$(\mathbf{1}_{s})_{\vec{x},\vec{y}}=\delta_{\vec{x},\vec{y}}$, ${\rm T}_{j}$
are here the spatial translation operators $({\rm
T}_{j})_{\vec{x},\vec{y}}=\delta_{\vec{x}+\hat{\jmath},\vec{y}}$ (with
periodic boundary conditions understood), $\mathcal{E}$ is an
$\vec{x}$-dependent $2N_{t}\times 2N_{t}$ diagonal matrix,

$\displaystyle\mathcal{E}(\vec{x})_{ka\,lb}$
$\displaystyle=\delta_{kl}\delta_{ab}e_{ka}(\vec{x})\,,$ $\displaystyle
e_{ka}(\vec{x})$ $\displaystyle=\eta_{4}(\vec{x})\sin\omega_{ka}(\vec{x})\,,$
(24)

and ${\cal V}_{j}$ are $\vec{x}$-dependent $2N_{t}\times 2N_{t}$ unitary
matrices,

$\displaystyle{\cal V}_{j}(\vec{x})_{ka\,lb}$
$\displaystyle=\frac{1}{N_{t}}\sum_{t=0}^{N_{t}-1}e^{-i\left(\omega_{ka}(\vec{x})-\omega_{lb}(\vec{x}+\hat{\jmath})\right)t}\,U_{j}^{\rm
tdg}(\vec{x},t)_{ab}\,,$ (25)

with $k,l=0,\ldots N_{t}-1$ and $a,b=1,2$. Here “tdg” stands for “temporal
diagonal gauge”, i.e., $U_{j}^{\rm tdg}$ are the spatial links in the temporal
gauge where all Polyakov loops are diagonal 333In the definition of
$U_{j}^{\rm tdg}$ in Eq. (B.8) of Ref. [16] the factors $u(\vec{x})$ and
$u(\vec{x}+\hat{\jmath})^{\dagger}$ are incorrectly missing.,

$\displaystyle U_{j}^{\rm tdg}(\vec{x},t)$
$\displaystyle=u(\vec{x})^{\dagger}P(\vec{x},t)U_{j}(\vec{x},t)P(\vec{x}+\hat{\jmath},t)^{\dagger}u(\vec{x}+\hat{\jmath})\,,$
(26) $\displaystyle P(\vec{x},t+1)$
$\displaystyle=P(\vec{x},t)U_{4}(\vec{x},t)\,,$

with $P(\vec{x},0)=\mathbf{1}$, and $u(\vec{x})$ a suitable unitary matrix
such that [notice $P(\vec{x})=P(\vec{x},N_{t})$]

$P(\vec{x})=u(\vec{x})\mathrm{diag}(e^{i\phi_{1}(\vec{x})},e^{i\phi_{2}(\vec{x})})u(\vec{x})^{\dagger}\,,$
(27)

with $\phi_{1,2}(\vec{x})\in[-\pi,\pi)$ and
$e^{i\left(\phi_{1}(\vec{x})+\phi_{2}(\vec{x})\right)}=1$. Moreover,
$\omega_{ka}(\vec{x})$ are effective Matsubara frequencies,

$\omega_{ka}(\vec{x})=\frac{\phi_{a}(\vec{x})+(2n_{ka}+1)\pi}{N_{t}}\,,$ (28)

with $n_{ka}\in\\{0,\ldots,N_{t}-1\\}$ chosen for each $a$ so that the
“energies” $e_{ka}$ satisfy $0\leq e_{1a}(\vec{x})\leq
e_{2a}(\vec{x})\leq\ldots\leq e_{\frac{N_{t}}{2}-1\,a}$, and
$e_{k+\frac{N_{t}}{2}\,a}(\vec{x})=-e_{ka}(\vec{x})$, for
$k=0,\ldots\frac{N_{t}}{2}-1$. Notice that thanks to the simple relation
between $\phi_{1}$ and $\phi_{2}$, one has $e_{k1}=e_{k2}$. This double
degeneracy is a consequence of the temporal hopping term being invariant under
the time-reversal transformation $T$ (see section II.2). With this choice for
$e_{ka}$, $H^{\mathrm{DA}}$ has the general structure

$\displaystyle H^{\mathrm{DA}}$ $\displaystyle=\begin{pmatrix}E&\mathbf{0}\\\
\mathbf{0}&-E\end{pmatrix}$ (29)
$\displaystyle\phantom{=}+\frac{1}{2i}\sum_{j=1}^{3}\eta_{j}\left[\begin{pmatrix}A_{j}&B_{j}\\\
B_{j}&A_{j}\end{pmatrix}{\rm T}_{j}-{\rm
T}_{j}{}^{\dagger}\begin{pmatrix}A_{j}{}^{\dagger}&B_{j}{}^{\dagger}\\\
B_{j}{}^{\dagger}&A_{j}{}^{\dagger}\end{pmatrix}\right]\,,$

where $E,A_{j},B_{j}$ are $N_{t}\times N_{t}$ matrices.

It was argued in Ref. [16] that sites where the diagonal blocks $A_{j}$ are
larger are the most favorable for the localization of low modes in a phase
where the Polyakov loops are ordered. In general, spatial regions with larger
$A_{j}$, which correspond to lower correlation among spatial links on
different time slices, are expected to be favored by low modes; in an ordered
phase such regions are localized, and so lead to low-mode localization. One
can check this by looking at the correlation between modes and the quantity

$A(\vec{x})=\frac{1}{6N_{t}}\sum_{j=1}^{3}{\rm
tr}\,A_{j}(\vec{x})^{\dagger}A_{j}(\vec{x})+{\rm
tr}\,A_{j}(\vec{x}-\hat{\jmath})^{\dagger}A_{j}(\vec{x}-\hat{\jmath})\,,$ (30)

i.e., using the observable

$\mathcal{A}_{l}=\sum_{\vec{x}}A(\vec{x})\sum_{t=0}^{N_{t}-1}\|\psi_{l}(\vec{x},t)\|^{2}\,,$
(31)

averaged according to Eq. (12) to get $\mathcal{A}(\lambda,N_{s})$, and
centered and rescaled according to Eq. (22) to get
$\widehat{\mathcal{A}}(\lambda,N_{s})$, i.e.,

$\displaystyle\widehat{\mathcal{A}}(\lambda,N_{s})$
$\displaystyle=\frac{\mathcal{A}(\lambda,N_{s})-\langle A\rangle}{\delta
A}\,,$ (32) $\displaystyle(\delta A)^{2}$ $\displaystyle=\langle
A(\vec{x})^{2}\rangle-\langle A(\vec{x})\rangle^{2}\,.$

##  III Phase diagram at finite temperature

In this section we report our results on the phase diagram of the model. We
worked at finite temperature, fixing the lattice temporal extension to
$N_{t}=4$, and performing numerical simulations with a standard heatbath
algorithm.

Theoretical arguments [36] and previous numerical studies [42] lead us to
expect three phases: a confined phase at small $\beta$ and small $\kappa$; a
deconfined phase at large $\beta$ and small $\kappa$; and a Higgs phase at
large $\kappa$. Based on the finite-temperature results of Ref. [42], and on
the observed weakening of the transition for smaller temporal extensions
reported there, we expect that the transitions between the three phases are
analytic crossovers. A detailed study of this issue is beyond the scope of
this paper, so we limited most of our simulations to a single lattice volume
with $N_{s}=20$, for 784 different $(\beta,\kappa)$ pairs, using 3000
configurations at each point. We took $\kappa\in[0,1.35]$ in steps of
$\Delta\kappa=0.05$ and $\beta\in[1.5,2.85]$ in steps of $\Delta\beta=0.05$. A
detailed volume-scaling study was done on a subset of these points: we discuss
this below.

Figure 1: Heatmap plot of the expectation value of $G$ (top panel), $U$
(center panel), and $P$ (bottom panel), see Eq. (5). Here $N_{s}=20$ and
$N_{t}=4$.

We show our results for $\langle G\rangle$, $\langle U\rangle$, and $\langle
P\rangle$ in Fig. 1 as heatmap plots, obtained by cubic interpolation of the
numerical results at the simulation points. These confirm our expectations,
and allow us to characterize the confined phase at small $\beta$ and $\kappa$
by small $\langle G\rangle$, $\langle U\rangle$, and $\langle P\rangle$; the
deconfined phase at large $\beta$ and small $\kappa$ by small $\langle
G\rangle$ and large $\langle U\rangle$ and $\langle P\rangle$; and the Higgs
phase at large $\kappa$ by large $\langle G\rangle$, $\langle U\rangle$, and
$\langle P\rangle$. We estimated errors with a standard jackknife procedure:
they are not shown, but relative errors are always within $7\cdot 10^{-5}$ for
$\langle U\rangle$; $2\cdot 10^{-3}$ for $\langle G\rangle$; and within
$1\cdot 10^{-3}$ for $\langle P\rangle$, except deep inside the confined phase
where the average becomes very small and indistinguishable from zero within
errors. More precisely, the expectation value of the gauge-Higgs coupling term
(Fig. 1, top panel) divides the phase diagram into two pieces: the Higgs phase
at large $\kappa$, with large $\langle G\rangle$, and the (undivided) confined
and deconfined phases at small $\kappa$, with similar and small values of
$\langle G\rangle$. The expectation value of the plaquette and of the Polyakov
loop (Fig. 1, center and bottom panel) divide the phase diagram into two parts
in a different way: the confined phase at low $\beta$ and $\kappa$, where both
$\langle U\rangle$ and $\langle P\rangle$ are small, and the (undivided) Higgs
and deconfined phases, where both $\langle U\rangle$ and $\langle P\rangle$
are large.

Figure 2: Heatmap plot of the susceptibility $\chi_{G}$ of the gauge-Higgs
coupling term $G$ (top panel), the plaquette susceptibility $\chi_{U}$ (center
panel), and the logarithm of the Polyakov-loop susceptibility $\chi_{P}$
(bottom panel), see Eqs. (5) and (6). Here $N_{s}=20$ and $N_{t}=4$. In the
top panel, the black point shows where the mobility edge
$\lambda_{c}=\lambda_{c}(\kappa)$ has an inflection point along the line at
constant $\beta=2.6$, see Fig. 17. In the center panel, it shows where the
mobility edge $\lambda_{c}=\lambda_{c}(\beta)$ vanishes along the line at
constant $\kappa=1.0$, see Fig. 16. In the bottom panel, it shows where the
mobility edge $\lambda_{c}=\lambda_{c}(\beta)$ vanishes along the line at
constant $\kappa=0.3$, see Fig. 15.

We show our results for the corresponding susceptibilities as heatmap plots in
Fig. 2. Also in this case we estimated errors (not shown in the figure) with a
standard jackknife procedure, finding them to be always within 3%. In the top
panel we show our results for $\chi_{G}$. This quantity has a narrow ridge,
visualized here as a bright line, providing a clear separation between the
Higgs phase and the rest in most of the explored parameter space; a weakening
of the transition is visible in the top left part of the phase diagram. In the
center panel we show the plaquette susceptibility $\chi_{U}$. This separates
clearly the confined phase from the Higgs phase, while the ridge broadens at
the transition between confined and deconfined phase (as well as in the top
left part of the phase diagram). In the bottom panel we show the logarithm of
the Polyakov-loop susceptibility. This plot shows a bright line of strong
transitions separating the confined and deconfined phases. This line continues
in the top left part of the plot, still clearly separating the confined and
Higgs phases, but it is much dimmer there as the signal is two orders of
magnitude weaker than at the transition from the confined to the deconfined
phase (see Figs. 4, top and 5, top). At the transition between the deconfined
and Higgs phase $\chi_{P}$ shows an inflection point instead of a peak (see
Fig. 6), with a sizeable decrease in susceptibility corresponding here to a
noticeable darkening of the plot.

A sketch of the resulting phase diagram is shown in Fig. 3, obtained by
merging the various transition lines, defined by the peaks of the
suceptibilities. The dashed line at low $\beta$ and large $\kappa$ signals a
sizeable reduction in the strength of the transition there, as shown by all
three observables. Except in this region, where they slighly deviate from each
other, the transition lines between confined and Higgs phase obtained from the
three different susceptibilities agree with each other, so we drew a single
line. Similarly, the transition lines between confined and deconfined phase
obtained from the plaquette and the Polyakov loop susceptibility agree with
each other, so we drew a single line in this case as well.

Figure 3: Schematic drawing of the phase diagram, obtained combining the
maxima of the susceptibilities shown in Fig. 2. A dashed line is used to
indicate the weakening of the transition.

To verify the expected crossover nature of the transitions, we studied the
volume dependence of the various susceptibilities in detail on three lines,
one at constant $\kappa=0.5$ and two at constant $\beta=2.0$ and $\beta=2.6$,
using lattices with $N_{s}=22,28,34,40$. For each simulation point and each
lattice volume we used 4500 configurations. We estimated errors by first
averaging over configurations in blocks of size $b_{\mathrm{size}}$ and
computing the standard jackknife error on the blocked ensemble, and then
increasing $b_{\mathrm{size}}$ until the error stabilized. For our final
estimates we used samples of size $b_{\mathrm{size}}=20$, except at
$\kappa=0.5$ where we used $b_{\mathrm{size}}=50$, although this was really
needed only around $\beta=2.3$. We show our results in Figs. 4–6.

In Fig. 4 we show $\chi_{P}$, $\chi_{U}$ and $\chi_{G}$ along a line of
constant $\kappa=0.5$ across the transition from the confined to the
deconfined phase. The signal is very strong in $\chi_{P}$, and a small peak is
visible also in $\chi_{U}$. The location of these peaks is not far from the
critical point $\beta_{c}\approx 2.3$ of the pure gauge theory at $\kappa=0$
[52, 53, 54, 55]. The relatively large error bars found for $N_{s}=22$ between
$\beta=2.28$ and $\beta=2.31$, especially at $\beta=2.3$, are most likely a
finite-size effect due to the vicinity of the critical point of the pure gauge
theory, and are not observed on larger volumes. On the other hand, no peak is
visible in $\chi_{G}$, which is constant within errors across the transition.
This makes the gauge-Higgs coupling term $G$ unsuitable to detect this
transition.

Figure 4: Polyakov-loop (top), plaquette (center), and gauge-Higgs coupling
term (bottom) susceptibility across the transition between the confined and
the deconfined phase at $\kappa=0.5$. Here $N_{t}=4$. The volume scaling is
consistent with an analytic crossover.

Figure 5: Polyakov-loop (top), plaquette (center), and gauge-Higgs coupling
term (bottom) susceptibility near the transition between the confined and the
Higgs phase at $\beta=2.0$. Here $N_{t}=4$. The volume scaling is consistent
with an analytic crossover.

Figure 6: Polyakov-loop (top), plaquette (center), and gauge-Higgs coupling
term (bottom) susceptibility near the transition between the deconfined and
the Higgs phase at $\beta=2.6$. Here $N_{t}=4$. The volume scaling is
consistent with an analytic crossover.

In Fig. 5 we show $\chi_{P}$, $\chi_{U}$ and $\chi_{G}$ along a line of
constant $\beta=2.0$ across the transition from the confined to the Higgs
phase. A clear peak is visible in all three observables, with $\chi_{P}$ two
orders of magnitude smaller than in Fig. 4 (top), and $\chi_{U}$ a factor of 2
larger than in Fig. 4 (bottom).

Finally, in Fig. 6 we show $\chi_{P}$, $\chi_{U}$ and $\chi_{G}$ along a line
of constant $\beta=2.6$ across the transition from the deconfined to the Higgs
phase. We observe a peak in $\chi_{G}$, of similar magnitude as the one in
Fig. 5 (bottom) for the transition from the confined to the Higgs phase.
Neither $\chi_{U}$ nor $\chi_{P}$ show any significant peak: $\chi_{U}$
changes slope at the transition, while $\chi_{P}$ shows an inflection point.
This makes $U$ and $P$ not quite suitable observables to detect this
transition.

While these results do not logically exclude the possibility of genuine phase
transitions at some points in the phase diagram, combined with the results of
Ref. [42] they make it implausible.

##  IV Localization properties of Dirac eigenmodes

$(\beta=1.9,\kappa=1.0)$
---
$N_{s}$ | #configurations | #eigenvalues
16 | 3000 | 33
20 | 1500 | 63

$(\beta=2.1,\kappa=1.0)$ and $(\beta=2.6,\kappa=0.3)$
---
$N_{s}$ | #configurations | #eigenvalues
20 | 8970 | 63
24 | 8000 | 110
28 | 3000 | 174
32 | 1150 | 260

Table 1: Configuration statistics and number of (non-degenerate) eigenvalues
used to study the volume scaling of the localization properties of staggered
eigenmodes in the confined phase (top table) and in the deconfined and Higgs
phases (bottom table).

In this section we discuss the localization properties of the eigenmodes of
the staggered operator and how these correlate with the gauge and Higgs
fields, and we present a detailed test of the sea/islands mechanism. We
obtained the lowest modes of $D^{\textrm{stag}}$ using the PRIMME package [56,
57] for sparse matrices, exploiting Chebyshev acceleration for faster
convergence. The use of algorithms for sparse matrices allows us to reduce the
scaling of computational time from $N_{s}^{9}$, expected for full
diagonalization, down to $N_{s}^{6}$.

We first analyzed the eigenmodes in detail at three points of the phase
diagram, using several lattice volumes to study the scaling of eigenvector and
eigenvalue observables with the system size. These points are
$\beta=1.9,\kappa=1.0$, in the confined phase, right below the transition to
the Higgs phase at constant $\kappa$ ($\beta/\beta_{c}\approx 0.97$, with
$\beta_{c}\approx 1.95$ corresponding to the peak in the Polyakov-loop
susceptibility); $\beta=2.1,\kappa=1.0$, in the Higgs phase, not far above the
transition between the two phases ($\beta/\beta_{c}\approx 1.08$); and
$\beta=2.6,\kappa=0.3$, deep in the deconfined phase. We looked at two lattice
volumes in the confined phase, and at four lattice volumes in the deconfined
and Higgs phases; see Tab. 1 for details about system size, configuration
statistics, and number of eigenmodes. We then computed the relevant
observables locally in the spectrum, approximating Eq. (12) by averaging over
spectral bins of size $\Delta\lambda=0.0025$ at $\beta=1.9$, $\kappa=1.0$
(confined phase), $\Delta\lambda=0.01$ at $\beta=2.6$, $\kappa=0.3$
(deconfined phase), and $\Delta\lambda=0.0075$ at $\beta=2.1$, $\kappa=1.0$
(Higgs phase). Our results, reported in sections IV.1 and IV.2, demonstrate
low-mode localization in the deconfined and in the Higgs phase.

This detailed study also allowed us to estimate the critical value of
$I_{s_{0}}$, which we could then use to efficiently determine the dependence
of the mobility edge, $\lambda_{c}$, on the parameters $\beta$ and $\kappa$.
We did this on two lines at constant $\kappa$: one in the deconfined phase
with $\kappa=0.3$, changing $\beta$ in the interval $[2.35,2.60]$ with
increments $\Delta\beta=0.05$; and one in the Higgs phase with $\kappa=1.0$,
changing $\beta$ in $[2.1,2.4]$ with increments $\Delta\beta=0.05$. We also
studied one line at constant $\beta=2.6$, changing $\kappa$ in $[0.35,1.0]$ in
increments of $\Delta\kappa=0.05$. Here we used a single volume
($N_{s}=20,N_{t}=4$) and 3000 configurations at each point (except for the
three points already discussed above). In all these calculations we computed
$I_{s_{0}}$ locally in the spectrum averaging over bins of size
$\Delta\lambda=0.008$. Our results, reported in section IV.3, show that along
both lines at constant $\kappa$ the mobility edge disappears at a critical
$\beta$ near the crossover to the confined phase; and that along the line at
constant $\beta$ the mobility edge is always nonzero, but it changes behavior
at the crossover between the deconfined and the Higgs phase.

We then studied the correlation between localized modes and the fluctuations
of the gauge and Higgs fields, and tested the refined sea/islands picture of
Ref. [16]. Our results, reported in section IV.4, show a strong correlation
with Polyakov-loop and plaquette fluctuations, and an even stronger
correlation with the fluctuations identified in Ref. [16] as the most relevant
to localization.

###  IV.1 Eigenvector observables

Figure 7: Participation ratio, Eq. (11), of the low staggered eigenmodes at
$\beta=1.9$ and $\kappa=1.0$ in the confined phase for two different spatial
volumes (top panel) and corresponding fractal dimension estimated using Eq.
(33) with $N_{s_{1}}=16$, $N_{s_{2}}=20$ (bottom panel). Here $N_{t}=4$.

In the top panel of Fig. 7 we show the PR of the modes in the confined phase.
The PR is slightly larger for $N_{s}=16$ than for $N_{s}=20$, signaling that
the fractal dimension is smaller than 3. This is shown explicitly in the
bottom panel, where we plot $\alpha(\lambda)$, see Eq. (13). This is estimated
numerically from a pair of volumes as

$\alpha_{\mathrm{num}}(\lambda;N_{s1},N_{s2})=3+\frac{\log\frac{\mathrm{PR}(\lambda,N_{s1})}{\mathrm{PR}(\lambda,N_{s2})}}{\log\frac{N_{s1}}{N_{s2}}}\,.$
(33)

The fractal dimension of near-zero modes is slightly below 3, and approaches 3
as one moves up in the spectrum. Taken at face value, this means that these
modes are only slightly short of being fully delocalized. Clearly, this effect
could be just a finite-size artifact due to the small volumes employed here.
However, it could also signal that a “geometric” transition is approaching,
where a mobility edge and, correspondingly, critical modes appear at the
origin.

Figure 8: The mode size $N_{t}V\cdot\mathrm{PR}=\mathrm{IPR}^{-1}$, Eq. (11),
of the low staggered eigenmodes for different volumes (top panel), and
corresponding fractal dimension estimated using Eq. (33) with three different
volume pairs (bottom panel), at $\beta=2.6$ and $\kappa=0.3$ in the deconfined
phase. Here $N_{t}=4$. The vertical solid line shows the position of the
mobility edge, vertical dashed lines indicate the corresponding error band. In
the bottom panel, horizontal dotted lines mark the values $\alpha=0$,
corresponding to localized modes, and $\alpha=3$, corresponding to totally
delocalized modes.

Figure 9: As in Fig. 8, but at $\beta=2.1$ and $\kappa=1.0$ in the Higgs
phase.

Figure 10: Ratio of generalized IPRs, Eq. (34), at $\beta=2.6$ and
$\kappa=0.3$ in the deconfined phase (top panel), and at $\beta=2.1$ and
$\kappa=1.0$ in the Higgs phase (bottom panel). The vertical solid line shows
the position of the mobility edge, vertical dashed lines indicate the
corresponding error band. A nontrivial volume scaling indicates nontrivial
multifractal properties of the eigenmodes at the mobility edge.

In the top panels of Figs. 8 and 9 we show the size
$N_{t}V\cdot\mathrm{PR}=\mathrm{IPR}^{-1}$ of the modes in the deconfined and
in the Higgs phase, respectively. In both cases the size of the lowest modes
does not change with the volume, showing that they are localized. Higher up
towards the bulk of the spectrum the mode size shows a strong volume
dependence. Above a certain point in the spectrum this is compatible with a
linear scaling in the volume, indicating that these modes are delocalized. The
point where this starts to happen is consistent with the mobility edge,
determined below in section IV.2 using spectral statistics, and marked in
these plots by a solid vertical line (with an error band shown with dashed
lines). The localization properties of low and bulk modes in the deconfined
and in the Higgs phase are made quantitative in the bottom panels of Figs. 8
and 9, where we show their fractal dimension. For low modes this is zero
within errors. Near the mobility edge our estimates for $\alpha$ increase
towards 3, which they almost reach at the upper end of the available spectral
range. The rise should become steeper when using pairs of larger volumes,
leading to a jump from 0 to 3 at the mobility edge in the infinite-volume
limit. Such a tendency is visible in the Higgs phase. Our results are also
consistent with modes at the mobility edge displaying critical localization
properties, with a fractal dimension between 1 and 2.

The nontrivial multifractal properties of the eigenmodes at the mobility edge
are made evident in Fig. 10, where we show the ratio

$\frac{\mathrm{IPR}_{2}(\lambda,N_{s})}{\sqrt{\mathrm{IPR}_{3}(\lambda,N_{s})}}\sim
N_{s}^{-\left(D_{2}(\lambda)-D_{3}(\lambda)\right)}\,,$ (34)

where the generalized IPRs have been defined in Eq. (14). This quantity tends
to a constant both in the localized ($D_{q}=0$) and in the delocalized regime
($D_{q}=3$), while it has a nontrivial volume scaling for modes displaying
multifractality, i.e, with $q$-dependent $D_{q}$. This is expected to be a
feature of the critical modes found at the mobility edge. This point in the
spectrum is indeed characterized by a nontrivial volume scaling of the ratio
in Eq. (34), which also reaches its minimum in the vicinity of the mobility
edge.

Comparing results in the confined and in the Higgs phase, that lie on the same
line at constant $\kappa$ near the transition, one sees that the rapid change
in the localization properties of the low modes takes place precisely in the
crossover region. This issue is studied in more detail below in section IV.3.

###  IV.2 Eigenvalue observables and mobility edge

Figure 11: The spectral density at $\beta=1.9$ and $\kappa=1.0$ in the
confined phase (top panel; here $N_{s}=20$), at $\beta=2.6$ and $\kappa=0.3$
in the deconfined phase (center panel; here $N_{s}=32$), and at $\beta=2.1$
and $\kappa=1.0$ in the Higgs phase (bottom panel; here $N_{s}=32$). In all
plots $N_{t}=4$.

We now discuss eigenvalue observables, starting from the spectral density, Eq.
(17), shown in Fig. 11. In the confined phase (top panel) the spectral density
is practically constant in the lowest bins (except for the very lowest, which
is depleted due to the smallness of the lattice volume), and grows as one
moves towards the bulk of the spectrum. If we were in the chiral limit of
massless fermions, a nonzero spectral density near the origin would indicate
the spontaneous breaking of chiral symmetry [11]. Being in the opposite limit
of infinitely massive fermions, we can speak of spontaneous chiral symmetry
breaking only in a loose sense. In the deconfined and in the Higgs phase
(bottom panels) we see instead that the spectral density is close to zero for
near-zero modes, corresponding (again, loosely speaking) to the restoration of
chiral symmetry. As we increase $\lambda$ the spectral density increases, and
does so faster as one approaches the mobility edge. However, no sign of
critical behavior is visible along the spectrum.

Figure 12: The integrated unfolded level spacing distribution $I_{s_{0}}$, Eq.
(19), at $\beta=1.9$ and $\kappa=1.0$ in the confined phase. Here $N_{t}=4$.
The horizontal dotted line shows the value of $I_{s_{0}}$ expected for RMT
statistics. Figure 13: The integrated unfolded level spacing distribution
$I_{s_{0}}$, Eq. (19), at $\beta=2.6$ and $\kappa=0.3$ in the deconfined
phase. Here $N_{t}=4$. The upper and lower horizontal dotted lines show the
value of $I_{s_{0}}$ expected for Poisson statistics and for RMT statistics,
respectively. The vertical solid and dashed lines indicate the position and
the error band of the mobility edge. The horizontal solid and dashed lines
correspond to the estimate for the critical value $I_{s_{0},c}$ of $I_{s_{0}}$
at the mobility edge and its error band. Figure 14: As in Fig. 13, but at
$\beta=2.1$ and $\kappa=1.0$ in the Higgs phase. Here $N_{t}=4$.

We now move on to discuss the spectral statistic $I_{s_{0}}$, Eq. (19), for
the low modes in the three different phases of the system. To estimate this
quantity numerically we unfolded the spectrum, averaging then $I_{s_{0}}$ in
small spectral bins and over gauge configurations. More precisely, we defined
the unfolded spacings using Eq. (15), using for
$\langle\lambda_{l+1}-\lambda_{l}\rangle_{\lambda}$ the average level spacing
found in a given spectral bin, including all pairs of eigenvalues for which
the smaller one fell in the bin.

In Fig. 12 we show $I_{s_{0}}$ in the confined phase. As expected, $I_{s_{0}}$
is compatible with the value predicted by RMT in the whole available spectral
range for both volumes, further confirming that these modes are delocalized.
In Figs. 13 and 14 we show the value of $I_{s_{0}}$ in the deconfined and in
the Higgs phase. For modes near the origin $I_{s_{0}}$ approaches the value
expected for Poisson statistics as we increase the volume, signaling that
these are localized modes. For higher modes the value of $I_{s_{0}}$ tends
instead to the RMT prediction as the volume increases, showing that modes are
delocalized in this spectral region. Between these two regimes, we can find
the mobility edge $\lambda_{c}$ as the point where $I_{s_{0}}$ is scale-
invariant and the curves cross each other.

To find $\lambda_{c}$ and the critical value $I_{s_{0},c}$ of the spectral
statistic we interpolated the numerical data with natural cubic splines, and
determined the crossing point for the various pairs of system sizes using
Cardano’s formula. The statistical error on each determination of
$\lambda_{c}$ and $I_{s_{0},c}$ originating in the numerical uncertainty on
$I_{s_{0}}$ in the various bins is estimated by obtaining the interpolating
splines and their crossing point for a set of synthetic data, generating 100
data sets by drawing for each bin a number from a Gaussian distribution with
mean equal to the average $I_{s_{0}}$ in the bin and variance equal to the
square of the corresponding error. The systematic errors on $\lambda_{c}$ and
$I_{s_{0},c}$ due to finite-size effects are estimated as the variance of the
set of values for the crossing point and corresponding value of $I_{s_{0}}$
obtained from all the pairs of volumes. We finally estimated the mobility edge
and the critical $I_{s_{0}}$ as those obtained from the crossing point of the
biggest volume pair ($N_{s}=28,32$), as it should be the closest to the actual
value in the infinite-volume limit, and the corresponding error by adding
quadratically its statistical error with the finite-size systematic error
discussed above. The total error is largely dominated by the finite-size
contribution. We did this separately for the configurations in the deconfined
and in the Higgs phase. The results for $\lambda_{c}$ and $I_{s_{0},c}$ are
reported in Tab. 2, and shown in Figs. 13 and 14 as solid lines, with dashed
lines marking the corresponding error bands. The two determinations of
$I_{s_{0},c}$, obtained in the deconfined and in the Higgs phase, agree within
errors. Despite the uncertainty on $I_{s_{0},c}$ being 10–15%, we could
determine $\lambda_{c}$ with a 1–2% uncertainty thanks to the steepness of
$I_{s_{0}}$ near the mobility edge.

$\beta$ | $\kappa$ | phase | $\lambda_{c}$ | $I_{s_{0},c}$
---|---|---|---|---
$2.6$ | $0.3$ | deconfined | 0.2493(20) | 0.164(16)
$2.1$ | $1.0$ | Higgs | 0.1367(24) | 0.177(26)
Table 2: Mobility edge and critical value of $I_{s_{0}}$ estimated at two
points of the phase diagram, one in the deconfined and one in the Higgs phase.

###  IV.3 $\beta$ and $\kappa$ dependence of the mobility edge

Figure 15: The dependence of the mobility edge on $\beta$ in the deconfined
phase on the line at constant $\kappa=0.3$. The solid line is a power-law fit,
Eq. (35), to the numerical data; the band corresponds to the finite-size
systematic uncertainty discussed in the text. The point where the mobility
edge vanishes is estimated at $\beta_{\mathrm{loc}}=2.2997(57)$, in the
crossover region between the confined and deconfined phases, see Fig. 2
(bottom).

Having obtained estimates of $I_{s_{0},c}$ we can now use them to efficiently
determine $\lambda_{c}$ throughout the phase diagram using a single lattice
volume at each point, and looking for the point in the spectrum where
$I_{s_{0}}$ takes the value $I_{s_{0},c}$. We use again natural cubic splines
to interpolate the numerical data, using the more precise determination of
$I_{s_{0},c}$ obtained in the deconfined phase and generating synthetic data
as discussed above to estimate the statistical error. To estimate the
magnitude of finite size effects, we determined also the crossing points
$\lambda_{c,\pm}$ of $I_{s_{0}}$ with $I_{s_{0},c}\pm\delta I_{s_{0},c}$, with
$\delta I_{s_{0},c}$ the uncertainty on $I_{s_{0},c}$. This is meant to
determine just how much the crossing point of $I_{s_{0}}$ may change with the
volume, as the error band on $I_{s_{0},c}$ is determined by the fluctuations
of the crossing point of the various pairs of volumes used to find the
mobility edge and the critical statistics in section IV.2, and has nothing to
do with the fact that $I_{s_{0},c}$ is not known exactly. As explained in
section II.2, one could in fact use any value intermediate between the RMT and
the Poisson expectations to give an estimate of the mobility edge in a finite
volume, and this would converge to the correct value in the thermodynamic
limit.

Figure 16: The dependence of the mobility edge on $\beta$ in the Higgs phase on the line at constant $\kappa=1.0$. The solid line is a power-law fit, Eq. (35), to the numerical data; the band corresponds to the finite-size systematic uncertainty discussed in the text. The point where the mobility edge vanishes is estimated at $\beta_{\mathrm{loc}}=2.0101(28)$, in the crossover region between the confined and Higgs phases, see Fig. 2 (center). | deconfined | Higgs
---|---|---
$\beta_{\mathrm{loc}}$ | 2.2997(22) | 2.0101(25)
$A$ | 0.3836(17) | 0.3851(14)
$B$ | 0.3592(54) | 0.4344(59)
$\chi^{2}/\mathrm{dof}$ | 1.48 | 1.64
Table 3: Parameters of a best fit of the $\beta$ dependence of the mobility
edge in the deconfined ($\kappa=0.3$) and Higgs ($\kappa=1.0$) phases, with
the fitting function in Eq. (35). Only statistical errors are reported. Figure
17: The dependence of the mobility edge on $\kappa$ on the line at constant
$\beta=2.6$ in the deconfined and Higgs phases. The solid line is a fit to the
data with Eq. (36); the band corresponds to the finite-size systematic
uncertainty discussed in the text. A change of behavior is found at
$\kappa_{\mathrm{loc}}=0.7303(59)$ in the crossover region between the two
phases, see Fig. 2 (top), marked here by a vertical solid line, with dashed
lines giving the corresponding error band.

We can then study how $\lambda_{c}$ depends on $\kappa$ and $\beta$. In Fig.
15 we show how $\lambda_{c}$ changes in the deconfined phase as one decreases
$\beta$ towards the confined phase at fixed $\kappa$. We expect that the
mobility edge disappears as we enter the confined phase and the Polyakov loop
loses its strong ordering. To estimate the value
$\beta_{\mathrm{loc}}(\kappa)$ of $\beta$ where this happens we fitted our
results with a power-law function,

$\lambda_{c}(\beta)=A\cdot(\beta-\beta_{\mathrm{loc}})^{B}\,,$ (35)

using the MINUIT library [58] to minimize the $\chi^{2}$, computed using only
the statistical errors on $\lambda_{c}$. We then repeated the fit using
$\lambda_{c\pm}(\beta)$ to find the corresponding $\beta_{\mathrm{loc}\pm}$
where they extrapolate to zero, and used these to estimate the systematic
uncertainty due to finite-size effects as
$\frac{1}{2}|\beta_{\mathrm{loc}+}-\beta_{\mathrm{loc}-}|$. We obtained for
the critical value
$\beta_{\mathrm{loc}}(0.3)=2.2997(22)_{\mathrm{stat}}(53)_{\mathrm{syst}}=2.2997(57)$,
where the total error is the sum in quadrature of the statistical error from
the fit and of the systematic error. The other fit parameters and the
$\chi^{2}$ per degree of freedom,
$\chi^{2}/\mathrm{dof}=\chi^{2}/(n_{\mathrm{data}}-n_{\mathrm{parameters}})$,
are reported in Tab. 3. The critical point is shown also in Fig. 2 (bottom),
where we see that the vanishing of the mobility edge matches well with the
crossover between the phases.

In Fig. 16 we show how $\lambda_{c}$ changes in the Higgs phase as one
decreases $\beta$ towards the confined phase at fixed $\kappa$. Again, we
expect the mobility edge to disappear at the crossover. For the critical
$\beta_{\mathrm{loc}}(\kappa)$ we find
$\beta_{\mathrm{loc}}(1.0)=2.0101(25)_{\mathrm{stat}}(13)_{\mathrm{syst}}=2.0101(28)$,
again from a fit with a power law, Eq. (35), using statistical errors only
(see Tab. 3 for the other fit parameters), and estimating systematic effects
by fitting $\lambda_{c\pm}$, as discussed above. This is shown also in Fig. 2
(center), where one sees that the vanishing of $\lambda_{c}$ takes place again
in the crossover region.

$\kappa_{\mathrm{loc}}$ | 0.7303(56)
---|---
$a$ | 0.24915(13)
$b$ | 0.1185(53)
$c$ | 0.1874(52)
$d$ | 26.4(2.0)
$\chi^{2}/\mathrm{dof}$ | 2.05
Table 4: Parameters of a best fit of the $\kappa$ dependence of the mobility
edge in the deconfined and Higgs phases at $\beta=2.6$, with the fitting
function in Eq. (36). Only statistical errors are reported.

The third case we examined is the transition from the Higgs phase to the
deconfined phase as we decrease $\kappa$ at fixed $\beta$. This is shown in
Fig. 17. One can see that at first $\lambda_{c}$ decreases quickly with
$\kappa$, but below a critical value $\kappa_{\mathrm{loc}}(\beta)$ it becomes
practically constant. The critical value is defined here as the point where
the behavior changes from approximately constant to approximately linear, as
obtained by fitting with the following function,

$\displaystyle\lambda_{c}(\kappa)$
$\displaystyle=a\cdot\left(1-\sigma\left(d\cdot(\kappa-\kappa_{\mathrm{loc}})\right)\right)$
(36)
$\displaystyle\phantom{=}+\left(b\kappa+c\right)\sigma(d\cdot(\kappa-\kappa_{\mathrm{loc}}))\,,$

where $\sigma(x)={1}/{(1+e^{-x})}$ is the sigmoid function. Following the same
procedure discussed above to estimate errors, we found
$\kappa_{\mathrm{loc}}(2.6)=0.7303(57)_{\mathrm{stat}}(17)_{\mathrm{syst}}=0.7303(59)$
for the critical point (see Tab. 4 for the other fit parameters). As shown in
Fig. 2 (top), also in this case the critical value matches well with the
position of the crossover. Notice that here the critical point is not as
sharply defined as in the previous two cases, as it simply corresponds to a
change in the $\kappa$ dependence of the mobility edge, rather than its very
appearance. However, it is possible that the change in the behavior of
$\lambda_{c}(\kappa)$ becomes singular in the infinite-volume limit, e.g., due
to a discontinuity in its derivative. If so, one would find a sharply defined
critical point for the geometric transition also in this case. At the present
stage this is only speculation, and a more careful determination of the
mobility edge is needed to test this possibility, either by a proper finite-
size scaling analysis, or by checking the volume dependence of the crossing
point of $I_{s_{0}}$ with $I_{s_{0},c}$.

Figure 18: Mode size averaged up to $\lambda_{c}/2$, Eq. 37, at $\kappa=0.3$ in the deconfined phase (top panel) and at $\kappa=1.0$ in the Higgs phase (center panel), as a function of $\beta$, and at $\beta=2.6$ across the transition from the deconfined to the Higgs phase (bottom panel), as a function of $\kappa$. In all plots $N_{s}=20$ and $N_{t}=4$. The solid line in the top and center panels is a fit with a power-law function. The vertical and dashed lines in the bottom panel mark the critical value $\kappa_{\mathrm{loc}}$ and the corresponding error band [see after Eq. (36)]. | deconfined | Higgs
---|---|---
$\beta_{\mathrm{loc}}$ | $2.3318(15)$ | $2.0499(92)$
$a$ | $67.2(2.1)$ | $52.0(2.9)$
$b$ | $0.501(17)$ | $0.424(43)$
$\chi^{2}/\mathrm{dof}$ | 1.96 | 0.34
Table 5: Parameters of a best fit of the $\beta$ dependence of the average
size of the lowest modes, $\langle
N_{t}V\cdot\mathrm{PR}\rangle_{\lambda<\frac{\lambda_{c}}{2}}$, Eq. (37), with
the fitting function in Eq. (38).

It is interesting to compare the estimates of $\beta_{\mathrm{loc}}$ and
$\kappa_{\mathrm{loc}}$ obtained from eigenvalue observables to similar
estimates obtained from eigenvector observables. In particular, if
$\lambda_{c}$ vanishes continuously at $\beta_{\mathrm{loc}}$, then in the
thermodynamic limit the localization length of the low modes should
correspondingly diverge. We have then looked at the size of the low modes
averaged over the lowest half of the localized spectral region,

$\displaystyle
N_{t}V\cdot\langle\mathrm{PR}\rangle_{\lambda<\frac{\lambda_{c}}{2}}$
$\displaystyle=\frac{1}{\mathcal{N}({\textstyle\frac{\lambda_{c}}{2}})}\int_{0}^{\frac{\lambda_{c}}{2}}d\lambda\,\rho(\lambda)\,N_{t}V\cdot\mathrm{PR}(\lambda,N_{s})\,,$
(37) $\displaystyle\mathcal{N}(\lambda_{0})$
$\displaystyle=\int_{0}^{\lambda_{0}}d\lambda\,\rho(\lambda)\,.$

In Fig. 18 we show this quantity as a function of $\beta$ for constant
$\kappa=0.3$ in the deconfined phase (top panel) and $\kappa=1.0$ in the Higgs
phase (center panel). This quantity does indeed grow large as one approaches
the confined phase. Fits with a power-law function,

$N_{t}V\cdot\langle\mathrm{PR}\rangle_{\lambda<\frac{\lambda_{c}}{2}}=a\cdot(\beta-\beta_{\mathrm{loc}})^{-b}\,,$
(38)

yield $\beta_{\mathrm{loc}}=2.3318(15)$ in the deconfined phase, and
$\beta_{\mathrm{loc}}=2.0499(92)$ in the Higgs phase, both in the crossover
region, and in reasonable agreement with the determinations based on the
extrapolation of the mobility edge. Here one should take into account that the
functional form Eq. (38) is not fully justified, as the mode size cannot
diverge in a finite volume, and there is no reason to assume that the mode
size goes to zero at large $\beta$. Nonetheless, one obtains decent fits (see
the resulting fit parameters and $\chi^{2}$ in Tab. 5); adding a costant term
makes them worse. On top of this, the error estimates do not include any
uncertainty due to finite-size effects, which are large near
$\beta_{\mathrm{loc}}$. For completeness, in the bottom panel of Fig. 18 we
show $N_{t}V\cdot\langle\mathrm{PR}\rangle_{\lambda<\frac{\lambda_{c}}{2}}$ as
a function of $\kappa$ at constant $\beta=2.6$ across the two phases. Here the
data indicate a finite mode size at all $\kappa$, with a change from a
constant to a steadily decreasing trend taking place at the crossover between
the deconfined and the Higgs phase, showing that localized modes shrink
rapidly as one moves deeper in the Higgs phase and the Polyakov-loop
expectation value increases (see Fig. 1).

###  IV.4 Correlation with bosonic observables and sea/island mechanism

Figure 19: Polyakov loop weighted by Dirac modes, centered to its average and
rescaled by the square root of its susceptibility, Eq. (21), at $\beta=1.9$
and $\kappa=1.0$ in the confined phase (top panel; here $N_{s}=20$), at
$\beta=2.6$ and $\kappa=0.3$ in the deconfined phase (center panel; here
$N_{s}=32$), and at $\beta=2.1$ and $\kappa=1.0$ in the Higgs phase (bottom
panel; here $N_{s}=32$). In all plots $N_{t}=4$. In the center and bottom
panels the solid line shows the value of the mobility edge, and the dashed
lines indicate the corresponding error band.

Figure 20: Plaquette weighted by Dirac modes, centered to its average and
rescaled by the square root of its susceptibility, Eq. (21), at $\beta=1.9$
and $\kappa=1.0$ in the confined phase (top panel; here $N_{s}=20$), at
$\beta=2.6$ and $\kappa=0.3$ in the deconfined phase (center panel; here
$N_{s}=32$), and at $\beta=2.1$ and $\kappa=1.0$ in the Higgs phase (bottom
panel; here $N_{s}=32$). In all plots $N_{t}=4$. In the center and bottom
panels the solid line shows the value of the mobility edge, and the dashed
lines indicate the corresponding error band.

Figure 21: Gauge-Higgs coupling term weighted by Dirac modes, centered to its
average and rescaled by the square root of its susceptibility, Eq. (21), at
$\beta=1.9$ and $\kappa=1.0$ in the confined phase (top panel; here
$N_{s}=20$), at $\beta=2.6$ and $\kappa=0.3$ in the deconfined phase (center
panel; here $N_{s}=32$), and at $\beta=2.1$ and $\kappa=1.0$ in the Higgs
phase (bottom panel; here $N_{s}=32$). In all plots $N_{t}=4$. In the center
and bottom panels the solid line shows the value of the mobility edge, and the
dashed lines indicate the corresponding error band.

We now proceed to discuss our results on the correlation of staggered
eigenmodes with the gauge and Higgs fields. To this end, the most informative
quantities are the centered and normalized observables
$\widehat{\mathcal{U}}$, $\widehat{\mathcal{P}}$ and $\widehat{\mathcal{G}}$,
defined in Eq. (21), that take into account the width of the distribution of
the relevant bosonic observables. Our results for these quantities are shown
in Figs. 19–21. The statistical error on the numerical estimate of these
quantities is obtained by first determining the jackknife error on
$\mathcal{U}$, $\mathcal{P}$ and $\mathcal{G}$, and correspondingly on
$\langle U\rangle$, $\langle P\rangle$, $\langle G\rangle$ and on $(\delta
U)^{2}$, $(\delta P)^{2}$, $(\delta G)^{2}$, followed by linear error
propagation. Correlations with Polyakov-loop and plaquette fluctuations are
always negative, showing that low modes prefer locations where these
quantities fluctuate to values below their average. Correlations with gauge-
Higgs coupling term fluctuations are again negative in the confined and in the
Higgs phase, while they are essentially compatible with zero in the deconfined
phase.

The correlation of low modes with Polyakov-loop fluctuations is shown in Fig.
19. In the confined phase this is small but significant, and decreasing very
little in magnitude as one goes up in the spectral region that we explored.
The strength of this correlation is considerably larger in the Higgs phase,
and even larger in the deconfined phase. Since Polyakov-loop fluctuations are
typically localized in these phases, this increased correlation is possible
only if the low modes tend to localize on the corresponding locations. In both
the deconfined and the Higgs phase one sees also a more rapid decrease in the
magnitude of the correlation as one moves up in the spectrum. This, however,
remains stronger than for the lowest modes in the confined phase also above
the mobility edge.

The correlation of low modes with plaquette fluctuations is shown in Fig. 20.
Also in this case a significant correlation is found in all three phases,
generally stronger (and comparable in size) in the deconfined and Higgs phases
than in the confined phase. Compared to the correlation with Polyakov-loop
fluctuations, one finds a similar magnitude in the deconfined phase, and a
larger magnitude in the Higgs phase. Since also plaquette fluctuations are
typically localized, this means that they are at least as relevant as
Polyakov-loop fluctuations for the localization of low modes. A clear upturn
is visible for the lowest modes in the deconfined phase and, to a much smaller
extent, also in the Higgs phase. We do not have an explanation for this. Even
though the density of near-zero modes is very small in both cases, leading to
large fluctuations, this upturn might be significant, as the mode size
displays a similar behavior (see Figs. 8 and 9), with an increase in size for
the lowest modes. (The downturn seen in $I_{s_{0}}$, Figs. 13 and 14, may also
be related, but could also be a finite-size artifact caused by the low and
rapidly changing density of modes, that makes our unfolding procedure not
fully reliable in that spectral region.) The same upturn in the mode size is
observed also in QCD [4], where it can be explained by the topological origin
of the near-zero modes [59, 60]. Such modes are in fact expected to originate
in the mixing of the localized zero modes associated with topological lumps in
the gauge configuration at finite temperature, so extending over more than one
such lump. While they fail to become delocalized due to the low density of
lumps at high temperature, they nonetheless should display a larger size than
localized modes not of topological origin. This picture is consistent with the
strong correlation between localized near-zero modes and the local topology of
the gauge configuration, demonstrated in Ref. [7], and with the lumpy nature
of near-zero Dirac modes in high-temperature QCD, demonstrated in Ref. [61]. A
similar mechanism could explain the larger size of the lowest modes observed
here. Interestingly, no upturn in the size of the lowest modes is observed in
2+1 dimensional pure $\mathrm{SU}(3)$ gauge theory [24] or in 2+1 dimensional
discrete gauge theories [27, 16], where the topology of gauge field
configurations is trivial.

Finally, the correlation of low modes with fluctuations of the gauge-Higgs
coupling term is shown in Fig. 21. A very mild correlation is visible in the
confined phase, no significant correlation is found in the deconfined phase,
and a clear but small correlation is found in the Higgs phase, weaker than the
correlation with Polyakov-loop and plaquette fluctuations. This leads us to
conclude that these fluctuations are much less relevant to low-mode
localization.

Figure 22: The quantity $\widehat{\mathcal{A}}$, Eq. (32), measuring the
correlation of staggered modes with fluctuations of $A(\vec{x})$, Eq. (31), at
$\beta=1.9$ and $\kappa=1.0$ in the confined phase (top panel; here
$N_{s}=16$), at $\beta=2.6$ and $\kappa=0.3$ in the deconfined phase (center
panel; here $N_{s}=20$), and at $\beta=2.1$ and $\kappa=1.0$ in the Higgs
phase (bottom panel; here $N_{s}=20$). In all plots $N_{t}=4$. In the center
and bottom panels the solid line shows the value of the mobility edge, and the
dashed lines indicate the corresponding error band.

We then studied the sea/island mechanism directly by looking at the
correlation of the staggered eigenmodes with the local fluctuations of the
hopping term in the Dirac-Anderson Hamiltonian, measured by the quantity $A$
of Eq. (30). To this end we analyzed 450 configurations with $N_{s}=16$ in the
confined phase, and 1400 configurations with $N_{s}=20$ in the deconfined and
Higgs phases, with $N_{t}=4$ in both cases. The average value of $A$ drops
substantially as one moves from the confined to the deconfined or to the Higgs
phase: for the given lattice sizes (but this quantity is not expected to show
a strong volume dependence), $\langle A\rangle=0.2761(11)$ at
$\beta=1.9,\kappa=1.0$ in the confined phase; $\langle A\rangle=0.15828(64)$
at $\beta=2.6,\kappa=0.3$ in the deconfined phase; and $\langle
A\rangle=0.20518(86)$ at $\beta=2.1,\kappa=1.0$ in the Higgs phase. This is
expected to happen, as a consequence of the ordering of the Polyakov loop and
the resulting strong correlation in the temporal direction [16].

The centered and normalized quantity $\widehat{\mathcal{A}}$ defined in Eq.
(32) is shown in Fig. 22. This quantity correlates positively with the spatial
density of low modes in all phases, in agreement with the refined sea/islands
picture of Ref. [16]. In the confined phase the magnitude of the correlation
with fluctuations in this quantity is comparable with the correlation with
plaquette fluctuations, and independent of the position in the spectrum in the
available region, within errors. In the Higgs and, especially, in the
deconfined phase this correlation is much stronger than those with Polyakov-
loop and with plaquette fluctuations. Although it remains strong also at the
beginning of the bulk region, it reduces by about a third when going from the
lowest modes to the first delocalized modes right above the mobility edge.
Since fluctuations of $A(\vec{x})$ are typically localized in the deconfined
and Higgs phases, this result strongly suggests that they are the ones mainly
responsible for trapping the eigenmodes in space.

##  V Conclusions

A strong connection has emerged in recent years between the deconfinement
phase transition in gauge theories with or without fermionic matter, and the
change in the localization properties of low Dirac modes [3, 4, 5, 6, 7, 8, 9,
10, 18, 19, 20, 21, 22, 12, 13, 14, 15, 23, 28, 24, 25, 26, 29, 27, 16]. In
this paper we extended this line of research by studying the lattice
$\mathrm{SU}(2)$-Higgs model with a Higgs field of fixed length [36, 37, 38,
39, 40, 41, 42] at finite temperature, probed with external static fermions.
The extension is twofold. On the one hand, this model has dynamical scalar
rather than fermionic matter: while one still expects localized modes in the
deconfined phase of the model, as the nature of the dynamical matter does not
affect the general argument for localization [12, 13, 14, 15, 10, 16], it is
nonetheless useful to verify this explicitly. On the other hand, and more
interestingly, the two-parameter phase diagram of this model displays a third
phase besides the confined and deconfined phases, i.e., the Higgs phase: one
can then check whether or not modes are localized in this phase, and if so
whether the onset of localization is related in any way to the thermodynamic
transition.

A survey of the phase diagram shows the expected tripartition into a confined,
a deconfined, and a Higgs phase, separated by analytic crossovers [42]. The
deconfined and the Higgs phases are distinguished from the confined phase by a
much larger expectation value of the Polyakov loop, and from each other by the
expectation value of the Higgs-coupling term, much larger in the Higgs phase
than in the deconfined and in the confined phases. Since the Polyakov loop is
strongly ordered, one expects localization of low Dirac modes to take place in
both phases [12, 13, 14, 15, 10, 16].

By means of numerical simulations, we have demonstrated that localized modes
are indeed present both in the deconfined and in the Higgs phase. In both
cases, the mobility edge separating localized and delocalized modes in the
spectrum decreases as one moves towards the confined phase, and disappears as
one reaches the crossover region. At the transition between the deconfined and
the Higgs phase, instead, the dependence of the mobility edge on the gauge-
Higgs coupling constant changes from almost constant to steadily increasing.
These findings provide further support to the universal nature of the
sea/islands picture of localization [12, 13, 14, 15, 10, 16] in a previously
unexplored setup in the presence of dynamical scalar matter.

We have then studied the sea/islands mechanism in more detail, measuring the
correlation between localized modes and fluctuations of the gauge and Higgs
fields. We found a strong correlation with Polyakov-loop and plaquette
fluctuations both in the deconfined and in the Higgs phase, and a mild but
significant correlation with fluctuations of the gauge-Higgs coupling term
only in the Higgs phase. Moreover, we found in both phases a very strong
correlation (stronger than that with Polyakov-loop or plaquette fluctuations)
with the type of gauge-field fluctuations identified in Ref. [16] as the most
relevant to localization. This provides further evidence for the validity of
the refined sea/islands picture proposed in Ref. [16].

A possible extension of this work would be a study of the low $\beta$, large
$\kappa$ corner of the phase diagram, where the crossover becomes very weak,
in order to check if the line of “geometric” transitions where the mobility
edge in the Dirac spectrum vanishes extends all the way to $\beta=0$, or if
instead it has an endpoint. This is interesting also in connection with the
“spin glass” approach of Ref. [44]: since in that region of parameter space
this predicts a transition line clearly distinct from the one found with more
traditional approaches based on gauge fixing, one would like to compare this
line with the one defined by the vanishing of the mobility edge (if the latter
exists). A different direction would be the study of the localization
properties of the eigenmodes of the covariant Laplacian, extending to finite
temperature and dynamical scalar matter the work of Refs. [62, 63].

## Acknowledgments

We thank T.G. Kovács for useful discussions and a careful reading of the
manuscript. MG was partially supported by the NKFIH grant KKP-126769.

## References

  * Aoki  _et al._ [2006] Y. Aoki, G. Endrődi, Z. Fodor, S. D. Katz, and K. K. Szabó, Nature 443, 675 (2006), arXiv:hep-lat/0611014 .
  * Bazavov  _et al._ [2012] A. Bazavov _et al._ , Phys. Rev. D 85, 054503 (2012), arXiv:1111.1710 [hep-lat] .
  * García-García and Osborn [2007] A. M. García-García and J. C. Osborn, Phys. Rev. D 75, 034503 (2007), arXiv:hep-lat/0611019 [hep-lat] .
  * Kovács and Pittler [2012] T. G. Kovács and F. Pittler, Phys. Rev. D 86, 114515 (2012), arXiv:1208.3475 [hep-lat] .
  * Giordano  _et al._ [2014] M. Giordano, T. G. Kovács, and F. Pittler, Phys. Rev. Lett. 112, 102002 (2014), arXiv:1312.1179 [hep-lat] .
  * Ujfalusi  _et al._ [2015] L. Ujfalusi, M. Giordano, F. Pittler, T. G. Kovács, and I. Varga, Phys. Rev. D 92, 094513 (2015), arXiv:1507.02162 [cond-mat.dis-nn] .
  * Cossu and Hashimoto [2016] G. Cossu and S. Hashimoto, J. High Energy Phys. 06, 056 (2016), arXiv:1604.00768 [hep-lat] .
  * Holicki  _et al._ [2018] L. Holicki, E.-M. Ilgenfritz, and L. von Smekal, PoS LATTICE2018, 180 (2018), arXiv:1810.01130 [hep-lat] .
  * Kehr  _et al._ [2023] R. Kehr, D. Smith, and L. von Smekal, arXiv:2304.13617 [hep-lat] (2023).
  * Giordano and Kovács [2021] M. Giordano and T. G. Kovács, Universe 7, 194 (2021), arXiv:2104.14388 [hep-lat] .
  * Banks and Casher [1980] T. Banks and A. Casher, Nucl. Phys. B 169, 103 (1980).
  * Bruckmann  _et al._ [2011] F. Bruckmann, T. G. Kovács, and S. Schierenberg, Phys. Rev. D 84, 034505 (2011), arXiv:1105.5336 [hep-lat] .
  * Giordano  _et al._ [2015] M. Giordano, T. G. Kovács, and F. Pittler, J. High Energy Phys. 04, 112 (2015), arXiv:1502.02532 [hep-lat] .
  * Giordano  _et al._ [2016] M. Giordano, T. G. Kovács, and F. Pittler, J. High Energy Phys. 06, 007 (2016), arXiv:1603.09548 [hep-lat] .
  * Giordano  _et al._ [2017a] M. Giordano, T. G. Kovács, and F. Pittler, Phys. Rev. D 95, 074503 (2017a), arXiv:1612.05059 [hep-lat] .
  * Baranka and Giordano [2022] G. Baranka and M. Giordano, Phys. Rev. D 106, 094508 (2022), arXiv:2210.00840 [hep-lat] .
  * Note [1] A notable exception is when the (untraced) Polyakov loops order along minus the identity, in which case low modes remain delocalized also in the deconfined phase [27].
  * Göckeler  _et al._ [2001] M. Göckeler, P. E. L. Rakow, A. Schäfer, W. Söldner, and T. Wettig, Phys. Rev. Lett. 87, 042001 (2001), arXiv:hep-lat/0103031 [hep-lat] .
  * Gattringer  _et al._ [2001] C. Gattringer, M. Göckeler, P. E. L. Rakow, S. Schaefer, and A. Schäfer, Nucl. Phys. B 618, 205 (2001), arXiv:hep-lat/0105023 [hep-lat] .
  * Gavai  _et al._ [2008] R. V. Gavai, S. Gupta, and R. Lacaze, Phys. Rev. D 77, 114506 (2008), arXiv:0803.0182 [hep-lat] .
  * Kovács [2010] T. G. Kovács, Phys. Rev. Lett. 104, 031601 (2010), arXiv:0906.5373 [hep-lat] .
  * Kovács and Pittler [2010] T. G. Kovács and F. Pittler, Phys. Rev. Lett. 105, 192001 (2010), arXiv:1006.1205 [hep-lat] .
  * Kovács and Vig [2018] T. G. Kovács and R. Á. Vig, Phys. Rev. D 97, 014502 (2018), arXiv:1706.03562 [hep-lat] .
  * Giordano [2019] M. Giordano, J. High Energy Phys. 05, 204 (2019), arXiv:1903.04983 [hep-lat] .
  * Vig and Kovács [2020] R. Á. Vig and T. G. Kovács, Phys. Rev. D 101, 094511 (2020), arXiv:2001.06872 [hep-lat] .
  * Bonati  _et al._ [2021] C. Bonati, M. Cardinali, M. D’Elia, M. Giordano, and F. Mazziotti, Phys. Rev. D 103, 034506 (2021), arXiv:2012.13246 [hep-lat] .
  * Baranka and Giordano [2021] G. Baranka and M. Giordano, Phys. Rev. D 104, 054513 (2021), arXiv:2104.03779 [hep-lat] .
  * Giordano  _et al._ [2017b] M. Giordano, S. D. Katz, T. G. Kovács, and F. Pittler, J. High Energy Phys. 02, 055 (2017b), arXiv:1611.03284 [hep-lat] .
  * Cardinali  _et al._ [2022] M. Cardinali, M. D’Elia, F. Garosi, and M. Giordano, Phys. Rev. D 105, 014506 (2022), arXiv:2110.10029 [hep-lat] .
  * Ghanbarpour and von Smekal [2022] M. Ghanbarpour and L. von Smekal, Phys. Rev. D 106, 054513 (2022), arXiv:2206.11697 [hep-lat] .
  * Myers and Ogilvie [2008] J. C. Myers and M. C. Ogilvie, Phys. Rev. D 77, 125030 (2008), arXiv:0707.1869 [hep-lat] .
  * Ünsal and Yaffe [2008] M. Ünsal and L. G. Yaffe, Phys. Rev. D 78, 065035 (2008), arXiv:0803.0344 [hep-th] .
  * Bonati  _et al._ [2018] C. Bonati, M. Cardinali, and M. D’Elia, Phys. Rev. D 98, 054508 (2018), arXiv:1807.06558 [hep-lat] .
  * Bonati  _et al._ [2020] C. Bonati, M. Cardinali, M. D’Elia, and F. Mazziotti, Phys. Rev. D 101, 034508 (2020), arXiv:1912.02662 [hep-lat] .
  * Athenodorou  _et al._ [2021] A. Athenodorou, M. Cardinali, and M. D’Elia, Phys. Rev. D 104, 074510 (2021), arXiv:2010.03618 [hep-lat] .
  * Fradkin and Shenker [1979] E. H. Fradkin and S. H. Shenker, Phys. Rev. D 19, 3682 (1979).
  * Lang  _et al._ [1981] C. B. Lang, C. Rebbi, and M. Virasoro, Phys. Lett. B 104, 294 (1981).
  * Montvay [1985] I. Montvay, Phys. Lett. B 150, 441 (1985).
  * Montvay [1986] I. Montvay, Nucl. Phys. B 269, 170 (1986).
  * Langguth and Montvay [1985] W. Langguth and I. Montvay, Phys. Lett. B 165, 135 (1985).
  * Campos [1998] I. Campos, Nucl. Phys. B 514, 336 (1998), arXiv:hep-lat/9706020 .
  * Bonati  _et al._ [2010] C. Bonati, G. Cossu, M. D’Elia, and A. Di Giacomo, Nucl. Phys. B 828, 390 (2010), arXiv:0911.1721 [hep-lat] .
  * Lautrup and Nauenberg [1980] B. E. Lautrup and M. Nauenberg, Phys. Rev. Lett. 45, 1755 (1980).
  * Greensite and Matsuyama [2022] J. Greensite and K. Matsuyama, Symmetry 14, 177 (2022), arXiv:2112.06421 [hep-lat] .
  * Mehta [2004] M. L. Mehta,  _Random matrices_ , 3rd ed., Pure and Applied Mathematics, Vol. 142 (Academic Press, 2004).
  * Verbaarschot and Wettig [2000] J. J. M. Verbaarschot and T. Wettig, Ann. Rev. Nucl. Part. Sci. 50, 343 (2000), arXiv:hep-ph/0003017 [hep-ph] .
  * Note [2] Since $\delimiter 2536205T\psi_{l}(n)\delimiter 2536205^{2}=\delimiter 2536205\psi_{l}(n)\delimiter 2536205^{2}$, eigenmodes in a degenerate pair have identical $(\mathrm{IPR}_{q})_{l}$.
  * Evers and Mirlin [2008] F. Evers and A. D. Mirlin, Rev. Mod. Phys. 80, 1355 (2008), arXiv:0707.4378 [cond-mat.mes-hall] .
  * Al’tshuler and Shklovskiĭ [1986] B. L. Al’tshuler and B. I. Shklovskiĭ, Sov. Phys. JETP 64, 127 (1986).
  * Shklovskiĭ  _et al._ [1993] B. I. Shklovskiĭ, B. Shapiro, B. R. Sears, P. Lambrianides, and H. B. Shore, Phys. Rev. B 47, 11487 (1993).
  * Note [3] In the definition of $U_{j}^{\rm tdg}$ in Eq. (B.8) of Ref. [16] the factors $u(\mathaccentV{vec}17E{x})$ and $u(\mathaccentV{vec}17E{x}+\mathaccentV{hat}05E{\jmath})^{\dagger}$ are incorrectly missing.
  * Engels  _et al._ [1990] J. Engels, J. Fingberg, and M. Weber, Nucl. Phys. B 332, 737 (1990).
  * Engels  _et al._ [1992] J. Engels, J. Fingberg, and D. E. Miller, Nucl. Phys. B 387, 501 (1992).
  * Engels  _et al._ [1993] J. Engels, J. Fingberg, and V. K. Mitrjushkin, Phys. Lett. B 298, 154 (1993).
  * Engels  _et al._ [1996] J. Engels, S. Mashkevich, T. Scheideler, and G. Zinovev, Phys. Lett. B 365, 219 (1996), arXiv:hep-lat/9509091 .
  * Stathopoulos and McCombs [2010] A. Stathopoulos and J. R. McCombs, ACM Transactions on Mathematical Software 37, 21:1 (2010).
  * Wu  _et al._ [2017] L. Wu, E. Romero, and A. Stathopoulos, SIAM Journal on Scientific Computing 39, S248 (2017), arXiv:1607.01404 [cs.MS] .
  * James and Roos [1975] F. James and M. Roos, Comput. Phys. Commun. 10, 343 (1975).
  * Edwards  _et al._ [2000] R. G. Edwards, U. M. Heller, J. E. Kiskis, and R. Narayanan, Phys. Rev. D 61, 074504 (2000), arXiv:hep-lat/9910041 .
  * Vig and Kovács [2021] R. Á. Vig and T. G. Kovács, Phys. Rev. D 103, 114510 (2021), arXiv:2101.01498 [hep-lat] .
  * Meng  _et al._ [2023] X.-L. Meng, P. Sun, A. Alexandru, I. Horváth, K.-F. Liu, G. Wang, and Y.-B. Yang ($\chi$QCD, CLQCD), arXiv:2305.09459 [hep-lat] (2023).
  * Greensite  _et al._ [2005] J. Greensite, Š. Olejník, M. Polikarpov, S. Syritsyn, and V. Zakharov, Phys. Rev. D 71, 114507 (2005), arXiv:hep-lat/0504008 [hep-lat] .
  * Greensite  _et al._ [2006] J. Greensite, A. V. Kovalenko, Š. Olejník, M. I. Polikarpov, S. N. Syritsyn, and V. I. Zakharov, Phys. Rev. D 74, 094507 (2006), arXiv:hep-lat/0606008 [hep-lat] .