# Quantum excitations of static charges in the Ginzburg-Landau model of
superconductivity

Jeff Greensite and Kazue Matsuyama  Physics and Astronomy Department
San Francisco State University
San Francisco, CA 94132, USA

###### Abstract

We point out that in superconductors there may exist localized quantum
excitations of the electric and condensate fields surrounding a static charge,
which cannot be interpreted as simply the ground state of the screened charge
plus some number of massive photons. This is illustrated via a lattice Monte
Carlo calculation of the energy spectrum of a pair of separated static charges
in an effective Ginzburg-Landau model of superconductivity.

Confinement,lattice gauge theories

###### pacs:

11.15.Ha, 12.38.Aw

##  I  Introduction

Composite quantum systems typically have a discrete spectrum of excitations,
and although certain objects such as electrons and muons are referred to as
“elementary” particles, this terminology can be a little misleading, since in
any interacting quantum field theory these elementary particles are inevitably
surrounded by a field of some kind. In this sense, elementary particles are
also composite. The question we would like to address here is whether there
can exist a discrete spectrum of localized excitations in the fields
surrounding a static charge, and in particular whether this might be the case
for the fields surrounding static ions in a superconductor.

Let us begin with the simplest case: a static charge coupled to the free
quantized Maxwell field. Gauge invariance of physical states under
infinitesimal gauge transformations, embodied in the Gauss Law, insists that
the static charge must be the source of an electric field. The ground state of
the static charge + gauge field system, as shown long ago by Dirac
Dirac:1955uv , is given by

$\displaystyle|\Psi_{\bm{x}}\rangle=\overline{\psi}({\bm{x}})\rho({\bm{x}};A)|\Psi_{0}\rangle\
,$ (1)

where $\overline{\psi}({\bm{x}})$, operating on the vacuum, creates the static
charge, and

$\displaystyle\rho({\bm{x}};A)$ $\displaystyle=$
$\displaystyle\exp\left[-i{e\over 4\pi}\int
d^{3}z~{}A_{i}(\bm{z}){\partial\over\partial
z_{i}}{1\over|{\bm{x}}-\bm{z}|}\right]\ .$ (2)

In the state $|\Psi_{\bm{x}}\rangle$ the expectation value of the electric
field is simply the Coulomb field of the static charge. The operator
$\rho(x;A)$, which depends only on the gauge field, is an example of what we
call a “pseudomatter” operator.111It is worth noting that $\rho^{*}(x;A)$ is
also the gauge transformation to Coulomb gauge. In fact, if $g(x;A)$ is the
gauge transformation to a physical gauge defined by some condition $F[A]=0$
imposed on each time slice, then it is not hard to see that $g^{*}(x;A)$ is
pseudomatter operator. This statement can be readily generalized to non-
abelian theories. Operators of this kind, functionals of only the gauge field,
transform under an infinitesimal gauge transformation like a static charge,
but are invariant under global gauge transformations $g(x)=g$ in the center of
the gauge group, which in this case is $g(x)=\exp(i\theta)\in$ global U(1). As
a result, the physical state (1) is not entirely gauge invariant; in fact it
transforms covariantly under such global U(1) transformations, i.e.

$\displaystyle|\Psi_{\bm{x}}\rangle\rightarrow
e^{-i\theta}|\Psi_{\bm{x}}\rangle\ .$ (3)

This is characteristic of an isolated charge in electrodynamics, unscreened by
any matter fields. It should be emphasized that covariance under the global
U(1) subgroup of the gauge group is in no way a violation of the physical
state condition (i.e. the Gauss law), which does not require invariance under
global transformations which affect neither the gauge field nor the charge
densities. Of course this global symmetry of the action may or may not be a
symmetry of the vacuum. As we have discussed extensively elsewhere
Greensite:2020nhg ; *Greensite:2021fyi, it is the global center subgroup of
the gauge group which is broken spontaneously in the Higgs phase of a gauge
Higgs theory, with a corresponding gauge-invariant order parameter closely
analogous to the Edwards-Anderson order parameter Edward_Anderson  for a spin
glass. Symmetry breaking of this kind is not a violation of Elitzur’s theorem,
which forbids the spontaneous breaking of local, rather than global
symmetries. While the transition to the Higgs phase is not necessarily a
thermodynamic transition, it is nonetheless accompanied by a change in
physical behavior, which is either the loss of metastable flux tube states and
Regge trajectories (transition from the confinement phase), or the
disappearance of massless vector bosons (transition from the Coulomb phase).

For our purposes the free field example is not so interesting. Of course there
are excitations of the charge+quantized field combination, but these
excitations consist simply of some number of photons, of arbitrary positive
energies, superimposed on a background Coulombic field; they are not the
localized excitations with a discrete energy spectrum that are of interest
here. We therefore move on to gauge Higgs theories, including those which,
like the Ginzburg-Landau model, may simply be the effective theory of some
more fundamental physics in which the Higgs field is composite. There is
already evidence of excitations of static charges in SU(3) gauge Higgs theory
Greensite:2020lmh , and in the abelian Higgs model Matsuyama:2020tvt . But the
theories of phenomenological interest would be the Ginzburg-Landau model of
superconductivity, and the electroweak sector of the standard model. A
spectrum of quark and lepton excitations, or more precisely a spectrum of
localized excitations of the fields surrounding quarks and leptons, would
obviously be of great interest, but here we face the formidable complication
that the electroweak theory is a chiral gauge theory, which resists a lattice
formulation.222See, however, the attempt to find excitations in a chiral U(1)
theory in Greensite:2021jef . For that reason we turn our attention to quantum
behavior in the Ginzburg-Landau model with static ions.

##  II the calculation

The effective Ginzburg-Landau action is (cf. coleman_2015 )

$\displaystyle S$ $\displaystyle=$ $\displaystyle\int
d^{4}x\bigg{\\{}\frac{1}{2}\rho_{s}\bigg{(}{1\over\upsilon^{2}}(\partial_{0}\xi+2eA_{0})^{2}$
(4) $\displaystyle\qquad-(\nabla\xi-2e{\bf
A})^{2}\bigg{)}+\frac{1}{2}(E^{2}-B^{2})\bigg{\\}}\ ,$

where $\xi$ is the Goldstone mode, $\rho_{s}=n_{s}/2M$ where $n_{s},M,2e$ are
the Cooper pair density, mass, and charge respectively, and $\upsilon\sim
10^{-2}$ is on the order of the ratio of the Fermi velocity in a metal to the
speed of light. The mass of the transverse photon in the continuum formulation
is, in natural units,

$M_{ph}={1\over\lambda_{L}}=2e\sqrt{\rho_{s}}\ ,$ (5)

where $\lambda_{L}$ is the London penetration depth. The transition to the
lattice formulation in Euclidean time, which is the non-relativistic version
of the charge $q=2$ lattice abelian Higgs model, is straightforward:

$\displaystyle S_{eff}$ $\displaystyle=$
$\displaystyle-\beta\sum_{plaq}\text{Re}[UUU^{*}U^{*}]$ (6)
$\displaystyle-\gamma\sum_{x}\sum_{k=1}^{3}\text{Re}[\phi^{*}(x)U^{2}_{k}(x)\phi(x+\hat{k})]$
$\displaystyle-{\gamma\over\upsilon^{2}}\sum_{x}\text{Re}[\phi^{*}(x)U^{2}_{0}(x)\phi(x+\hat{t})\
,$

where $\beta=1/e^{2}\approx 10.9,~{}\phi=e^{i\xi},~{}\rho_{s}=\gamma/a^{2},$
and $a$ is the lattice spacing. A choice of gamma, together with an observed
London penetration depth and the tree-level relationship (5), requires a
lattice spacing

$a=2e\gamma^{\frac{1}{2}}\lambda_{L}\ .$ (7)

Obviously the continuum limit would require $\gamma\rightarrow 0$, but this is
impossible because of a transition to the massless phase at finite $\gamma$.
At $\beta=10.9$ we find, from investigation of the susceptibility of
$\langle\text{Re}[\phi^{*}(x)U^{2}_{k}(x)\phi(x+\hat{k})\rangle$, a transition
close to $\gamma=0.017$ (see the Appendix). So it must be emphasized that the
lattice formulation is an effective theory, valid beyond some inherent short
distance cutoff which, since Cooper pairs are composite, would most naturally
be the diameter of a Cooper pair. In most texts this effective theory is
treated classically, but here we would like to treat the effective theory as a
quantum theory in its own right, with a short distance cutoff $a$. In lattice
Monte Carlo simulations of $S_{eff}$ it is convenient to fix to unitary gauge,
in which case , since $1/\upsilon^{2}\sim 10^{4}$, it is a very good
approximation to take $U_{0}(x)=\pm 1$.

The aim is to search for excitations around pairs of widely separated static
$q=\pm 1$ (e) charges, having fixed ions in mind. We are obliged to consider
pairs of charges for technical reasons discussed below, which are related to
the fact that one cannot create a single charge in a periodic volume. We
therefore consider physical states of the following form:

$|\Phi_{n}({\bm{x}},{\bm{y}})\rangle=Q_{n}({\bm{x}},{\bm{y}})|\Psi_{0}\rangle\
,$ (8)

with

$\displaystyle Q_{2n-1}({\bm{x}},{\bm{y}})$ $\displaystyle=$
$\displaystyle\overline{\psi}({\bm{x}})\zeta_{n}({\bm{x}})\zeta_{n}^{*}({\bm{y}})\psi({\bm{y}})$
$\displaystyle Q_{2n}({\bm{x}},{\bm{y}})$ $\displaystyle=$
$\displaystyle\overline{\psi}({\bm{x}})\phi({\bm{x}})\zeta^{*}_{n}({\bm{x}})\zeta_{n}({\bm{y}})\phi^{*}({\bm{y}})\psi({\bm{y}})\
,$ (9)

where $\overline{\psi},\psi$ are static fermion operators, and the
pseudomatter operators $\zeta_{n}(x)$ are eigenstates of the covariant
Laplacian operator

$\sum_{{\bm{y}}}(-D^{2})_{{\bm{x}}{\bm{y}}}\zeta_{n}({\bm{y}})=\lambda_{n}\zeta({\bm{x}})\
,$ (10)

where

$\displaystyle(-D^{2})_{{\bm{x}}{\bm{y}}}=\sum_{k=1}^{3}\left[2\delta_{{\bm{x}}{\bm{y}}}-U_{k}({\bm{x}})\delta_{{\bm{y}},{\bm{x}}+\hat{k}}-U_{k}^{\dagger}({\bm{x}}-\hat{k})\delta_{{\bm{y}},{\bm{x}}-\hat{k}}\right]\
.$ (11)

Note that $\zeta_{n}({\bm{x}})$ is a pseudomatter operator, and
$\phi({\bm{x}})\zeta_{n}^{*}({\bm{x}})$ transforms like a pseudomatter
operator. We have found it important, in the calculations discussed below, to
include both.

Now let us consider a subspace of Hilbert space spanned by $N$ physical states
$\\{|\Phi_{n}({\bm{x}},{\bm{y}})\rangle,n=1,..,N\\}$, and let $\tau$ be the
transfer matrix. Define the rescaled transfer matrix

$\mathcal{T}=\tau e^{\mathcal{E}_{0}}\ ,$ (12)

where $\mathcal{E}_{0}$ is the vacuum energy (or, more precisely, if
$\kappa_{0}$ is the largest eigenvalue of $\tau$, then
$\mathcal{E}_{0}=-\log(\kappa_{0})$). We would like to construct a set of
eigenstates of $\mathcal{T}$ in the truncated basis. For this non-orthogonal
basis, we first calculate numerically the matrix elements and overlaps,

$\displaystyle[\mathcal{T}]_{\alpha\beta}(R)$ $\displaystyle=$
$\displaystyle\langle\Phi_{\alpha}|\mathcal{T}|\Phi_{\beta}\rangle$
$\displaystyle=$ $\displaystyle\langle
Q^{\dagger}_{\alpha}({\bm{x}},{\bm{y}},1)Q_{\beta}({\bm{x}},{\bm{y}},0)\rangle$
$\displaystyle\left[O\right]_{\alpha\beta}(R)$ $\displaystyle=$
$\displaystyle\langle\Phi_{\alpha}|\Phi_{\beta}\rangle$ (13) $\displaystyle=$
$\displaystyle\langle
Q^{\dagger}_{\alpha}({\bm{x}},{\bm{y}},0)Q_{\beta}({\bm{x}},{\bm{y}},0)\rangle\
.$

where $R=|{\bm{x}}-{\bm{y}}|$, and $Q({\bm{x}},{\bm{y}},t)$, in Euclidean time
path-integral formulation, is the $Q({\bm{x}},{\bm{y}})$ operator acting at
time $t$. We obtain the eigenvalues of $\mathcal{T}$ in the subspace by
solving the generalized eigenvalue problem,

$\displaystyle[\mathcal{T}]\bm{\upsilon}^{(n)}$ $\displaystyle=$
$\displaystyle\lambda_{n}[O]\bm{\upsilon}^{(n)}$
$\displaystyle|\Psi_{n}({\bm{x}},{\bm{y}})\rangle$ $\displaystyle=$
$\displaystyle\sum_{\alpha=1}^{N}\upsilon^{(n)}_{\alpha}|\Phi_{\alpha}({\bm{x}},{\bm{y}})\rangle\
.$ (14)

Each $|\Psi_{n}({\bm{x}},{\bm{y}})\rangle$ is a linear combination of the non-
orthogonal basis states $|\Phi_{\alpha}({\bm{x}},{\bm{y}})\rangle$, and the
set of states $\\{|\Psi_{n}({\bm{x}},{\bm{y}})\rangle\\}$ are the energy
eigenstates (i.e. eigenstates of the transfer matrix) of the isolated static
pair in the subspace.

Let us define, for $M$ pseudomatter operators $\zeta_{\alpha}$, the $N=2M$
operators

$\displaystyle V_{2\alpha-1}({\bm{x}},{\bm{y}},U)$ $\displaystyle=$
$\displaystyle\zeta_{\alpha}({\bm{x}})\zeta^{*}_{\alpha}({\bm{y}})$
$\displaystyle V_{2\alpha}({\bm{x}},{\bm{y}},U)$ $\displaystyle=$
$\displaystyle\zeta^{*}_{\alpha}({\bm{x}})\phi({\bm{x}})\phi^{*}({\bm{y}})\zeta_{\alpha}({\bm{y}})$
(15)

In order to calculate the required matrix elements numerically, we integrate
out the heavy fermions (via a hopping parameter expansion Gattringer:2010zz )
to obtain

$\displaystyle[\mathcal{T}^{T}]_{\alpha\beta}(R)$ $\displaystyle=$
$\displaystyle\langle\Phi_{\alpha}|\mathcal{T}^{T}|\Phi_{\beta}\rangle$
$\displaystyle=$ $\displaystyle\langle
Q_{\alpha}^{\dagger}({\bm{x}},{\bm{y}},T)Q_{\beta}({\bm{x}},{\bm{y}},0)\rangle$
$\displaystyle=$ $\displaystyle\langle
V^{\dagger}_{\alpha}({\bm{x}},{\bm{y}};U(t+T))P^{\dagger}({\bm{x}},t,T)$
$\displaystyle~{}~{}\times
V_{\beta}({\bm{x}},{\bm{y}};U(t))P({\bm{y}},t,T)\rangle$
$\displaystyle\left[O\right]_{\alpha\beta}(R)$ $\displaystyle=$
$\displaystyle\langle
V^{\dagger}_{\alpha}({\bm{x}},{\bm{y}};U(t))V_{\beta}({\bm{x}},{\bm{y}};U(t))\rangle$
(16)

where indices $\alpha,\beta$ range from 1 to $N$, and

$P({\bm{x}},t,T)=U_{0}({\bm{x}},t)U_{0}({\bm{x}},t+1)...U_{0}({\bm{x}},t+T-1)\
.$ (17)

is a timelike Wilson line of length $T$. In (16) we have dropped powers of the
fermion mass in the hopping parameter expansion, which only contribute an
overall constant to the energy of the fermion-antifermion system, and are
therefore irrelevant to the question of excitations.

The eigenstates $\\{|\Psi_{n}\rangle\\}$ of the transfer matrix in the
$N$-dimensional subspace are in general not eigenstates of the transfer matrix
in the full Hilbert space. But it may happen that $\Psi_{1}$, the state with
the lowest energy expectation value in the subspace, has a very large overlap
with the static charge ground state in the full Hilbert space, and it follows
that the $\Psi_{n>1}$ would have a correspondingly small overlap. On general
grounds

$\displaystyle\mathcal{T}_{nn}(R,T)$ $\displaystyle\equiv$
$\displaystyle\langle\Psi_{n}|\mathcal{T}^{T}|\Psi_{n}\rangle$ (18)
$\displaystyle=$
$\displaystyle\sum_{\alpha\beta}\upsilon^{*(n)}_{\alpha}\langle
Q_{\alpha}^{\dagger}({\bm{x}},{\bm{y}},T)Q_{\beta}({\bm{x}},{\bm{y}},0)\rangle\upsilon^{(n)}_{\beta}$
$\displaystyle=$ $\displaystyle\sum_{k}|c_{k}(R)|^{2}e^{-E_{k}(R)T}\ ,$

where $E_{k}$ is the energy above the ground state of the ${k\mbox{-th}}$
energy eigenstate in the full Hilbert space, containing a static fermion-
antifermion pair separated by a distance $R$. If
$\Psi_{n>1}({\bm{x}},{\bm{y}})$ has a large overlap with one excited energy
eigenstate $\Psi^{exact}_{i}$, and a very small overlap with the ground state,
then we may expect that for some range of $T_{min}\leq T\leq T_{max}$

$\mathcal{T}_{nn}(R,T)\approx|c_{i}(R)|^{2}e^{-E_{i}(R)T}~{}~{}~{},~{}~{}~{}T_{min}\leq
T\leq T_{max}\ ,$ (19)

and in that case we may extract the excitation energy $E_{i}(R)$ from a
logarithmic plot of $\mathcal{T}_{nn}(R,T)$ vs. $T$.

Of course there is no guarantee that a strategy of this kind will work. It
relies on the conjectures that the overlap of $|\Psi_{1}\rangle$ with the true
$\overline{\psi}\psi$ ground state is very large. Even if this conjecture is
true, it might be the case that the excited states are nothing more than the
ground state plus one or more gauge bosons, as in pure QED. But the general
idea is testable, and our results for the Ginzburg-Landau model in the Higgs
phase are reported in the next section.

###  II.1 Why not single fermion states?

Before presenting numerical results, we would like to address this question:
why not consider single fermion states of the form

$\Omega_{\alpha}=\overline{\psi}({\bm{x}})\zeta_{\alpha}({\bm{x}})\Psi_{0}\ ,$
(20)

where $\zeta_{\alpha}$ is a pseudomatter field. We assume that there is some
algorithm which determines $\zeta({\bm{x}},t)$ uniquely on time slice $t$ from
the spacelike link variables on that time slice. Such numerical algorithms
exist for Laplacian eigenstates, and also for numerical fixing to Coulomb
gauge.333The iterative gauge-fixing algorithms used in computer simulations
are deterministic, and fix to a unique Gribov copy satisfying the gauge fixing
condition. The latter determines the gauge transformation to Coulomb gauge,
which is itself a pseudomatter field, in a finite periodic volume.

Now consider the transfer matrix elements

$\langle\Omega_{\alpha}|\mathcal{T}|\Omega_{\beta}\rangle=\langle\zeta_{\alpha}^{\dagger}({\bm{x}},1)U^{\dagger}_{0}({\bm{x}},0)\zeta_{\beta}({\bm{x}},0)\rangle\
,$ (21)

which is essential in our discussion of energy expectation values. The problem
here is that the operator on the right hand side of (21) is not invariant
under gauge transformations $g({\bm{x}},t)=e^{i\theta(t)}$ or, in an SU(N)
gauge theory, under transformations $g({\bm{x}},t)=z(t)\in Z_{N}$.
Transformations of this kind do not alter spacelike links, and hence do not
alter the pseudomatter fields $\zeta_{\alpha}(x)$. Nor do these
transformations, which in each time slice belong to the global $Z_{N}$ center
subgroup of the gauge group, alter the gauge-invariant action. But they do
transform the timelike links $U_{0}({\bm{x}},t)$. As a result, the matrix
elements (21) all vanish, at least in a finite periodic volume. This is very
general, it holds both in pure gauge theory and gauge Higgs theories, and it
is one way of understanding why, on the lattice, it is impossible to place a
single charge in a periodic volume, despite the fact that one can numerically
transform the gauge field to some unique Gribov copy of Coulomb gauge. If
there were a single-charged matter field in the theory, call it $\varphi$,
then the same objection would not hold for the neutral state

$\Psi=\overline{\psi}({\bm{x}})\varphi({\bm{x}})\Psi_{0}\ ,$ (22)

as the operator $\overline{\psi}({\bm{x}})\varphi({\bm{x}})$ is fully gauge
invariant, including under global gauge transformations, and the expression in
(21), with $\zeta$ replaced by $\varphi$, is invariant under
$g({\bm{x}},t)=e^{i\theta(t)}$. But there are no such single-charged dynamical
fields in the Ginzburg-Landau model under consideration.

It is for these reasons that we must consider pseudomatter states with two
static charges,

$\displaystyle\Phi_{\alpha\beta}({\bm{x}},{\bm{y}})=\overline{\psi}({\bm{x}})\zeta_{\alpha}({\bm{x}})\zeta^{*}_{\beta}({\bm{y}})\psi({\bm{y}})\Psi_{0}\
,$ (23)

where the $\zeta_{\alpha}$ are eigenstates of the covariant Laplacian
operator. But here there is still an ambiguity, in that if
$\zeta_{\alpha}({\bm{x}})$ is a Laplacian eigenstate, then so is

$\zeta^{\prime}_{\alpha}({\bm{x}};U)=e^{iF_{\alpha}[U]}\zeta_{\alpha}({\bm{x}};U)\
,$ (24)

where $F_{\alpha}[U]$ is any real-valued functional of the link variables. So
in general we may expect that any $\zeta_{\alpha}({\bm{x}};U)$ determined by
the Arnoldi algorithm will have wild fluctuations in a global phase factor as
$U$ is varied, and fluctuations of that sort will cause

$\langle\Phi_{\alpha\beta}|\mathcal{T}^{T}|\Phi_{\alpha\beta}\rangle$ (25)

to vanish, for any $T>0$, and $\alpha\neq\beta$. And in fact this is what is
observed in numerical simulations. For this reason we must restrict our
considerations to states $\Phi_{n}$ of the form (8,9), where there is no
remaining ambiguity, and any global phase factor as in (24) cancels out.

###  II.2 Photon mass

The photon mass in the Ginzburg-Landau model is determined from the time
correlation of space-averaged gauge-invariant link operators. We define

$G(t)={1\over
3}\sum_{i=1}^{3}\langle\mathcal{A}_{i}(0)\mathcal{A}_{i}(t)\rangle\ ,$ (26)

where

$\mathcal{A}_{i}(t)={1\over
L^{3}}\sum_{\mathbf{x}}\mbox{Im}[\phi^{\dagger}({\bm{x}},t)U^{2}_{i}({\bm{x}},t)\phi({\bm{x}}+\hat{i})]\
.$ (27)

A typical result, at $\beta=10.9$ and $\gamma=0.60$ is shown in Fig. 1. The
data was taken on a $16^{3}\times 36$ lattice volume, with 400 lattices
separated by 100 sweeps. An exponential fit in the range $1\leq t\leq 10$
gives a value of $m_{ph}=0.446(3)$ for the photon mass in lattice units. A
similar fit with the same parameters, except for $\gamma=0.25$, results in
$m_{ph}=0.288(1)$. We note that both of these values are quite close to the
tree-level expression $m_{ph}=2e\sqrt{\gamma}$ (with $m_{ph}$ in lattice
units), obtained by multiplying both sides of eq. (5) (where $M_{ph}$ is in
physical units) by the lattice spacing $a$.

Figure 1: Determination of the photon mass at $\beta=10.9,\gamma=0.6$ from the
time correlator $G(t)$ of space averaged gauge invariant link variables on a
$16^{3}\times 36$ lattice volume.

##  III Excitations

We set $\beta=10.9$, corresponding to ${e^{2}/4\pi}=1/137$ in natural units,
and begin with Higgs coupling $\gamma=0.6$ on a $12^{3}\times 36$ lattice
volume. We consider static charge $q=\pm 1$ pairs at all pairs of lattice
coordinates ${\bm{x}}=(x_{1},x_{2},x_{3}),{\bm{y}}=(y_{1},y_{2},y_{3})$ with
$y_{i}=x_{i}+\Delta_{i}$ and $-4\leq\Delta_{i}\leq 4$. This allows a maximum
off-axis separation of $R=|{\bm{x}}-{\bm{y}}|=6.93$. Four Laplacian
eigenstates $\zeta_{n}({\bm{x}})$, and therefore eight operators
$Q_{\alpha}(R)$ at each $R$, were used in the computation.

Fig. 2 shows an example of our results for $\mathcal{T}_{nn}(R,T)$ at
$R=5.83$. We see that $\mathcal{T}_{11}(R,T)=1$ , independent of $T$, to a
very high degree of precision. This means that the ground state energy is
$E_{1}(R)=0$, and it turns out that this result is obtained at all $R$, not
just at $R=5.83$. The $\mathcal{T}_{22}$ and $\mathcal{T}_{33}$ data are fit
to an exponential

$c_{n}(e^{-E_{n}T}+e^{-E_{n}(36-T)})\ .$ (28)

It can be seen that the data for $\mathcal{T}_{22},\mathcal{T}_{33}$ fall off
linearly on the logarithmic plot, with fits taken in the range $[2:7]$, and
the slopes are slightly different, corresponding to a small energy difference
between $E_{2}$ and $E_{3}$. Beyond $T=8$ or so there are rather large error
bars, so those points are not included in the fit.

Figure 2: $\mathcal{T}_{nn}(R,T)$ vs. $T$ at $R=5.83$, for $n=1,2,3$ together
with their best fits. The corresponding energies $E_{n}(R)$ are extracted from
the slope of exponential fits on a log plot. This data was obtained on a
$12^{3}\times 36$ lattice volume at $\beta=10.9$ and $\gamma=0.60$.

The energies $E_{2}(R),E_{3}(R)$ vs. $R$ are displayed in Fig. 3. Also shown
is the mass of a static photon (blue line) at ${\gamma=0.6}$, as obtained in
the previous section, along with the energy of the massive photon
$\sqrt{m_{ph}^{2}+(2\pi/12)^{2}}$ at the minimal non-zero momentum on a
periodic lattice of length 12. The ground state energy, not shown here, is at
$E_{1}(R)=0$.

The energy $E_{2}(R)$ of the first excited state above the ground state pretty
nearly coincides with the photon mass. So this state is easy to interpret: it
is simply the ground state of the static charges plus a static photon. The
second excited state, of energy $E_{3}(R)$, cannot be interpreted in that way,
because (with the possible exception of one outlier) it lies above the one
photon mass, but below the energy of a photon with minimal momentum. So there
are two possibilities. One is that this state represents the energy of a
static photon plus a small excitation, of energy $\Delta E=E_{3}-m_{ph}$, of
the field surrounding the static charge pair. But in that case we might expect
to see this energy $\Delta E$ as the energy of the first excited state, i.e.
an excitation without the massive photon, which is not seen. The second
possibility is that $E_{3}$ represents an excitation energy of the static
charges with no extra photons, which happens to lie above the threshold for
photon production, and this seems more likely. It of course implies that in
the infinite volume limit, where the photon momentum can take on any value,
this excitation is metastable, and could decay via massive photon emission. On
the other hand, if the charges are very far apart, this option might be
strongly suppressed, since the excitation energy of the field surrounding
either charge is only half the total excitation energy, $\frac{1}{2}E_{3}$.

Figure 3: Energies $E_{n}(R)$ vs. $R$ for $n=2,3$ again at $\gamma=0.60$ on a
$12^{3}\times 16$ lattice volume. The lower solid line represents the mass of
a massive photon, and the upper line is the energy of a massive photon at the
lowest possible momentum on this finite periodic lattice.

From the data, it appears that $\Psi_{1}({\bm{x}},{\bm{y}})$ and
$\Psi_{2}({\bm{x}},{\bm{y}})$ are very close to the true ground state, and the
ground state plus a massive photon respectively. We note in passing that this
result required use of both types of operators shown in (9). Use of only the
$Q_{2n-1}$ operators results in data that is not easily fit by a single
exponential.

###  III.1 Other Higgs couplings, volumes, and excitations

There is no visible dependence on $R$ in $E_{1}(R)$, and very little in
$E_{2}(R)$ in Fig. 3. However, we see significant scatter in the values shown
for $E_{3}(R)$, which cannot be attributed to error bars in the exponential
fit. We conjecture that the reason for this scatter is that
$|\Psi_{3}({\bm{x}},{\bm{y}})\rangle$, while close to the true energy
eigenstate, may contain enough (probably random) admixtures of higher excited
states to show a rather small, mainly random dependence on $R$.

Figure 4 is a comparison of the previous data in Fig. 3, at volume
$12^{3}\times 36$, with data obtained at the same $\beta,\gamma$ parameters on
a $16^{3}\times 36$ lattice volume. The data for $E_{2}$ coincide, while the
data points for $E_{3}$ at the larger $L=16$ spatial extension are roughly
compatible, sometimes coinciding, with the corresponding data points at
$L=12$, but there is much less scatter in the data at the larger volume. This
presumbably indicates a smaller admixture of higher excitations at large
spatial volume. A similar reduction in scatter, upon enlarging the spatial
volume, is seen in the data for the energies $E_{4}(R)$ of the next excited
state, displayed in Fig. 5.

At spatial extension $L=16$ the energy of a photon with smallest non-vanishing
momentum is at $E=0.597$, which is very close to to the $E_{3}$ data points.
If we had only the data on the $16^{3}\times 36$ volume, we would probably
conclude that $E_{3}$ corresponds to the ground state plus a massive photon of
that kind. But this interpretation is inconsistent with the data found on the
smaller $12^{3}\times 36$, and since increasing the volume has only a minimal
effect on the values of $E_{3}$, we conclude that we are seeing the same state
in both volumes, i.e. an excitation above the ground state of the static ions,
rather than a photon of non-zero momentum. The data for $E_{4}$ is likewise
almost independent of the lattice spatial extension $L$, which argues against
an interpretation involving one or more photons of finite momentum.

Figure 4: Comparison of $E_{2,3}(R)$ vs $R$ obtained on $12^{3}\times 36$ and
$16^{3}\times 36$ lattice volumes at $\beta=10.9,\gamma=0.60$. The solid line
represents the mass of a massive photon. Figure 5: Same comparison of
different volume data, as in Fig. 4, but this time for the excitation energy
$E_{4}(R)$.

Figure 6 shows the results of the same calculation of $E_{2-4}$ on a
$16^{3}\times 36$ volume, with the same exponential fit in the range
$T\in[2,7]$, but this time with $\gamma=0.25$. The mass of the massive photon
is of course reduced (in lattice units) as $\gamma$ is reduced, as one
expects, but the results are similar. The next question is: how similar?

Figure 6: Energies $E_{2,3,4}(R)$, but this time at $\gamma=0.25$, with
$\beta=10.9$ and lattice volume $16^{3}\times 36$. The lower and upper solid
lines represent the energies of a static massive photon, and a photon of
minimal momentum for this periodic volume.

###  III.2 Scaling

We have computed the photon mass and excitations $E_{n}$ in lattice units for
two different $\gamma$ values, namely $\gamma=0.25,0.60$. For a given London
penetration depth $\lambda_{L}$ we can compute the lattice spacing, and from
there convert the charge separation $R$, photon mass, and excitation energies
in physical units. Let $m_{ph}(\gamma)$ be the photon mass in lattice units
found at a particular $\gamma$ value, and $M_{ph}(\gamma)=m_{ph}(\gamma)/a$ be
the corresponding photon mass in physical units, where $a$ is the lattice
spacing given in eq. (7) . We then have

${M_{ph}(\gamma_{1})\over
M_{ph}(\gamma_{2})}=\sqrt{\gamma_{2}\over\gamma_{1}}{m_{ph}(\gamma_{1})\over
m_{ph}(\gamma_{2})}$ (29)

Inserting the computed values of $m_{ph}(\gamma=0.25)=0.288(1)$ and
$m_{ph}(\gamma=0.60)=0.446(3)$ in lattice units, we find the ratio of photon
masses in physical units to be

${M_{ph}(0.25)\over M_{ph}(0.60)}=1.000(8)$ (30)

If we choose a typical London penetration depth, say ${\lambda_{L}=50}$ nm, we
can compare data for $E_{n}$ vs. $R$ in physical units, obtained at the two
$\gamma$ values. The result, for data taken on a $16^{3}\times 36$ lattice
volume, is shown in Fig. 7. $E_{1}$ is consistent with zero for both $\gamma$
values, and the excitations $E_{2},E_{3}$ essentially coincide. Values for
$E_{4}$ differ at small $R$, but appear to converge at larger $R$. The overall
conclusion is that the results in physical units do not really depend on
$\gamma$, at least at larger charge separations.

Figure 7: A test of scaling. Using a fixed London penetration depth of
$\lambda_{L}=50$ nm to set the scale, we convert both $R$ and $E_{2,3,4}(R)$
to physical units at both $\gamma=0.25$ and $\gamma=0.60$, with $\beta=10.9$
fixed.

##  IV Discussion

The Ginzburg-Landau model is of course a simplified effective theory of
superconductivity, presumably relevant at scales beyond the diameter of a
Cooper pair. But if the Ginzburg-Landau model is not misleading us, then the
results presented here do suggest a possible experimental test via X-ray
photoemission spectroscopy (XPS).

We have argued that the self-interacting field surrounding a static charge may
have a spectrum of localized excitations. In a certain sense, this has already
been seen in normal metals. The electric field of a static charge which is
suddenly inserted into a normal metal is of course screened by electrons in
the conduction band, and one can ask whether these screening electrons
themselves have a spectrum of excitations. The answer appears to be yes, and
this is the origin of the asymmetric lineshape in the core electron XPS
spectrum, as pointed out long ago by Doniach and Sunjic Doniach_1970  (for a
simplified discussion, cf. huffner ). When an electron is ejected from the
valence band, this corresponds to a sudden appearance of an isolated charge in
the metal, and the process of screening can leave the charge plus Fermi sea in
one of a near-continuum of excitations slightly above the screened ground
state, since very little energy is required to create electron-hole
excitations near the Fermi level. This results in the Doniach-Sunjic
lineshape, which closely fits the XPS data.

According to our results in the Ginzburg-Landau model, things may be a little
different in the XPS spectrum of core electron emission in the superconducting
phase of a conventional superconductor. We have found a discrete spectrum of
excitations of the condensate, with energies on the order of mass of the
massive photon. This mass, which in natural units is the inverse of the London
penetration depth, depends on the metal, but in general is on the order of a
few electron volts. So in the superconducting phase we would expect to see a
number of additional peaks in the core electron emission spectrum, separated
from the main peaks by a few ev. Unfortunately the experimental situation is a
little different from our idealized setup of two static charges. One of the
charges, namely the hole left by the emitted core electron, is indeed static,
and we would expect the condensate distribution around the hole to be no
different from the idealized situation. The emitted electron, however, is by
no means static, which complicates making a precise prediction for the
location of the additional peaks. If this complication is ignored, then given
the Landau penetration depth $\lambda_{L}$, the positions of those peaks are
calculable. For, e.g., $\lambda_{L}=50$ nm, the excitation energies for widely
separated charges are the $E_{3},E_{4}$ energies seen in Fig. 7 at the larger
separations. The relative heights and widths of these extra peaks depends on
microscopic dynamics determining transition amplitudes and lifetimes, which
are beyond the scope of the simplfied effective model.

It appears that a comparison of XPS core electron spectra above and below the
superconducting transition, for conventional (or, for that matter, high Tc)
superconductors, has not yet been done. At least, we have not been able to
find a comparison of this type in the literature. An experimental
investigation along these lines could, for present purposes, be very helpful.

An obvious question is whether quarks and leptons in the electroweak sector of
the Standard Model, which is also a gauge Higgs theory, would have a spectrum
of excitations analogous to what we find in the Ginzburg-Landau model. If so,
these would appear as “elementary” particles in their own right.
Unfortunately, in the absence of a lattice formulation of chiral gauge theory
having a positive transfer matrix and a sensible continuum limit, we are
unable to make any predictions. Should such a formulation ever become
available, it would be possible to compute an excitation spectrum along the
lines presented here.

###### Acknowledgements.

We thank Peter Johnson and John Tranquada for helpful correspondence. This
research is supported by the U.S. Department of Energy under Grant No. DE-
SC0013682. *

##  Appendix A Numerical details

In all cases we have constructed, on each time slice, the Laplacian
eigenstates $\zeta_{n}$ with the four lowest eigenvalues, leading to eight
$|\Psi_{n}\rangle$ states, according to the procedures described above. The
Laplacian eigenstates were computed numerically via the Arnoldi algorithm, as
implemented in the ARPACK software
package.444https://www.caam.rice.edu/software/ARPACK/. Data for
$[\mathcal{T}^{T}]_{\alpha\beta}(R)$, along with error bars for those values,
were obtained by averaging the results of ten independent Monte Carlo
simulations at $\gamma=0.25,0.60$ on both $12^{3}\times 36$ and $16^{3}\times
36$ lattice volumes. In each simulation, 120 lattice configurations were
generated after 5000 thermalizing sweeps, with data taking sweeps separated by
100 Monte Carlo update sweeps. The numerical solution of the generalized
eigenvalue problem was carried out by the Matlab eig routine, which derived
values and error bars for $\mathcal{T}_{nn}(R,T)$. The determination of
$\\{E_{n}\\}$ from fits to a single exponential, and corresponding error bars,
were obtained from the gnuplot software.

Fitting data points over a finite range raises the question of what is the
“best” range, and how the answers would differ if the range is slightly
modified. Although the lattice was large enough in the time direction to
compute $\mathcal{T}_{nn}(R,T)$ up to $T=18$, in practice we found the
statistical errors were significant for data points beyond $T=8$. Excluding
points beyond those limits, and dropping the first data point (which would be
the most susceptible to mixing from higher excitations) we can compare the
values for $E_{2,3,4}$ using slightly different fitting intervals. Typical
results are shown in Table 1, for $R=5.83$, and lattice size $16^{3}\times
36$, where $E_{n}^{0.25},E_{n}^{0.60}$ refer to excitation energies computed
at $\gamma=0.25$ and $\gamma=0.60$ respectively. Error bars for $E_{2},E_{3}$
were approximately one percent, and 3-4% for $E_{4}$, which, given the small
dependence on $R$, could indicate a slight overestimate of error bars on the
fitted data points. The excitation energies shown in our figures were derived
for a fitting interval of $R$ in the range $2-7$, but it is clear that
choosing a different interval would not affect our conclusions.

(a)

(b)

Figure 8: (a) Link susceptibility $\chi$ vs. $\gamma$; and (b) gauge-invariant link $\langle L\rangle$ vs. $\gamma$, on a variety of lattice volumes. fitting interval | $E_{2}^{0.6}$ | $E_{3}^{0.6}$ | $E_{4}^{0.6}$ | $E_{2}^{0.25}$ | $E_{3}^{0.25}$ | $E_{4}^{0.25}$
---|---|---|---|---|---|---
2-4 | 0.472 | 0.543 | 1.18 | 0.288 | 0.363 | 0.83
2-5 | 0.472 | 0.540 | 1.13 | 0.286 | 0.357 | 0.81
2-6 | 0.453 | 0.538 | 1.13 | 0.282 | 0.354 | 0.81
2-7 | 0.453 | 0.537 | 1.13 | 0.282 | 0.353 | 0.82
2-8 | 0.453 | 0.537 | 1.13 | 0.282 | 0.350 | 0.81
Table 1: Comparison of $E_{2,3,4}(R)$ at $\beta=10.9$ and $\gamma=0.25,0.60$
at $R=5.83$, obtained from fitting the corresponding $\mathcal{T}_{nn}(R,T)$
vs. $T$ data in the different fitting intervals shown.

As mentioned in the text, at $\beta=10.9$ we located a phase transition,
presumably to the massless phase, at the rather low $\gamma$ value of
$\gamma\approx 0.017$. This was determined from inspection of the link
susceptibility

$\displaystyle\chi=V(\langle L^{2}\rangle-\langle L\rangle^{2})$ (31)

where $V$ is the lattice volume and

$L={1\over
3V}\sum_{x}\sum_{k=1}^{3}\text{Re}[\phi^{*}(x)U^{2}_{k}(x)\phi(x+\hat{k})]$
(32)

with the result shown in Fig. 8(a). At each volume, data was taken on 1400
lattices separated by 100 sweeps, after 20,000 thermalizing sweeps. In
previous work on the relativistic abelian Higgs model we have seen that the
transition to the massless phase corresponds to a discontinuity (a “kink”) in
the slope of $L$ in the infinite volume limit (cf. Fig. 1 in Matsuyama:2019lei
). Although this is not as obvious in our Ginzburg-Landau data, there is some
evidence of such a kink developing, as the volume increases, in Fig. 8(b).

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