[a]D. K. Sinclair

# Chiral Symmetry Breaking in QED induced by an External Magnetic Field

J. B. Kogut

###### Abstract

We simulate Lattice QED in a constant and homogeneous external magnetic field
using the Rational Hybrid Monte-Carlo (RHMC) algorithm developed for Lattice
QCD. Our current simulations are directed towards observing chiral symmetry
breaking in the limit of zero electron bare mass as predicted by approximate
(Schwinger-Dyson) methods. Our earlier simulations were performed on a
$36^{4}$ lattice at the fine structure constant $\alpha=1/137$, close to its
physical value, with ‘safe’ electron masses $m=0.1$ and $m=0.2$. At this
$\alpha$, the dynamical electron mass produced by the external magnetic field,
which is an order parameter for this chiral symmetry breaking, is predicted to
be far too small to be measurable. Hence we are now simulating at the larger
$\alpha=1/5$, where the predicted dynamical electron mass at strong external
magnetic fields accessable on the lattice is large enough to be measurable.
However this requires electron masses down to $m=0.001$. Such a small $m$
requires lattices larger than $36^{4}$, but at magnetic fields large enough to
produce measurable dynamical electron masses, $36$ is an adequate spatial
extent for the lattice in the plane orthogonal to the magnetic field because
the electrons preferentially occupy the lowest Landau level. We are therefore
performing finite size analyses using $36^{2}\times N_{\parallel}^{2}$
lattices with $N_{\parallel}\geq 36$. We measure the chiral condensate
$\langle\bar{\psi}\psi\rangle$ as our order parameter for chiral symmetry
breaking, since it should remain finite as $m\rightarrow 0$ if chiral symmetry
is broken by the magnetic field, but vanish otherwise. Our preliminary results
strongly suggest that chiral symmetry is broken by the external magnetic
field. In all our simulations, as well as measuring other observables during
these simulations, we are storing configurations at regular intervals for
further analysis. One such measurement planned for these stored configurations
is the determination of the effects that an external magnetic field has on the
coulomb field of a charged particle placed in this magnetic field.

##  1 Introduction

We study Lattice QED in external electromagnetic fields using methods
developed for Lattice QCD. Since QED in background electric fields has a
complex action, because the vacuum is unstable against decays into electron-
positron pairs – the Sauter-Schwinger effect, [1, 2] – standard simulation
methods which rely on importance sampling cannot be used. Because of this we
start by considering Lattice QED in background magnetic fields where the
action is real and bounded below. This enables us to perform simulations using
standard methods. We use the Rational Hybrid Monte-Carlo (RHMC) method of
Clark and Kennedy [3] whose implementation we describe in the appendix.

Relativistic quantum mechanical studies of electrons in external
electromagnetic fields and the modified actions this produces for those
fields, by Sauter, Euler and Heisenberg [1, 4] and formalized by Schwinger
[2], are some of the earliest QED calculations. See Dunne [5] for some of the
cases where exact solutions are known.

Phenomena such as the Sauter-Schwinger effect only become significant when the
electric field $E\sim E_{cr}=m^{2}/e$ or larger and/or the magnetic field
$B\sim B_{cr}=m^{2}/e$ or larger. Interest in extending such studies to full
QED including non-perturbative effects have been revived by planned
experiments colliding electron beams with intense beams of light from petawatt
lasers [6, 7], where the electromagnetic fields are of this magnitude, or
larger at LBNL, SLAC and possibly ELI. In addition it has been realized that
compact astronomical X- and $\gamma$-ray sources are probably neutron-stars
with magnetic fields of order $B_{cr}$ or larger (magnetars). See for example
the review [8]. Finally it has been noted that beam-beam interactions in the
next generation of electron/positron colliders could produce electromagnetic
fields orders of magnitude larger than their critical values [9]. Here, multi-
electron-loop contributions become as, or more important than, single loop
contributions and all conventional QED calculations break down.

For our simulations of QED in an external magnetic field, we choose a magnetic
field which is constant over all space and time. For definiteness, we choose a
magnetic field oriented in the $z$ ($3$) direction. Classically the orbit of a
charged particle (electron) in such a magnetic field is a helix around a fixed
magnetic field line, whose projection on the $(x,y)$ ($(1,2)$) plane is a
circle, while the motion in the $z$ direction is free. Quantum mechanically
the motion in the $(x,y)$ plane is quantized into a set of levels whose
transverse energies squared are evenly spaced with spacing $|2eB|$ – the
Landau levels[10]. The lowest level has a single helicity, while the higher
levels have both helicities. The the motion along the $z$ direction is free
field so that the $z$ momenta have a continuous spectrum. Including QED means
adding a dynamical photon field. The electron field feels both the external
and the dynamical photon fields, while only the dynamical photon field has a
kinetic term. For details of the lattice transcription of QED in this external
field and its simulation, see the appendix. The most important feature is that
at large $|eB|$ all electrons preferentially occupy the lowest Landau level
whose orbit has a finite extent (proportional to $1/\sqrt{|eB|}$, leading to
an effective dimensional reduction from $3+1$ dimensions to $1+1$ dimensions.

One of the most theoretically interesting non-perturbative effects predicted
by truncated Schwinger-Dyson analyses of QED in such constant magnetic fields
is that, in the limit $m\rightarrow 0$, chiral symmetry is broken by the
magnetic field leading to a dynamical electron mass $\propto\sqrt{|eB|}$ [11,
12, 13, 14, 15, 16, 17, 18] and a chiral condensate
$\langle\bar{\psi}\psi\rangle\propto|eB|^{3/2}$ [19, 20]. This non-
perturbative effect is often referred to as ‘magnetic catalysis’. For a good
review article with a more complete set of references see Miransky and
Shovkovy [21]. We note that in $3+1$ dimensions, for massless electrons in an
external magnetic field without QED, i.e. without internal photons, chiral
symmetry is unbroken for all $eB$. In fact, as $m\rightarrow 0$ the chiral
condensate vanishes $\propto m\log(m)$, so QED is essential for the breaking
of chiral symmetry in a magnetic field. This contrasts with the situation in
$2+1$ dimensions where chiral symmetry is broken with a finite chiral
condensate $\propto eB$ for massless electrons in an external magnetic field,
even without QED.

In our RHMC lattice QED simulations, we measure this chiral condensate, since
as a local operator, it is easier to measure than the dynamical mass, which
would require measuring the electron propagator itself. At physical
$\alpha=e^{2}/(4\pi)\approx 1/137$, the predicted dynamical mass is more than
30 orders of magnitude less than any value we could possibly measure. We
therefore perform simulations with a stronger $\alpha=1/5$, which appears to
be in the perturbative regime for $eB=0$, and for which the predicted
dynamical electron mass and chiral condensate although small, should be
measurable for the $eB$ value we choose. Following our earlier simulations at
$\alpha=1/137$ we simulate on a $36^{4}$ lattice at $\alpha=1/5$ and with
masses in the range $0.001\leq m\leq 0.2$. We choose $|eB|=2\pi\times
100/36^{2}=0.4848...$, which is large but safely in the range $|eB|<0.65$
required to keep discretization errors under control. For these parameters a
lattice of size $36$ in both the $x$ and $y$ directions is considerably larger
than the projection of the lowest Landau level on the $(x,y)$ plane and
therefore adequate. However, $36$ is too small a value in the $z$ and $t$
directions to accommodate the smallest masses, so a finite size scaling
analysis is needed, however it is only necessary to increase the lattice sizes
in the $z$ and $t$ directions. Preliminary results of such a finite size
scaling analysis are presented in the next section, and strongly suggest that
there is chiral symmetry breaking in the $m\rightarrow 0$ limit.

##  2 Simulations and Results

Here we discuss only the simulations and results from our simulations with
$\alpha=1/5$. For simulations and results with $\alpha=1/137$ and for ‘free’
electrons in an external magnetic field see the proceedings from our talk at
Lattice 2021 [22].

We perform RHMC simulations with $\alpha=1/5$ aimed primarily at searching for
evidence for chiral symmetry breaking in the presence of a constant and
uniform magnetic field in the limit $m\rightarrow 0$. For this we measure the
chiral condensate $\langle\bar{\psi}\psi\rangle$, which should remain finite
and non zero as $m\rightarrow 0$ if chiral symmetry is broken in this limit.

Figure 1: $\langle\bar{\psi}\psi\rangle$ as a function of mass at $eB=0$,
showing lattice size dependence.

If chiral symmetry is unbroken in the $m=0$ limit, the chiral condensate is
dominated by the short distance (ultraviolet) regime and should vanish
proportional to $m$ possibly times some power of $\log(m^{2})$ as
$m\rightarrow 0$. In this case it should be insensitive to the size of the
lattice. Therefore we should be able to run with arbitrarily small masses
without observing finite lattice size effects. To test this we perform
$\alpha=1/5$ simulations with $0.001<m<0.2$ on a $36^{4}$ lattice at $eB=0$,
where chiral symmetry is believed to be unbroken at $m=0$. Note that the
normal requirement that $mN_{\mu}>>1$ ($\mu=1,2,3,4$) to avoid finite size
effects is not true for the lower part of this range. In figure 1 we plot the
condensate as a function of mass and see that it does appear to be approaching
zero for small $m$. We repeat these simulations on a $48^{4}$ lattice at the
lowest mass $m=0.001$. The condensate on the $36^{4}$ lattice is
$4.3259(7)\times 10^{-4}$, while that on the $48^{4}$ lattice is
$4.3294(7)\times 10^{-4}$, a mere less than $0.1\%$ difference. Such
insensitivity to this finite size scaling analysis we take as evidence that
the condensate is zero and chiral symmetry remains unbroken in the chiral
($m=0$) limit, at least to within the precision of our simulations. In fact, a
linear extrapolation to $m=0$ from the points at $m=0.005$ and $m=0.001$
which, based on the curvature of this graph, should yield an upper estimate of
the value of the condensate at $m=0$, gives
$\langle\bar{\psi}\psi\rangle\approx 10^{-6}$ which is only about 3 standard
deviations from zero, giving further evidence that chiral symmetry remains
unbroken as $m\rightarrow 0$.

We now turn to the case of large $eB$ and choose $|eB|=2\pi\times
100/36^{2}=0.4848...$, near the upper end of the range of $|eB|$ values where
discretization errors are small. Again we run our RHMC simulations with
$\alpha=1/5$ and $0.001<m<0.2$ on a $36^{4}$ lattice. As indicated above, the
extent of the lattice in the $x$ and $y$ directions is large enough to contain
the lowest Landau levels. However the extent of the lattice in the $z$ and $t$
directions is insufficient to prevent finite size effects. If chiral symmetry
is broken in the limit $m\rightarrow 0$ then this indicates that there are
modes with momenta of order $\sqrt{|eB|}$ or less which contribute to the
condensate at small $m$. These modes will make a contribution of order
$|eB|^{3/2}$ to the chiral condensate, keeping it non-zero as $m\rightarrow 0$
and making it sensitive to increases in the lattice extent in the $z$ and $t$
directions [19, 20].

Figure 2: $\langle\bar{\psi}\psi\rangle$ as a function of mass at
$eB=2\pi\times 100/36^{2}=0.4848...$, showing dependence on lattice size in
the $z$ and $t$ directions.

Since the condensate on the $36^{4}$ lattice appears to be headed towards zero
as $m\rightarrow 0$, a sign that chiral symmetry is broken in this limit is
that the chiral condensate at small enough $m$ should increase when the
lattice sizes in the $z$ and $t$ directions are increased. We have therefore
performed simulations with the same parameters on $36^{2}\times 64^{2}$
lattices. For $m=0.025$ the change in the chiral condensate in going from the
$36^{4}$ lattice to the $36^{2}\times 64^{2}$ lattice is very small,
indicating that we need only simulate on the larger lattice at masses less
than $0.025$. At $m=0.0125$ there is a small but significant increase in the
condensate in going to the larger lattice, while at $m=0.005$ the increase in
the condensate in going to the larger lattice is relatively large. At m=0.001
the condensate on the larger lattice is approximately twice that on the
smaller lattice. At $m=0.005$, we have increased our lattice size even further
to $36^{2}\times 96^{2}$. While this leads to a further significant increase
in the condensate, it is only by $\approx 5\%$, and so going to an even larger
lattice is unnecessary. Our next task is to increase the lattice size at
$m=0.001$ to $36^{2}\times 96^{2}$ and possibly go to a lattice with $z$ and
$t$ extents of $128$, which should make the case for chiral symmetry breaking
even more compelling and allow us to estimate the value of the chiral
condensate at $m=0$. Figure 2 shows the mass dependence of the chiral
condensate as a function of mass $m$ from these simulations at
$|eB|=2\pi\times 100/36^{2}$ on our chosen lattice sizes. Even with
simulations on only $36^{4}$ and $36^{2}\times 64^{2}$ lattices at $m=0.001$,
this graph does suggest that the condensate remains finite and non-zero as
$m\rightarrow 0$.

##  3 Summary, Discussion and Conclusions

We simulate lattice QED on lattice sizes $36^{4}$, $36^{2}\times 64^{2}$, and
$36^{2}\times 96^{2}$ with $\alpha=1/5$, in an external magnetic field
$eB=2\pi\times 100/36^{2}$ and masses in the range $0.001\leq m\leq 0.2$. The
simulations on the $36^{2}\times 96^{2}$ lattice at $m=0.001$ have yet to be
performed. Preliminary results of this finite size scaling analysis show that
for $m\leq 0.0125$, the chiral condensates increase as the lattice size is
increased, which is evidence that the condensate remains finite in the chiral
($m\rightarrow 0$) limit. The forthcoming larger lattice simulations at
$m=0.001$ should confirm this and predict the value of this condensate at
$m=0$. We also perform simulations with the same $\alpha$ and masses, but with
$eB=0$ on $36^{4}$ lattices and for $m=0.001$ on a $48^{4}$ lattice. Here the
condensate shows no significant finite size effects, and the data strongly
indicate that this condensate vanishes for $m\rightarrow 0$.

The size of the chiral condensate at $m=0$ appears to be larger than that
predicted by Schwinger-Dyson methods. This might mean that $\alpha=1/5$ is too
large for the approximations made in the Schwinger-Dyson calculations or the
derivation of the condensate from the dynamical electron mass predicted by
these methods. It could also be due to the limitations of the lattice methods
at these parameters.

Because the size of the $(x,y)$ projection of lowest Landau levels for the
chosen $eB$ are appreciably smaller than $36^{2}$, we are simulating with the
same parameters on lattices with dimensions $18^{2}$ in the $(x,y)$ plane for
comparison, in particular comparing the chiral condensates for small $m$.

We plan to perform further analyses on stored configurations. Of particular
interest is the measurement of the distortion and screening of the coulomb
fields of charged particles in the presence of external magnetic fields [23,
24, 25, 26]

It is of interest to repeat our simulations in different external magnetic
fields to test if the $m=0$ chiral condensate scales as $|eB|^{3/2}$. We
should also study the chiral behavior of QED extended to include multiple
electron ‘flavours’. If chiral symmetry is broken in the $m\rightarrow 0$
limit, this includes the spontaneous breaking of chiral flavour symmetry,
complete with flavoured massless Goldstone bosons. These are allowed because,
being uncharged, they do not feel the magnetic field and are thus not subject
to the dimensional reduction from $3+1$ to $1+1$ dimensions, and hence are
massless $3+1$ dimensional excitations.

We will explore the possibility of designing an effective action which
incorporates the assumption that only the lowest Landau level contributes, but
is otherwise fully $3+1$ dimensional.

We will explore the inclusion of external electric fields. Because these make
the action complex, we will need to resort to simulation methods such as the
complex Langevin equation (CLE). Because the non-compact gauge action
describes a free field which is a collection of harmonic oscillators then, in
the absence of fermions, the real Lagrangian is an attractive fixed point of
the CLE, rather than a repulsive fixed point as is the case for QCD and
probably lattice QED with a compact action. Therefore one might hope that this
might remain true when fermions are included, possibly with a modified fermion
action, at least for weak coupling. This would allow the study of the Sauter-
Schwinger effect in full QED in an external electric field using Lattice QED
simulations.

## Acknowledgments

DKS’s research is supported in part U.S. Department of Energy, Division of
High Energy Physics, under Contract No. DE-AC02-06CH11357. The high
performance computing was provided by the LCRC at Argonne National Laboratory
on their Bebop cluster. Access to Stampede-2 at TACC, Expanse at UCSD and
Bridges-2 at PSC was provided under an XSEDE allocation. Time on Cori at NERSC
was provided through an ERCAP allocation. DKS thanks G. T. Bodwin for
inciteful discussions, while JBK would like to thank V. Yakimenko for
discussions which helped inspire this project, and I. A. Shovkovy for helpful
discourse on magnetic catalysis.

##  Appendix A Lattice QED in an external magnetic field

We simulate using the non-compact gauge action

$S(A)=\frac{\beta}{2}\sum_{n,\mu<\nu}[A_{\nu}(n+\hat{\mu})-A_{\nu}(n)-A_{\mu}(n+\hat{\nu})+A_{\mu}(n)]^{2}$

where $n$ is summed over the lattice sites, and $\mu$ and $\nu$ run from $1$
to $4$ subject to the restriction $\mu<\nu$. $\beta=1/e^{2}$. The expectation
value of an observable ${\cal O}(A)$ is then

$\langle{\cal
O}\rangle=\frac{1}{Z}\int_{-\infty}^{\infty}\Pi_{n,\mu}dA_{\mu}(n)e^{-S(A)}[\det{\cal
M}(A+A_{ext})]^{1/8}{\cal O}(A)$

where ${\cal M}=M^{\dagger}M$, $A$ is the dynamic photon field and $A_{ext}$
is the external photon field while

$M(A+A_{ext})=\sum_{\mu}D_{\mu}(A+A_{ext})+m$

where the operator $D_{\mu}$ is defined by

$\displaystyle[D_{\mu}(A+A_{ext})\psi](n)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\eta_{\mu}(n)\\{e^{i(A_{\mu}(n)+A_{ext,\mu}(n))}\psi(n+\hat{\mu})$
$\displaystyle-$ $\displaystyle
e^{-i(A_{\mu}(n-\hat{\mu})+A_{ext,\mu}(n-\hat{\mu}))}\psi(n-\hat{\mu})\\}$

and $\eta_{\mu}$ are the staggered phases.

We use the RHMC simulation method of Clark and Kennedy, using rational
approximations to ${\cal M}^{-1/8}$ and ${\cal M}^{\pm 1/16}$. To account for
the range of normal modes of the non-compact gauge action, we randomly vary
the trajectory lengths over the range of periods of these modes [27].
$A_{ext}$ are chosen in the symmetric gauge in the x-y plane so that the
magnetic fields from each plaquette are in the +z-direction and have the value
$eB$ modulo $2\pi$. This requires $eB=2\pi n/(n_{1}n_{2})$, where $n_{1}$ and
$n_{2}$ are the lattice dimensions in the x and y directions, and $n$ is an
integer in the range $[0,n_{1}n_{2}/2]$ [28].

One of the observables we calculate is the electron contribution to the
effective gauge action per site $\frac{-1}{8V}{\rm trace}[\ln({\cal M})]$. For
this we use a rational approximation to $\ln$ following Kelisky and Rivlin
[29], and a stochastic approximation to the trace.

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