Flux unwinding in the lattice Schwinger model
Chris Nagele,
1,2
J. Eduardo Cejudo,
1
Tim Byrnes,
1,3,4,5,6
and Matthew Kleban
2,6
1
New York University Shanghai, 1555 Century Avenue, Pudong, Shanghai 200122, China
2
Center for Cosmology and Particle Physics, New York University, New York, 10003, USA
3
State Key Laboratory of Precision Spectroscopy, School of Physical and Material Sciences,
East China Normal University, Shanghai 200062, China
4
NYU-ECNU Institute of Physics at NYU Shanghai, 3663 Zhongshan Road North,
Shanghai 200062, China
5
National Institute of Informatics, 2-1-2 Hitotsubas hi, Chiyoda-ku, Tokyo 101-8430, Japan
6
Department of Physics, New York University, New York, New York 10003, USA
(Received 11 January 2019; published 22 May 2019)
We study the dynamics of the massive Schwinger model on a lattice using exact diagonalization. When
periodic boundary conditions are imposed, analytic arguments indicate that a nonzero electric flux in the
initial state can “unwind” and decrease to a minimum value equal to minus its initial value, due to the
effects of a pair of charges that repeatedly traverse the spatial circle. Our numerical results support
the existence of this flux unwinding phenomenon, both for initial states containing a charged pair inserted
by han d, and when the charges are produced by Schwinger pair production. We also study boundary
conditions where charges are confined to an interval and flux unwinding cannot occur, and the massless
limit, where our resu lts agree with the predictions of the bosonized description of the Schwinger model.
DOI: 10.1103/PhysRevD.99.094501
I. INTRODUCTION
The massive Schwinger model [1]—quantum electro-
dynamics in one space and one time dimension—is a
fascinating quantum field theory that has been studied
intensively since the 1950s. It has a wide set of applica-
tions: as a simple example of a quantum gauge theory, as an
Abelian theory that nevertheless exhibits a linearly growing
potential between charges and hence a kind of confinement,
as a theory that exhibits a prototype strong/weak duality via
bosonization, and even to models of cosmic inflation in
string theory [2–4].
Most work on the Schwinger model has focused on its
static properties, such as its spectrum of excitations, the
value of the chiral condensate, etc. There has been
relatively little work, either analytical or numerical, on
time-dependent phenomena in the Schwinger model. Two
recent works include Hebenstreit et al., who considered the
dynamics of string breaking in the massive Schwinger
model using a numerical technique where the gauge field is
treated classically/statistically [5], and Buyens et al. who
studied real-time evolution of the wave function using the
matrix product states formalism in the thermodynamic
limit [6].
Despite the absence of electromagnetic waves in one
spatial dimension, the electric field in the Schwinger model
is generally time-dependent because charged particles
move and affect its value. These particles can be sponta-
neously produced by Schwinger pair production in the
quantum theory [7], or simply be present in the initial state.
In Ref. [8], a new time-dependent phenomenon was
discovered in the Schwinger model (and a broad class of
other theories) with spatially periodic boundary conditions.
A solution to the classical theory with no charges is a
homogeneous, time-independent electric field that winds
around the spatial circle. If a pair of equal and opposite
charges is present, the field accelerates the charges in
opposite directions until they collide at some point on the
opposite side of the circle (see Fig. 1). If the charges
transmit through each other, they will continue in the same
direction, unwinding two units of charge on each circuit
(charge and field strength have the same units in one
dimension). As a result, the initial value of the field will
steadily decrease. In the absence of any other dynamics, the
momentum of the charges causes the electric field to
overshoot zero, decreasing to a value with equal magnitude
and opposite sign as the initial field. This is sharply in
contrast with the case of the infinite line or boundary
conditions on an interval that forbid charges from crossing,
where a single charged pair can at most reduce the field by
two units.
Published by the American Physical Society under the terms of
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PHYSICAL REVIEW D 99, 094501 (2019)
2470-0010=2019=99(9)=094501(9) 094501-1 Published by the American Physical Society