Massless Schwinger model with a 4-fermi interaction at topological angle qp Dominic Hirtlera Christof Gattringerb

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Massless Schwinger model with a 4-fermi

interaction at topological angle θ=π

Dominic Hirtlera*, Christof Gattringerb†

aUniversity of Graz, 8010 Graz, Austria‡

bFWF - Austrian Science Fund, 1090 Vienna, Austria

dominic.hirtler@edu.uni-graz.at

christof.gattringer@fwf.ac.at

We study the massless Schwinger model with an additional 4-fermi interaction and a topological

term. For topological angle θ=πcharge conjugation symmetry is implemented in a non-trivial

way and we study the possibility of its spontaneous breaking. For the lattice discretization we

use staggered fermions and the Villain action for the gauge ﬁelds, where the topological term is

an integer and charge conjugation at θ=πis an exact symmetry. The complex action problem

is overcome by a suitable worldline/worldsheet representation. We ﬁnd that as a function of the

4-fermi coupling the system shows a critical point separating a weak coupling phase where charge

conjugation symmetry is intact from a strong coupling phase with spontaneously broken charge

conjugation symmetry.

The 39th International Symposium on Lattice Field Theory,

8th-13th August, 2022,

Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, Germany

*Speaker.

†Currently on leave of absence from University of Graz, 8010 Graz, Austria.

‡Member of NAWI Graz.

Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/

arXiv:2210.13787v1 [hep-lat] 25 Oct 2022

Schwinger model at θ=π

1. Introductory comments

Topological terms are an interesting ingredient in quantum ﬁeld theories, as they may alter the

symmetry content of a theory in a non-local way. This non-local character implies that a non-

perturbative regularization has to be used. An interesting non-perturbative regularization is the

lattice, which, however, poses two challenges: a suitable discretization of the topological charge,

and a way to overcome the complex action problem that is caused by the topological term.

In recent work [1] it was shown that a generalization of the Villain action [2] gives rise to an

integer-valued deﬁnition of the topological charge in terms of the Villain variables. As a conse-

quence, charge conjugation (C) symmetry at θ=πis implemented exactly and its spontaneous

breaking was studied in gauge Higgs models [3,4], while a study of C symmetry breaking in the

same model but with Wilson gauge action and a ﬁeld theoretic non-integer deﬁnition of the topo-

logical charge led to less conclusive results [5,6]. We remark that also the Atiyah Singer index

theorem [7] is relevant for the physics of the fermionic system studied here, which emerges in the

continuum limit of the Villain formulation [8]. Generalized Villain formulations were also used to

map the gauged XY model in the strong coupling limit at θ=πto the Ising model [9], to construct

a lattice discretization for fracton theories [10] and to explore non-invertible duality defects [11].

As was already mentioned, the topological term also generates a complex action problem,

which, however, may be solved by switching to a worldline/worldsheet representation [3,4] that

can be simulated efﬁciently [12,13]. For fermions there is also a potential sign problem coming

from the Grassmann nature of the fermionic variables and the γ-algebra. For the case of massless

staggered fermions in 2d, i.e., the discretization we use here, the sign problem is known to be absent

[14,15], and also the quartic fermion self interaction does not alter this result.

The ingredients outlined in the last two paragraphs, i.e., the integer-valued Villain-based deﬁ-

nition of the topological charge, the worldline/worldsheet representation for overcoming the com-

plex action problem and the absence of a fermionic sign problem for massless staggered fermions

in 2d allow one for the ﬁrst time to study the spontaneous breaking of charge conjugation in a

fermionic system: The 2d massless Schwinger model with a quartic self interaction and a topolog-

ical term at θ=π. Charge conjugation appears as a Z2symmetry and we explore whether it can

be broken spontaneously as a function of the quartic coupling parameter J.

Using Monte Carlo simulations of the system in its worldline/worldsheet representation we

study various bulk quantities, in particular the C symmetry breaking topological charge density hqi

and the corresponding susceptibility. For weak coupling Jthe symmetry remains unbroken, while

at strong coupling we observe breaking of C symmetry. We ﬁnd strong evidence for a critical point

near J∼0.9, which seems compatible with the 2d Ising universality class as expected. Varying θ

in the strong coupling phase we observe a ﬁrst order jump in the order parameter hqiwhen crossing

θ=π, which is a further indication that the system implements the 2d Ising phenomenology.

2. The Schwinger Model and its worldline/worldsheet representation

The partition sum of our model is given by,

Z=ZD[A]ZDψ,ψBβ,θ[A]e−SFψ,ψ,A,(2.1)

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摘要：

MasslessSchwingermodelwitha4-fermiinteractionattopologicalangleq=pDominicHirtlera*,ChristofGattringerbaUniversityofGraz,8010Graz,AustriabFWF-AustrianScienceFund,1090Vienna,Austriadominic.hirtler@edu.uni-graz.atchristof.gattringer@fwf.ac.atWestudythemasslessSchwingermodelwithanadditional4-fermiinte...

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Massless Schwinger model with a 4-fermi interaction at topological angle qp Dominic Hirtlera Christof Gattringerb

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