Chiral current induced by torsional Weyl anomaly Chong-Sun Chu234 Rong-Xin Miao1 1School of Physics and Astronomy Sun Yat-Sen University 2 Daxue Road Zhuhai 519082

2025-04-27 0 0 462.09KB 15 页 10玖币

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Chiral current induced by torsional Weyl anomaly

Chong-Sun Chu 2,3,4∗, Rong-Xin Miao 1†

1School of Physics and Astronomy, Sun Yat-Sen University, 2 Daxue Road, Zhuhai 519082,

China1

2Department of Physics, National Tsing-Hua University, Hsinchu 30013, Taiwan

3Center of Theory and Computation, National Tsing-Hua University, Hsinchu 30013, Taiwan

4National Center for Theoretical Sciences, Taipei 10617, Taiwan

Abstract

Torsion can be realized as dislocation in the crystal lattice of material. It is particularly

interesting if the material has fermions in the spectrum, such as graphene, topological insula-

tors, Dirac and Weyl semimetals, as it’s transport properties can be aﬀected by the torsion.

In this letter, we ﬁnd that, due to Weyl anomaly, torsion in Dirac and Weyl semimetals can

induce novel chiral currents, either near a boundary or in a “conformally ﬂat space”. We

brieﬂy discuss how to measure this interesting eﬀect in experiment. It is remarkable that

these experiments can help to clarify the theoretical controversy of whether an imaginary

Pontryagin density could appear in the Weyl anomaly.

∗Email: cschu@phys.nthu.edu.tw

†Email: miaorx@mail.sysu.edu.cn

1All the Institutes of authors contribute equally to this work, the order of Institutes is adjusted for the

assessment policy of SYSU.

arXiv:2210.01382v3 [cond-mat.mes-hall] 13 Jun 2023

Contents

1 Introduction 1

2 Torsion and Weyl Anomaly 4

3 Chiral Current 5

3.1 BoundaryTheory.................................... 5

3.2 Conformally Flat Spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4 Weyl Fermions 9

1 Introduction

The study of anomaly induced transport is an interesting subject (see [1] for a recent review).

Although anomaly was originally discovered in particle physics, due to its universal nature,

anomaly has non-trivial implications to a large number of physical phenomena ranging over

vastly diﬀerent scales. For example, the chiral anomaly of nonabelian gauge theory imposes

nontrivial constraints on the fundamental interaction of chiral fermions in the standard model [2].

Chiral anomaly also aﬀects the transport dynamics of systems with chiral fermions [3–12] due to

the well-known chiral magnetic and chiral vortical eﬀects (see [13–15] for review). Interestingly,

this kind of anomalous transport occurs only in a material system since nonvanishing chemical

potentials are required. As anomaly itself is intrinsic to the quantum vacuum, it is an interesting

question to ask if anomaly induced transport can occur independent of the chemical potentials.

Recently, chiral response associated with chiral anomaly in a torsional background has been

a subject of intensive study. In general, a curved spacetime is equipped with a metric which ﬁxes

the causal structure and metric relations, and a connection which deﬁnes the parallel transport

of tensors on the manifold. While the path dependence of parallel transport is measured by the

curvature, the non-closure of parallelism of parallel transport is measure by the torsion. For the

parallel transport

∇µVρ=∂µVρ+ Γρµν Vν,(1)

the torsion

Tρµν := Γρµν −Γρνµ,(2)

is given by the antisymmetric part of the connection. In Einstein general relativity (GR), the

geometry of spacetime is taken to be torsion free since there is seemingly no observational

evidence for torsion in the spacetime of our universe [16]. However, perhaps unexpectedly,

torsion ﬁnds a legitimate position in condensed matter physics since torsion appears to be

naturally suited for the geometrical description of dislocation defects in crystals [17–20]. Torsion

has been realized and studied in diverse material systems such as graphene [21, 22], topological

insulators [23–25], Dirac and Weyl semimetal [26–31]. As a result of the speciﬁc manner fermion

is coupled to torsion, chiral anomaly could emerge and give rise to novel chiral response in

torsional material systems [32–40].

Just as a system may possess a chiral anomaly which characterizes the quantum chiral depen-

dence of the system, generally a system may also possess a Weyl anomaly which characterizes

the quantum scale dependence of the system. In general, the Weyl anomaly is deﬁned as a

diﬀerence between the trace of renormalized stress tensor and the renormalized trace of stress

tensor [41, 42]

A=ZM

√−ghgµν ⟨Tµν ⟩−⟨gµν Tµν ⟩i.(3)

In the presence of a background gauge ﬁeld, the Weyl anomaly receives a contribution

A=ZM

√−g b1Fµν Fµν (4)

whose form is universal and is entirely determined by the coeﬃcient b1, a bulk central charge

of the theory. For the normalization of the gauge ﬁeld kinetic term S=−1/4RF2,b1is given

by the beta function of the theory as b1=−β/2. As Weyl anomaly is independent of the chiral

anomaly, it is interesting to ask if and how it give rises to any transport phenomena in a system.

The answer is positive. Recently, a new kind of induced transport was discovered for boundary

vacuum system as a result of the Weyl anomaly. It was found that [43,44] for any renormalizable

quantum ﬁeld theory with a current coupled to an external electromagnetic (EM) ﬁeld

SA=ZM

√−g JµAµ,(5)

the Weyl anomaly give rises to an induced magnetization current in the vicinity of the boundary

of the vacuum system

⟨Jµ⟩=−2βFµν nν

x+··· , x ∼0.(6)

Here xis the proper distance to the boundary, nµis the inner normal vector, ... denote higher

order terms in O(x) and βis the beta function. Hereafter we will drop the symbol ⟨ ⟩ for the

expectation value. It is instructive to review the derivation of this result to appreciate how it

could be derived from the Weyl anomaly. In general, for a boundary quantum ﬁeld theory, the

renormalized current is generally singular near the boundary and the expectation value takes

the asymptotic form near x∼0:

Jµ=1

x3J(3)

µ+1

x2J(2)

µ+1

xJ(1)

µ+J(0) log x+··· ,(7)

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ChiralcurrentinducedbytorsionalWeylanomalyChong-SunChu2,3,4∗,Rong-XinMiao1†1SchoolofPhysicsandAstronomy,SunYat-SenUniversity,2DaxueRoad,Zhuhai519082,China12DepartmentofPhysics,NationalTsing-HuaUniversity,Hsinchu30013,Taiwan3CenterofTheoryandComputation,NationalTsing-HuaUniversity,Hsinchu30013,Taiwan...

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Chiral current induced by torsional Weyl anomaly Chong-Sun Chu234 Rong-Xin Miao1 1School of Physics and Astronomy Sun Yat-Sen University 2 Daxue Road Zhuhai 519082

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