Dynamical Eects from Anomaly Modied Electrodynamics in Weyl Semimetal Xuzhe Ying1 2A. A. Burkov1 2and Chong Wang2 1Department of Physics and Astronomy University of Waterloo Waterloo ON N2L 3G1 Canada

2025-04-24 0 0 752.82KB 8 页 10玖币

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Dynamical Eﬀects from Anomaly: Modiﬁed Electrodynamics in Weyl Semimetal

Xuzhe Ying,1, 2 A. A. Burkov,1, 2 and Chong Wang2

1Department of Physics and Astronomy, University of Waterloo, Waterloo, ON, N2L 3G1, Canada

2Perimeter Institute for Theoretical Physics, Waterloo, ON, N2L 2Y5, Canada

We discuss the modiﬁed quantum electrodynamics from a time-reversal-breaking Weyl semimetal

coupled with a U(1)gauge (electromagnetic) ﬁeld. A key role is played by the soft dispersion of

the photons in a particular direction, say ˆz, due to the Hall conductivity of the Weyl semimetal.

Due to the soft photon, the fermion velocity in ˆzis logarithmically reduced under renormalization

group ﬂow, together with the ﬁne structure constant. Meanwhile, fermions acquire a ﬁnite lifetime

from spontaneous emission of the soft photon, namely the Cherenkov radiation. At low energy E,

the inverse of the fermion lifetime scales as τ−1∼E/PolyLog(E). Therefore, even though fermion

quasiparticles are eventually well-deﬁned at very low energy, over a wide intermediate energy window

the Weyl semimetal behaves like a marginal Fermi liquid. Phenomenologically, our results are more

relevant for emergent Weyl semimetals, where the fermions and photons all emerge from strongly

correlated lattice systems. Possible experimental implications are discussed.

PACS numbers:

I. INTRODUCTION

Weyl fermions, since their original proposal, have been

widely studied due to the chiral nature [1–9]. In the

recent decades, much focus has been put on the con-

densed matter realization, namely the Weyl semimetal

(WSM) [7–10]. In Weyl semimetals, due to the separa-

tion of Weyl fermions in momentum space, various in-

triguing phenomena have been observed, e.g., Fermi arc

[8,9], anomalous Hall eﬀect [7,11], quantized circular pho-

togalvanic eﬀect [12], etc. The dynamical properties of

WSM also attract much attention [13–16]. While a mag-

netic Weyl semimetal is typically subject to weak inter-

action of particular form (short-ranged or Coulomb), an

emergent WSM is a more versatile playground for study-

ing interaction eﬀects, e.g., topological orders in three

dimension 17–19], generalizations of the standard QED

20, etc.

An emergent WSM is a strongly interacting lattice sys-

tem of spins or electrons, of which the low energy eﬀec-

tive theory is described by an emergent U(1)gauge ﬁeld

(or some Zmdescendent) coupled to a WSM formed by

emergent fermions. In spin liquid terminology, these are

U(1)(or Zm) spin liquids with spinon Weyl semi-metals.

Unlike in ordinary Weyl semimetals, the emergent Weyl

fermions could naturally have velocity close to that of the

U(1)gauge ﬁeld, and the gauge coupling strength (ﬁne

structure constant) does not have to be small at a given

energy scale. The possibility of an emergent WSM phase

has been demonstrated in Ref. [17, 21–24]. The emer-

gent WSM phase was further proposed to be the parent

state of topological orders in three dimension [17–19].

While the descendent topologically ordered phases are

stable by the formation of many-body gap, the proper-

ties of the emergent WSM phase itself is largely studied

at the mean-ﬁeld level. In particular, the dynamical con-

sequences of gauge ﬂuctuations in emergent WSM remain

unexplored.

In this work, we focus on the case with the emergent

U(1) gauge ﬁeld, also referred to as emergent electro-

magnetic (EM) ﬁeld. One important notion in studying

WSM phase is the unquantized anomaly, which guaran-

tees the gaplessness of WSM [11, 25–27]. When an EM

ﬁeld emerges, the dynamical aspect of the anomalies is

also an important piece of information. The unquan-

tized anomaly appears as a Chern-Simons-like action in

3+1D [11, 25–27]. Together with the Maxwell action,

the modiﬁed electrodynamics is usually referred to as

Carroll-Field-Jackiw electrodynamics [20]. In the mod-

iﬁed electrodynamics, the physical polarization of prop-

agating photons is diﬀerent from those in the vacuum.

Another important feature is the anisotropy. In particu-

lar, one of the photon modes becomes soft in a particular

direction [20]. Emergent photons with similar features

were also found in the coupled layers of Laughlin states

[28].

In this article, we study the interplay between the

fermionic degrees of freedom and the modiﬁed electro-

dynamics in emergent WSM. The situation under con-

sideration is really a simple, non-Lorentz-symmetric gen-

eralization of textbook quantum electrodynamics (QED),

however with unconventional outcomes. Indeed, we will

show that due to the interaction with the soft photons,

the emergent WSM represents an unconventional quan-

tum liquid.

More speciﬁcally, the presence of soft photons sig-

niﬁcantly inﬂuences the low-energy properties of the

fermions. There are two major results. First, the fermion

dispersion is strongly dressed by the photons. Namely,

the fermion’s velocity in the soft photon direction is re-

duced to zero under the rernormalization group (RG)

ﬂow. Besides, the system ﬂows to a non-interacting limit

under RG. Second, fermions can spontaneously emit pho-

tons. As a result, the fermions acquire a ﬁnite lifetime,

due to the Cherenkov radiation of the soft photons, that

is inversely proportional to the ﬁne structure constant

and the fermion’s energy. The two eﬀects just men-

tioned make the emergent WSM signiﬁcantly diﬀerent

from the free WSM or the standard QED. Indeed, over

arXiv:2210.06641v1 [cond-mat.str-el] 13 Oct 2022

a wide energy window, the emergent WSM behaves like

a marginal Fermi liquid due to the ﬁnite lifetime [29].

Meanwhile, the IR behavior shows an asymptotic two di-

mensional character and is well-controlled under the RG

ﬂow. Hence, the emergent WSM represents an unconven-

tional type of quantum liquid. Lastly, we propose that

the reported feature of a emergent WSM can be observed

from the speciﬁc heat measurement at low temperature

[30,31].

The rest of the article is organized as follows. Sec-

tion. II reviews the low energy description of a Weyl

semimetal as well as the modiﬁed electrodynamics. A

physical picture is developed for the dynamical eﬀects.

Section. III is devoted to quantum mechanical one-loop

diagram calculations, from which the RG ﬂow and the

fermion lifetime can be obtained. Section. IV concludes

the article and discusses the possible experimental impli-

cations.

II. (EMERGENT) WEYL SEMIMETAL AND

MODIFIED ELECTRODYNAMICS

In this section, we review the low energy description of

the fermions and the modiﬁed electrodynamics in emer-

gent Weyl semimetal (WSM) at mean ﬁeld level. A qual-

itative picture for the dynamical eﬀects is provided.

The mean ﬁeld description of the emergent WSM starts

with a parton decomposition, which formally corresponds

to fractionalizing the electron annihilation operator cinto

a neutral fermion (spinon, ψ) and a charged boson (char-

geon, eiθc): c=eiθcψ[32]. The local U(1)gauge ambi-

guity of this decomposition

ψ→eiαψ, eiθc→e−iαeiθc,(1)

dictates the necessity of the emergence of a dynamical

U(1)gauge ﬁeld [33]. In other words, there is a dynami-

cal U(1)gauge ﬁeld that couples to both the the spinon

and the chargeon with gauge charge q=±1. We then

consider mean ﬁeld states in which the chargeons eiθc

are gapped (and therefore can be integrated out at low

energy), while the spinons ψform a Weyl semimetal band

structure, with two Weyl cones of the opposite chirality

separated in momentum space by 2Q(see Fig. 1 (a)).

At low energy, the fermionic excitations of a Weyl

semimetal is eﬀectively described by a Dirac Lagrangian

[11]:

Lf=¯

ψVµi∂µψ, with Vµ=[γ0, γ1, γ2, v3γ3](2)

where γ0,1,2,3are the usual 4 ×4 gamma matrices. Here,

we assumed the velocity in the xy-plane to be one while

the velocity in the z-direction being diﬀerent, namely

v3. The energy of the fermions is E±(k)=±E(k)=

±k2

x+k2

y+v2

3k2

z. We should comment that in this

article, we interchangeably use µ=(0,1,2,3)or µ=

(t, x, y, z)to indicate the time and space directions. The

Figure 1. Low energy dispersions for (a) free WSM; (b)

photon in the modiﬁed electrodynamics with qx=qy=0.

(a) Around two Weyl nodes, fermions have linear dispersion.

The two Weyl nodes are separated in momentum space by

2Q=λz3in z-direction. On the sample boundary, there is

Fermi arc connecting two Weyl nodes, an indication of anoma-

lous Hall eﬀect. (b) There are two photon modes. The gapped

mode (green) has a gap given by e2λz3

2π. The other mode is soft

(red), with a quadratic dispersion, Eq. (5). The grey dahsed

line corresponds to ω=e

e3qz, namely the photon dispersion

when λ=0. (c) Cartoon of fermions moving in a soft elec-

tromagentic environment. Fast moving fermions constantly

interact with virtual photons (red dashed-dot) and sponta-

neously emit soft photons (red solid). The emitted soft pho-

tons are roughly conﬁned around the z-axis indicated by the

red dashed lines.

former is algebraically convenient, while the latter more

descriptive and intuitive.

One should notice that the simplicity in Eq. (2) is de-

ceptive. Indeed, when a lattice model of WSM is con-

sidered, the two Weyl components of a Dirac fermion are

separated in Brillouin zone [7], as shown in Fig 1(a). As a

result, there will be chiral edge states for a ﬁnite size sys-

tem. The intersection of chiral edge state and the Fermi

level is the Fermi arc, which is a hallmark of WSM [8,9].

In terms of transport, the features just mentioned im-

plies that WSM shows an anomalous Hall conductivity

[7,11,34,10]. The anomaly argument requires the value

of the Hall conductivity to be given by 2Q, even though

high-energy fermions may be strongly interacting with

no well-deﬁned quasi-particles [18,26].

Due to the Hall eﬀect, the Lagrangian for the emergent

U(1) gauge ﬁeld has a Chern-Simons(CS)-like term, in

addition to the Maxwell term [20]:

LCS-like =−λz3

4π(φBz−axEy+ayEx)(3a)

LM=1

2e2E2

x+E2

y−B2

z+1

2e2

3E2

z−B2

x−B2

y(3b)

where φ=a0and ais the scalar and vector potential of

the gauge ﬁeld; Eand Bare the emergent electric and

magnetic ﬁeld strength respectively.

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DynamicalEectsfromAnomaly:ModiedElectrodynamicsinWeylSemimetalXuzheYing,1,2A.A.Burkov,1,2andChongWang21DepartmentofPhysicsandAstronomy,UniversityofWaterloo,Waterloo,ON,N2L3G1,Canada2PerimeterInstituteforTheoreticalPhysics,Waterloo,ON,N2L2Y5,CanadaWediscussthemodiedquantumelectrodynamicsfromatime-...

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Dynamical Eects from Anomaly Modied Electrodynamics in Weyl Semimetal Xuzhe Ying1 2A. A. Burkov1 2and Chong Wang2 1Department of Physics and Astronomy University of Waterloo Waterloo ON N2L 3G1 Canada.pdf

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Dynamical Eects from Anomaly Modied Electrodynamics in Weyl Semimetal Xuzhe Ying1 2A. A. Burkov1 2and Chong Wang2 1Department of Physics and Astronomy University of Waterloo Waterloo ON N2L 3G1 Canada

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