Pis’ma v ZhETF, vol. 110, iss. 12, pp. 797 - 798

On thermal Nieh-Yan anomaly in topological Weyl materials

J.Nissinen⁺¹⁾, G.E.Volovik^+∗1)

⁺Low Temperature Laboratory, Department of Applied Physics, Aalto University, P. O. Box 15100, FI-00076 AALTO, Finland

^∗Landau Institute for Theoretical Physics Russian Academy of Sciences, 119334 Moscow, Russia

Submitted 7 November 2019

Resubmitted 7 November 2019

Accepted 8 November 2019

DOI: 10.1134/S0370274X19240056

We discuss the possibility of a gravitional Nieh-Yan

There has been several attempts to consider

anomaly of the type ∂_µj^µ5 = γT²T^a ∧ T_a in topological

the NY anomaly in condensed matter systems with

Weyl materials, where T is temperature and T^a is the

Weyl fermions, see, e.g.,

[13, 16-18]. However, in

effective or emergent torsion. As distinct from the non-

non-relativistic systems the relativistic high-energy

universal parameter Λ in the conventional (zero temper-

cut-off Λ is not a well defined parameter. The complete

ature) Nieh-Yan (NY) anomaly [1-4] - with canonical

UV theory is non-Lorentz invariant and the linear,

dimensions of momentum - the parameter γ is dimen-

quasirelativistic Weyl regime is valid at much lower

sionless. This suggests that the dimensionless parame-

scales. Moreover, the anomalous hydrodynamics of

ter is fundamental, being determined by the geometry,

superfluid³He at zero temperature suggests that the

topology and the number of chiral quantum fields with-

chiral anomaly is completely exhausted by the emergent

out any explicit non-universal ultraviolet (UV) scales.

axial gauge field corresponding to the shift of the node

In non-relativistic topological matter, quasi-

or, conversely, the NY anomaly term. Nevertheless, it

relativistic description of low-energy quasiparticles

was shown in [13] that the low-energy theory satisfies

with linear spectrum phenomena may emerge [5, 6]. In

the symmetries and conservations laws related to an

three spatial dimensions at a generic (two-fold) fermion

emergent quasirelativistic spacetime with torsion and

band crossing at momentum p_W , the Hamiltonian is

Λ is determined from the UV-scale where the linear

of the Weyl form [5,7-9] H_W = σ^ae^ia(p - p_W )_i + · · · ,

Weyl approximation breaks down.

where the e^ia are the linear coefficients of the Pauli

The fully relativistic responses work unambiguously

matrices σ^a, playing the role of background spacetime

only for terms in the effective action with dimensionless

tetrad fields. The shift of the Weyl node p_W acts as an

coefficients. An example is the 2+1-dimensional topo-

emergent (axial) gauge field. These background fields

logical Chern-Simons (CS) terms describing the quan-

imply the chiral anomaly for the low-energy massless

tum Hall effect. Gravitational CS terms similarly are

quasiparticles, see, e.g., [5, 9-11]. In particular, the

quantized in terms of chiral central charge which has

non-trivial coordinate dependence (torsion) related to

relation to thermal transport and the gapless boundary

the tetrads e^µa(x) can lead to the gravitational NY

modes [19, 20]. The CS action was recently generalized

anomaly [1-4, 12, 13]. Here we discuss this anomaly in

to crystalline topological insulators in odd space dimen-

the presence of finite temperature [14, 15].

sions. The CS term is expressed via elasticity tetrads

For spacetimes with torsion T (and curvature R) the

E with dimension [E] = [M] as the topological term

4-dimensional invariant was introduced [1, 2, 4]:

E ∧ A ∧ dA with quantized dimensionless coefficients

[21-23].

N =T^a ∧T_a -e^a ∧e^b ∧R_ab

(1)

Another such example is the temperature correction

∫

and can be associated with a difference of two topolog-

to curvature effects, with δSeff =

T²R in the low-

ical terms [3]. It was also suggested that N contributes

energy action [24]. This represent the analog of the grav-

to the anomalous chiral current j^µ5:

itational coupling (Newton constant) in the low-energy

action where the curvature scalar R is some analog of

∂_µj^µ5 =

N (r, t),

(2)

scalar spacetime curvature. Since [T]²[R] = [M]⁴, the

4π²

coefficient of this term in the low-energy theory is di-

mensionless, and thus can be given in terms of universal

where the parameter Λ has dimension of mass [Λ] = [M]

constants. It is fully determined by the number of the

and is determined by an UV scale.

fermionic and bosonic species and works both in rela-

¹⁾e-mail: jaakko.k.nissinen@gmail.com; volovik@boojum.hut.fi

tivistic and non-relativistic systems [24].

Письма в ЖЭТФ том 110 вып. 11 - 12

2019

797

798

J.Nissinen, G.E.Volovik

The same universal behavior takes place with the

This work has been supported by the European Re-

terms describing the chiral magnetic and chiral vorti-

search Council (ERC) under the European UnionтАЩs

cal effects in Weyl superfluids, where the coefficients

Horizon

2020

research and innovation programme

are dimensionless [5, 25]. Similarly, the coefficient of the

(Grant Agreement # 694248).

R∧R gravitional anomaly in chiral Weyl systems affects

Full text of the paper is published in JETP Letters

the thermal transport coefficients in flat space [26, 27].

journal. DOI: 10.1134/S0021364019240020

These coefficients are fundamental, being determined by

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(3)

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2019