Dirac and Weyl semimetals are three-dimensional electronic systems with the Fermi level at or near a band crossing. Their low energy quasi-particles are described by a relativistic Dirac Hamiltonian with zero effective mass, challenging the standard Fermi liquid (FL) description of metals. In FL systems, electrical and thermo–electric transport coefficient are linked by very robust relations. The Mott relation links the thermoelectric and conductivity transport coefficients. In a previous publication, the thermoelectric coefficient was found to have an anomalous behavior originating in the quantum breakdown of the conformal anomaly by electromagnetic interactions. We analyze the fate of the Mott relation in the system. We compute the Hall conductivity of a Dirac metal as a function of the temperature and chemical potential and show that the Mott relation is not fulfilled in the conformal limit.

Dirac semimetals are three-dimensional (3D) crystals with band crossings near the Fermi level. In a low energy continuum description their quasi particles obey the massless Dirac equation and the interacting system is identical to massless quantum electrodynamics (QED). The low energy bands in Dirac semimetals are four fold degenerate (two spins, two chiralities). Breaking inversion or time-reversal symmetry gives rise to Weyl semimetals subjected to many interesting transport phenomena related to the chiral anomaly [

Leaving aside the important technological applications, thermal and electro–thermal transport are very useful tools to characterise the physical properties of new materials [

In a recent publication [

We perform an explicit calculation of the Hall conductivity in the conformal limit and, combining it with the thermoelectric coefficient of ref. [

Applying an external electric field

The best known phenomenological laws used in thermo–electrical transport are the Wiedemann-Franz (WF) law and the Mott relation [

The first one establishes that the ratio of the thermal to the electrical conductivity is the temperature times a universal number, the Lorenz number

An interesting question arose associated to the thermoelectric relations in topological materials. These materials have anomalous conductivities (particularly Hall conductivity) similar to that occurring in ferromagnetic materials induced by the Berry curvature of the bands. The question of whether or not these anomalous transport coefficients obeyed the WF and Mott relations, arose soon after the recognition of topological properties. The validity of the Mott relation for the anomalous transport phenomena was observed experimentally in films of

Typically Dirac materials in two and three dimensions are expected to follow the standard relations in the low T regime and deviate from it at larger temperatures [

In the next section we will analyze the Mott relation at the light of the results in [

The action associated to this Hamiltonian in the presence of a background electromagnetic potential

Chosing the magnetic field to point in the OZ direction (

The coefficient

In order to analyze the Mott relation (second equation in (

In most works on topological metals, the Hall conductivity is calculated using a Boltzmann formalism for the electronic transport. Since in Dirac semimetals the density of states at the neutrality point is zero, a Boltzmann approach does not seem reliable.

In this work the Hall conductivity is computed with a Kubo formulation as the one done in [

The magnetic field is coupled to the Hamiltonian (

The presence of the zeroth Landau level (LL) (

The green straight line represents the chiral zeroth order LL. The inset shows the zero temperature thermoelectric coefficient

The Landau eigenfunctions are:

Capital letters refer to the absolute value of Landau levels,

In the Landau level basis and using the Lehman representation of the Green’s function, the Hall conductivity in the local and zero frequency limit is given by the expression:

In (

For completeness we have also performed the calculation at finite chemical potential and temperature. These variables enter the Kubo expression (

In the next section we will analyze the Mott relation.

The Mott relation (

We have analyzed the situation at finite chemical potential and finite temperature performing a numerical calculation of the expression (

Inserting the thermoelectric conductivity

The ratio

Restoring the units we get, away from

The main conclusion of this work is the violation of the Mott relation at the conformal point of Dirac matter. In particular, a previous calculation [

Our results show that, as also happens in graphene, the

All the authors contributed equally to this work. All authors have read and agreed to the published version of the manuscript.

This work has been supported by the PIC2016FR6/PICS07480, Spanish MECD grant FIS2014-57432-P, European Union structural funds and the Comunidad Autónoma de Madrid (CAM) NMAT2D-CM Program (S2018-NMT-4511). J. B. acknowledges the financial support from the European Research Council (ERC-2015-AdG-694097) and Spanish Ministry (MINECO) Grant No. FIS2016-79464-P.

We thank B. Bradlyn, A. Cortijo and Y. Ferreiros, for interesting discussions.

The authors declare no conflict of interest.

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Landau level structure of a single chirality in a Dirac semi-metal. The green straight line represents the chiral zeroth LL. The inset shows the thermoelectric coefficient

The Hall conductivity

Behaviour of

Temperature dependence of the Mott ratio between the thermoelectric response function