# Thermoelectric Relations in the Conformal Limit in Dirac and Weyl Semimetals

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## Abstract

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## 1. Introduction

## 2. Thermoelectric Relations

## 3. Thermoelectric Coefficient in the Conformal Invariant Point in Dirac Semimetals

## 4. The Hall Conductivity

## 5. The Mott Relation

## 6. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Xiong, J.; Kushwaha, S.K.; Liang, T.; Krizan, J.W.; Hirschberger, M.; Wang, W.; Cava, R.J.; Ong, N.P. Evidence for the chiral anomaly in the Dirac semimetal Na
_{3}Bi. Science**2015**, 350, 413. [Google Scholar] [CrossRef][Green Version] - Li, C.; Wang, L.X.; Liu, H.; Wang, J.; Liao, Z.M.; Yu, D.P. Giant negative magnetoresistance induced by the chiral anomaly in individual Cd3As2 nanowires. Nat. Commun.
**2015**, 6, 10137. [Google Scholar] [CrossRef] - Zhang, C.; Xu, S.-Y.; Belopolski, I.; Yuan, Z.; Lin, Z.; Tong, B.; Bian, G.; Alidoust, N.; Lee, C.-C.; Huang, S.-M.; et al. Signatures of the Adler Bell Jackiw chiral anomaly in a Weyl fermion semimetal. Nat. Commun.
**2016**, 7, 10735. [Google Scholar] [CrossRef][Green Version] - Landsteiner, K. Anomalous transport of Weyl fermions in Weyl semimetals. Phys. Rev. B
**2014**, 89, 075124. [Google Scholar] [CrossRef][Green Version] - Gooth, J.; Schierning, G.; Felser, C.; Nielsch, K. Quantum materials for thermoelectricity. MRS Bull.
**2018**, 43, 187. [Google Scholar] [CrossRef] - Behnia, K.; Aubin, H. Nernst effect in metals and superconductors: A review of concepts and experiments. Rep. Prog. Phys.
**2016**, 79, 046502. [Google Scholar] [CrossRef][Green Version] - Sondheimer, E.H. The theory of the galvanomagnetic and thermomagnetic effects in metals. Proc. R. Soc. Lon. Ser. A Math. Phys. Sci.
**1948**, 193, 484. [Google Scholar] - Cutler, M.; Mott, N.F. Observation of Anderson Localization in an Electron Gas. Phys. Rev.
**1969**, 181, 1336–1340. [Google Scholar] [CrossRef] - Kittel, C. Introduction to Solid State Physics; John Wiley and Sons: Hoboken, NJ, USA, 2005. [Google Scholar]
- Xiao, D.; Yao, Y.; Fang, Z.; Niu, Q. Berry-Phase Effect in Anomalous Thermoelectric Transport. Phys. Rev. Lett.
**2006**, 97, 026603. [Google Scholar] [CrossRef] [PubMed][Green Version] - Checkelsky, J.G.; Ong, N.P. Thermopower and Nernst effect in graphene in a magnetic field. Phys. Rev. B
**2009**, 80, 081413. [Google Scholar] [CrossRef][Green Version] - Wei, P.; Bao, W.; Pu, Y.; Lau, C.N.; Shi, J. Anomalous Thermoelectric Transport of Dirac Particles in Graphene. Phys. Rev. Lett.
**2009**, 102, 166808. [Google Scholar] [CrossRef] [PubMed] - Proskurin, I.; Ogata, M. Thermoelectric Transport Coefficients for Massless Dirac Electrons in Quantum Limit. J. Phys. Soc. Jpn.
**2013**, 82, 063712. [Google Scholar] [CrossRef][Green Version] - Ghahari, F.; Xie, H.Y.; Taniguchi, T.; Watanabe, K.; Foster, M.S.; Kim, P. Enhanced Thermoelectric Power in Graphene: Violation of the Mott Relation by Inelastic Scattering. Phys. Rev. Lett.
**2016**, 116, 136802. [Google Scholar] [CrossRef][Green Version] - Liang, T.; Gibson, Q.; Xiong, J.; Hirschberger, M.; Koduvayur, S.P.; Cava, R.J.; Ong, N.P. Evidence for massive bulk Dirac fermions in Pb
_{1−x}Sn_{x}Se from Nernst and thermopower experiments. Nat. Commun.**2013**, 4, 2696. [Google Scholar] [CrossRef] [PubMed][Green Version] - Lundgren, R.; Laurell, P.; Fiete, G.A. Thermoelectric properties of Weyl and Dirac semimetals. Phys. Rev. B
**2014**, 90, 165115. [Google Scholar] [CrossRef][Green Version] - Lucas, A.; Davison, R.A.; Sachdev, S. Hydrodynamic theory of thermoelectric transport and negative magnetoresistance in Weyl semimetals. Proc. Natl. Acad. Sci. USA
**2016**, 113, 9463–9468. [Google Scholar] [CrossRef][Green Version] - Gorbar, E.V.; Miransky, V.A.; Shovkovy, I.A.; Sukhachov, P.O. Anomalous thermoelectric phenomena in lattice models of multi-Weyl semimetals. Phys. Rev. B
**2017**, 96, 155138. [Google Scholar] [CrossRef][Green Version] - Manna, K.; Muechler, L.; Muechler, L.; Kao, T.-H.; Stinshoff, R.; Zhang, Y.; Gooth, J.; Kumar, N.; Kreiner, G.; Koepernik, K.; et al. From colossal to zero: Controlling the Anomalous Hall Effect in Magnetic Heusler Compounds via Berry Curvature Design. Phys. Rev. X
**2018**, 8, 041045. [Google Scholar] [CrossRef][Green Version] - Nakai, R.; Nagaosa, N. Nonreciprocal thermal and thermoelectric transport of electrons in noncentrosymmetric crystals. arXiv
**2018**, arXiv:1812.02372. [Google Scholar] [CrossRef][Green Version] - Skinner, B.; Fu, L. Large, nonsaturating thermopower in a quantizing magnetic field. Sci. Adv.
**2018**, 4, 2621. [Google Scholar] [CrossRef][Green Version] - Muñoz, E.; Soto-Garrido, R. Thermoelectric transport in torsional strained Weyl semimetals. J. Appl. Phys.
**2019**, 125, 082507. [Google Scholar] [CrossRef][Green Version] - Bandurin, D.A.; Torre, I.; Kumar, R.K.; Ben Shalom, M.; Tomadin, A.; Principi, A.; Auton, G.H.; Khestanova, E.; Novoselov, K.S.; Grigorieva, I.V.; et al. Negative local resistance caused by viscous electron backflow in graphene. Science
**2016**, 351, 1055–1058. [Google Scholar] [CrossRef] [PubMed][Green Version] - Crossno, J.; Shi, J.K.; Wang, K.; Liu, X.; Harzheim, A.; Lucas, A.; Sachdev, S.; Kim, P.; Taniguchi, T.; Watanabe, K.; et al. Observation of the Dirac fluid and the breakdown of the Wiedemann-Franz law in graphene. Science
**2016**, 351, 1058–1061. [Google Scholar] [CrossRef] [PubMed][Green Version] - Moll, P.J.W.; Kushwaha, P.; Nandi, N.; Schmidt, B.; Mackenzie, A.P. Evidence for hydrodynamic electron flow in PdCoO
_{2}. Science**2016**, 351, 1061–1064. [Google Scholar] [CrossRef][Green Version] - Bandurin, D.A.; Shytov, A.V.; Levitov, L.S.; Kumar, R.K.; Berdyugin, A.I.; Ben Shalom, M.; Grigorieva, I.V.; Geim, A.K.; Falkovich, G. Fluidity onset in graphene. Nat. Commun.
**2018**, 9, 4533. [Google Scholar] [CrossRef] - Gooth, J.; Menges, F.; Kumar, N.; Süß, V.; Shekhar, C.; Sun, Y.; Drechsler, U.; Zierold, R.; Felser, C.; Gotsmann, B. Thermal and electrical signatures of a hydrodynamic electron fluid in tungsten diphosphide. Nat. Commun.
**2018**, 9, 4093. [Google Scholar] [CrossRef][Green Version] - Jaoui, A.; Fauqué, B.; Rischau, C.W.; Subedi, A.; Fu, C.; Gooth, J.; Kumar, N.; Süß, V.; Maslov, D.L.; Felser, C.; et al. Departure from the Wiedemann–Franz law in WP2 driven by mismatch in T-square resistivity prefactors. NJP Quantum Mater.
**2018**, 3, 64. [Google Scholar] [CrossRef] - Chernodub, M.N.; Cortijo, A.; Vozmediano, M.A.H. Generation of a Nernst Current from the Conformal Anomaly in Dirac and Weyl Semimetals. Phys. Rev. Lett.
**2018**, 120, 206601. [Google Scholar] [CrossRef][Green Version] - Chernodub, M.N. Anomalous Transport Due to the Conformal Anomaly. Phys. Rev. Lett.
**2016**, 117, 141601. [Google Scholar] [CrossRef] - Arjona, V.; Chernodub, M.N.; Vozmediano, M.A.H. Fingerprints of the conformal anomaly in the thermoelectric transport in Dirac and Weyl semimetals: Result from a Kubo formula. Phys. Rev. B
**2019**, 99, 235123. [Google Scholar] [CrossRef][Green Version] - Ziman, J.M. Electrons and Phonons: The Theory of Transport Phenomena in Solids; International Series of Monographs on Physics; Clarendon Press: Oxford, UK, 1960. [Google Scholar]
- Jonson, M.; Mahan, G.D. Mott’s formula for the thermopower and the Wiedemann-Franz law. Phys. Rev. B
**1980**, 21, 4223–4229. [Google Scholar] [CrossRef] - Smrcka, L.; Streda, P. Transport coefficients in strong magnetic fields. J. Phys. C Solid State Phys.
**1977**, 10, 2153–2161. [Google Scholar] [CrossRef] - Jonson, M.; Girvin, S.M. Thermoelectric effect in a weakly disordered inversion layer subject to a quantizing magnetic field. Phys. Rev. B
**1984**, 29, 1939–1946. [Google Scholar] [CrossRef] - Cooper, N.R.; Halperin, B.I.; Ruzin, I.M. Thermoelectric response of an interacting two-dimensional electron gas in a quantizing magnetic field. Phys. Rev. B
**1997**, 55, 2344–2359. [Google Scholar] [CrossRef][Green Version] - Qin, T.; Niu, Q.; Shi, J. Energy Magnetization and the Thermal Hall Effect. Phys. Rev. Lett.
**2011**, 107, 236601. [Google Scholar] [CrossRef] - Andreev, A.V.; Kivelson, S.A.; Spivak, B. Hydrodynamic Description of Transport in Strongly Correlated Electron Systems. Phys. Rev. Lett.
**2011**, 106, 256804. [Google Scholar] [CrossRef][Green Version] - Pu, Y.; Chiba, D.; Matsukura, F.; Ohno, H.; Shi, J. Mott Relation for Anomalous Hall and Nernst Effects in Ga
_{1−x}Mn_{x}As Ferromagnetic Semiconductors. Phys. Rev. Lett.**2008**, 101, 117208. [Google Scholar] [CrossRef][Green Version] - Ryu, S.; Moore, J.E.; Ludwig, A.W.W. Electromagnetic and gravitational responses and anomalies in topological insulators and superconductors. Phys. Rev. B
**2012**, 85, 045104. [Google Scholar] [CrossRef][Green Version] - Kim, K.S. Role of axion electrodynamics in a Weyl metal: Violation of Wiedemann-Franz law. Phys. Rev. B
**2014**, 90, 121108. [Google Scholar] [CrossRef][Green Version] - Liang, T.; Gibson, Q.; Ali, M.N.; Liu, M.; Cava, R.J.; Ong, N.P. Ultrahigh mobility and giant magnetoresistance in the Dirac semimetal Cd
_{3}As_{2}. Nat. Mater.**2015**, 14, 280–284. [Google Scholar] [CrossRef][Green Version] - Liang, T.; Lin, J.; Gibson, Q.; Gao, T.; Hirschberger, M.; Liu, M.; Cava, R.J.; Ong, N.P. Anomalous Nernst Effect in the Dirac Semimetal Cd
_{3}As_{2}. Phys. Rev. Lett.**2017**, 118, 136601. [Google Scholar] [CrossRef] [PubMed][Green Version] - Watzman, S.J.; McCormick, T.M.; Shekhar, C.; Wu, S.C.; Sun, Y.; Prakash, A.; Felser, C.; Trivedi, N.; Heremans, J.P. Dirac dispersion generates unusually large Nernst effect in Weyl semimetals. Phys. Rev. B
**2018**, 97, 161404. [Google Scholar] [CrossRef][Green Version] - Sakai, A.; Mizuta, Y.P.; Nugroho, A.A.; Sihombing, R.; Koretsune, T.; Suzuki, M.-T.; Takemori, N.; Ishii, R.; Nishio-Hamane, D.; Arita, R.; et al. Giant anomalous Nernst effect and quantum-critical scaling in a ferromagnetic semimetal. Nat. Phys.
**2018**, 14, 1119. [Google Scholar] [CrossRef] - Armitage, N.P.; Mele, E.J.; Vishwanath, A. Weyl and Dirac semimetals in three-dimensional solids. Rev. Mod. Phys.
**2018**, 90, 015001. [Google Scholar] [CrossRef][Green Version] - Collins, J.C.; Duncan, A.; Joglekar, S.D. Trace and dilatation anomalies in gauge theories. Phys. Rev. D
**1977**, 16, 438–449. [Google Scholar] [CrossRef] - Nielsen, H.B.; Ninomiya, M. The Adler-Bell-Jackiw anomaly and Weyl fermions in a crystal. Phys. Lett. B
**1983**, 130, 389. [Google Scholar] [CrossRef] - Kumar, G.S.; Prasad, G.; Pohl, R.O. Experimental determinations of the Lorenz number. J. Mater. Sci.
**1993**, 28, 4261–4272. [Google Scholar] [CrossRef] - Gooth, J.; Niemann, A.C.; Meng, T.; Grushin, A.G.; Landsteiner, K.; Gotsmann, B.; Menges, F.; Schmidt, M.; Shekhar, C.; Süß, V.; et al. Experimental signatures of the mixed axial gravitational anomaly in the Weyl semimetal NbP. Nature
**2017**, 547, 23005. [Google Scholar] [CrossRef] - Schindler, C.; Galeski, S.; Schnelle, W.; Wawrzyńczak, R.; Abdel-Haq, W.; Guin, S.N.; Kroder, J.; Kumar, N.; Fu, C.; Borrmann, H.; et al. Anisotropic electrical and thermal magnetotransport in the magnetic semimetal GdPtBi. Phys. Rev. B
**2020**, 101, 125119. [Google Scholar] [CrossRef][Green Version] - Vu, D.; Zhang, W.; Şahin, C.; Flatté, M.; Trivedi, N.; Heremans, J.P. Thermal chiral anomaly in the magnetic-field induced ideal Weyl phase of Bi89Sb11. arXiv
**2019**, arXiv:1906.02248. [Google Scholar]

**Figure 1.**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 ${\chi}^{xy}={\alpha}^{xy}/T$ computed in ref. [31].

**Figure 2.**The Hall conductivity ${\sigma}^{xy}$ computed in this work as a function of the chemical potential in the range $-\hslash {\omega}_{c}\le \mu \le \hslash {\omega}_{c}$ for various values of the temperature. At $T=0$, the conductivity is linear in $\mu $.

**Figure 3.**Behaviour of ${\chi}^{ij}/{\partial}_{\mu}{\sigma}^{ij}$ as a function of the temperature. The Mott relation is not satisfied for $T=0$, where the ratio presents a finite value.

**Figure 4.**Temperature dependence of the Mott ratio between the thermoelectric response function ${\chi}^{xy}$ and the derivative of the electric conductivity ${\partial}_{\mu}{\sigma}^{xy}$ at $\mu =0$. Red dots are the numerical calculation and the blue line is the fit to the function $f\left(\tilde{T}\right)=1+9.27{\tilde{T}}^{2}$. The deviation is smaller than the size of the points.

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**MDPI and ACS Style**

Arjona, V.; Borge, J.; Vozmediano, M.A.H.
Thermoelectric Relations in the Conformal Limit in Dirac and Weyl Semimetals. *Symmetry* **2020**, *12*, 814.
https://doi.org/10.3390/sym12050814

**AMA Style**

Arjona V, Borge J, Vozmediano MAH.
Thermoelectric Relations in the Conformal Limit in Dirac and Weyl Semimetals. *Symmetry*. 2020; 12(5):814.
https://doi.org/10.3390/sym12050814

**Chicago/Turabian Style**

Arjona, Vicente, Juan Borge, and María A. H. Vozmediano.
2020. "Thermoelectric Relations in the Conformal Limit in Dirac and Weyl Semimetals" *Symmetry* 12, no. 5: 814.
https://doi.org/10.3390/sym12050814