Quantization of the Thermal Hall Conductivity at Small Hall Angles
Abstract
We consider the effect of coupling between phonons and a chiral Majorana edge in a gapped chiral spin liquid with Ising anyons (e.g., Kitaev’s nonAbelian spin liquid on the honeycomb lattice). This is especially important in the regime in which the longitudinal bulk heat conductivity κ x x due to phonons is much larger than the expected quantized thermal Hall conductance $κ$ $$^{q}_{xy}$$ = (πT/12)($k$ $$^{2}_{B}$$/ℏ) of the ideal isolated edge mode, so that the thermal Hall angle, i.e., the angle between the thermal current and the temperature gradient, is small. By modeling the interaction between a Majorana edge and bulk phonons, we show that the exchange of energy between the two subsystems leads to a transverse component of the bulk current and thereby an effective Hall conductivity. Remarkably, the latter is equal to the quantized value when the edge and bulk can thermalize, which occurs for a Hall bar of length L >> ℓ, where ℓ is a thermalization length. We obtain ℓ ~ T^{–5} for a model of the Majoranaphonon coupling. We also find that the quality of the quantization depends on the means of measuring the temperature and, surprisingly, a more robust quantization is obtained when the lattice, not the spin, temperature is measured. We present general hydrodynamic equations for the system, detailed results for the temperature and current profiles, and an estimate for the coupling strength and its temperature dependence based on a microscopic model Hamiltonian. Furthermore, our results may explain recent experiments observing a quantized thermal Hall conductivity in the regime of small Hall angle, κ_{xy}/κ_{xx} ~ 10^{–3}, in α–RuCl_{3}.
 Authors:

 Univ. of Minnesota, Minneapolis, MN (United States); Univ. of California, Santa Barbara, CA (United States)
 Univ. of California, Santa Barbara, CA (United States)
 Univ. de Lyon (France)
 Publication Date:
 Research Org.:
 Univ. of California, Santa Barbara, CA (United States)
 Sponsoring Org.:
 USDOE Office of Science (SC)
 OSTI Identifier:
 1610005
 Alternate Identifier(s):
 OSTI ID: 1475084
 Grant/Contract Number:
 FG0208ER46524
 Resource Type:
 Accepted Manuscript
 Journal Name:
 Physical Review Letters
 Additional Journal Information:
 Journal Volume: 121; Journal Issue: 14; Journal ID: ISSN 00319007
 Publisher:
 American Physical Society (APS)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY; physics; anyons; frustrated magnetism; quantum spin liquid; thermal Hall effect
Citation Formats
Ye, Mengxing, Halász, Gábor B., Savary, Lucile, and Balents, Leon. Quantization of the Thermal Hall Conductivity at Small Hall Angles. United States: N. p., 2018.
Web. doi:10.1103/physrevlett.121.147201.
Ye, Mengxing, Halász, Gábor B., Savary, Lucile, & Balents, Leon. Quantization of the Thermal Hall Conductivity at Small Hall Angles. United States. doi:10.1103/physrevlett.121.147201.
Ye, Mengxing, Halász, Gábor B., Savary, Lucile, and Balents, Leon. Mon .
"Quantization of the Thermal Hall Conductivity at Small Hall Angles". United States. doi:10.1103/physrevlett.121.147201. https://www.osti.gov/servlets/purl/1610005.
@article{osti_1610005,
title = {Quantization of the Thermal Hall Conductivity at Small Hall Angles},
author = {Ye, Mengxing and Halász, Gábor B. and Savary, Lucile and Balents, Leon},
abstractNote = {We consider the effect of coupling between phonons and a chiral Majorana edge in a gapped chiral spin liquid with Ising anyons (e.g., Kitaev’s nonAbelian spin liquid on the honeycomb lattice). This is especially important in the regime in which the longitudinal bulk heat conductivity κ x x due to phonons is much larger than the expected quantized thermal Hall conductance $κ$ $^{q}_{xy}$ = (πT/12)($k$ $^{2}_{B}$/ℏ) of the ideal isolated edge mode, so that the thermal Hall angle, i.e., the angle between the thermal current and the temperature gradient, is small. By modeling the interaction between a Majorana edge and bulk phonons, we show that the exchange of energy between the two subsystems leads to a transverse component of the bulk current and thereby an effective Hall conductivity. Remarkably, the latter is equal to the quantized value when the edge and bulk can thermalize, which occurs for a Hall bar of length L >> ℓ, where ℓ is a thermalization length. We obtain ℓ ~ T–5 for a model of the Majoranaphonon coupling. We also find that the quality of the quantization depends on the means of measuring the temperature and, surprisingly, a more robust quantization is obtained when the lattice, not the spin, temperature is measured. We present general hydrodynamic equations for the system, detailed results for the temperature and current profiles, and an estimate for the coupling strength and its temperature dependence based on a microscopic model Hamiltonian. Furthermore, our results may explain recent experiments observing a quantized thermal Hall conductivity in the regime of small Hall angle, κxy/κxx ~ 10–3, in α–RuCl3.},
doi = {10.1103/physrevlett.121.147201},
journal = {Physical Review Letters},
number = 14,
volume = 121,
place = {United States},
year = {2018},
month = {10}
}
Web of Science
Works referenced in this record:
Anyons in an exactly solved model and beyond
journal, January 2006
 Kitaev, Alexei
 Annals of Physics, Vol. 321, Issue 1
Incommensurate counterrotating magnetic order stabilized by Kitaev interactions in the layered honeycomb $\alpha \text{\u2212}{\mathrm{Li}}_{2}{\mathrm{IrO}}_{3}$
journal, May 2016
 Williams, S. C.; Johnson, R. D.; Freund, F.
 Physical Review B, Vol. 93, Issue 19
NonAbelian anyons and topological quantum computation
journal, September 2008
 Nayak, Chetan; Simon, Steven H.; Stern, Ady
 Reviews of Modern Physics, Vol. 80, Issue 3
$\alpha {\mathrm{RuCl}}_{3}$ : A spinorbit assisted Mott insulator on a honeycomb lattice
journal, July 2014
 Plumb, K. W.; Clancy, J. P.; Sandilands, L. J.
 Physical Review B, Vol. 90, Issue 4
Paired states of fermions in two dimensions with breaking of parity and timereversal symmetries and the fractional quantum Hall effect
journal, April 2000
 Read, N.; Green, Dmitry
 Physical Review B, Vol. 61, Issue 15
Faulttolerant quantum computation by anyons
journal, January 2003
 Kitaev, A. Yu.
 Annals of Physics, Vol. 303, Issue 1
Challenges in design of Kitaev materials: Magnetic interactions from competing energy scales
journal, June 2016
 Winter, Stephen M.; Li, Ying; Jeschke, Harald O.
 Physical Review B, Vol. 93, Issue 21
Quantum spin liquids: a review
journal, November 2016
 Savary, Lucile; Balents, Leon
 Reports on Progress in Physics, Vol. 80, Issue 1
Majorana zero modes in superconductor–semiconductor heterostructures
journal, May 2018
 Lutchyn, R. M.; Bakkers, E. P. A. M.; Kouwenhoven, L. P.
 Nature Reviews Materials, Vol. 3, Issue 5
Majorana quantization and halfinteger thermal quantum Hall effect in a Kitaev spin liquid
journal, July 2018
 Kasahara, Y.; Ohnishi, T.; Mizukami, Y.
 Nature, Vol. 559, Issue 7713
Mott Insulators in the Strong SpinOrbit Coupling Limit: From Heisenberg to a Quantum Compass and Kitaev Models
journal, January 2009
 Jackeli, G.; Khaliullin, G.
 Physical Review Letters, Vol. 102, Issue 1
Observation of halfinteger thermal Hall conductance
journal, June 2018
 Banerjee, Mitali; Heiblum, Moty; Umansky, Vladimir
 Nature, Vol. 559, Issue 7713
Direct evidence for dominant bonddirectional interactions in a honeycomb lattice iridate Na2IrO3
journal, May 2015
 Hwan Chun, Sae; Kim, JongWoo; Kim, Jungho
 Nature Physics, Vol. 11, Issue 6
Nonabelions in the fractional quantum hall effect
journal, August 1991
 Moore, Gregory; Read, Nicholas
 Nuclear Physics B, Vol. 360, Issue 23
Works referencing / citing this record:
Hydrodynamics of threedimensional skyrmions in frustrated magnets
journal, August 2019
 Zarzuela, Ricardo; Ochoa, Héctor; Tserkovnyak, Yaroslav
 Physical Review B, Vol. 100, Issue 5
Theory of the fieldrevealed Kitaev spin liquid
journal, June 2019
 Gordon, Jacob S.; Catuneanu, Andrei; Sørensen, Erik S.
 Nature Communications, Vol. 10, Issue 1
Unfolding the complexity of phonon quasiparticle physics in disordered materials
journal, January 2020
 Mu, Sai; Olsen, Raina J.; Dutta, Biswanath
 npj Computational Materials, Vol. 6, Issue 1
Heisenberg–Kitaev physics in magnetic fields
journal, July 2019
 Janssen, Lukas; Vojta, Matthias
 Journal of Physics: Condensed Matter, Vol. 31, Issue 42
Thermaltransport studies of kagomé antiferromagnets
journal, November 2019
 Yamashita, Minoru; Akazawa, Masatoshi; Shimozawa, Masaaki
 Journal of Physics: Condensed Matter, Vol. 32, Issue 7
Bulk thermal transport coefficients in a quantum Hall system and the fundamental difference between thermal and charge response
journal, July 2019
 VinklerAviv, Yuval
 Physical Review B, Vol. 100, Issue 4
Finite field regime for a quantum spin liquid in $\alpha \text{\u2212}{\mathrm{RuCl}}_{3}$
journal, August 2019
 Balz, Christian; LampenKelley, Paula; Banerjee, Arnab
 Physical Review B, Vol. 100, Issue 6
Terahertz excitations in $\alpha \text{\u2212}\mathrm{RuC}{\mathrm{l}}_{3}$ : Majorana fermions and rigidplane shear and compression modes
journal, September 2019
 Reschke, S.; Tsurkan, V.; Do, S. H.
 Physical Review B, Vol. 100, Issue 10
Quantized thermal Hall conductance from edge current calculations in lattice models
journal, October 2019
 Tang, Wei; Xie, X. C.; Wang, Lei
 Physical Review B, Vol. 100, Issue 15
Chiral spin order in some purported Kitaev spinliquid compounds
journal, November 2018
 Modic, K. A.; Ramshaw, B. J.; Shekhter, A.
 Physical Review B, Vol. 98, Issue 20
Phenomenological theory of heat transport in the fractional quantum Hall effect
journal, January 2019
 AharonSteinberg, Amit; Oreg, Yuval; Stern, Ady
 Physical Review B, Vol. 99, Issue 4
NonKitaev spin liquids in Kitaev materials
journal, May 2019
 Li, YaoDong; Yang, Xu; Zhou, Yi
 Physical Review B, Vol. 99, Issue 20
Impact of offdiagonal exchange interactions on the Kitaev spinliquid state of $\alpha \text{\u2212}{\mathrm{RuCl}}_{3}$
journal, June 2019
 Takikawa, Daichi; Fujimoto, Satoshi
 Physical Review B, Vol. 99, Issue 22
Large Contribution of Fermi Arcs to the Conductivity of Topological Metals
journal, August 2019
 Breitkreiz, M.; Brouwer, P. W.
 Physical Review Letters, Vol. 123, Issue 6
Thermal Hall signatures of nonKitaev spin liquids in honeycomb Kitaev materials
journal, August 2019
 Gao, Yong Hao; Hickey, Ciarán; Xiang, Tao
 Physical Review Research, Vol. 1, Issue 1
Fieldinduced ${\mathrm{QCD}}_{3}$ ChernSimons quantum criticalities in Kitaev materials
journal, January 2020
 Zou, Liujun; He, YinChen
 Physical Review Research, Vol. 2, Issue 1
Quantum spin liquids
journal, January 2020
 Broholm, C.; Cava, R. J.; Kivelson, S. A.
 Science, Vol. 367, Issue 6475