Energy Magnetization and Thermal Hall Effect

Transcription

1 Energy Magnetization and Thermal Hall Effect Qian Niu University of Texas at Austin International Center for Quantum Materials at Peking University NQS2011 YITP, Kyoto November 25, 2011 In collaboration with: Tao Qin, Junren Shi arxiv: , to appear in PRL soon.

2 Outline Introduction - Thermal Hall Effect and Issue Formulation of Magnetizations Corrections to the Thermal Hall Coefficients Application to Non-interacting electrons -> Wiedemann-Franz Law Summary

3 Thermal Hall (Righi-Leduc) Effects Phonon Hall Effect Magnon Hall Effect Anomalous Hall System and Wiedemann-Franz Law Tb3Ga5O12 Strohm et al., PRL95, (2006) Lu2V2O7 Onose et al., Science 329, 297(2010) Onose et al., PRL100, (2008)

4 Kubo Formula Mahan, Many Particle Physics Kubo, Toda and Hashitsume, Statistical Physics II

5 Applied to non-interacting carriers - Bloch Hamiltonian - Bloch wave function (periodic part) - Dispersion However The same divergence also occurs for the phonon Hall effect, and others!

6 " The Central Issue and its solution Kubo formula yields divergent thermal Hall conductivity even at zero temperature. It violates the Einstein relation. It violates the Wiedemann-Franz law for noninteracting electrons. This is partly answered by wave packet semiclassical theory and Streda type linear response as demonstrated by Murakami and collaborators. But, Einstein relation is assumed. The results are valid only for non-interacting carriers. Here we present a general solution: Valid even for interacting systems. Einstein relation is established rather than assumed. Derived general formulas for energy magnetization---circulating energy current in equilibrium which turns into transport current in non- equilibrium.

7 Transport Energy Current, Energy Magnetization Energy current is only defined up to a curl: Energy current may have non-zero expectation value even when the system is in equilibrium: Transport energy current: which vanishes in equilibrium, but in non-equilibrium, Energy magnetization giving extra contribution from the magnetization to the transport coefficients.

8 Energy Magnetization Equilibrium energy magnetization is defined by: " Unfortunately, energy magnetization cannot be uniquely determined:" " " or -- Hirst, RMP69, 607(1997) Which gauge is appropriate for the purpose of calculating the thermal Hall coefficient? How to evaluate the DC component of energy magnetization for an extended system? Issues: What energy magnetization is, and how to calculate it? Determining its correction to the thermal Hall coefficient

9 Preliminaries A general interacting electronic system: Introducing external mechanic forces: potential and gravitational field - mechanic counterpart of temperature gradient (Luttinger, PR135, A1505(1964)) Particle and Energy current definitions: Scaling laws of currents: Definitions of Magnetizations:

10 Our Magnetization Formulas---Results

11 Thermodynamic Interpretation Shi et al., PRL 99, (2007)

12 Gravitomagnetism -- Wikipedia, Gravitomagnetism Thermal magnetization is gravitomagnetic response. Ryu, Moore, and Ludwig, arxiv: (2010)

13 Sketch of Proof Introducing the static response functions: We can prove, when the scaling laws of current operators are valid: Then, it must have the decomposition: We can identify and do integral over r.

14 Magnetization Corrections to Thermal Transport Transport Responses: Cooper-Halperin-Ruzin, PRB 55, 2344(1997) Flux Force Kubo formula: Transport Currents:

15 Onsager Relations and Einstein Relations Onsager Relations: Einstein Relations: drifting diffusion Vanishing when the system is in global equilibrium:

16 Sketch of Proof Density matrix: Expectation values of currents: Local equilibrium currents: Static response theory: Kubo, Toda and Hashitsume, Statistical Physics II

17 Sketch of Proof, continued Combine both: Define the transport currents Combine All

18 Applied to AHS: Scaling of Energy Current In the presence of

19 Proper Energy Current Operator To apply the formula, the energy current operator must scale with: However: Redefine:

20 Thermal Hall Coefficient of Bloch Electrons Wiedemann-Franz Law

21 Summary A set of general formulas for calculating electromagnetic orbital magnetization as well as gravitomagnetic energy magnetization Explicitly demonstrating the magnetization corrections to the thermal transport coefficients, recovering the Onsager and Einstein relations General theoretical approach easily extendable to other transports: e.g. spincaloric transport: Proper definition of spin current: Shi et al., PRL96, (2005) Eliminating the unphysical divergence, and recovering the Wiedemann-Franz law for the non-interacting anomalous Hall system.

22 Thank You!

23 Mechanic vs. Thermodynamic Forces Transport can be driven either by mechanic force:, or thermodynamic force: In Equilibrium: Equilibrium means: Non-equilibrium drives transport current: conductance and diffusion constants are related conduction diffusion Einstein Relations

24 Luttinger s Approach for Thermal Transport Gravitational field Equilibrium means: Non-equilibrium drives transport energy current: Luttinger, Phys. Rev. 135, A1505 (1964)

25 Cooper-Halperin-Ruzin Theory For magnetic systems: In the presence of the gravitational field: Linear response: Cooper-Halperin-Ruzin, PRB 55, 2344(1997)

26 Our Theory Logic: There is no unique definition of energy current. The transport energy current should vanish when the system is in equilibrium -- This is also its definition. The above requirement defines

27 Local Equilibrium Local equilibrium state: local temperature, external gravitational field Local equilibrium Dynamic correction L Linear Response Theory

28 Local Energy Current Kubo formula Static response theory: Kubo, Toda and Hashitsume, Statistical Physics II

29 Gradient Expansion Spatially slow varying temperature field:

30 Energy Magnetization System in equilibrium When To cancel

31 Formula: Thermal Hall Coefficient The formula is valid only when the energy current operator obeys the scaling law:

32 Generalized to Electronic System Generalization to spin-caloric transport: Orbital magnetization: Shi et al., PRL 99, (2007) Proper definition of spin current: Shi et al., PRL96, (2005)