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:1108.3879, to appear in PRL soon.

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

Thermal Hall (Righi-Leduc) Effects

Phonon Hall Effect

Strohm et al., PRL95, 155901(2006)

Tb3Ga5O12

Magnon Hall Effect

Onose et al., Science 329, 297(2010)

Lu2V2O7

Anomalous Hall System and Wiedemann-Franz Law

Onose et al., PRL100, 016601 (2008)

Kubo Formula

Mahan, Many Particle Physics

Kubo, Toda and Hashitsume, Statistical Physics II

Applied to non-interacting carriers

- Bloch Hamiltonian

- Dispersion - Bloch wave function (periodic part)

However

The same divergence also occurs for the phonon Hall effect, and others!

"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.

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:

giving extra contribution from the magnetization to the transport coefficients.

which vanishes in equilibrium, but in non-equilibrium,

Energy magnetization

Energy Magnetization • Equilibrium energy magnetization is defined by: "

• Unfortunately, energy magnetization cannot be uniquely determined:"""-- 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

or

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:

Our Magnetization Formulas---Results

Thermodynamic Interpretation

Shi et al., PRL 99, 197202 (2007)

Gravitomagnetism

-- Wikipedia, Gravitomagnetism

Ryu, Moore, and Ludwig, arXiv:1010.0936 (2010)

Thermal magnetization is gravitomagnetic response.

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.

Magnetization Corrections to Thermal Transport

Flux Force

Transport Responses:

Transport Currents:

Kubo formula:

Cooper-Halperin-Ruzin, PRB 55, 2344(1997)

Onsager Relations and Einstein Relations • Onsager Relations:

• Einstein Relations:

• Vanishing when the system is in global equilibrium:

drifting diffusion

Sketch of Proof • Density matrix:

• Expectation values of currents:

•  Local equilibrium currents:

Static response theory: Kubo, Toda and Hashitsume, Statistical Physics II

Sketch of Proof, continued • 

• Combine both:

• Define the transport currents

• Combine All

Applied to AHS: Scaling of Energy Current

In the presence of

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

However:

Redefine:

Thermal Hall Coefficient of Bloch Electrons

Wiedemann-Franz Law

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. spin-caloric transport:

• Eliminating the unphysical divergence, and recovering the Wiedemann-Franz law for the non-interacting anomalous Hall system.

Proper definition of spin current: Shi et al., PRL96, 076604 (2005)

Thank You!

Mechanic vs. Thermodynamic Forces

Equilibrium means:

Non-equilibrium drives transport current:

conduction diffusion

conductance and diffusion constants are related Einstein Relations

Transport can be driven either by mechanic force: , or thermodynamic force:

In Equilibrium:

Luttinger’s Approach for Thermal Transport

Equilibrium means:

Non-equilibrium drives transport energy current:

Gravitational field

Luttinger, Phys. Rev. 135, A1505 (1964)

Cooper-Halperin-Ruzin Theory

For magnetic systems:

In the presence of the gravitational field:

Linear response:

Cooper-Halperin-Ruzin, PRB 55, 2344(1997)

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

Local Equilibrium • Local equilibrium state: local temperature , external gravitational field

Local equilibrium Dynamic correction

L Linear Response Theory

Local Energy Current

Kubo formula

Static response theory: Kubo, Toda and Hashitsume, Statistical Physics II

Gradient Expansion

Spatially slow varying temperature field:

Energy Magnetization

System in equilibrium

To cancel

When

Formula: Thermal Hall Coefficient

The formula is valid only when the energy current operator obeys the scaling law:

Generalized to Electronic System

Orbital magnetization: Shi et al., PRL 99, 197202 (2007)

Generalization to spin-caloric transport: Proper definition of spin current: Shi et al., PRL96, 076604 (2005)