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arXiv:2004.13222 · Axioms

Dissipative Dynamics of Non-Interacting Fermion Systems and Conductivity

Kazuki Yamaga

Published Nov 3, 2020• 1 citation
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Abstract

In this paper, Non-Equilibrium Steady State that is induced by electric field and the conductivity of non-interacting fermion systems under the dissipative dynamics is discussed. The dissipation is taken into account within a framework of the quantum dynamical semigroup introduced by Davies (1977). We obtain a formula of the conductivity for the stationary state, which is applicable to arbitrary potentials. Our formula gives a justification of an adiabatic factor that is often introduced in practical calculation while using the Kubo formula. In addition, the conductivity of crystals (i.e., periodic potentials) is also discussed.

DOI
10.3390/axioms9040128

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In this paper, Non-Equilibrium Steady State that is induced by electric field and the conductivity of non-interacting fermion systems under the dissipative dynamics is discussed. The dissipation is taken into account within a framework of the quantum dynamical semigroup introduced by Davies (1977). We obtain a formula of the conductivity for the stationary state, which is applicable to arbitrary potentials. Our formula gives a justification of an adiabatic factor that is often introduced in practical calculation while using the Kubo formula. In addition, the conductivity of crystals (i.e., periodic potentials) is also discussed.
Novel physics arises when strongly correlated system is driven out of equilibrium by external fields. Dramatic changes in physical properties, such as conductivity, are empirically observed in strongly correlated materials under high electric field. In particular, electric-field driven metal-insulator transitions are well-known as the resistive switching effect in a variety of materials, such as VO2_22​, V2_22​O3_33​ and other transition metal oxides. To satisfactorily explain both the phenomenology and its underlying mechanism, it is required to model microscopically the out-of-equilibrium dissipative lattice system of interacting electrons. In this thesis, we developed a systematic method of modeling non-equilibrium steady states for dissipative lattice systems by means of Non-equilibrium Green's function and Dynamical Mean Field Theory. We firstly establish a "minimum model" to formulate the strong-field transport in non-interacting dissipative electron lattice. This model is exactly soluble and convenient for discussing energy dissipation and steady-state properties. The formalism is then combined with Dynamical Mean Field Theory to provide a systematic framework describing the nonequilibrium steady-state of correlated materials. We use the formalism to study the strong-field transport properties of correlated materials, Mott insulators and Dirac electrons in graphene. We concentrate on the microscopic description of resistive switching. Of particular interest is the filament formation during the dynamical phase transition, which has been interpreted as a result of the delicate interplay between dissipation and Mott physics. We will also examine IVIVIV characteristics and particularly the current saturation of Dirac electrons in graphene. The arXiv version has been updated with minor modifications and corrections.
Abstract The optical conductivity is the basic defining property of materials characterizing the current response toward time-dependent electric fields. In this work, following the approach of Kubo’s response theory, we study the general properties of the nonlinear optical conductivities of quantum many-body systems both in equilibrium and non-equilibrium. We obtain an expression of the second- and the third-order optical conductivity in terms of correlation functions and present a perturbative proof of the generalized Kohn formula proposed recently. We also discuss a generalization of the f-sum rule to a non-equilibrium setting by focusing on the instantaneous response.
In this note we consider lattice fermions on Z2 with a locally unique gapped ground state and show how to apply the non-equilibrium almost stationary state (NEASS) approach to linear response to derive a formula for the Hall conductance in terms of the ground state expectation of a commutator of modified step functions. This formula is usually derived by a charge pumping argument going back to Laughlin. Here we show that it can also be obtained as the linear response coefficient of the microscopic current response to an adiabatic increase of the chemical potential on a half plane (or more generally on any cone-like region). Indeed, in a manner reminiscent of the bulk-boundary correspondence, we show that raising the chemical potential in any cone-like region gives rise to a current that flows along its boundary and is nearly linear in the increase in chemical potential. We also discuss the connection with the double commutator formula with modified position operators for the Hall conductivity derived in [Wesle et al., Commun. Math. Phys. 406, 199 (2025)] as the linear response coefficient of the macroscopic current response to the adiabatic application of a constant electric field.
We first review the problem of a rigorous justification of Kubo’s formula for transport coefficients in gapped extended Hamiltonian quantum systems at zero temperature. In particular, the theoretical understanding of the quantum Hall effect rests on the validity of Kubo’s formula for such systems, a connection that we review briefly as well. We then highlight an approach to linear response theory based on non-equilibrium almost-stationary states (NEASS) and on a corresponding adiabatic theorem for such systems that was recently proposed and worked out by one of us in [51] for interacting fermionic systems on finite lattices. In the second part of our paper, we show how to lift the results of [51] to infinite systems by taking a thermodynamic limit.
Correlated systems are a wide class of materials in which the strong electron-electron repulsion is the origin of very fascinating and unusual properties, among which metal-to-insulator transitions and high temperature superconductivity are the most striking examples. In the recent years, the fast development of non-adiabatic probing techniques opened new interesting perspectives for the investigation of such materials in non-equilibrium conditions. In this thesis, we discuss the theoretical description of few relevant cases which represent different examples of non-equilibrium phenomena in correlated materials. In particular, we will focus on the dynamics following a sudden excitation and the coupling to an external driving field. As a first example we consider the dynamics across a phase transition, namely we explore the possibility of driving a phase transition as the result of a sudden excitation, as e.g. the coupling with a short light pulse. We consider systems showing different equilibrium phases and study the conditions under which the off-equilibrium dynamics may lead to non-trivial dynamical phase transitions. A different case is represented by the dynamics induced by a driving electric field. This problem is particularly relevant for the possible applications of correlated materials in electronic devices. Here we consider the paradigmatic case of a correlated material coupled to external sources which impose a finite bias across the system. We analyze the formation and the properties of the non-equilibrium stationary states in which a finite current flows through the system. This allows us to study the non-linear response properties of a correlated system. In this context, a particularly relevant aspect is the problem of the dielectric breakdown of a Mott insulator, namely the formation of conducting states in the Mott insulating phase. In this thesis we explore different mechanisms leading to such possibility. First we discuss a quantum tunneling mechanism of carriers driven across the insulating gap by the effect of strong electric-fields. Eventually, we discuss the possibility of a resistive transition from an insulating to a metallic state induced by the application of an external electric-field.
We review some recent results from the mathematical theory of transport of charge and spin in gapped crystalline quantum systems. The emphasis will be on transport coefficients, such as conductivities and conductances. As for the former, those are computed as appropriate expectations of current operators in a non-equilibrium almost-stationary state (NEASS), which arises from the perturbation of an equilibrium state by an external electric field. While for charge transport the usual double-commutator Kubo formula is recovered (also beyond linear response), we obtain formulas for appropriately defined spin conductivities, which are still explicit but more involved. Certain “Kubo-like” terms in these formulas are also shown to agree with the corresponding contributions to the spin conductance. In addition to that, we employ similar techniques to show a new result, namely that even in systems with non-conserved spin, there is no generation of spin torque, that is, the spin torque operator has an expectation in the NEASS which vanishes faster than any power of the intensity of the perturbing field.