Berry Phase Correction to Electron Density of States in Solids

Abstract

Liouville's theorem on the conservation of phase space volume is violated by Berry phase in the semiclassical dynamics of Bloch electrons. This leads to a modification of the phase space density of states, whose significance is discussed in a number of examples: field modification of the Fermi-sea volume, connection to the anomalous Hall effect, and a general formula for orbital magnetization. The effective quantum mechanics of Bloch electrons is also sketched, where the modified density of states plays an essential role.PACS numbers: 73.43.-f, 72.15.-v, 75.20.-g Semiclassical dynamics of Bloch electrons in external fields has provided a powerful theoretical framework to account for various properties of metals, semiconductors and insulators [1]. In recent years, it has become increasingly clear that essential modification of the semiclassical dynamics is necessary for a proper understanding of a number of phenomena. It was known earlier that global geometric phase effects [2,3] on Bloch states are very important for insulators in our understanding of the quantum Hall effect [4], quantized adiabatic pumps [5], and electric polarization [6,7]. It was shown [8,9] later that geometric phase also modifies the local dynamics of Bloch electrons and thus affects the transport properties of metals and semiconductors. Recently these ideas have been successfully applied to the anomalous Hall effect in ferromagnetic semiconductors and metals [10,11,12,13], as well as spin transport [14,15].In this Letter, we reveal a general property of the Berry phase modified semiclassical dynamics which has been overlooked so far: the violation of Liouville's theorem for the conservation of phase space volume. Liouville's theorem was originally established for standard classical Hamiltonian dynamics, and its importance cannot be over emphasized as it serves as a foundation for classical statistical physics. The Berry phase makes, in general, the equations of motion non-canonical [8,9,16,17,18], rendering the violation of Liouville's theorem. Nevertheless, we are able to remedy the situation by modifying the density of states in the phase space.This modified phase-space density of states enters naturally in the semiclassical expression for the expectation value of physical quantities, and has profound effects on equilibrium as well as transport properties. We demonstrate this with several examples. First, we consider a Fermi sea of electrons in a weak magnetic field, and show that the Fermi sea volume can be changed linearly by the field. Second, we show how the Berry phase formula for the intrinsic anomalous Hall conductivity may be derived from equilibrium thermodynamics using the Středa formula [19]. Third, we provide a general derivation of an orbital-magnetization formula which is convenient for first-principles calculations.In addition, we present an effective quantum mechanics for Bloch electrons in solids by quantizing the semiclassical dynamics with the geometric phase. The density of states enters in a nontrivial manner into...

Berry Phase Correction to Electron Density in Solids and "Exotic" Dynamics

“…Much of the transport phenomena rely on the Liouville's theorem on the conservation of the phase-space volume element. Recently, [4] Xiao et al observed that, due to the Berry curvature term, the naive phase-space volume form drdk is no more conserved. They attribute this fact to the non-Hamiltonian character of the equations (1)- (2), and suggest to redefine the phase-space density by including a pre-factor,…”

Semiclassical Approximations for Hamiltonians with Operator-Valued Symbols

“…To our best knowledge the statement of Theorem 1 on stationary expectations was used only recently by Gat and Avron [GA03] and later in [XSN05]. As far as we understand, there the statement is taken for granted, but not systematically derived or proved.…”

“…This formula for the magnetization in solids is rather recent even on a heuristic level, see [GA03,XSN05]. It was proved by a completely different approach as a special case of a more general formula in [SBT12].…”

Classical version of the non-Abelian gauge anomaly

“…This measure with its √ G modification was originally regarded as "non-canonical" [18], but it is precisely the canonical phase-space volume associated with the unconventional symplectic structure [19].…”