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arXiv:1909.12128 · Physical review. D/Physical review. D.

Note on the Bloch theorem

C.X. Zhang, Mikhail A. Zubkov

Published Dec 27, 2019• 14 citations
arXivHTMLPDF

Abstract

The Bloch theorem in ordinary quantum mechanics means the absence of the total electric current in equilibrium. In the present paper, we analyze the possibility that this theorem remains valid within quantum field theory relevant for the description of both high-energy physics and condensed matter physics phenomena. First of all, we prove that the total electric current in equilibrium is the topological invariant for the gapped fermions that are subject to periodical boundary conditions; i.e., it is robust to the smooth modification of such systems. This property remains valid when the interfermion interactions due to the exchange by bosonic excitations are taken into account perturbatively. We give the proof of this statement to all orders in perturbation theory. Thus, we prove the weak version of the Bloch theorem and conclude that the total current remains zero in any system, which is obtained by smooth modification of the one with the gapped charged fermions, periodical boundary conditions, and vanishing total electric current. We analyze several examples, in which the fermions are gapless. In some of them, the total electric current vanishes. At the same time, we propose the counterexamples of the equilibrium gapless systems, in which the total electric current is nonzero.

DOI
10.1103/physrevd.100.116021

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The Bloch theorem in ordinary quantum mechanics means the absence of the total electric current in equilibrium. In the present paper, we analyze the possibility that this theorem remains valid within quantum field theory relevant for the description of both high-energy physics and condensed matter physics phenomena. First of all, we prove that the total electric current in equilibrium is the topological invariant for the gapped fermions that are subject to periodical boundary conditions; i.e., it is robust to the smooth modification of such systems. This property remains valid when the interfermion interactions due to the exchange by bosonic excitations are taken into account perturbatively. We give the proof of this statement to all orders in perturbation theory. Thus, we prove the weak version of the Bloch theorem and conclude that the total current remains zero in any system, which is obtained by smooth modification of the one with the gapped charged fermions, periodical boundary conditions, and vanishing total electric current. We analyze several examples, in which the fermions are gapless. In some of them, the total electric current vanishes. At the same time, we propose the counterexamples of the equilibrium gapless systems, in which the total electric current is nonzero.
We review the proofs of a theorem of Bloch on the absence of macroscopic stationary currents in quantum systems. The standard proof shows that the current in 1D vanishes in the large volume limit under rather general conditions. In higher dimensions, the total current across a cross-section does not need to vanish in gapless systems but it does vanish in gapped systems. We focus on the latter claim and give a self-contained proof motivated by a recently introduced index for the many-body charge transport in quantum lattice systems having a conserved [Formula: see text]-charge.
Abstract The Bloch theorem is a powerful theorem stating that the expectation value of the U(1) current operator averaged over the entire space vanishes in large quantum systems. The theorem applies to the ground state and to the thermal equilibrium at a finite temperature, irrespective of the details of the Hamiltonian as far as all terms in the Hamiltonian are finite ranged. In this work we present a simple yet rigorous proof for general lattice models. For large but finite systems, we find that both the discussion and the conclusion are sensitive to the boundary condition one assumes: under the periodic boundary condition, one can only prove that the current expectation value is inversely proportional to the linear dimension of the system, while the current expectation value completely vanishes before taking the thermodynamic limit when the open boundary condition is imposed. We also provide simple tight-binding models that clarify the limitation of the theorem in dimensions higher than one.
Bloch’s theorem, which is one of the cornerstones of condensed matter physics, has played a fundamental role in the development of many theories, such as the band theory, Fermi liquid theory, and BCS theory. It states that when the system has discrete translational symmetry, the corresponding eigenstate can be labeled by a conserved quantity, or a good quantum number — Crystal momentum. Based on Bloch’s theorem, many important physical quantities can be expressed as the integral over the entire Brillouin zone or the Fermi surface. In realistic macroscopic materials, the translational symmetry is explicitly broken down due to the existence of boundaries. It is natural to ask why we can still use Bloch’s theorem to understand the physical properties of real materials? In traditional textbooks of solid-state physics, a thermodynamic limit argument is used to explain the above question. Since the lattice size of the macroscopic material is very large, e.g. N≈1023 , its asymptotic behaviors can be described by the thermodynamic limit N→∞ . Therefore, if we fix the lattice constant, i.e., a, the system length will extend to infinity, i.e. L=aN→∞ . Since wave functions, e.g. ψ(x)∝eikx , should be bounded at the infinity, e.g. |ψ(x→±∞)|<∞ , the corresponding momentum is restricted to be real numbers. This revives Bloch’s theorem. Physically, for a finite-size system with open boundary conditions, the boundary only acts the role of scattering potential and can be regarded as a perturbation to the Bloch Hamiltonian. Therefore, the eigenstate of the OBC Hamiltonian is a superposition of all the scattered Bloch waves with the same energy. In this sense, one can say that, although the existence of boundary definitely breaks the translational symmetry, Bloch’s theorem is perturbatively broken down, and one can still use Bloch’s theorem to understand the physical properties in real materials. Even for interacting systems, Bloch’s theorem is also approximately preserved. Indeed, in a many-body system, the elementary excitations (or quasiparticles) are referred to the eigenmodes with well-defined energy, momentum, and dispersion relation. This means the excitations of the system are extended Bloch-like states in the bulk, which is consistent with the physical intuition in a disorder-free system. In this paper, we show that all the above arguments are challenged in some Hermitian bosonic systems, in which all the eigenstates, or quasiparticles are localized at the boundary, indicating that Bloch’s theorem is nonperturbatively broken down. This phenomenon generalizes the concept of the non-Hermitian skin effect from dissipative systems to non-dissipative systems, and can be understood as the Hermitian skin effect in general. Here the physical origin of the Hermitian skin effect is that although the bosonic Hamiltonian is Hermitian, its elementary excitations are determined by a non-Hermitian matrix. In order to understand the quasiparticles in such systems, the so-called generalized Brillouin zone theory is necessary. Finally, based on the Bogoliubov theory, the quench dynamic is also studied to illustrate the non-perturbative breakdown of Bloch’s theorem.
A long time ago, Bloch showed that in a system of interacting nonrelativistic particles the net particle-number current must vanish in any equilibrium state. Bloch's argument does not generalize easily to the energy current. We devise an alternative argument which proves the vanishing of the net energy currents in equilibrium states of lattice systems as well as systems of nonrelativistic particles with finite-range potential interactions. We discuss some applications of these results. In particular, we show that neither a one-dimensional (1D) lattice system nor a 1D system of nonrelativistic particles with finite-range potential interactions can flow to a conformal field theory with unequal left-moving and right-moving central charges.
Abstract We consider an infinitely extended system of fermions on a d-dimensional lattice with (magnetic) translation-invariant short-range interactions. We further assume that the system has a gapped ground state. Physically, this is a model for the bulk of a generic topological insulator at zero temperature, and we are interested in the current response of such a system to a constant external electric field. Using the non-equilibrium almost-stationary states approach, we prove that the longitudinal current density induced by a constant electric field of strength ε\varepsilon ε ε is of order O(ε∞)\mathcal {O}(\varepsilon ^\infty )O(ε∞) O ( ε ∞ ) , i.e. the system is an insulator in the usual sense. For the Hall current density we show instead that it is linear in ε\varepsilon ε ε up to terms of order O(ε∞)\mathcal {O}(\varepsilon ^\infty )O(ε∞) O ( ε ∞ ) . The proportionality factor σH\sigma _\textrm{H}σH​ σ H is by definition the Hall conductivity, and we show that it is given by a generalization of the well known double commutator formula to interacting systems. As a by-product of our results, we find that the Hall conductivity is constant within gapped phases, and that for d=2d=2d=2 d = 2 the relevant observable that “measures” the Hall conductivity in experiments, the Hall conductance, not only agrees with σH \sigma _{\textrm{H}}σH​ σ H in expectation up to O(ε∞)\mathcal {O}(\varepsilon ^\infty )O(ε∞) O ( ε ∞ ) , but also has vanishing variance. A notable difference to several existing results on the current response in interacting fermion systems is that we consider a macroscopic system exposed to a small constant electric field, rather than to a small voltage drop.
Abstract We prove that the magnetization is equal to the edge current in the thermodynamic limit for a large class of models of lattice fermions with finite-range interactions satisfying local indistinguishability of the Gibbs state, a condition known to hold for sufficiently high temperatures. Our result implies that edge currents in such systems are determined by bulk properties and are therefore stable against large perturbations near the boundaries. Moreover, the equality persists also after taking the derivative with respect to the chemical potential. We show that this form of bulk-edge correspondence is essentially a consequence of homogeneity in the bulk and locality of the Gibbs state. An important intermediate result is a new version of Bloch’s theorem for two-dimensional systems, stating that persistent currents vanish in the bulk.