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.
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.
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.
Abstract We show that whenever the Gibbs state of a quantum spin system satisfies decay of correlations, then it is stable, in the sense that local perturbations affect the Gibbs state only locally, and it satisfies local indistinguishability, i.e. it exhibits local insensitivity to system size. These implications hold in any dimension, require only locality of the Hamiltonian, and are based on Lieb–Robinson bounds and on a detailed analysis of the locality properties of the quantum belief propagation for Gibbs states. To demonstrate the versatility of our approach, we explicitly apply our results to several physically relevant models in which the decay of correlations is either known to hold or is proved by us. These include Gibbs states of one-dimensional spin chains with polynomially decaying interactions at any temperature, and high-temperature Gibbs states of quantum spin systems with finite-range interactions in any dimension. We also prove exponential decay of correlations above a threshold temperature for Gibbs states of one-dimensional finite spin chains with translation-invariant and exponentially decaying interactions, and then apply our general results.
This thesis analyses several locality properties of extended quantum lattice systems, considering both spins and fermions. We begin by assuming locality of the Hamiltonian and then discuss how this translates into locality of the time evolution and the thermal states. Starting from a local Hamiltonian, it is well known that the Heisenberg time evolution satisfies Lieb-Robinson bounds, which quantify its locality. We strengthen these bounds for long-range interacting fermions and for commuting interactions. Furthermore, we establish the first Lieb-Robinson bounds for Hamiltonians whose local terms can grow linearly away from the origin. We also prove stability of Lieb-Robinson bounds with respect to local perturbations in the Hamiltonian. For the thermal states we focus on three types of locality known as decay of correlations, the local perturbations perturb locally (LPPL) principle and local indistinguishability. As a key tool we use spectral filters, which were introduced by Hastings in the 2000s for various applications. It enable us to use Lieb-Robinson bounds to obtain information about locality properties of states. In particular, we use quantum belief propagation to prove an equivalence of decay of correlations, LPPL and local indistinguishability for Gibbs states at any positive temperature. Using the improved Lieb-Robinson bounds mentioned before, we obtain improvements for decay of correlations in the ground state, the locality of the spectral flow and LPPL for long-range interacting systems and commuting Hamiltonians. Furthermore, we adjust the filter functions to achieve exponential decay for certain applications that were previously limited to stretched exponential decay. Spectral filters are also used to rigorously justify response theory, which usually requires a gapped system to begin with. To relax this assumption, we introduce various notions of local gaps, discuss their relations and use one to justify response theory and Kubo’s formula in the absence of a global gap. In another result we use local indistinguishability as an input to prove bulk-edge correspondence at finite temperature, in the sense that the bulk magnetisation is equal to the edge current. This result has mostly been studied for non-interacting models or at zero temperature before. Finally, we present a class of weakly interacting spin systems with an on-site gap and prove that the ground state satisfies a strong form of LPPL. Moreover, we show that these systems satisfy decay of correlations and local indistinguishability uniformly in temperature.
We consider the edge transport properties of interacting quantum Hall systems on a cylinder, in the infinite volume and zero temperature limit. We prove that the edge conductance is universal, and equal to the sum of the chiralities of the non-interacting edge modes. With respect to previous work, our result allows to consider a generic class of quantum Hall systems, displaying arbitrarily many edge modes. Our proof quantifies the validity and the limitations of the Luttinger liquid effective description for the edge currents. In particular, due to edge states scattering, the effective description alone is not able to predict the universality of the edge conductance. The exact quantization follows after fully taking into account the bulk degrees of freedom, whose precise contribution to the edge transport is determined thanks to lattice conservation laws.
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.
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 ε is of order 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 ε up to terms of order O(ε∞) O ( ε ∞ ) . The proportionality factor 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=2 d = 2 the relevant observable that “measures” the Hall conductivity in experiments, the Hall conductance, not only agrees with σH σ H in expectation up to 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.