Real-space entanglement spectrum of quantum Hall states

Abstract

We investigate the entanglement spectra arising from sharp real-space partitions of the system for quantum Hall states. These partitions differ from the previously utilized orbital and particle partitions and reveal complementary aspects of the physics of these topologically ordered systems. We show, by constructing one to one maps to the particle partition entanglement spectra, that the counting of the real-space entanglement spectra levels for different particle number sectors versus their angular momentum along the spatial partition boundary is equal to the counting of states for the system with a number of (unpinned) bulk quasiholes excitations corresponding to the same particle and flux numbers. This proves that, for an ideal model state described by a conformal field theory, the real-space entanglement spectra level counting is bounded by the counting of the conformal field theory edge modes. This bound is known to be saturated in the thermodynamic limit (and at finite sizes for certain states). Numerically analyzing several ideal model states, we find that the real-space entanglement spectra indeed display the edge modes dispersion relations expected from their corresponding conformal field theories. We also numerically find that the real-space entanglement spectra of Coulomb interaction ground states exhibit a series of branches, which we relate to the model state and (above an entanglement gap) to its quasiparticle-quasihole excitations. We also numerically compute the entanglement entropy for the ν = 1 integer quantum Hall state with real-space partitions and compare against the analytic prediction. We find that the entanglement entropy indeed scales linearly with the boundary length for large enough systems, but that the attainable system sizes are still too small to provide a reliable extraction of the sub-leading topological entanglement entropy term.

Edge-state inner products and real-space entanglement spectrum of trial quantum Hall states

“…Although the work of LH relies on a bipartition that is a cut in momentum! space (the so-called "orbital partition" [15]), which is not local [16], their main observations remain valid for the more natural cut in real-space [16][17][18]. The real-space ES is expected to be the spectrum of a local field theory along the cut, in agreement with the insightful suggestion of KP: not only the multiplicities but also the eigenvalues of H E = − log ρ A are the ones of a (perturbed) CFT along the cut, when the length of the cut is large compared to the mean particle spacing (which plays the role of an UV cutoff).…”

“…In that case, the Schmidt eigenvalues/eigenvectors can be organized in SO(3) multiplets. It can be shown easily that the Schmidt rank is the same in each L Â z subsector both for RSP and PP [16,17]. For PP, one can define L A 0 as the maximum total angular momentum for the subsystem A, and N A0 the corresponding number of particles.…”

“…In quantum Hall systems the RSP [16][17][18] is obtained by dividing the complex plane C where the coordinates z i are defined (see section I B) into two complementary parts C = A ∪ B. As is customary in the literature, the bipartition of the plane C = A ∪ B is usually taken such that the subsystem A is rotationally invariant (and simply-connected for simplicity): A is then a disc of radius R centered on the origin.…”

“…The ES is related to S b (A) and S b (B) as follows. Let us consider the operator S ES defined by 17) which acts in the Hilbert space of the chiral CFT. This operator is nothing but the "pseudo-Hamiltonian" we are looking for: its spectrum is the ES, up to the global additive constant (4.14).…”

Exact matrix product states for quantum Hall wave functions

“…III, the computational complexity of the MPS representation is on the order O(b L ) for b ∼ O (1). However, to achieve the same type of scaling in the traditional Hilbert space representation (say on a sphere 11,[30][31][32][33][34] would require N ∼ O(L 2 ) particles and a Hilbert space dimension scaling as b L 2 . We note that previously a conceptually distinct approach found an MPS for the Laughlin state in which there is one matrix per particle, rather than per orbital.…”