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arXiv:1006.5435 · Physical Review Letters

Extracting Excitations from Model State Entanglement

Antoine Sterdyniak, Nicolas Regnault, Bogdan Andrei Bernevig

Published Mar 8, 2011• 149 citations
arXivHTMLPDF

Abstract

We extend the concept of an entanglement spectrum from the geometrical to the particle bipartite partition. We apply this to several fractional quantum Hall wave functions on both sphere and torus geometries to show that this new type of entanglement spectra completely reveals the physics of bulk quasihole excitations. While this is easily understood when a local Hamiltonian for the model state exists, we show that the quasihole wave functions are encoded within the model state even when such a Hamiltonian is not known. As a nontrivial example, we look at Jain's composite fermion states and obtain their quasiholes directly from the model state wave function. We reach similar conclusions for wave functions described by Jack polynomials.

DOI
10.1103/physrevlett.106.100405

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We extend the concept of an entanglement spectrum from the geometrical to the particle bipartite partition. We apply this to several fractional quantum Hall wave functions on both sphere and torus geometries to show that this new type of entanglement spectra completely reveals the physics of bulk quasihole excitations. While this is easily understood when a local Hamiltonian for the model state exists, we show that the quasihole wave functions are encoded within the model state even when such a Hamiltonian is not known. As a nontrivial example, we look at Jain's composite fermion states and obtain their quasiholes directly from the model state wave function. We reach similar conclusions for wave functions described by Jack polynomials.
We study the entanglement spectra of many-particle systems in states which are closely related to products of Slater determinants or products of permanents, or combinations of the two. Such states notably include the Laughlin and Jain composite fermion states which describe most of the observed conductance plateaus of the fractional quantum Hall effect. We identify a set of entanglement wave functions (EWFs), for subsets of the particles, which completely describe the entanglement spectra of such product wave functions, both in real space and in particle space. A subset of the EWFs for the Laughlin and Jain states can be recognized as composite fermion states. These states provide an exact description of the low angular-momentum sectors of the real-space entanglement spectrum (RSES) of these trial wave functions and a physical explanation of the branches of excitations observed in the RSES of the Jain states.
A quantum phase transition is usually achieved by tuning physical parameters in a Hamiltonian at zero temperature. Here, we show that the ground state of a topological phase itself encodes critical properties of its transition to a trivial phase. To extract this information, we introduce an extensive partition of the system into two subsystems both of which extend throughout the bulk in all directions. The resulting bulk entanglement spectrum has a low-lying part that resembles the excitation spectrum of a bulk Hamiltonian, which allows us to probe a topological phase transition from a single wave function by tuning either the geometry of the partition or the entanglement temperature. As an example, this remarkable correspondence between the topological phase transition and the entanglement criticality is rigorously established for integer quantum Hall states.
The entanglement spectroscopy, initially introduced by Li and Haldane in the context of the fractional quantum Hall effects, has stimulated an extensive range of studies. The ent anglement spectrum is the spectrum of the reduced density matrix, when we partition the system into two. For many quantum systems, it unveils a unique feature: Computed from the bulk ground state wave function, the entanglement spectrum give access to the physics of edge excitations. In this manuscript, we give an overview of the entanglement spectroscopy. We introduce the basic concepts through the case of the quantum spin chains. We present an extensive rev iew of the entanglement spectra applied to the fractional quantum Hall phases, showing how much information is encoded within the ground state and how different partitions probe different type of excitations. As a practical application of this technique, we discuss how it can discriminate between the different phases that emerge in the strongly interacti ng Chern insulators.
Materials exhibiting exotic quantum phenomena at a macroscopic level are of great interest because of its intrinsic value to Physics, and also because of its potential applications in technology. Strongly correlated electronic systems form interesting examples of these. Exploring them calls for a good understanding of its physics, and development of new tools that are applicable to their analysis. Robustness of the fractional quantum Hall effect (FQHE) makes it an ideal stage to study strongly correlated electron systems. The complexity of the FQHE problem stems directly from the strong interactions between the constituent particles and not from the specific nature of the material it is implemented in. This work attempts to gain a theoretical description of some of the less understood aspects of FQHE. FQHE in the lowest Landau level is well understood using the theory of weakly interacting emergent particles called composite fermions. Composite fermions formed in the second Landau level however appear to be strongly interacting. Of particular interest is the FQHE at 2 + 1/2 filling fraction, where the many body ground state is believed to be described by a paired state of composite fermions, represented by the Pfaffian wavefunction. Based on predicted properties of its excitations, there has been proposals for implementing topological quantum com- putations in this system. However a good understanding of the nature of these excitations is still lacking. This thesis introduces a class of many body wavefunctions called “bipartite composite fermion states” that provide a unified description of all excitations of the paired state at 5/2. In the bipartite composite fermion (BCF) picture, constituent composite fermions split into two partitions which are correlated to each other. Within each partition, they fill the lowest composite fermion Landau level of its partition. Properties of the BCF state can now be related to the properties of the individual partitions, which are well understood in terms of the composite fermion theory. In particular, we show that the excitations of BCF states are obtained by creating excitations in the individual partitions. The paired state with odd number of composite fermions should have an unpaired composite fermion. The BCF model predicts that this unpaired composite fermion behaves as an exciton in the quasiparticle and quasihole are in separate partitions. We confirm this prediction numerically in finite size systems. Generalizations of bipartite composite fermion functions to other filling fractions especially 2 + 4/7 and 2 + 3/5 are also studied. Idea of bipartite composite fermions can be generalized to the case of more partitions. Specifically, we look at tripartite composite fermion states which are similar to the bipartite states, except that these have three correlated partitions instead of two. Tripartite states have a filling fraction of ν = 3/5, and its particle hole conjugate is a candidate wavefunction for describing the ν = 2 + 2/5…
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 $\ensuremath{\nu}=1$ 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 subleading topological entanglement entropy term.