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arXiv:1710.09411 · Physical review. B./Physical review. B

Bulk-edge correspondence in fractional quantum Hall states

Bin Yan, Rudro R. Biswas, Chris H. Greene

Published Jan 28, 2019• 10 citations
arXivHTMLPDF

Abstract

We substantiate a complete picture of the ``bulk-edge correspondence'' conjecture for topological phases. By studying the eigenstates in the entanglement spectrum for both the ideal and realistic Coulomb ground states of the fractional quantum Hall system, it is verified that the eigenstates in the universal part of the entanglement spectrum purely lie in the Hilbert space of the edge excitations projected onto the physical Hilbert space of the subsystem itself. Hence, not only are the eigenlevels in the entanglement spectrum in one-to-one correspondence with the eigenenergies of an effective dynamical edge Hamiltonian, but all the eigenstates are confirmed to be the actual (projected) edge excitations of the subsystem. This result also reveals the possibility of extracting the full information of the edge excitations from the state of the subsystem reduced from a geometric cut of the pure ground state of the total system in topological phases.

DOI
10.1103/physrevb.99.035153

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We substantiate a complete picture of the ``bulk-edge correspondence'' conjecture for topological phases. By studying the eigenstates in the entanglement spectrum for both the ideal and realistic Coulomb ground states of the fractional quantum Hall system, it is verified that the eigenstates in the universal part of the entanglement spectrum purely lie in the Hilbert space of the edge excitations projected onto the physical Hilbert space of the subsystem itself. Hence, not only are the eigenlevels in the entanglement spectrum in one-to-one correspondence with the eigenenergies of an effective dynamical edge Hamiltonian, but all the eigenstates are confirmed to be the actual (projected) edge excitations of the subsystem. This result also reveals the possibility of extracting the full information of the edge excitations from the state of the subsystem reduced from a geometric cut of the pure ground state of the total system in topological phases.
Li and Haldane conjectured and numerically substantiated that the entanglement spectrum of the reduced density matrix of ground states of time-reversal-breaking topological phases [fractional quantum Hall (FQH) states] contains information about the counting of their edge modes when the ground state is cut in two spatially distinct regions and one of the regions is traced out. We analytically substantiate this conjecture for a series of FQH states defined as unique zero modes of pseudopotential Hamiltonians by finding a one-to-one map between the thermodynamic limit counting of two different entanglement spectra: the particle entanglement spectrum (PES), whose counting of eigenvalues for each good quantum number is identical to the counting of bulk quasiholes (up to accidental zero eigenvalues of the reduced density matrix), and the orbital entanglement spectrum (OES), considered by Li and Haldane. By using a set of clustering operators that have their origin in conformal-field-theory (CFT) operator expansions, we show that the counting of the OES eigenvalues in the thermodynamic limit must be identical to the counting of quasiholes in the bulk. The latter equals the counting of edge modes at a hard-wall boundary placed on the sample. Our results can be interpreted as a bulk-edge correspondence in entanglement spectra. Moreover, we show that the counting of the PES and OES is identical even for CFT states that are likely bulk gapless, such as the Gaffnian wave function.
The edge of the electronic fractional quantum Hall (FQH) system obeys the law of the chiral Luttinger liquid theory due to its intrinsic topological properties and the relation of bulk-edge correspondence. However, in a realistic experimental system, such as the usual Hall bar setup, the softening of the background confinement potential can induce the reconstruction of the edge spectrum, which breaks the chirality and universality of the FQH edge. The entanglement spectrum (ES) of the FQH ground state has the same counting structure as that of the energy spectrum indicating the topological characters of the quantum state. In this work, we report that the ES can also have an edge reconstruction while sweeping the area of the subsystem in real space cut. Moreover, we found that the critical area of the subsystem matches accurately with the intrinsic building block of the FQH liquids, namely the correlation hole of the FQH liquids. The above results seem to be universal after our study of a series of typical FQH states, such as two Laughlin states at $\ensuremath{\nu}=1/3$ and 1/51/51/5, and the Moore-Read state for $\ensuremath{\nu}=5/2$.
The entanglement spectra of fractional quantum Hall states contain universal fingerprints of their underlying topological order, as posited by the Li-Haldane conjecture. In this work, we uncover a finer universal structure within the entanglement spectra unravelled by projective measurements. Concretely, we study the projected ensemble of fractional quantum Hall states, defined as the collection of quantum states on a subsystem conditioned on measurement outcomes of its complement. We find that this ensemble exhibits a hidden hierarchy inside the Li-Haldane edge manifold: by conditioning on measurement outcomes, the entanglement spectrum's support is split into measurement-dependent sectors whose ranks we demonstrate are fixed by conformal field theory counting, an observation we dub the measurement-resolved Li-Haldane conjecture. For the non-Abelian Moore-Read state, this hierarchy is particularly rich: each parity-resolved edge manifold contains internal subspaces whose dimensions reproduce the conformal field theory counting of the opposite-parity sector. This structure persists even in realistic Coulomb-interacting ground states, establishing the projected ensemble as a sharp new probe of topological order beyond what the entanglement spectrum alone can detect.
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.
Abstract Entanglement spectroscopy has stimulated an extensive range of studies. The entanglement spectrum is the spectrum of the reduced density matrix when the system is partitioned into two. For many quantum systems, when computed from the bulk ground-state wavefunction, it gives access to the physics of edge excitations. It is thus a valuable tool to diagnose topological ordering. These lecture notes provide an overview of entanglement spectroscopy, mostly in the context of the fractional quantum Hall effect. The basic concepts are introduced through the example of quantum spin chains. The connection with entanglement entropy and the matrix product state representation is discussed. The entanglement spectrum can be computed for non-interacting topological phases, revealing edge excitation from the ground state. Application to fractional quantum Hall phases shows how much information is encoded within the ground state and how different partitions probe different types of excitations. Application to fractional Chern insulators is discussed.
It has been recently realized that strong interactions in topological Bloch bands give rise to the appearance of novel states of matter. Here we study connections between these systems---fractional Chern insulators and the fractional quantum Hall states---via generalization of a gauge-fixed Wannier-Qi construction in the cylinder geometry. Our setup offers a number of important advantages compared to the earlier exact diagonalization studies on a torus. Most notably, it gives access to edge states and to a single-cut orbital entanglement spectrum, hence to the physics of bulk-edge correspondence. It is also readily implemented in the state-of-the-art density matrix renormalization group method that allows for numerical simulations of significantly larger systems. We demonstrate our general approach on examples of flat-band models on ruby and kagome lattices at bosonic filling fractions $\ensuremath{\nu}=1/2$ and $\ensuremath{\nu}=1$, which show the signatures of (non)-Abelian phases, and establish the correspondence between the physics of edge states and the entanglement in the bulk. Notably, we find that the non-Abelian $\ensuremath{\nu}=1$ phase can be stabilized by purely on-site interactions in the presence of a confining potential.