Bulk-edge correspondence in entanglement spectra

Chandran, Anushya; Hermanns, Maria; Regnault, Nicolas; Bernevig, B. Andrei

doi:10.1103/physrevb.84.205136

Cited by 153 publications

(196 citation statements)

References 69 publications

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“…In that sense, and for N A /N → 1/2 when N, N A → ∞, the part A (B) corresponds to the northern (southern) hemisphere. Thus, for these states, parts A and B may roughly be seen as spatial region, just like in the case of RSP (this limit is also considered in [25]). This observation can be made precise, and it has the consequence that the particle ES-the ES with PP-at large!…”

Section: Particle Partition (Pp)mentioning

confidence: 99%

“…Previous steps towards an analytic understanding of the ES in quantum Hall systems include direct calculations for the integer quantum Hall effect [16,20,21] or other free fermion systems [22][23][24] and rigorous results on the multiplicities for a large class of trial wavefunctions [25,26]. Some general arguments for a correspondence between the entanglement and edge spectra have been proposed previously.…”

Section: A Motivationmentioning

confidence: 99%

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

2012

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We consider the trial wavefunctions for the fractional quantum Hall effect that are given by conformal blocks, and construct their associated edge excited states in full generality. The inner products between these edge states are computed in the thermodynamic limit, assuming generalized screening (i.e. short-range correlations only) inside the quantum Hall droplet, and using the language of boundary conformal field theory (boundary CFT). These inner products take universal values in this limit: they are equal to the corresponding inner products in the bulk 2d chiral CFT which underlies the trial wavefunction. This is a bulk/edge correspondence; it shows the equality between equal-time correlators along the edge and the correlators of the bulk CFT up to a Wick rotation.This approach is then used to analyze the entanglement spectrum of the ground state obtained with a bipartition A ∪ B in real-space. Starting from our universal result for inner products in the thermodynamic limit, we tackle corrections to scaling using standard field-theoretic and renormalization group arguments. We prove that generalized screening implies that the entanglement Hamiltonian HE = − log ρA is isospectral to an operator that is local along the cut between A and B. We also show that a similar analysis can be carried out for particle partition. We discuss the close analogy between the formalism of trial wavefunctions given by conformal blocks and Tensor Product States, for which results analogous to ours have appeared recently. Finally, the edge theory and entanglement spectrum of px ± ipy paired superfluids are treated in a similar fashion in the appendix.

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Section: Particle Partition (Pp)mentioning

confidence: 99%

Section: A Motivationmentioning

confidence: 99%

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

2012

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“…In the same manner, TRI 2D and 3D insulators are also classified by the corresponding homotopy group, Z 2 , of continuous mappings from the d = 2, 3-dimensional BZ to a target space of projectors to occupied states under the constraint of TRI. Besides the bulk topological indices defined in insulators with translational invariance and with disorder, a bulkedge correspondence exists which guarantees presence of gapless modes on the boundary between two insulators having different values of Z or Z 2 index 25,29,30 .…”

mentioning

confidence: 99%

“…The entanglement spectrum has been successfully applied to identify topological orders in a variety of condensed matter systems such as fractional quantum Hall systems 35 , spin chains 36 , Chern insulators 37 , and topological band insulators 30,[38][39][40] . Unlike the energy spectrum, the quantum entanglement spectrum solely depends on the many-body ground state, and is, for this reason, more suitable for identifying any non-trivial topology in the ground state.…”

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confidence: 99%

Entanglement spectrum classification ofCn-invariant noninteracting topological insulators in two dimensions

2013

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We study the single particle entanglement spectrum in 2D topological insulators which possess n-fold rotation symmetry. By defining a series of special choices of subsystems on which the entanglement is calculated, or real space cuts, we find that the number of protected in-gap states for each type of these real space cuts is a quantum number indexing (if any) non-trivial topology in these insulators. We explicitly show the number of protected in-gap states is determined by a Z n -index, (z1, ..., zn), where zm is the number of occupied states that transform according to m-th one-dimensional representation of the Cn point group. We find that the entanglement spectrum contains in-gap states pinned in an interval of entanglement eigenvalues [1/n, 1 − 1/n]. We determine the number of such in-gap states for an exhaustive variety of cuts, in terms of the Zm quantum numbers. Furthermore, we show that in a homogeneous system, the Z n index can be determined through an evaluation of the eigenvalues of point group symmetry operators at all high-symmetry points in the Brillouin zone. When disordered n-fold rotationally symmetric systems are considered, we find that the number of protected in-gap states is identical to that in the clean limit as long as the disorder preserves the underlying point group symmetry and does not close the bulk insulating gap.The study of novel topological phases of matter has become one of the most active fields in condensed matter physics [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] . Chronologically speaking, the first of these phases to be experimentally realized is the integer quantum Hall (IQH) state, which is a striking departure from the traditional theory of conductivity due to its quantized Hall conductance and chiral edge modes 21 . Shortly after its experimental discovery, Thouless demonstrated that the special properties of the IQH state came down to its non-trivial bandstructure topology 22 . For the IQH state, one can prove that the quantized Hall conductance is equivalent to a momentum space integral of the Berry curvature, which is a non-zero integer (known as the Chern number in topology), multiplied by the conductance quantum e 2 /h. Beyond this, Haldane demonstrated that the IQH state may be further generalized to a system which does not require an external overall magnetic flux, yet still possesses a non-zero Chern number and quantized Hall conductance. This system is generally referred to as a Chern insulator 23 .Further as systems which preserve time-reversal invariance (TRI) pose an interesting problem as their Chern number vanishes in the presence of non-trivial topology. This necessitates the definition of a new quantum number capable of distinguishing between trivial and non-trivial topological states in systems which possess TRI. In 2D 1-3 and 3D 7,24 , one can define a Z 2 -number, which here we call γ 0 , as a quantity which is uniquely determined by the matrix representations of the time-reversal symmetry operator at all time-reversal inv...

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“…When the boundary phase Φ changes from 0 to 3 × 2π, the three ground states evolve into each other, being always separated from excited states by a gap, and finally return to the initial configuration. In order to discard competing possibilities, such as charge density waves (CDWs), we investigate the particlecut entanglement spectrum (PES) [20,30,44,45], which can probe the excitation structure of the ground states [ Fig. 3(c)].…”

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confidence: 99%

Tunable Band Topology Reflected by Fractional Quantum Hall States in Two-Dimensional Lattices

et al. 2013

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Two-dimensional lattice models subjected to an external effective magnetic field can form nontrivial band topologies characterized by nonzero integer band Chern numbers. In this Letter, we investigate such a lattice model originating from the Hofstadter model and demonstrate that the band topology transitions can be realized by simply introducing tunable longer-range hopping. The rich phase diagram of band Chern numbers is obtained for the simple rational flux density and a classification of phases is presented. In the presence of interactions, the existence of fractional quantum Hall states in both |C| = 1 and |C| > 1 bands is confirmed which can reflect the band topologies in different phases. In contrast, when our model reduces to a one-dimensional lattice, the ground states are crucially different from fractional quantum Hall states. Our results may provide insights into the study of new fractional quantum Hall states and experimental realizations of various topological phases in optical lattices.

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Bulk-edge correspondence in entanglement spectra

Cited by 153 publications

References 69 publications

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

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

Entanglement spectrum classification ofCn-invariant noninteracting topological insulators in two dimensions

Tunable Band Topology Reflected by Fractional Quantum Hall States in Two-Dimensional Lattices

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