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

Dubail, Jérôme; Read, N.; Rezayi, E. H.

doi:10.1103/physrevb.86.245310

Cited by 102 publications

(172 citation statements)

References 91 publications

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“…We are grateful to Nick Read for making the observation that the extrapolation length may be interpreted directly as a boundary perturbation by the stress-tensor, in the context of [27]. We also wish to thank Pasquale Calabrese, Viktor Eisler, Paul Fendley, Moshe Goldstein, Grégoire Misguich and Vincent Pasquier for enlightening discussions.…”

Section: Acknowledgmentsmentioning

confidence: 99%

“…This equivalence between the idea of an extrapolation length and the perturbation by the stress-tensor along the boundary plays an important role in [27].…”

Section: Leading Order Of the Corner Free Energy At Criticality: The mentioning

confidence: 99%

“…This extrapolation length is a concept that is ubiquitous in the study of surface critical phenomena [25]. The importance of this length has been emphasized lately in the context of global quantum quenches [26], entanglement spectra in fractional quantum hall systems [27], and also, although in a slightly different form, in quantum impurity problems such as the Kondo screening cloud ‡. Its appearance in logarithmic corrections to the corner free energy provides a practically reliable way of measuring this length in numerical simulations; it also affects various computable quantities in one-dimensional quantum systems, such as the bipartite fidelity [15] or the emptiness formation probability [31].…”

Section: Universality and Free Energiesmentioning

confidence: 99%

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Logarithmic corrections to the free energy from sharp corners with angle 2π

Stéphan¹,

Dubail²

2013

J. Stat. Mech.

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Abstract. We study subleading corrections to the corner free energy in classical two-dimensional critical systems, focusing on a generic boundary perturbation by the stress-tensor of the underlying conformal field theory (CFT). In the particular case of an angle 2π, we find that there is an unusual correction of the form L −1 log L, where L is a typical length scale in the system. This correction also affects the one-point function of an operator near the corner. The prefactor can be seen as semi-universal, in the sense that it depends on a single non-universal quantity, the extrapolation length. Once this ultraviolet cutoff is known, the term is entirely fixed by the geometry of the system, and the central charge of the CFT. Such a corner appears for example in the bipartite fidelity of a one-dimensional quantum system at criticality, which allows for several numerical checks in free fermions systems. We also present an exact result in the XX and Ising chains that confirms this analysis. Finally, we consider applications to the time evolution of the (logarithmic) Loschmidt echo and the entanglement entropy following a local quantum quench. Due to subtle issues in analytic continuation, we find that the logarithmic term in imaginary time transforms into a time-dependent L −2 correction for the entanglement entropy, and a L −2 log L term for the Loschmidt echo.

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Section: Acknowledgmentsmentioning

confidence: 99%

“…This equivalence between the idea of an extrapolation length and the perturbation by the stress-tensor along the boundary plays an important role in [27].…”

Section: Leading Order Of the Corner Free Energy At Criticality: The mentioning

confidence: 99%

Section: Universality and Free Energiesmentioning

confidence: 99%

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Logarithmic corrections to the free energy from sharp corners with angle 2π

Stéphan¹,

Dubail²

2013

J. Stat. Mech.

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“…Because of the inherent boundary-local nature [23,24], the entanglement spectrum from the leftright partition does not directly reveal properties of the bulk.…”

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confidence: 99%

Bulk Entanglement Spectrum Reveals Quantum Criticality within a Topological State

Hsieh

2014

Phys. Rev. Lett.

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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. DOI: 10.1103/PhysRevLett.113.106801 PACS numbers: 73.43.Cd, 03.67.Mn Topological phases of matter are characterized by quantized physical properties that arise from topological quantum numbers. For instance, the quantized Hall conductance of an integer quantum Hall state is determined by its Chern number [1], the quantized magnetoelectric response of a topological insulator is governed by its Z 2 topological invariant [2][3][4], and the quasiparticle charge in a fractional quantum Hall state is deeply related to its topological degeneracy [5]. Remarkably, the complete set of topological quantum numbers, or topological order [6], is entirely encoded in the ground state wave function, which can be computed either directly [1,7] or from the topological entanglement entropy [8][9][10][11][12][13].To extract more information about a topological phase from its ground state wave function, Li and Haldane [14] considered the full entanglement spectrum of the reduced density matrix upon tracing out a subsystem. When a topologically nontrivial ground state is spatially divided into two halves, the resulting entanglement spectrum bears a remarkable similarity to the edge state spectrum of the system in the presence of a physical boundary [16][17][18][19][20][21][22]. Given this capability of the entanglement spectrum to simulate edge excitations, one may wonder if universal bulk properties of topological phases can also be obtained via entanglement.In this work, we show that the ground state of a topological phase (a single wave function) encodes information on its phase transition to a trivial product state, despite the fact that the system itself is away from the phase transition. To expose this "hidden" topological phase transition, we introduce a new type of real-space partitions, which divide the system into two parts that are extensive with system size in all directions, as shown in Fig. 1. The entanglement Hamiltonian obtained from such an extensive partition is a bulk entity, which we term the bulk entanglement Hamiltonian. The corresponding bulk entanglement spectrum (BES) has a low-lying part that resembles the excitation spectrum of a physical system in...

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“…A potentially powerful method in this context is the density matrix renormalization group (DMRG); 31 initially designed for strongly correlated one-dimensional systems, it has been successfully applied in the simulations of a variety of twodimensional states of matter including geometrically frustrated magnets [32][33][34] and FQH systems. [35][36][37][38][39][40] From modern developments in the theory of quantum entanglement, which brought new ideas of the area law, 41 the entanglement spectrum, 28 and matrix product states, [42][43][44][45] it has now become clear that a cylinder geometry plays a very special role in the DMRG studies of strongly correlated systems. 42,46 Recently two pioneering papers 33,47 suggested using this geometry to extract information about topological properties of correlated states, including the FCIs.…”

Section: Introduction Fractional Quantum Hall (Fqh) Statesmentioning

confidence: 99%

Bulk-edge correspondence in fractional Chern insulators

2013

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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 renormalisation 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 ν = 1/2 and ν = 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 ν = 1 phase can be stabilized by purely on-site interactions in the presence of a confining potential.

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

Cited by 102 publications

References 91 publications

Logarithmic corrections to the free energy from sharp corners with angle 2π

Logarithmic corrections to the free energy from sharp corners with angle 2π

Bulk Entanglement Spectrum Reveals Quantum Criticality within a Topological State

Bulk-edge correspondence in fractional Chern insulators

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