Entanglement Gap and a New Principle of Adiabatic Continuity

Abstract

We give a complete definition of the entanglement gap separating low-energy, topological levels, from high-energy, generic ones, in the "entanglement spectrum" of Fractional Quantum Hall (FQH) states. By removing the magnetic length inherent in the FQH problem -a procedure which we call taking the "conformal limit", we find that the entanglement spectrum of an incompressible ground-state of a generic (i.e. Coulomb) lowest Landau Level Hamiltonian re-arranges into a low-(entanglement) energy part separated by a full gap from the high energy entanglement levels. As previously observed [1], the counting of these levels starts off as the counting of modes of the edge theory of the FQH state, but quickly develops finite-size effects which we show can also serve as a fingerprint of the FQH state. As the sphere manifold where the FQH resides grows, the level spacing of the states at the same angular momentum goes to zero, suggestive of the presence of relativistic gapless edge-states. By using the adiabatic continuity of the low entanglement energy levels, we investigate whether two states are topologically connected.PACS numbers: 03.67. Mn, 05.30.Pr, 73.43.f Topological phases of matter generally lack local order parameters that can distinguish them from trivial ones. Moreover, extracting the topological order directly from the ground-state wavefunction is a nontrivial task. For incompressible states, several non-local indicators of the topological nature of the phase, such as ground-state degeneracy on compact high genus manifolds, the structure of edge modes and their scaling exponents, as well as quantum dimension analysis exist, but still do not fully describe the topological phase. The measure of choice has so far been the entanglement entropy (EE), especially its topological part [2,3]. For a given state |Ψ 0 and according density matrix ρ = |Ψ 0 Ψ 0 |, let the Hilbert space be decomposed as a direct product H = H A ⊗ H B . Defining ρ A ≡ Tr B [ρ], the EE with respect to the partitioning (A, B) is defined by S A = −Tr A [ρ A lnρ A ]. For twodimensional quantum systems, except in special cases where analytical solutions can be found [4], extracting the topological part of the EE becomes a highly nontrivial (and almost impossible) task.While the EE is just one number, it was recently proposed and numerically substantiated [1] that the entanglement spectrum (ES), i.e. the full set of eigenvalues of ρ A , understood as a geometric partition of the quantum Hall sphere [5], is a better indicator of topological order in the ground state of FQH systems. Writing the eigenvalues as the spectrum of a fictitious Hamiltonian, ρ A = exp(−H), where one can think of the H eigenvalues ξ as a quasi-energy (or entanglement energy), Li and Haldane [1] showed that the low quasi-energy spectrum for generic gapped ν = 5/2 states exhibits a universal structure, related to conformal field theory. A few of the eigenvalues displaying this CFT counting are separated from a non-universal high energy spectrum by an entanglement gap which ...

Exact matrix product states for quantum Hall wave functions

“…In contrast, in the orbital cut each entanglement eigenvalue scales as E a − E 0 ∼ L −ζ a , which precludes the possibility of collapsing the spectrum. This is contrary to earlier indications that the orbital spectrum showed the same linear dispersion, 32 though the case studied there was the "conformal limit" of bosonic ν = 1 2 wave function on a finite sphere.…”

“…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.…”

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

“…[24] (see formula (13) and (14) in that reference where we have defined the boundary perturbation as…”

“…[13], and observed that it contains a low-lying part in which the multiplicities are related in a universal way to the ones of the conformal field the-ory (CFT) describing the low-energy edge excitations. This low-lying universal part is usually well-separated from the rest of the ES by an entanglement gap (see also the extended discussion in [14]). LH suggested that this low-lying part could be used as a diagnostic tool when comparing two ground state wavefunctions.…”

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

“…The single particle entanglement spectrum is closely related to the entanglement entropy 42,[44][45][46] by…”