Universal entanglement spectra in critical spin chains

Abstract

We advocate that in critical spin chains, and possibly in a larger class of 1D critical models, a gap in the momentum-space entanglement spectrum separates the universal part of the spectrum, which is determined by the associated conformal field theory, from the non-universal part, which is specific to the model. To this end, we provide affirmative evidence from multicritical spin chains with low energy sectors described by the SU(2)2 or the SU(3)1 Wess-Zumino-Witten model.Introduction.-Quantum entanglement has become a key concept in contemporary condensed matter physics. This is due in part to its ability to probe intrinsic topological order [1][2][3]. Consider a density matrix, ρ, represented by the projector onto a many-body ground state. If the associated Hilbert space is partitioned into two non-overlapping regions A and B, the entanglement between the regions A and B can be quantified either through the entanglement entropy (EE) or the entanglement spectrum (ES) obtained by the reduced density matrix ρ A = Tr B ρ. The EE is given by S A = −Tr(ρ A lnρ A ), and the ES is defined as the spectrum of the entanglement Hamiltonian H E = −lnρ A [4]. By definition, EE and ES depend on the chosen basis to partition (cut) the manybody Hilbert space. To resolve bulk and edge features of topological order, some form of spatial cut [4-8] (along with a particle cut [9,10]) is the predominantly used choice. This works well in systems with a bulk energy gap and hence an associated length scale. Upon partitioning, the ES then mimics the spectral features located along the cut of an edge termination [7]. In particular, a set of universal entanglement modes related to the edge can be identified as distinct from generic entanglement weight through the entanglement gap (EG), which can be employed to define topological adiabaticity in the entanglement spectral flow [5]. The spatial EG evolves in a way similar to the physical bulk gap of the topologically ordered phase, even though bulk gap closures occur at points of parameter space different from EG closures due to the unit fictitious temperature in H E [11].

Momentum-space entanglement after smooth quenches

“…We will consider the same choice, which in our case translates to mass-increasing or mass-decreasing quenches, as well as another class of quench profiles that recovers the initial mass at late times. For previous work on momentum space entanglement in quantum field theory we refer the reader to [17,18,19]; see also [20,21] for similar work with spin chains.…”

Schmidt gap in random spin chains

“…Originally introduced to study the transition to a topologically ordered state in the quantum Hall effect [1], ES has been used for the characterization of spin chains and other one-dimensional (1D) models in real and momentum space [4][5][6][7][8][9][10][11][12][13]. The distribution of the Schmidt eigenvalues in the middle of the spectrum has been studied by means of conformal field theory [14].…”

Entanglement spectrum of fermionic bilayer honeycomb lattice: Hofstadter butterfly

“…14 , the Kitaev model 15 , one dimensional quantum spin systems [16][17][18][19][20][21][22][23][24][25] , Hofstadter poblem [26][27][28] , interacting fermions on honeycomb lattice 29 and bosonic critical system in three dimensions 30 . As a result of these studies, the ES depends on the chosen basis to partition the many body Hilbert space.…”