Quantum critical chains are well-described and understood by virtue of conformal field theory. Still, the meaning of the real space entanglement spectrum—the eigenvalues of the reduced density matrix—of such systems remains elusive in general, even when there is an additional quantum number available such as the spin or particle number. In this paper, we explore in detail the properties and structure of the reduced density matrix of critical XXZ spin- chains. We investigate the quantum/thermal correspondence between the reduced density matrix of a T = 0 pure quantum state and the thermal density matrix of an effective entanglement Hamiltonian. Using large scale DMRG and QMC simulations, we investigate the conformal structure of the spectra, the entanglement Hamiltonian, and temperature. We then introduce the notion of spin-resolved entanglement entropies, which display interesting scaling features.
We investigate the entanglement spectrum near criticality in finite quantum spin chains. Using finite size scaling we show that when approaching a quantum phase transition, the Schmidt gap, i.e., the difference between the two largest eigenvalues of the reduced density matrix λ(1), λ(2), signals the critical point and scales with universal critical exponents related to the relevant operators of the corresponding perturbed conformal field theory describing the critical point. Such scaling behavior allows us to identify explicitly the Schmidt gap as a local order parameter.
Entanglement entropy and entanglement spectrum have been widely used to characterize quantum entanglement in extended many-body systems. Given a pure state of the system and a division into regions
A and
B, they can be obtained in terms of the
Schmidt values, or eigenvalues
λα of the reduced density matrix
ρA for region
A. In this paper we draw attention instead to the
Schmidt vectors, or eigenvectors
∣vα⟩ of
ρA. We consider the ground state of critical quantum spin chains whose low energy/long distance physics is described by an emergent conformal field theory (CFT). We show that the Schmidt vectors
∣vα⟩ display an emergent universal structure, corresponding to a realization of the Virasoro algebra of a boundary CFT (a chiral version of the original CFT). Indeed, we build weighted sums
Hn of the lattice Hamiltonian density
hj,j+1 over region
A and show that the matrix elements
⟨vαHn∣vα′⟩ are universal, up to finite-size corrections. More concretely, these matrix elements are given by an analogous expression for
HnCFT=21(Ln+L−n) in the boundary CFT, where
Ln's are (one copy of) the Virasoro generators. We numerically confirm our results using the critical Ising quantum spin chain and other (free-fermion equivalent) models.
We discuss entanglement and critical properties of the spin-
3/2 XXZ chain in its entire gapless region. Employing density-matrix renormalization-group calculations combined with different methods based on level spectroscopy, correlation functions, and entanglement entropies, we determine the sound velocity and the Luttinger parameter of the model as a function of the anisotropy parameter. Then, we focus on entanglement properties by systematically studying the behavior of R\'enyi entropies under both open and periodic boundary conditions, providing further evidence of recent findings about entanglement entropies of excited states in conformal field theory.
We carry out a systematic study of the exact block entanglement in the XXZ spin chain at Δ = −1/2. We present the first analytic expressions for reduced density matrices for n spins in a chain of length L (for n ≤6 and arbitrary but odd L ) for a truly interacting model. The entanglement entropy and the moments of the reduced density matrix and its spectrum are then easily derived. We explicitly construct the 'entanglement Hamiltonian' as the logarithm of this matrix. Exploiting the degeneracy of the ground state, we find the scaling behavior of the entanglement of the zero-temperature mixed state.
Numerical studies of the reduced density matrix of a gapped spin-1/2 Heisenberg antiferromagnet on a two-leg ladder find that it has the same form as the Gibbs density matrix of a gapless spin-1/2 Heisenberg antiferromagnetic chain at a finite temperature determined by the spin gap of the ladder. We investigate this interesting result by considering a model of free fermions on a two-leg ladder (gapped by the inter-chain tunneling operator) and in spin systems on a ladder with a gapped ground state using exact solutions and several controlled approximations. We calculate the reduced density matrix and the entanglement entropy for a leg of the ladder (i.e. a cut made between the chains). In the fermionic system we find the exact form of the reduced density matrix for one of the chains and determine the entanglement spectrum explicitly. Here we find that in the weak tunneling limit of the ladder the entanglement entropy of one chain of the gapped ladder has a simple and universal form dictated by conformal invariance. In the case of the spin system, we consider the strong coupling limit by using perturbation theory and get the reduced density matrix by the Schmidt decomposition. The entanglement entropies of a general gapped system of two coupled conformal field theories (in 1 + 1 dimensions) are discussed using the replica trick and scaling arguments. We show that (1) for a system with a bulk gap the reduced density matrix has the form of a thermal density matrix, (2) the long-wavelength modes of one subsystem (a chain) of a gapped coupled system are always thermal, (3) the von Neumann entropy equals the thermodynamic entropy of one chain, and (4) the bulk gap plays the role of effective temperature.
The entanglement spectrum (i.e., the full distribution of Schmidt eigenvalues of the reduced density matrix) contains more information than the conventional entanglement entropy and has been studied recently in several many-particle systems. We compute the disorder-averaged entanglement spectrum in the form of the disorder-averaged moments $\stackrel{\underline{}}{\mathrm{Tr}{\mathrm{\ensuremath{\rho}}}_{\mathrm{A}}^{\mathrm{\ensuremath{\alpha}}}}$ of the reduced density matrix ${\ensuremath{\rho}}_{A}$ for a contiguous block of many spins at the random-singlet quantum critical point in one dimension. The result compares well in the scaling limit with numerical studies on the random
XX model and is also expected to describe the (interacting) random Heisenberg model. Our numerical studies on the
XX case reveal that the dependence of the entanglement entropy and spectrum on the geometry of the Hilbert space partition is quite different than for conformally invariant critical points.