We numerically investigate the low-lying entanglement spectrum of the ground state of random one-dimensional spin chains obtained after partition of the chain into two equal halves. We consider two paradigmatic models: the spin-1/2 random transverse-field Ising model, solved exactly, and the spin-1 random Heisenberg model, simulated using the density matrix renormalization group. In both cases we analyze the mean Schmidt gap, defined as the difference between the two largest eigenvalues of the reduced density matrix of one of the two partitions, averaged over many disorder realizations. We find that the Schmidt gap detects the critical point very well and scales with universal critical exponents.
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.
We study the dynamics of the entanglement spectrum, that is the time evolution of the eigenvalues of the reduced density matrices after a bipartition of a one-dimensional spin chain. Starting from the ground state of an initial Hamiltonian, the state of the system is evolved in time with a new Hamiltonian. We consider both instantaneous and quasi adiabatic quenches of the system Hamiltonian across a quantum phase transition. We analyse the Ising model that can be exactly solved and the XXZ for which we employ the time-dependent density matrix renormalisation group algorithm. Our results show once more a connection between the Schmidt gap, i.e. the difference of the two largest eigenvalues of the reduced density matrix and order parameters, in this case the spontaneous magnetisation.
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.
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.
Two electronic channels competing to screen a single impurity spin, as in the two-channel Kondo model, are expected to generate a ground state with a nontrivial entanglement structure. We exploit a spin-chain representation of the two-channel Kondo model to probe the ground-state block entropy, negativity, tangle, and Schmidt gap, using a density matrix renormalization group approach. In the presence of symmetric coupling to the two channels, we confirm field-theory predictions for the boundary entropy difference
ln(gUV/gIR)=ln(2)/2 between the ultraviolet and infrared limits and the leading
ln(x)/x impurity correction to the block entropy. The impurity entanglement
Simp is shown to scale with the characteristic length ${\ensuremath{\xi}}_{2\text{CK}}$. We show that both the Schmidt gap and the entanglement of the impurity with one of the channels---as measured by the negativity---faithfully serve as order parameters for the impurity quantum phase transition appearing as a function of channel asymmetry, allowing for explicit determination of critical exponents, $\ensuremath{\nu}\ensuremath{\approx}2$ and $\ensuremath{\beta}\ensuremath{\approx}0.2$. Remarkably, we find the emergence of tripartite entanglement only in the vicinity of the critical channel-symmetric point.
We investigate the logarithmic negativity in strongly disordered spin chains in the random-singlet phase. We focus on the spin-
21 random Heisenberg chain and the random
XX chain. We find that for two arbitrary intervals, the disorder-averaged negativity and the mutual information are proportional to the number of singlets shared between the two intervals. Using the strong-disorder renormalization group (SDRG), we prove that the negativity of two adjacent intervals grows logarithmically with the intervals' length. In particular, the scaling behavior is the same as in conformal field theory, but with a different prefactor. For two disjoint intervals the negativity is given by a universal simple function of the cross ratio, reflecting scale invariance. As a function of the distance of the two intervals, the negativity decays algebraically in contrast with the exponential behavior in clean models. We confirm our predictions using a numerical implementation of the SDRG method. Finally, we also implement density matrix renormalization group simulations for the negativity in open spin chains. The chains accessible in the presence of strong disorder are not sufficiently long to provide a reliable confirmation of the SDRG results.