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arXiv:1104.4005 · Physical Review A

Even-odd entanglement in boson and spin systems

R. Rossignoli, N. Canosa, J. M. Matera

Published Apr 22, 2011• 6 citations
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Abstract

We examine the entanglement entropy of the even half of a translationally invariant finite chain or lattice in its ground state. This entropy measures the entanglement between the even and odd halves (each forming a ``comb'' of $n/2$ sites) and can be expected to be extensive for short-range couplings away from criticality. We first consider bosonic systems with quadratic couplings, where analytic expressions for arbitrary dimensions can be provided. The bosonic treatment is then applied to finite spin chains and arrays by means of the random-phase approximation. Results for first-neighbor anisotropic $\mathit{XY}$ couplings indicate that, while at strong magnetic fields this entropy is strictly extensive, at weak fields important deviations arise, stemming from parity-breaking effects and the presence of a factorizing field (in the vicinity of which it becomes size-independent and identical to the entropy of a contiguous half). Exact numerical results for small spin $s$ chains are shown to be in agreement with the bosonic random-phase approximation prediction.

DOI
10.1103/physreva.83.042328

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We examine the entanglement entropy of the even half of a translationally invariant finite chain or lattice in its ground state. This entropy measures the entanglement between the even and odd halves (each forming a ``comb'' of n/2n/2n/2 sites) and can be expected to be extensive for short-range couplings away from criticality. We first consider bosonic systems with quadratic couplings, where analytic expressions for arbitrary dimensions can be provided. The bosonic treatment is then applied to finite spin chains and arrays by means of the random-phase approximation. Results for first-neighbor anisotropic XY\mathit{XY}XY couplings indicate that, while at strong magnetic fields this entropy is strictly extensive, at weak fields important deviations arise, stemming from parity-breaking effects and the presence of a factorizing field (in the vicinity of which it becomes size-independent and identical to the entropy of a contiguous half). Exact numerical results for small spin sss chains are shown to be in agreement with the bosonic random-phase approximation prediction.
The area law relates the bipartite entanglement entropy of a quantum many-body ground state to the size of the boundary between the subsystems, but the geometry of this boundary is rarely discussed. We inspect this in the 1D Bose-Hubbard model at fixed density by comparing four spatial bipartitions of the periodic lattice: first half, second half, even sites, and odd sites; sharing the same number of sites but differing in how the boundary is arranged. We find analytical limits with perturbation theory: in the Mott insulator the contiguous cut obeys the area lay while the alternating cut obeys a volume law S∝NsS\propto N_sS∝Ns​, in this sense an anomaly, and for the superfluid both cuts colapse to the binomial saturation due to delocalization of the state. We formulate these limits as a statement about the many-body problem using a generalized slave-boson approach based on mean-field with quantum fluctuations while verifying with Exact Diagonalization (ED) for small lattice sizes and Densitiy Matrix Renormalization Group (DMRG) simulations for Ns≫1N_s\gg 1Ns​≫1. The slave-boson Gaussian ground state allows to compute the entanglement entropy from a reduced correlation matrix for any desired bipartition consistent with ED and DMRG results. Using slave bosons the computational cost is set by the local cutoff nmax⁡n_{\max}nmax​ rather than the Hilbert space dimension, so we can reach lattice sizes far beyond ED. Our method is capable of establishing the partition-dependent scaling laws as a many-body feature, not only a finite-size effect, in great agreement with the ED for Ns∈[4,10]N_s\in[4,10]Ns​∈[4,10] and DMRG for larger lattice sizes.
Entanglement measures are useful tools in characterizing otherwise unknown quantum phases and indicating transitions between them. Here we examine the concurrence and entanglement entropy in quantum spin chains with random long-range couplings, spatially decaying with a power-law exponent $\ensuremath{\alpha}$. Using the strong disorder renormalization group (SDRG) technique, we find by analytical solution of the master equation a strong disorder fixed point, characterized by a fixed point distribution of the couplings with a finite dynamical exponent, which describes the system consistently in the regime $\ensuremath{\alpha}>\frac{1}{2}$. A numerical implementation of the SDRG method yields a power-law spatial decay of the average concurrence, which is also confirmed by exact numerical diagonalization. However, we find that the lowest-order SDRG approach is not sufficient to obtain the typical value of the concurrence. We therefore implement a correction scheme which allows us to obtain the leading-order corrections to the random singlet state. This approach yields a power-law spatial decay of the typical value of the concurrence, which we derive both by a numerical implementation of the corrections and by analytics. Next, using numerical SDRG, the entanglement entropy (EE) is found to be logarithmically enhanced for all $\ensuremath{\alpha}$, corresponding to a critical behavior with an effective central charge c=ln(2)c=\mathrm{ln}(2)c=ln(2), independent of $\ensuremath{\alpha}$. This is confirmed by an analytical derivation. Using numerical exact diagonalization (ED), we confirm the logarithmic enhancement of the EE and a weak dependence on $\ensuremath{\alpha}$. For a wide range of partition size lll, the EE fits a critical behavior with a central charge close to c=1c=1c=1, which is the same as for the clean Haldane-Shastry model with a power-law-decaying interaction with $\ensuremath{\alpha}=2$. Only for small $l\ensuremath{\ll}L$, in a range which increases with the number of spins NNN, we find deviations which are rather consistent with the strong disorder fixed point central charge c=ln(2)c=\mathrm{ln}(2)c=ln(2). Furthermore, we find using ED that the concurrence shows power-law decay, albeit with smaller power exponents than obtained by SDRG. We also present results obtained with DMRG and find agreement with ED for sufficiently small $\ensuremath{\alpha}<2$, whereas for larger $\ensuremath{\alpha}$ DMRG tends to underestimate the entanglement entropy and finds a faster decaying concurrence.
Consider a generic quantum spin chain that can be mapped to free quadratic fermions via Jordan-Wigner (JW) transformation. In the presence of arbitrary boundary magnetic fields, this Hamiltonian is no longer a quadratic Hamiltonian after JW transformation. Using ancillary sites and enlarging the Hamiltonian we first introduce a bigger quadratic Hamiltonian. Then we diagonalize this enlarged Hamiltonian in its most generic form and show that all the states are degenerate because of the presence of a zero mode. The eigenstates of the original spin chain with boundary magnetic fields can be derived after appropriate projection. We study in depth the properties of the eigenstates of the enlarged Hamiltonian. In particular we find: 1) the eigenstates in configuration bases, 2) calculate all the correlation functions, 3) find the reduced density matrices, 4) calculate the entanglement entropy. We show that the generic eigenstate of the enlarged Hamiltonian (including the eigenstates of the original spin chain) breaks the parity number symmetry and consequently one needs to take care of some technicalities regarding the calculation of the reduced density matrix and entanglement entropy. Interestingly we show that the entanglement structure of these eigenstates is quite universal and independent of the Hamiltonian. We support our results by applying them to a couple of examples.
Using free-fermionic techniques we study the entanglement entropy of a block of contiguous spins in a large finite quantum Ising chain in a transverse field, with couplings of different types: homogeneous, periodically modulated and random. We carry out a systematic study of finite-size effects at the quantum critical point, and evaluate subleading corrections both for open and for periodic boundary conditions. For a block corresponding to a half of a finite chain, the position of the maximum of the entropy as a function of the control parameter (e.g. the transverse field) can define the effective critical point in the finite sample. On the basis of homogeneous chains, we demonstrate that the scaling behavior of the entropy near the quantum phase transition is in agreement with the universality hypothesis, and calculate the shift of the effective critical point, which has different scaling behaviors for open and for periodic boundary conditions.
We discuss a general treatment based on the mean field plus random-phase approximation (RPA) for the evaluation of subsystem entropies and negativities in ground states of spin systems. The approach leads to a tractable general method that becomes straightforward in translationally invariant arrays. The method is examined in arrays of arbitrary spin with XYZ\mathit{XYZ}XYZ couplings of general range in a uniform transverse field, where the RPA around both the normal and parity-breaking mean-field state, together with parity-restoration effects, is discussed in detail. In the case of a uniformly connected XYZ\mathit{XYZ}XYZ array of arbitrary size, the method is shown to provide simple analytic expressions for the entanglement entropy of any global bipartition, as well as for the negativity between any two subsystems, which become exact for large spin. The limit case of a spin sss pair is also discussed.
We investigate the entanglement entropy of a block of LLL sites in quasifree translation-invariant spin chains concentrating on the effect of reflection-symmetry breaking. The Majorana two-point functions corresponding to the Jordan-Wigner transformed fermionic modes are determined in the most general case; from these, it follows that reflection symmetry in the ground state can only be broken if the model is quantum critical. The large LLL asymptotics of the entropy are calculated analytically for general gauge-invariant models, which have, until now, been done only for the reflection-symmetric sector. Analytical results are also derived for certain nongauge-invariant models (e.g., for the Ising model with Dzyaloshinskii-Moriya interaction). We also study numerically finite chains of length NNN with a nonreflection-symmetric Hamiltonian and report that the reflection symmetry of the entropy of the first LLL spins is violated but the reflection-symmetric Calabrese-Cardy formula is recovered asymptotically. Furthermore, for noncritical reflection-symmetry-breaking Hamiltonians, we find an anomaly in the behavior of the saturation entropy as we approach the critical line. The paper also provides a concise but extensive review of the block-entropy asymptotics in translation-invariant quasifree spin chains with an analysis of the nearest-neighbor case and the enumeration of the yet unsolved parts of the quasifree landscape.