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arXiv:0911.4660 · Physical Review Letters

Parity Effects in the Scaling of Block Entanglement in Gapless Spin Chains

Pasquale Calabrese, Massimo Campostrini B, Fabian H. L. Eßler, Bernard Nienhuis

Published Mar 5, 2010• 194 citations
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Abstract

We consider the R\'enyi $\ensuremath{\alpha}$ entropies for Luttinger liquids (LL). For large block lengths $\ensuremath{\ell}$, these are known to grow like $\mathrm{ln}\ensuremath{\ell}$. We show that there are subleading terms that oscillate with frequency $2{k}_{F}$ (the Fermi wave number of the LL) and exhibit a universal power-law decay with $\ensuremath{\ell}$. The new critical exponent is equal to $K/(2\ensuremath{\alpha})$, where $K$ is the LL parameter. We present numerical results for the anisotropic XXZ model and the full analytic solution for the free fermion (XX) point.

DOI
10.1103/physrevlett.104.095701

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We consider the R'enyi \ensuremathα\ensuremath{\alpha}\ensuremathα entropies for Luttinger liquids (LL). For large block lengths \ensuremathℓ\ensuremath{\ell}\ensuremathℓ, these are known to grow like ln\ensuremathℓ\mathrm{ln}\ensuremath{\ell}ln\ensuremathℓ. We show that there are subleading terms that oscillate with frequency 2kF2{k}_{F}2kF​ (the Fermi wave number of the LL) and exhibit a universal power-law decay with \ensuremathℓ\ensuremath{\ell}\ensuremathℓ. The new critical exponent is equal to K/(2\ensuremathα)K/(2\ensuremath{\alpha})K/(2\ensuremathα), where KKK is the LL parameter. We present numerical results for the anisotropic XXZ model and the full analytic solution for the free fermion (XX) point.
Using the density matrix renormalization group, we investigate the R\'enyi entropy of the anisotropic spin-sss Heisenberg chains in a zzz-magnetic field. We considered the half-odd-integer spin-sss chains, with s=1/2s=1/2s=1/2, 3/2, and 5/25/25/2, and periodic and open boundary conditions. In the case of the spin-1/2 chain we were able to obtain accurate estimates of the new parity exponents ${p}_{\ensuremath{\alpha}}^{(p)}$ and ${p}_{\ensuremath{\alpha}}^{(o)}$ that gives the power-law decay of the oscillations of the $\ensuremath{\alpha}$-R\'enyi entropy for periodic and open boundary conditions, respectively. We confirm the relations of these exponents with the Luttinger parameter KKK, as proposed by Calabrese et al. [Phys. Rev. Lett. 104, 095701 (2010)]. Moreover, the predicted periodicity of the oscillating term was also observed for some nonzero values of the magnetization mmm. We show that for s>1/2s>1/2s>1/2 the amplitudes of the oscillations are quite small and get accurate estimates of ${p}_{\ensuremath{\alpha}}^{(p)}$ and ${p}_{\ensuremath{\alpha}}^{(o)}$ become a challenge. Although our estimates of the new universal exponents ${p}_{\ensuremath{\alpha}}^{(p)}$ and ${p}_{\ensuremath{\alpha}}^{(o)}$ for the spin-3/2 chain are not so accurate, they are consistent with the theoretical predictions.
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We present a study of the scaling behavior of the R\'enyi entanglement entropy (REE) in SU(N)\mathrm{SU}(N)SU(N) spin chain Hamiltonians, in which all of the spins transform under the fundamental representation. These SU(N)\mathrm{SU}(N)SU(N) spin chains are known to be quantum critical and described by a well known Wess-Zumino-Witten (WZW) nonlinear sigma model in the continuum limit. Numerical results from our lattice Hamiltonian are obtained using stochastic series expansion quantum Monte Carlo for both closed and open boundary conditions. As expected for this 1D critical system, the REE shows a logarithmic dependence on the subsystem size with a prefactor given by the central charge of the SU(N)\mathrm{SU}(N)SU(N) WZW model. We study in detail the subleading oscillatory terms in the REE under both periodic and open boundaries. Each oscillatory term is associated with a WZW field and decays as a power law with an exponent proportional to the scaling dimension of the corresponding field. We find that the use of periodic boundaries (where oscillations are less prominent) allows for a better estimate of the central charge, while using open boundaries allows for a better estimate of the scaling dimensions. We also present numerical data on the thermal R\'enyi entropy which equally allows for extraction of the central charge.
We investigate the scaling of the entanglement spectrum and of the R\'enyi block entropies and determine its universal aspects in the ground state of critical and noncritical one-dimensional quantum spin models. In all cases, the scaling exhibits an oscillatory behavior that terminates at the factorization point and whose frequency is universal. Parity effects in the scaling of the R\'enyi entropies for gapless models at zero field are thus shown to be a particular case of such universal behavior. Likewise, the absence of oscillations for the Ising chain in transverse field is due to the vanishing value of the factorizing field for this particular model. In general, the transition occurring at the factorizing field between two different scaling regimes of the entanglement spectrum corresponds to a quantum transition to the formation of finite-range, ordered structures of quasidimers, quasitrimers, and quasipolymers. This entanglement-driven transition is superimposed to and independent of the long-range magnetic order in the broken symmetry phase. Therefore, it conforms to recent generalizations that identify and classify the quantum phases of matter according to the structure of ground-state entanglement patterns. We characterize this form of quantum order by a global order parameter of entanglement defined as the integral, over blocks of all lengths, of the R\'enyi entropy of infinite order. Equivalently, it can be defined as the integral of the bipartite single-copy or geometric entanglement. The global entanglement order parameter remains always finite at fields below the factorization point and vanishes identically above it.
We consider a system of two coupled Tomonaga-Luttinger liquids (TLL's) on parallel chains and study the R\'enyi entanglement entropy Sn{S}_{n}Sn​ between the two chains. Here the entanglement cut is introduced between the chains, not along the perpendicular direction, as has been done in previous studies of one-dimensional systems. The limit $n\ensuremath{\rightarrow}1$ corresponds to the von Neumann entanglement entropy. The system is effectively described by two-component bosonic field theory with different TLL parameters in the symmetric and antisymmetric channels as far as the coupled system remains in a gapless phase. We argue that in this system, Sn{S}_{n}Sn​ is a linear function of the length of the chains (boundary law) followed by a universal subleading constant ${\ensuremath{\gamma}}_{n}$ determined by the ratio of the two TLL parameters. The formulas of ${\ensuremath{\gamma}}_{n}$ for integer $n\ensuremath{\geqslant}2$ are derived using (a) ground-state wave functionals of TLL's and (b) boundary conformal field theory, which lead to the same result. These predictions are checked in a numerical diagonalization analysis of a hard-core bosonic model on a ladder. Although the analytic continuation of ${\ensuremath{\gamma}}_{n}$ to $n\ensuremath{\rightarrow}1$ turns out to be a difficult problem, our numerical result suggests that the subleading constant in the von Neumann entropy is also universal. Our results may provide useful characterization of inherently anisotropic quantum phases such as the sliding Luttinger liquid phase via qualitatively different behaviors of the entanglement entropy with the entanglement partitions along different directions.
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