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arXiv:1512.08597 · Physical review. B./Physical review. B

Entanglement and corner Hamiltonian spectra of integrable open spin chains

Panjin Kim, Hosho Katsura, Nandini Trivedi, Jung Hoon Han

Published Nov 7, 2016• 31 citations
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Abstract

We investigate the entanglement entropy (EE) and entanglement spectra (ES) of critical $\mathrm{SU}(N) (2\ensuremath{\le}N\ensuremath{\le}4)$ spin chains and other integrable models of finite length with the density matrix renormalization group method. For all models under investigation, we find a remarkable agreement of the level spacings and the degeneracy structure of the ES with the spectrum of the corner Hamiltonian (CS), defined as the generator of the associated corner transfer matrix. The correspondence holds between ${\mathrm{ES}}^{(n)}$ at the $n\mathrm{th}$ cut position from the edge of the spin model, and the spectrum ${\mathrm{CS}}^{(n)}$ of the corner Hamiltonian of length $n$, for all values of $n$ that we have checked. The cut position dependence of the ES shows a period-$N$ oscillatory behavior for a given $\mathrm{SU}(N)$ chain, reminiscent of the oscillatory part of the entanglement entropy observed in the past for the same models. However, the oscillations of the ES do not die out in the bulk of the chain, in contrast to the asymptotically vanishing oscillation of the entanglement entropy. We present a heuristic argument based on Young tableaux construction that can explain the period-$N$ structure of the ES qualitatively.

DOI
10.1103/physrevb.94.195110

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We investigate the entanglement entropy (EE) and entanglement spectra (ES) of critical SU(N)(2\ensuremath≤N\ensuremath≤4)\mathrm{SU}(N) (2\ensuremath{\le}N\ensuremath{\le}4)SU(N)(2\ensuremath≤N\ensuremath≤4) spin chains and other integrable models of finite length with the density matrix renormalization group method. For all models under investigation, we find a remarkable agreement of the level spacings and the degeneracy structure of the ES with the spectrum of the corner Hamiltonian (CS), defined as the generator of the associated corner transfer matrix. The correspondence holds between ES(n){\mathrm{ES}}^{(n)}ES(n) at the nthn\mathrm{th}nth cut position from the edge of the spin model, and the spectrum CS(n){\mathrm{CS}}^{(n)}CS(n) of the corner Hamiltonian of length nnn, for all values of nnn that we have checked. The cut position dependence of the ES shows a period-NNN oscillatory behavior for a given SU(N)\mathrm{SU}(N)SU(N) chain, reminiscent of the oscillatory part of the entanglement entropy observed in the past for the same models. However, the oscillations of the ES do not die out in the bulk of the chain, in contrast to the asymptotically vanishing oscillation of the entanglement entropy. We present a heuristic argument based on Young tableaux construction that can explain the period-NNN structure of the ES qualitatively.
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.
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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.
The power of matrix product states to describe infinite-size translational-invariant critical spin chains is investigated. At criticality, the accuracy with which they describe ground-state properties of a system is limited by the size $\ensuremath{\chi}$ of the matrices that form the approximation. This limitation is quantified in terms of the scaling of the half-chain entanglement entropy. In the case of the quantum Ising model, we find $S\ensuremath{\sim}\frac{1}{6}\text{log}\text{ }\ensuremath{\chi}$ with high precision. This result can be understood as the emergence of an effective finite correlation length ${\ensuremath{\xi}}_{\ensuremath{\chi}}$ ruling all the scaling properties in the system. We produce six extra pieces of evidence for this finite-$\ensuremath{\chi}$ scaling, namely, the scaling of the correlation length, the scaling of magnetization, the shift of the critical point, the scaling of the entanglement entropy for a finite block of spins, the existence of scaling functions, and the agreement with analogous classical results. All our computations are consistent with a scaling relation of the form ${\ensuremath{\xi}}_{\ensuremath{\chi}}\ensuremath{\sim}{\ensuremath{\chi}}^{\ensuremath{\kappa}}$, with $\ensuremath{\kappa}=2$ for the Ising model. In the case of the Heisenberg model, we find similar results with the value $\ensuremath{\kappa}\ensuremath{\sim}1.37$. We also show how finite-$\ensuremath{\chi}$ scaling allows us to extract critical exponents. These results are obtained using the infinite time evolved block decimation algorithm which works in the thermodynamical limit and are verified to agree with density-matrix renormalization-group results and their classical analog obtained with the corner transfer-matrix renormalization group.
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