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arXiv:1005.0787 · Physical Review B

Valence-bond entanglement entropy of frustrated spin chains

Fabien Alet, Ian P. McCulloch, Sylvain Capponi, Matthieu Mambrini

Published Sep 29, 2010• 19 citations
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Abstract

We extend the definition of the recently introduced valence-bond entanglement entropy to arbitrary SU(2) wave functions of $S=1/2$ spin systems. Thanks to a reformulation of this entanglement measure in terms of a projection, we are able to compute it with various numerical techniques for frustrated spin models. We provide extensive numerical data for the one-dimensional ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ spin chain where we are able to locate the quantum phase transition by using the scaling of this entropy with the block size. We also systematically compare with the scaling of the von Neumann entanglement entropy. We finally underline that the valence-bond entropy definition does depend on the choice of bipartition so that, for frustrated models, a ``good'' bipartition should be chosen, for instance, according to the Marshall sign.

DOI
10.1103/physrevb.82.094452

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We extend the definition of the recently introduced valence-bond entanglement entropy to arbitrary SU(2) wave functions of S=1/2S=1/2S=1/2 spin systems. Thanks to a reformulation of this entanglement measure in terms of a projection, we are able to compute it with various numerical techniques for frustrated spin models. We provide extensive numerical data for the one-dimensional J1\ensuremath-J2{J}_{1}\text{\ensuremath{-}}{J}_{2}J1​\ensuremath-J2​ spin chain where we are able to locate the quantum phase transition by using the scaling of this entropy with the block size. We also systematically compare with the scaling of the von Neumann entanglement entropy. We finally underline that the valence-bond entropy definition does depend on the choice of bipartition so that, for frustrated models, a ``good'' bipartition should be chosen, for instance, according to the Marshall sign.
The von Neumann entanglement entropy of exact valence-bond ground states is studied in two frustrated one-dimensional spin chains: the spin-1/2 Majumdar-Ghosh (MG) model and the spin-3/2 J1-J2-J3 chain in its fully dimerized (FD) and partially dimerized (PD) phases. Using matrix-product-state representations, the entropy is computed as a function of system size for three complementary bipartitions - half-chain, single-site, and pairwise - under both open and periodic boundary conditions. In all cases, the entropy saturates to a finite constant in the thermodynamic limit, confirming area-law behavior. The saturation values, extracted via finite-size scaling, are directly related to the underlying virtual-spin bond structure. The MG model and FD phase exhibit similar entanglement behavior, differing primarily in saturation magnitude determined by spin value and bond multiplicity, and both display even-odd oscillations and exponential convergence with system size. In contrast, the PD phase shows qualitatively distinct signatures, including multiple half-chain saturation values depending on the bond type at the cut, asymmetric edge contributions in the single-site entropy, and a multi-band structure in the pairwise entropy reflecting the coexistence of single- and double-singlet bonds. These results establish entanglement entropy as a robust signature of frustrated bond architecture, enabling clear distinction among dimerized phases with different spin magnitude, bond multiplicity, and dimerization patterns.
In this thesis we examine methods for measuring entanglement entropy in spin-1/2 Heisenberg systems using quantum Monte Carlo in the valence bond basis. We begin by presenting the quantum Monte Carlo techniques used in this research. We then use these techniques to directly compare the recently proposed valence bond entanglement entropy to the standard definition of entanglement entropy: the von Neumann entanglement entropy. We find that the valence bond entanglement entropy does not give a bound on the von Neumann entanglement entropy, and that it exhibits a multiplicative logarithmic correction to the area law that is not present in the scaling of the von Neumann entanglement entropy. We then present a method to measure higher orders of the generalized Renyi entanglement entropies using valence bond quantum Monte Carlo, and show results for the second Renyi entropy. We find the results converge to the exact results for one dimensional Heisenberg spin-1/2 chains, and see that the scaling of the second Renyi entropy follows an area law in the two dimensional Heisenberg ground state.
The valence-bond structure of spin-1/2 Heisenberg antiferromagnets is closely related to quantum entanglement. We investigate measures of entanglement entropy based on transition graphs, which characterize state overlaps in the overcomplete valence-bond basis. The transition graphs can be generated using projector Monte Carlo simulations of ground states of specific Hamiltonians or using importance sampling of valence-bond configurations of amplitude-product states. We consider definitions of entanglement entropy based on the bonds or loops shared by two subsystems (bipartite entanglement). Results for the bond-based definition agree with a previously studied definition using valence-bond wave functions (instead of the transition graphs, which involve two states). For the one-dimensional Heisenberg chain, with uniform or random coupling constants, the prefactor of the logarithmic divergence with the size of the smaller subsystem agrees with exact results. For the ground state of the two-dimensional Heisenberg model (and also N\'eel-ordered amplitude-product states), there is a similar multiplicative violation of the area law. In contrast, the loop-based entropy obeys the area law in two dimensions while still violating it in one dimension---both behaviors in accord with expectations for proper measures of entanglement entropy.
We introduce for SU(2) quantum spin systems the valence bond entanglement entropy as a counting of valence bond spin singlets shared by two subsystems. For a large class of antiferromagnetic systems, it can be calculated in all dimensions with quantum Monte Carlo simulations in the valence bond basis. We show numerically that this quantity displays all features of the von Neumann entanglement entropy for several one-dimensional systems. For two-dimensional Heisenberg models, we find a strict area law for a valence bond solid state and multiplicative logarithmic corrections for the Néel phase.
We present a direct comparison of the recently proposed valence bond entanglement entropy and the von Neumann entanglement entropy on spin-1/2 Heisenberg systems using quantum Monte Carlo and density-matrix renormalization group simulations. For one-dimensional chains we show that the valence bond entropy can be either less or greater than the von Neumann entropy; hence, it cannot provide a bound on the latter. On ladder geometries, simulations with up to seven legs are sufficient to indicate that the von Neumann entropy in two dimensions obeys an area law, even though the valence bond entanglement entropy has a multiplicative logarithmic correction.
By relating the ground state of Temperley-Lieb hamiltonians to partition functions of 2D statistical mechanics systems on a half plane, and using a boundary Coulomb gas formalism, we obtain in closed form the valence bond entanglement entropy as well as the valence bond probability distribution in these ground states. We find in particular that for the XXX spin chain, the number N_c of valence bonds connecting a subsystem of size L to the outside goes, in the thermodynamic limit, as = (4/pi^2) ln L, disproving a recent conjecture that this should be related with the von Neumann entropy, and thus equal to 1/(3 ln 2) ln L. Our results generalize to the Q-state Potts model.