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Register a new userExact valence bond entanglement entropy and probability distribution in the XXX spin chain and the Potts model
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Jesper Lykke Jacobsen
Publication date
2007
fields
Physics
and research's language is
English
Authors
Jesper Lykke Jacobsen
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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 <N_c> = (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.
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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 Neel phase.
Recently, a non-trivial relation between the quasi-particle spectrum and entanglement entropy production was discovered in non-integrable quenches in the paramagnetic Ising quantum spin chain. Here we study the dynamics of analogous quenches in the quantum Potts spin chain. Tuning the parameters of the system, we observe a sudden increase in the entanglement production rate, which is shown to be related to the appearance of new quasiparticle excitations in the post-quench spectrum. Our results demonstrate the generality of the effect and support its interpretation as the non-equilibrium version of the well-known Gibbs paradox related to mixing entropy which appears in systems with a non-trivial quasi-particle spectrum.
We consider the isotropic spin-1/2 Heisenberg spin chain weakly perturbed by a local translationally- and SU(2)-invariant perturbation. Starting from the local integrals of motion of the unperturbed model, we modify them in order to obtain quasi-conserved integrals of motion (charges) for the perturbed model. Such quasi-conserved quantities are believed to be responsible for the existence of the prethermalization phase at intermediate timescales. We find that for a sufficiently local perturbation only the first few integrals of motion can be promoted to the quasi-conserved charges, whereas higher-order integrals of motion do not survive.
The $q$-state Potts model has stood at the frontier of research in statistical mechanics for many years. In the absence of a closed-form solution, much of the past efforts have focused on locating its critical manifold, trajectory in the parameter ${q, e^J}$ space where $J$ is the reduced interaction, along which the free energy is singular. However, except in isolated cases, antiferromagnetic (AF) models with $J<0$ have been largely neglected. In this paper we consider the Potts model with AF interactions focusing on deducing its critical manifold in exact and/or closed-form expressions. We first re-examine the known critical frontiers in light of AF interactions. For the square lattice we confirm the Potts self-dual point to be the sole critical point for $J>0$. We also locate its critical frontier for $J<0$ and find it to coincide with a solvability condition observed by Baxter in 1982. For the honeycomb lattice we show that the known critical point holds for {all} $J$, and determine its critical $q_c = frac 1 2 (3+sqrt 5) = 2.61803$ beyond which there is no transition. For the triangular lattice we confirm the known critical point to hold only for $J>0$. More generally we consider the centered-triangle (CT) and Union-Jack (UJ) lattices consisting of mixed $J$ and $K$ interactions, and deduce critical manifolds under homogeneity hypotheses. For K=0 the CT lattice is the diced lattice, and we determine its critical manifold for all $J$ and find $q_c = 3.32472$. For K=0 the UJ lattice is the square lattice and from this we deduce both the $J>0$ and $J<0$ critical manifolds and find $q_c=3$ for the square lattice. Our theoretical predictions are compared with recent tensor-based numerical results and Monte Carlo simulations.
We have calculated the energy per site for the ground state of antiferromagnetic quantum spin chain with variable range exchange $h(j-k)propto sinh^2 a sinh^{-2}a(j-k)$ in the framework of the asymptotic Bethe ansatz. By expanding it in powers of $e^{-2a}$, we have confirmed the value of the second-neighbor correlator for the model with nearest-neighbor exchange obtained earlier in the atomic limit of the Hubbard chain.
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