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Register a new userSolving frustrated Ising models using tensor networks
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Motivated by the recent success of tensor networks to calculate the residual entropy of spin ice and kagome Ising models, we develop a general framework to study frustrated Ising models in terms of infinite tensor networks %, i.e. tensor networks that can be contracted using standard algorithms for infinite systems. This is achieved by reformulating the problem as local rules for configurations on overlapping clusters chosen in such a way that they relieve the frustration, i.e. that the energy can be minimized independently on each cluster. We show that optimizing the choice of clusters, including the weight on shared bonds, is crucial for the contractibility of the tensor networks, and we derive some basic rules and a linear program to implement them. We illustrate the power of the method by computing the residual entropy of a frustrated Ising spin system on the kagome lattice with next-next-nearest neighbour interactions, vastly outperforming Monte Carlo methods in speed and accuracy. The extension to finite-temperature is briefly discussed.
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Estimating the expected value of an observable appearing in a non-equilibrium stochastic process usually involves sampling. If the observables variance is high, many samples are required. In contrast, we show that performing the same task without sampling, using tensor network compression, efficiently captures high variances in systems of various geometries and dimensions. We provide examples for which matching the accuracy of our efficient method would require a sample size scaling exponentially with system size. In particular, the high variance observable $mathrm{e}^{-beta W}$, motivated by Jarzynskis equality, with $W$ the work done quenching from equilibrium at inverse temperature $beta$, is exactly and efficiently captured by tensor networks.
We consider two fully frustrated Ising models: the antiferromagnetic triangular model in a field of strength, $h=H T k_B$, as well as the Villain model on the square lattice. After a quench from a disordered initial state to T=0 we study the nonequilibrium dynamics of both models by Monte Carlo simulations. In a finite system of linear size, $L$, we define and measure sample dependent first passage time, $t_r$, which is the number of Monte Carlo steps until the energy is relaxed to the ground-state value. The distribution of $t_r$, in particular its mean value, $< t_r(L) >$, is shown to obey the scaling relation, $< t_r(L) > sim L^2 ln(L/L_0)$, for both models. Scaling of the autocorrelation function of the antiferromagnetic triangular model is shown to involve logarithmic corrections, both at H=0 and at the field-induced Kosterlitz-Thouless transition, however the autocorrelation exponent is found to be $H$ dependent.
Recently, there has been significant progress in solving quantum many-particle problem via machine learning based on the restricted Boltzmann machine. However, it is still highly challenging to solve frustrated models via machine learning, which has not been demonstrated so far. In this work, we design a brand new convolutional neural network (CNN) to solve such quantum many-particle problems. We demonstrate, for the first time, of solving the highly frustrated spin-1/2 J$_1$-J$_2$ antiferromagnetic Heisenberg model on square lattices via CNN. The energy per site achieved by the CNN is even better than previous string-bond-state calculations. Our work therefore opens up a new routine to solve challenging frustrated quantum many-particle problems using machine learning.
We study the distribution of lengths and other statistical properties of worms constructed by Monte Carlo worm algorithms in the power-law three-sublattice ordered phase of frustrated triangular and kagome lattice Ising antiferromagnets. Viewing each step of the worm construction as a position increment (step) of a random walker, we demonstrate that the persistence exponent $theta$ and the dynamical exponent $z$ of this random walk depend only on the universal power-law exponents of the underlying critical phase, and not on the details of the worm algorithm or the microscopic Hamiltonian. Further, we argue that the detailed balance criterion obeyed by such worm algorithms and the power-law correlations of the underlying equilibrium system together give rise to two related properties of this random walk: First, the steps of the walk are expected to be power-law correlated in time. Second, the position distribution of the walker relative to its starting point is given by the equilibrium position distribution of a particle in an attractive logarithmic central potential of strength $eta_m$, where $eta_m$ is the universal power-law exponent of the equilibrium defect-antidefect correlation function of the underlying spin system. We derive a scaling relation, $z = (2-eta_m)/(1-theta)$, that allows us to express the dynamical exponent $z(eta_m)$ of this process in terms of its persistence exponent $theta(eta_m)$. Our measurements of $z(eta_m)$ and $theta(eta_m)$ are consistent with this relation over a range of values of the universal equilibrium exponent $eta_m$, and yield subdiffusive ($z>2$) values of $z$ in the entire range. Thus we demonstrate that the worms represent a discrete-time realization of a fractional Brownian motion characterized by these properties.
We present a fully analytically solvable family of models with many-body cluster interaction and Ising interaction. This family exhibits two phases, dubbed cluster and Ising phases, respectively. The critical point turns out to be independent of the cluster size $n+2$ and is reached exactly when both interactions are equally weighted. For even $n$ we prove that the cluster phase corresponds to a nematic ordered phase and in the case of odd $n$ to a symmetry protected topological ordered phase. Though complex, we are able to quantify the multi-particle entanglement content of neighboring spins. We prove that there exists no bipartite or, in more detail, no $n+1$-partite entanglement. This is possible since the non-trivial symmetries of the Hamiltonian restrict the state space. Indeed, only if the Ising interaction is strong enough (local) genuine $n+2$-partite entanglement is built up. Due to their analytically solvableness the $n$-cluster-Ising models serve as a prototype for studying non trivial-spin orderings and due to their peculiar entanglement properties they serve as a potential reference system for the performance of quantum information tasks.
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