22 research outputs found
Approaching the Kosterlitz-Thouless transition for the classical XY model with tensor networks
We apply variational tensor-network methods for simulating the Kosterlitz-Thouless phase transition in the classical two-dimensional XY model. In particular, using uniform matrix product states (MPS) with non-Abelian O(2) symmetry, we compute the universal drop in the spin stiffness at the critical point. In the critical low-temperature regime, we focus on the MPS entanglement spectrum to characterize the Luttinger-liquid phase. In the high-temperature phase, we confirm the exponential divergence of the correlation length and estimate the critical temperature with high precision. Our MPS approach can be used to study generic two-dimensional phase transitions with continuous symmetries
Scaling hypothesis for matrix product states
We study critical spin systems and field theories using matrix product states, and formulate a scaling hypothesis in terms of operators, eigenvalues of the transfer matrix, and lattice spacing in the case of field theories. The critical point, exponents, and central charge are determined by optimizing them to obtain a data collapse. We benchmark this method by studying critical Ising and Potts models, where we also obtain a scaling Ansatz for the correlation length and entanglement entropy. The formulation of those scaling functions turns out to be crucial for studying critical quantum field theories on the lattice. For the case of lambda phi(4) with mass parameter mu(2) and lattice spacing a, we demonstrate a double data collapse for the correlation length delta xi(mu, lambda, D) = (xi) over tilde((alpha - alpha(c))(delta/a)(-1/nu)) with D the bond dimension, delta the gap between eigenvalues of the transfer matrix, and alpha(c) = mu(2)(R)/lambda the parameter which fixes the critical quantum field theory
Solving frustrated Ising models using tensor networks
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
Tangent-space methods for truncating uniform MPS
A central primitive in quantum tensor network simulations is the problem of
approximating a matrix product state with one of a lower bond dimension. This
problem forms the central bottleneck in algorithms for time evolution and for
contracting projected entangled pair states. We formulate a tangent-space based
variational algorithm to achieve this for uniform (infinite) matrix product
states. The algorithm exhibits a favourable scaling of the computational cost,
and we demonstrate its usefulness by several examples involving the
multiplication of a matrix product state with a matrix product operator
Partial lifting of degeneracy in the Ising antiferromagnet on the kagome lattice
Motivated by dipolar-coupled artificial spin systems, we present a
theoretical study of the classical Ising antiferromagnet on the
kagome lattice. We establish the ground-state phase diagram of this model for
based on exact results for the ground-state energies. When
all the couplings are antiferromagnetic, the model has three macroscopically
degenerate ground-state phases, and using tensor networks, we can calculate the
entropies of these phases and of their boundaries very accurately. In two
cases, the entropy appears to be a fraction of that of the triangular lattice
Ising antiferromagnet, and we provide analytical arguments to support this
observation. We also notice that, surprisingly enough, the dipolar ground state
is not a ground state of the truncated model, but of the model with smaller
interactions, an indication of a very strong competition between
low-energy states in this model.Comment: Accepted version for PRB; 23 pages (16 main text), 20 figure
Variational methods for contracting projected entangled-pair states
The norms or expectation values of infinite projected entangled-pair states
(PEPS) cannot be computed exactly, and approximation algorithms have to be
applied. In the last years, many efficient algorithms have been devised -- the
corner transfer matrix renormalization group (CTMRG) and variational uniform
matrix product state (VUMPS) algorithm are the most common -- but it remains
unclear whether they always lead to the same results. In this paper, we
identify a subclass of PEPS for which we can reformulate the contraction as a
variational problem that is algorithm independent. We use this variational
feature to assess and compare the accuracy of CTMRG and VUMPS contractions.
Moreover, we devise a new variational contraction scheme, which we can extend
to compute general N-point correlation functions