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Register a new userScaling of entanglement support for Matrix Product States
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Added by
Thiago Rodrigues de Oliveira
Publication date
2008
fields
Physics
and research's language is
English
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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 $chi$ 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 $S sim {1/6}log chi$ with high precision. This result can be understood as the emergence of an effective finite correlation length $xi_chi$ ruling of all the scaling properties in the system. We produce five extra pieces of evidence for this finite-$chi$ scaling, namely, the scaling of the correlation length, the scaling of magnetization, the shift of the critical point, and the scaling of the entanglement entropy for a finite block of spins. All our computations are consistent with a scaling relation of the form $xi_chisim chi^{kappa}$, with $kappa=2$ for the Ising model. In the case of the Heisenberg model, we find similar results with the value $kappasim 1.37$. We also show how finite-$chi$ scaling allow 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.
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We investigate the use of matrix product states (MPS) to approximate ground states of critical quantum spin chains with periodic boundary conditions (PBC). We identify two regimes in the (N,D) parameter plane, where N is the size of the spin chain and D is the dimension of the MPS matrices. In the first regime MPS can be used to perform finite size scaling (FSS). In the complementary regime the MPS simulations show instead the clear signature of finite entanglement scaling (FES). In the thermodynamic limit (or large N limit), only MPS in the FSS regime maintain a finite overlap with the exact ground state. This observation has implications on how to correctly perform FSS with MPS, as well as on the performance of recent MPS algorithms for systems with PBC. It also gives clear evidence that critical models can actually be simulated very well with MPS by using the right scaling relations; in the appendix, we give an alternative derivation of the result of Pollmann et al. [Phys. Rev. Lett. 102, 255701 (2009)] relating the bond dimension of the MPS to an effective correlation length.
We revisit the question of describing 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. Critical exponents and central charge are determined by optimizing the exponents such as 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 $lambdaphi^4$ with mass $mu^2$ and lattice spacing $a$, we demonstrate a double data collapse for the correlation length $ delta xi(mu,lambda,D)=tilde{xi} left((alpha-alpha_c)(delta/a)^{-1/ u}right)$ with $D$ the bond dimension, $delta$ the gap between eigenvalues of the transfer matrix, and $alpha_c=mu_R^2/lambda$ the parameter which fixes the critical quantum field theory.
We adapt the time-evolving block decimation (TEBD) algorithm, originally devised to simulate the dynamics of 1D quantum systems, to simulate the time-evolution of non-equilibrium stochastic systems. We describe this method in detail; a systems probability distribution is represented by a matrix product state (MPS) of finite dimension and then its time-evolution is efficiently simulated by repeatedly updating and approximately re-factorizing this representation. We examine the use of MPS as an approximation method, looking at parallels between the interpretations of applying it to quantum state vectors and probability distributions. In the context of stochastic systems we consider two types of factorization for use in the TEBD algorithm: non-negative matrix factorization (NMF), which ensures that the approximate probability distribution is manifestly non-negative, and the singular value decomposition (SVD). Comparing these factorizations we find the accuracy of the SVD to be substantially greater than current NMF algorithms. We then apply TEBD to simulate the totally asymmetric simple exclusion process (TASEP) for systems of up to hundreds of lattice sites in size. Using exact analytic results for the TASEP steady state, we find that TEBD reproduces this state such that the error in calculating expectation values can be made negligible, even when severely compressing the description of the system by restricting the dimension of the MPS to be very small. Out of the steady state we show for specific observables that expectation values converge as the dimension of the MPS is increased to a moderate size.
Here we demonstrate that tensor network techniques - originally devised for the analysis of quantum many-body problems - are well suited for the detailed study of rare event statistics in kinetically constrained models (KCMs). As concrete examples we consider the Fredrickson-Andersen and East models, two paradigmatic KCMs relevant to the modelling of glasses. We show how variational matrix product states allow to numerically approximate - systematically and with high accuracy - the leading eigenstates of the tilted dynamical generators which encode the large deviation statistics of the dynamics. Via this approach we can study system sizes beyond what is possible with other methods, allowing us to characterise in detail the finite size scaling of the trajectory-space phase transition of these models, the behaviour of spectral gaps, and the spatial structure and entanglement properties of dynamical phases. We discuss the broader implications of our results.
We quantify how well matrix product states approximate exact ground states of 1-D quantum spin systems as a function of the number of spins and the entropy of blocks of spins. We also investigate the convex set of local reduced density operators of translational invariant systems. The results give a theoretical justification for the high accuracy of renormalization group algorithms, and justifies their use even in the case of critical systems.
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