Abstract
We analyze the problem of extracting the correlation length from infinite matrix product states (MPS) and corner transfer matrix (CTM) simulations. When the correlation length is calculated directly from the transfer matrix, it is typically significantly underestimated for finite bond dimensions used in numerical simulation. This is true even when one considers ground states at a distance from the critical point. We introduce extrapolation procedure to overcome this problem. To that end we quantify how much the dominant part of the MPS/CTM transfer matrix spectrum deviates from being continuous. The latter is necessary to capture the exact asymptotics of the correlation function where the exponential decay is typically modified by an additional algebraic term. By extrapolating such a refinement parameter to zero, we show that we are able to recover the exact value of the correlation length with high accuracy. In a generic setting, our method reduces the error by a factor of $\sim 100$ as compared to the results without extrapolation and a factor of $\sim 10$ as compared to simple extrapolation schemes using bond dimension. We test our approach in a number of solvable models both in 1d and 2d. Subsequently, we apply it to 1d XXZ spin-$\frac32$ and the Bose-Hubbard models in a massive regime in the vicinity of BKT critical point. We then fit the scaling form of the correlation length and extract the position of the critical point and obtain results comparable or better than those of other state-of-the-art numerical methods. Finally, we show how the algebraic part of the correlation function asymptotics can be directly recovered from the scaling of the dominant form factor within our approach. Our method provides the means for detailed studies of phase diagrams of quantum models in 1d and, through the finite correlation length scaling of projected entangled pair states, also in 2d.
Highlights
Tensor networks and related numerical renormalization group techniques allow us to efficiently approximate systems of exponentially many degrees of freedom (d.o.f.) with a manageable number of a few relevant ones, providing invaluable tools in the studies of strongly correlated systems
We argue that the exponent η of the algebraic part of the correlation function asymptotics in Eq (2) is directly related to how the relevant dominant form factor decays as we approach the exact solution, δ → 0
Away from the critical point, this would require the ability to fit the correlation function asymptotics for distances between the physical correlation length and the length scale that results from the discreteness of the matrix product state (MPS) transfer matrix
Summary
Tensor networks and related numerical renormalization group techniques allow us to efficiently approximate systems of exponentially many degrees of freedom (d.o.f.) with a manageable number of a few relevant ones, providing invaluable tools in the studies of strongly correlated systems. For the specific point in the XXZ spin-1=2 model which we discuss in detail later, MPS with bond dimension 4096 recovers the exact ground state energy with an error of the order of 10−12, but at the same time it is still underestimating the correlation length by a factor of 2. In order to benchmark our approach, we analyze a number of models where the correlation length, or some related properties like the position of the critical point and its universality class, are known analytically Based on this data we argue that the method proposed in this article is more reliable and produces more accurate results than the one that directly uses the bond dimension as a refinement parameter. In Appendix B, we argue that methods that provide good extrapolation of energy per lattice site are not suited to work well for the correlation length
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