Simulation of Quantum Many-Body Systems with Strings of Operators and Monte Carlo Tensor Contractions

We introduce string-bond states, a class of states obtained by placing strings of operators on a lattice, which encompasses the relevant states in quantum information. For string-bond states, expectation values of local observables can be computed efficiently using Monte Carlo sampling, making them suitable for a variational algorithm which extends the density matrix renormalization group to higher dimensional and irregular systems. Numerical results demonstrate the applicability of these states to the simulation of many-body systems.

Figure 1

Various string patterns. In (a), (c), and (d), the strings are long lines—in particular, (c) directly generalizes the DMRG ansatz, and since (d) contains (a), it gives a larger class of states. In (b) and (f), the strings form small loops; pattern (f) underlies the toric code state. In (e), the strings have length one, which suffices, e.g., for the cluster state or the coherent version of classical Gibbs states. Patterns (a), (b), and both atop of each other have been implemented numerically.

Figure 2

(a) A PEPS with factorizing projectors—connected by the maximally entangled bonds (blue)—yields a string-bond state. (b) To convert a general string state with many strings [we illustrate patterns Fig. 1a, 1b together] into a PEPS, each string is routed over a separate bond.

Figure 3

Magnetization for a frustrated $XX$ model (where each plaquette is frustrated) as a function of the transverse field on a $10\times 10$ lattice with PBC. Note that there is no other method available which can deal with such systems.

Figure 4

Magnetization for the $10\times 10$ PBC Ising model with transverse field: results for quantum Monte Carlo (QMC) and the string setups Fig. 1a (lines) and Fig. 1a, 1b (loops). The right inset shows the relative error in the ground state energy compared to QMC as a function of the field.