Tensor networks and their applications to many-body quantum systems

The simplest object in many-body quantum systems is a one-dimensional spin chain. Ground state is described by $2^N$ parameters, while Hamiltonian has $4^N$ parameters, where $N$ is a length of the chain. The exponential number of parameters makes it difficult to perform numerical calculations for long ($ > \sim 15$) spin chains. But one can decompose a tensor having support on $N$ lattice sites into a product of local tensors, having support on one site (tensor network). The key property arising in this procedure is a bond dimension: it regulates how much a local part of a tensor is connected to the parts located at neighbouring sites. The direct measure of this “interconnectedness” is entanglement entropy (of a tensor bipartition). Thus, for small entanglement entropy, one can effectively describe a tensor by a tensor network with a small bond dimension (and make a number of parameters not exponential, but linear in $N$).

In paper [1], this procedure is applied to finding slowest operators. In Heisenberg representation, these operators (being slower than others) describe the late-time dynamics of a spin chain. It is found that local slowest operators correspond to energy propagation and they smoothly change their properties when approaching an integrable point. But translationally invariant slowest operators play a different role: they appear in Generalized Gibbs Ensemble (GGE), that describes an intermediate step of relaxation of a system. In a separate work, faster local operators are considered. It is shown that they interchange their roles when moving away from the integrable point. Moreover, they convert one into another during the evolution.

References

1. E. Izotova, “Local versus translationally invariant slowest operators in quantum Ising spin chains”, Phys. Rev. E, 108:2 (2023), 024138