Matrix product states : imposing SU(2) x U(1) symmetry

Mykhailo V. Rakov

Tao Xiang

This chapter discusses the properties of matrix product state (MPS). It starts with a simple proof that the wave function generated by DMRG is an MPS. Then three different but equivalent canonical forms or representations of MPS are introduced. An MPS generally has redundant gauge degrees of freedom on each bond linking two neighboring local tensors. One can convert it into a canonical form by taking a canonical transformation to remove the gauge redundancy in the local tensors. Finally, the implementation of symmetries, including both the U(1) and SU(2) symmetries, is discussed.

Chu Guo, Dario Poletti

Quantum many-body physics simulations with matrix product states can often be accelerated if the quantum symmetries present in the system are explicitly taken into account. Conventionally, quantum symmetries have to be determined beforehand when constructing the tensors for the matrix product state algorithm. In this work, we present a matrix product state algorithm with an adaptive $U(1)$ symmetry. This algorithm can take into account or benefit from $U(1)$ or ${Z}_{2}$ symmetries when they are present or analyze the nonsymmetric scenario when the symmetries are broken without any external alteration of the code. To give some concrete examples we consider an XYZ model and show the insight that can be gained by (i) searching the ground state and (ii) evolving in time after a symmetry-changing quench. To show the generality of the method, we also consider an interacting bosonic system under the effect of a symmetry-breaking dissipation.

Andreas Weichselbaum, Jan von Delft

Chen Liu, Anders W. Sandvik, Yu-Cheng Su +1 more

Lin Zhang

The presence of symmetries can lead to nontrivial dynamics of operator entanglement in open quantum many-body systems, which characterizes the cost of a matrix product density operator representation of the density matrix in tensor-network methods and provides insights into the corresponding classical simulability. One example is $\mathrm{U}(1)$-symmetric open quantum systems with dephasing, in which the operator entanglement increases logarithmically at late times instead of being suppressed by the dephasing. Here we numerically study the far-from-equilibrium dynamics of operator entanglement in a dissipative quantum many-body system with the more complicated $\mathrm{SU}(2)$ symmetry and dissipations beyond dephasing. We show that after the initial rise and fall, the operator entanglement increases again in a logarithmic manner at late times in the $\mathrm{SU}(2)$-symmetric case. We find that this behavior can be fully understood from the corresponding $\mathrm{U}(1)$ subsymmetry by considering the symmetry-resolved operator entanglement. Especially, the probability distribution of different $\mathrm{U}(1)$ sectors also follows the Gaussian distribution observed in the $\mathrm{U}(1)$-symmetric case with dephasing, with the variance growing as a power law. But unlike the latter, both the classical Shannon entropy associated with the probabilities for the half system to be in different symmetry sectors and the corresponding symmetry-resolved operator entanglement now make nontrivial contributions to the late-time logarithmic growth of operator entanglement in our $\mathrm{SU}(2)$-symmetric case. Our results show evidence that the logarithmic growth of operator entanglement at long times is a generic behavior of dissipative quantum many-body dynamics with $\mathrm{U}(1)$ as the symmetry or subsymmetry and for broader dissipations beyond dephasing, although more analytical and numerical proof from future studies is still required for this conjecture. By breaking the $\mathrm{SU}(2)$ symmetry of our quantum many-body dynamics to $\mathrm{U}(1)$, we also show that the latter property---the logarithmic growth behavior of operator entanglement for broader dissipations beyond dephasing---is valid even for open quantum systems with only $\mathrm{U}(1)$ symmetry.