Classification of Gapped Symmetric Phases in 1D Spin Systems

TL;DR

Problem: classify gapped 1D spin phases with symmetry using LU and MPS. Approach: analyze SRC-MPS under symmetric LU RG to fixed-point forms, deriving cohomology-based classifications. Contributions: no 1D topological order without symmetry; symmetry produces SPTO phases labeled by H^2(G,C), with detailed TI, parity, and TR classifications; higher-dimensional partial insights. Significance: provides a unifying, constructive framework for SPTO in 1D and informs higher-dimensional topological phases.

Abstract

Quantum many-body systems divide into a variety of phases with very different physical properties. The question of what kind of phases exist and how to identify them seems hard especially for strongly interacting systems. Here we make an attempt to answer this question for gapped interacting quantum spin systems whose ground states are short-range correlated. Based on the local unitary equivalence relation between short-range correlated states in the same phase, we classify possible quantum phases for 1D matrix product states, which represent well the class of 1D gapped ground states. We find that in the absence of any symmetry all states are equivalent to trivial product states, which means that there is no topological order in 1D. However, if certain symmetry is required, many phases exist with different symmetry protected topological orders. The symmetric local unitary equivalence relation also allows us to obtain some simple results for quantum phases in higher dimensions when some symmetries are present.

Figures (7)

• Figure 1: (Color online) (a) The possible phases for class of Hamiltonians $H(g_1,g_2)$ without any symmetry restriction. (b) The possible phases for class of Hamiltonians $H_\text{symm}(g_1,g_2)$ with some symmetries. The shaded regions in (a) and (b) represent the phases with short range entanglement.
• Figure 2: (Color online) Disentangling fixed point state (upper layer, product of entangled pairs) into direct product state (lower layer) with LU transformations.
• Figure 3: A $\{n_i\}$-block TI LU transformation described by a quantum circuit of three layers. Here $n_1=4$, $n_2=3$, $n_3=2$. The unitary transformations $U_i$ on different blocks in the $i^{th}$ layer are the same.
• Figure 4: (Color online) Representative states of the four parity symmetric phases, each corresponding to (a) $\alpha(P)=1$, $\beta(P)=1$ (b) $\alpha(P)=-1$, $\beta(P)=1$ (c) $\alpha(P)=-1$, $\beta(P)=-1$ (d) $\alpha(P)=1$, $\beta(P)=-1$. $+$ stands for a parity even entangled pair (e.g. $|00\rangle+|11\rangle$), $-$ stands for a parity odd entangled pair (e.g. $|01\rangle-|10\rangle$). Each site contains four virtual spins.
• Figure 5: Tensor-network -- a graphic representation of the tensor-product wave function TNS on a 2D square lattice. The indices on the links are summed over.