The Lieb-Schultz-Mattis Theorem: A Topological Point of View

TL;DR

The article presents a topological reformulation of Lieb-Schultz-Mattis type no-go theorems for 1D quantum spin systems, focusing on translation invariance with $U(1)$ and $\mathbb{Z}_2\times\mathbb{Z}_2$ symmetry. It develops a local twist operator framework to define a topological index that matches the filling factor $\nu$, proving that a non-integer $\nu$ forbids a locally-unique gapped ground state in 1D and on the infinite cylinder. For discrete symmetry, it introduces the Ogata index via projective representations to extend LSM-type results, culminating in a constraint that $pS$ must be an integer; half-odd-integer values obstruct LUGGS. The work situates these 1D and quasi-1D results within the broader field of symmetry-protected topological phases and boundary-condition dependent phenomena, linking to higher-dimensional LSM theorems and paving the way for rigorous analyses of LSM-type constraints via topological indices.

Abstract

We review the Lieb-Schultz-Mattis theorem and its variants, which are no-go theorems that state that a quantum many-body system with certain conditions cannot have a locally-unique gapped ground state. We restrict ourselves to one-dimensional quantum spin systems and discuss both the generalized Lieb-Schultz-Mattis theorem for models with U(1) symmetry and the extended Lieb-Schultz-Mattis theorem for models with discrete symmetry. We also discuss the implication of the same arguments to systems on the infinite cylinder, both with the periodic boundary conditions and with the spiral boundary conditions. For models with U(1) symmetry, we here present a rearranged version of the original proof of Lieb, Schultz, and Mattis based on the twist operator. As the title suggests we take a modern topological point of view and prove the generalized Lieb-Schultz-Mattis theorem by making use of a topological index (which coincides with the filling factor). By a topological index, we mean an index that characterizes a locally-unique gapped ground state and is invariant under continuous (or smooth) modification of the ground state. For models with discrete symmetry, we describe the basic idea of the most general proof based on the topological index introduced in the context of symmetry-protected topological phases. We start from background materials such as the classification of projective representations of the symmetry group. We also review the notion that we call a locally-unique gapped ground state of a quantum spin system on an infinite lattice and present basic theorems. This notion turns out to be natural and useful from the physicists' point of view. We have tried to make the present article readable and almost self-contained. We only assume basic knowledge about quantum spin systems.

Figures (8)

• Figure 1: The spin-rotation angle $\theta_j$ increases gradually from 0 to $2\pi$ as the coordinate varies from $x$ to $x+\ell$. Otherwise it is equal to 0 or $2\pi$.
• Figure 2: (a) When $x$ is varied from $0$ to $p$, the profile (see Figure \ref{['f:theta']}) of the rotation angle is modified continuously. (b) Since $\omega(\hat{U}_{0,\ell})=\omega(\hat{U}_{p,\ell})$, the expectation value $\omega(\hat{U}_{x,\ell})\ne0$ for $x\in[0,p]$ defines a closed oriented path in the complex plane that does not touch the origin. One can then determine the winding number that characterizes the path or the ground state $\omega$. The winding number is two in this figure.
• Figure 3: A graphic representation of a part of the dimer state \ref{['e:dimerstate']}. A black dot represents a spin with $S=1/2$, and two dots connected by a line represents a spin-singlet. (a) The restriction of the state to the half-infinite chain $\{2k,2k+1,\ldots\}$ consists only of spin-singlets. It is invariant under the $\mathbb{Z}_2\times\mathbb{Z}_2$ transformation and should be characterized by the index $\operatorname{Ind}=0$. (b) The restriction of the state to the half-infinite chain $\{2k+1,2k+2,\ldots\}$ contains an extra $S=1/2$ at site $2k+1$, and should be characterized by the index $\operatorname{Ind}=1$.
• Figure 4: A graphic representation of the exact ground state, known as the Valence-Bond Solid (VBS) state, of the AKLT Hamiltonian \ref{['e:AKLT']}. As in Figure \ref{['f:dimer']}, a black dot represents a spin with $S=1/2$, and two dots connected by a line represents a spin-singlet. Here two black dots surrounded by an oval represent the symmetrization of two $S=1/2$'s, which is equivalent to a state of spin with $S=1$. If we examine the transformation property of the VBS state restricted onto a half-infinite chain, the situation is similar to that in Figure \ref{['f:dimer']} (b). One observes an emergent $S=1/2$ degree of freedom at the edge, which should be characterized by the index $\operatorname{Ind}=1$. See, e.g., Chapter 7 of TasakiBook for details about the VBS state.
• Figure 5: A graphic interpretation of the additivity \ref{['e:Oadd']}. (a) The Ogata index $\operatorname{Ind}_j^\omega$ characterizes the transformation property of the state $\omega$ restricted onto the half-infinite chain $\{j,j+1,\ldots\}$. (b) The same transformation property can be expressed by the sum of the index $\operatorname{ind}_j$ of the single spin at site $j$ and the Ogata index $\operatorname{Ind}^\omega_{j+1}$ for the half-infinite chain $\{j+1,j+2,\ldots\}$.

Theorems & Definitions (22)

• Definition 2.1: state
• Definition 2.2: ground state
• Definition 2.3: locally-unique gapped ground state
• Theorem 2.4
• Definition 2.5: unique gapped ground state
• Theorem 2.6
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• Theorem 3.4: a necessary condition for a locally-unique gapped ground state