Lieb-Schultz-Mattis in Higher Dimensions

TL;DR

This work extends the Lieb-Schultz-Mattis theorem to higher dimensions by combining loop-operator techniques with a twisted-boundary framework. It proves that a d-dimensional spin-1/2 system with an odd number of spins per unit cell cannot be both gapped and translation-symmetric in the thermodynamic limit, establishing a bound $\Delta E \leq c \log(L)/L$ for the first excited state when the ground state is unique. The authors construct a topologically twisted, approximately invariant state and use translation symmetry to show that a persistent gap across all twists would be inconsistent with odd-width boundary conditions, thereby ruling out long-range order and implying the presence of a topological excitation. The approach yields general locality bounds, provides cluster bounds on correlations, and offers a framework for extensions to SU(N) and Markov-type dynamics, with several conjectures for even-width geometries and finite-temperature behavior.

Abstract

A generalization of the Lieb-Schultz-Mattis theorem to higher dimensional spin systems is shown. The physical motivation for the result is that such spin systems typically either have long-range order, in which case there are gapless modes, or have only short-range correlations, in which case there are topological excitations. The result uses a set of loop operators, analogous to those used in gauge theories, defined in terms of the spin operators of the theory. We also obtain various cluster bounds on expectation values for gapped systems. These bounds are used, under the assumption of a gap, to rule out the first case of long-range order, after which we show the existence of a topological excitation. Compared to the ground state, the topologically excited state has, up to a small error, the same expectation values for all operators acting within any local region, but it has a different momentum.

Figures (3)

• Figure 1: Illustration of the bound on $|\tilde{f}^+(0)-f^+(0)|$, as described in the text. The vertical line describes a $\delta$-function spike in $f(\omega)$. This produces a Gaussian in $\tilde{f}(\omega)$. The integral over $\omega$ of the Gaussian is the same as the integral of the $\delta$-function; however, the shaded region of the curve of the Gaussian falls above $\omega=0$, and hence does not contribute to $f^+(0)$. This leads to the difference between $\tilde{f}^+(0)$ and $f^+(0)$.
• Figure 2: Plot of the system, showing the $x$-coordinate along the length axis. The $x$ is shown ranging from $x=0$ to $x=L$; due to the periodicity of the system, $x=0$ is identified with $x=L$. The $y$ coordinate specifies the position in the directions normal to the length, as well as specifying the particular site in each unit cell. The twist angles are noted; the twist $\theta$ changes the boundary condition near $x=L$, while the twist $\theta'$ changes the coupling between sites near $x=L/2$.
• Figure 3: Plot of the system, showing the twists and coordinates as before. The halves of the system have been shaded in. The shading at the left and right side of the system (diagonal lines going up and right) denotes sites in half (1), the shading in the middle (diagonal lines going up and left) denotes sites in half (2). The solid shading denotes sites in both halves; the length of the solid region is at least $2R$, so that the Hamiltonian can be written as a sum of terms, each of which is contained in only one half.