Locality in Quantum Systems

TL;DR

The notes study locality in quantum many-body systems via Lieb-Robinson bounds, showing how a finite light-cone controls dynamics and static correlations. The core approach combines LR bounds, Fourier analysis, and quasi-adiabatic continuation to prove exponential correlation decay in gapped systems, analyze topological order, and derive higher-dimensional generalizations of the Lieb-Schultz-Mattis theorem, as well as a non-relativistic Goldstone-type result. Key contributions include a robust framework for handling ground-state structure, phase classification, and stability of topological phases under local perturbations, with tools that extend to multiple ground states and topologically nontrivial manifolds. These results illuminate how locality constrains phase structure, enable rigorous proofs of topological memory under time evolution, and provide practical methods for assessing phase stability in lattice quantum systems.

Abstract

These lecture notes focus on the application of ideas of locality, in particular Lieb-Robinson bounds, to quantum many-body systems. We consider applications including correlation decay, topological order, a higher dimensional Lieb-Schultz-Mattis theorem, and a nonrelativistic Goldstone theorem. The emphasis is on trying to show the ideas behind the calculations. As a result, the proofs are only sketched with an emphasis on the intuitive ideas behind them, and in some cases we use techniques that give very slightly weaker bounds for simplicity. This is a preliminary version of the lecture notes, with the goal of getting the notes out close to the end of the school. Comments welcome.

Figures (7)

• Figure 1: Plot of a step function (solid line) and a sketch of a step function convolved with a Gaussian (solid line). Dashed line at $\Delta E$ shows that the difference between the functions is small at sufficiently large frequency.
• Figure 2: Illustration of the geometry we consider. $X,Y$ are shown as shaded regions, while $X'$ includes everything within the outer circle around $X$.
• Figure 3: A) One ground state of Majumdar-Ghosh model. Circles indicate lattice sites. Light line around circles indicate that they are in a singlet. B) Another ground state.
• Figure 4: Twist in boundary conditions are applied at two places, at $x(i)=0=L$ (along the boundary of the systems) and at $x(i)=L/2$ (along the dashed line). We want to approximate $W_1$ by an operator supported on the upward slanting grey lines (near $x(i)=0$) and to approximate the operator $W_2$ by an operator supported on the downward slanting grey lines, so that $W_1,W_2$ will approximately commute. This requires a gap sufficiently large compared to $1/L$.
• Figure 5: Energy of lowest two states as a function of parallel magnetic field $H$. This is a sketch. The crossing is an avoided crossing but the splitting between states is exponentially small at $H=0$. At $H$ of order $1/N$, there is another avoided crossing as the energy gap becomes of order unity.

Theorems & Definitions (9)

• Theorem 1
• Definition 1
• Theorem 2
• Lemma 1
• proof
• Definition 2
• Definition 3
• Theorem 3
• Theorem 4