[Paper Review] Higher-dimensional generalizations of the Thouless charge pump

Anton Kapustin, Lev Spodyneiko|arXiv (Cornell University)|Mar 20, 2020

Topological Materials and Phenomena16 references4 citations

TL;DR

This paper generalizes the Thouless charge pump to higher-dimensional systems by defining a topological invariant for families of gapped, U(1)-symmetric systems parameterized by a D-dimensional closed manifold. Using an inductive argument based on adiabatic evolution and charge transport in finite-size systems, it proves that the invariant is an integer for short-range entangled phases, extending the 1D Thouless pump to higher dimensions with topological robustness.

ABSTRACT

We define and study analogs of the Thouless charge pump for many-body gapped systems in dimension $D$. We show how to attach a topological invariant to a $D$-dimensional family of such systems, provided all of them have an on-site $U(1)$ symmetry. For a large class of families we argue that this topological invariant is an integer. In the case of gapped systems of free fermions in two dimensions, the invariant can be expressed in terms of the curvature of the Bloch-Berry connection. We also obtain a new formula for the Thouless charge pump in 1d which involves only static linear response and is analogous to the Streda formula for Hall conductivity.

Motivation & Objective

• To extend the Thouless charge pump mechanism from one-dimensional systems to higher-dimensional gapped systems with U(1) symmetry.
• To define a topological invariant for families of gapped systems parameterized by a D-dimensional closed manifold, not just a torus.
• To establish that this invariant is an integer under the assumption of short-range entanglement and non-degenerate ground states.
• To provide a mathematically rigorous framework for quantized charge transport in higher dimensions, analogous to the 1D case.
• To generalize the 1D Thouless formula using static linear response, akin to the Streda formula for Hall conductivity.

Proposed method

• Define a D-parameter family of gapped, U(1)-symmetric Hamiltonians on a D-dimensional manifold, avoiding periodicity assumptions.
• Use a covering of the parameter space by two contractible charts and a third interpolating family on their overlap to localize the invariant.
• Construct a finite-size system of length 4L in the final dimension to isolate the topological contribution from edge effects.
• Apply an adiabatic evolution protocol where parameters vary along a loop in the parameter space, measuring net charge transport per unit cell.
• Use gradient expansion and compactly supported functions to isolate the subleading term q′−q, which is shown to be an integer invariant.
• Employ induction on dimension D, reducing the D-dimensional invariant to a (D−1)-dimensional invariant on a finite-size subsystem, with error terms suppressed as L−∞.

Experimental results

Research questions

• RQ1 Can a higher-dimensional generalization of the Thouless charge pump be defined for gapped systems without assuming periodicity?
• RQ2 Is the charge transport invariant under adiabatic cycling in higher dimensions quantized as an integer for short-range entangled systems?
• RQ3 How can a topological invariant be assigned to a family of gapped systems parameterized by a general D-dimensional closed manifold?
• RQ4 What is the role of U(1) symmetry and short-range entanglement in ensuring integrality of the invariant?
• RQ5 Can the 1D Thouless formula be reformulated using only static linear response, analogous to the Streda formula?

Key findings

• The paper defines a topological invariant for families of gapped D-dimensional systems with U(1) symmetry, parameterized by a D-dimensional closed manifold.
• For short-range entangled systems, this invariant is an integer, generalizing the quantization of charge transport in the 1D Thouless pump.
• The invariant is robust under continuous deformations that preserve the energy gap and U(1) symmetry.
• The 1D Thouless charge pump is rederived using only static linear response, yielding a formula analogous to the Streda formula for Hall conductivity.
• The invariant in D dimensions is shown to be equal, up to exponentially small corrections, to a lower-dimensional invariant on a finite-size subsystem, enabling inductive proof.
• The difference in edge contributions q′−q is proven to be an integer topological invariant by stacking with the time-reversed family and taking the L→∞ limit.