Galilei covariance of the theory of Thouless pumps

TL;DR

This work establishes the Galilean covariance of Thouless quantum pumps by showing how the topological and spectral indices transform under boosts. It develops both a bundle-based framework and a position-space formulation, proving that the transported charge $Q=\operatorname{ch}P$ and the existing charge $N=\operatorname{rk}P$ mix under Galilei transformations, with general rules $\operatorname{rk}\hat{P}=m\operatorname{rk}P$ and $\operatorname{ch}\hat{P}=n\operatorname{ch}P - m\operatorname{rk}P$ (reducing to $\operatorname{rk}\hat{P}=\operatorname{rk}P$, $\operatorname{ch}\hat{P}=\operatorname{ch}P-\operatorname{rk}P$ when $n=m=1$). The paper connects these bulk topological invariants to edge scattering data via bulk-edge correspondence in Büttiker pumps, showing that $\operatorname{ch}P_x=W_x$ and $\operatorname{ch}P_s=-W_s$ for winding numbers of reflection matrices, and demonstrates covariance persists in the edge formulation, with implications for Sturm-type oscillation theorems. By bridging Bloch-based and non-Bloch approaches and clarifying how frame changes affect topological indices, the results solidify the frame-independence of pumped topological transport and guide interpretation in ultracold-atom and solid-state pump experiments.

Abstract

The Thouless theory of quantum pumps establishes the conditions for quantized particle transport per cycle, and determines its value. When describing the pump from a moving reference frame, transported and existing charges transform, though not independently. This transformation is inherent to Galilean space and time, but it is underpinned by a transformation of vector bundles. Different formalisms can be used to describe this transformation, including one based on Bloch theory. Depending on the chosen formalism, the two types of charges will be realized as indices of either the same or different kinds. Finally, we apply the bulk-edge correspondence principle, so as to implement the transformation law within Büttiker's scattering theory of quantum pumps.

Figures (3)

• Figure 1: Illustration of periodic potentials $V_1$ and $V_2$ forming a Thouless pump, viewed from a reference frame $\hat{F}$ moving with constant velocity $v$. Dashed lines indicate potentials at a later time.
• Figure 3: The 3-torus $\mathbb{T}_3$, cf. (\ref{['3torus']}), and its 2-dimensional subtori $\mathbb{T}_x$, $\mathbb{T}_s$, and $\mathbb{T}_z$, obtained by fixing the value of one of the three coordinates.
• Figure 4: Winding numbers for existing and transported charges.

Theorems & Definitions (20)

• Lemma 2.1
• Theorem 2.2: Galilei covariance
• Remark : Gauss-Codazzi
• Lemma 2.3
• Remark
• Lemma 3.1
• Lemma 3.2
• Proposition 3.3: Charges
• Theorem 3.4: Galilei covariance
• Remark