Bulk pumping in two-dimensional topological phases

The notion of topological (Thouless) pumping in topological phases is traditionally associated with Laughlin's pump argument for the quantization of the Hall conductance in two-dimensional (2D) quantum Hall systems. It relies on magnetic flux variations that thread the system of interest without penetrating its bulk, in the spirit of Aharonov-Bohm effects. Here we explore a different paradigm for topological pumping induced, instead, by magnetic flux variations $\delta \chi$ inserted through the bulk of topological phases. We show that $\delta \chi$ generically controls the analog of a topological pump, accompanied by robust physical phenomena. We demonstrate this concept of bulk pumping in two paradigmatic types of 2D topological phases: integer and fractional quantum Hall systems and topological superconductors. We show that bulk pumping provides a unifying connection between seemingly distinct physical effects such as density variations described by Streda's formula in quantum Hall phases and fractional Josephson currents in topological superconductors. More importantly, we argue that bulk pumping provides a generic tool for probing topological phases and inducing robust physical effects, similar in spirit yet crucially different from Laughlin's pump. We discuss its generalizations in other topological phases.

Figure 1

(a): Schematic setup for conventional (Laughlin's) topological pumping vs “bulk pumping.” A gapped topological phase with cylinder or Corbino disk geometry is exposed to two types of magnetic fluxes: an Aharonov-Bohm flux $\mathrm{\Phi }$ threading its hole and a transverse flux $\chi$ threading its bulk. In the examples considered here, phases exhibit counterpropagating gapless edge modes [red (L) and blue (R)] supporting quasiparticle excitations with charge $e/l$, where $l\ge 1$ is an odd integer. (b): Schematic anomalous spectral flow of low-energy (edge) modes due to small flux variations $\delta \mathrm{\Phi }$ and $\delta \chi$. While Laughlin's conventional pump corresponds to $\delta \mathrm{\Phi }=2\pi l$ ($l$ flux quanta) [2, 3], the bulk pump of interest corresponds to $\delta \chi =4\pi l$ (see text). Bulk pumping induces spectral flow at the right edge only [or opposite flows at opposite edges (shown in gray), in a gauge where $\delta \mathrm{\Phi }\to \delta \mathrm{\Phi }+\delta \chi /2$; see Eq. (7)]. Empty/filled dots represent empty/filled states (see text), and $\mu$ denotes the Fermi level energy.

Figure 2

Effects of bulk flux quanta $\delta \chi$ in 2D topological superconductors. (a) To insert $\delta \chi$ through the superconducting bulk, we split the system into two parts, “left” (L) and “right” (R). When these parts are fully disconnected, a pair of chiral Majorana modes ${\gamma }_{L,s}\left(p\right)$ and ${\gamma }_{R,s}\left(p\right)$ (shown with simplified notations) appears at the edges, with another pair ${\gamma }_{JL,s}\left(p\right)$ and ${\gamma }_{JR,s}\left(p\right)$ at the junction. (b) Accordingly, the spectra of left and right parts exhibit a pair of counterpropagating chiral Majorana modes centered around momentum $p=0$ and $p=q_s\delta \chi /L_x$, respectively (setting $\delta \mathrm{\Phi }=0$ as in the text), where $L_x$ is the system's length in the $x$ direction. Shown here is the integer case $l=1$ (see text), where dots represent nonfractionalized single-particle chiral Majorana states. Arrows illustrate the spectral flow induced by $\delta \chi =2\pi l$, which only acts on the right part of the system. (c) When introducing a weak coupling $H_J$ between junction modes (with low-energy Hamiltonian $H_{JL}$ and $H_{JR}$, respectively), the change in energy $E_J=〈H_{JL}+H_{JR}+H_J〉$ exhibits a $\delta \chi =4\pi l$ periodicity (assuming that parity is conserved, see text), corresponding to a fractional Josephson current $I_J=\partial E_J/\partial \chi$ with the same periodicity. The energy levels sketched here are the two fermionic modes resulting from the hybridization of the junction Majorana modes, for $l=1$.

Figure 3

Coupled-wire description of the system in a noninteracting integer quantum Hall phase with filling factor $\nu =1$. (a) Generic setup consisting of $N_y$ 1D wires (red to blue) wrapped around a cylinder threaded by Aharonov-Bohm and transverse fluxes $\mathrm{\Phi }$ and $\chi$. A constant background flux $\chi ={\chi }_0$ is present, tuned with the Fermi level $\mu$ so that $\nu =1$. Low-energy degrees of freedom are illustrated on the right: Each wire with index $y$ and energy dispersion ${\epsilon }_{k,y}$ exhibits a pair of left- and right-moving modes at the Fermi level (fermionic modes with unit charge). Neighboring wires are coupled by some tunneling ${\delta }_{\perp }$ [Eq. (A4)] so that right movers in wire $y$ couple to left movers in wire $y+1$, leaving a pair of uncoupled modes at the edges. (b) Resulting (schematic) band structure: Bands of individual wires (faint colors) are shifted with respect to each other due to $\chi$ [Eq. (A3)]. They resonantly couple at the Fermi level, creating a gapped phase with counterpropagating gapless edge modes.

Figure 4

Coupled-wire description of the system in a noninteracting (integer) topological superconducting phase. (a) Same setup as in Fig. 3, with two modifications: (i) no background flux ${\chi }_0$ and (ii) superconducting pairings induced, e.g., by an underlying superconductor (gray striped background). Each wire corresponds to a gapless Kitaev chain [6], i.e., to a 1D superconductor with a pair of left- and right-moving modes crossing at the Fermi level $\mu$ [Eq. (B5)]. Due to particle-hole (PH) symmetry, each mode represents a superposition of quasiparticles and quasiholes (fermionic ones, with unit charge). Neighboring wires are coupled by a combination ${\delta }_{\perp }$ of tunneling and pairing [Eq. (B7)] so that right movers in wire $y$ couple to left movers in wire $y+1$, leaving a pair of uncoupled modes at the edges. (b) Resulting (schematic) single-particle band structure: Bands of individual wires (shown in black) resonantly couple, leading to a gapped phase with PH-symmetric counterpropagating gapless edge modes (red and blue). (c) Josephson junction created by the insertion of bulk flux quanta $\delta \chi$ (see text).

Figure 5

Low-energy BdG spectrum of the noninteracting (integer) topological superconductor described by Eqs. (B4) and (B7), with flux $\delta \chi$ inserted between wires $Y$ and $Y+1$ where superconductivity is absent. The resulting Josephson junction is governed by the weak coupling $H_J$ between these two wires [chosen here as in Eq. (B9)], which couples the two chiral Majorana fermionic modes appearing at the junction, when $H_J=0$ (${\gamma }_{JL}$ and ${\gamma }_{JR}$ in Fig. 4). The low-energy spectrum visible here shows the two modes arising from the hybridization of ${\gamma }_{JL}$ and ${\gamma }_{JR}$. All other modes are gapped, appearing at higher energies of the order of the bulk interwire coupling strength ${\delta }_{\perp }$ [Eq. (B7)]. For clarity, low-energy edge modes (${\gamma }_L$ and ${\gamma }_R$ in Fig. 4) are also gapped out by a direct coupling of the form of Eq. (B7). For even $\delta \chi /\pi$, the pair of Majorana zero modes that appears at the junction is gapped by $H_J$. For odd $\delta \chi /\pi$, in contrast, a single Majorana zero mode is present at the junction, and, hence, the effect of $H_J$ is exponentially small. The above plot shows the actual spectrum corresponding, for $l=1$, to the schematic spectrum in Fig. 2. In general, energy levels cross the Fermi level at odd $\delta \chi /\pi$ in an exponential way. The energy-dispersion and pairing functions in Eq. (B2) are chosen here as ${\xi }_k=\mu -t_{\parallel }cos\left(k\right)$ and ${\mathrm{\Delta }}_k=i\mathrm{\Delta }sin\left(k\right)$ with $\mu =t_{\perp }$, such that decoupled wires correspond to gapless Kitaev chains [6].