Topological charge pumping in excitonic insulators

Controlling many-body systems, and in particular using appropriately applied external fields to ‘steer’ order parameters of symmetry broken phases, has emerged as a central theme in current physics Kirilyuk et al. (2010); Mentink et al. (2015); Byrnes et al. (2014); Zhang and Averitt (2014); Basov et al. (2017); Golež et al. (2016); Claassen et al. (2019); Sun and Millis (2020a). The excitonic insulator (EI) is state of matter first proposed in the 1960s Mott (1961); Kozlov and Maksimov (1965); Jérome et al. (1967); Portengen et al. (1996) with an order parameter defined as a condensate of bound electron hole pairs that activates a hybridization between two otherwise (in the simplest case) decoupled bands and opens a gap in the electronic spectrum. Several candidate materials including electron-hole bilayers Fogler et al. (2014); Li et al. (2017); Du et al. (2017), Ta₂NiSe₅ Lu et al. (2017); Werdehausen et al. (2018); Wakisaka et al. (2009); Kaneko et al. (2013); Sugimoto et al. (2018); Mazza et al. (2020), $1T$-TiSe₂ Kogar et al. (2017); Cercellier et al. (2007); Kaneko et al. (2018); Chen et al. (2018) and monolayer WTe₂ Jia et al. (2020) are objects of current intensive study; recent work Du et al. (2017); Hu et al. (2018); Wang et al. (2019); Hu et al. (2020); Varsano et al. (2020); Perfetto and Stefanucci (2020); Hou et al. (2019) has pointed out their possible topological aspects. While the early theories of EI considered a one component order parameter, typically of inversion symmetric $s$-wave type, realistic interactions also allow for electron-hole pairing in sub-dominant channels including $p$-wave (inversion-odd) ones. In equilibrium, the $s$-wave ground state is favored, with the potential for $p$-wave order revealed by its fluctuations accompanied by dipole moment oscillations: the ‘Bardasis-Schrieffer’ collective mode Sun and Millis (2020b).

In this paper we show that applied electric fields can steer the order parameter to rotate in the space of s and p symmetry components, as shown in Fig. 1(a), leading to a realization of the ‘Thouless charge pump’ Thouless (1983); Rice and Mele (1982); King-Smith and Vanderbilt (1993); Zhang et al. (2020), providing quantized charge transport across an insulating sample.

The minimal model of an EI involves two electron bands shown in Fig. 1(b): a valence band with energy $\xi_{v,k}$ that disperses downwards from a high symmetry point (taken to have zero momentum) and a conduction band ($\xi_{c,k}$) that disperses upwards. For simplicity we assume that their energies are equal and opposite ($\xi_{c}=-\xi_{v}=\xi$). Relaxing this assumption does not change our results in an essential way. Defining the overlap $G=2\xi_{v,0}$, we distinguish the ‘BCS’ case $G>0$ where the two bands cross at a fermi wavevector $k_{F}$ with fermi velocity $v_{F}$ as shown by the dashed lines, leading to electron and hole pockets, and the ‘BEC’ case where $G<0$ and the bands do not cross. Excitonic order corresponds to the spontaneous formation of a hybridization between the two bands due to the electron-electron interaction $V$, leading to an order parameter $\Delta(k)=\sum_{k^{\prime}}V_{k-k^{\prime}}\left<\psi^{\dagger}_{c,k^{\prime}}\psi_{v,k^{\prime}}\right>+c.c.$ where $\psi_{c/v,k}$ is the electron annihilation operator at momentum $k$ of the conduction/valence band and $V_{q}$ is the Fourier transform of the density-density interaction potential $V(r)$. The $s$-wave order parameter $\Delta_{s}(k)$ is invariant under crystal symmetry operations while $p$-wave order parameters are odd under inversion: $\Delta_{p}(k)=-\Delta_{p}(-k)$, and often transform as a multi-dimensional representation of the crystal symmetry group. For simplicity we neglect the $k$-dependence of $\Delta_{s}$, and define $\Delta_{p}(k)=\Delta_{p}f_{k}$ where the pairing function $f_{k}$ carries the momentum dependence and satisfies $\text{max}(|f_{k_{F}}|)=1$. We focus on the $p_{x}$ pairing channel which is induced by the x-direction electric fields we consider here. While the qualitative conclusions hold generically for all spatial dimensions, we will indicate the dimensionality if a specific d-dimensional system is discussed where the momentum $k$ means a d-dimensional vector.

Writing the partition function $Z$ as a path integral over fermion fields $\psi=(\psi_{c},\,\psi_{v})$, performing a Hubbard-Stratonovich transformation of the interaction term in the excitonic pairing channel and subsuming the intraband interaction into $\xi$ one obtains the action (see SI Sec. I)

as an integral over space-time $(r,\tau)$ and the partition function is $Z=\int D[\bar{\psi},\psi]D[\bar{\Delta},\Delta]e^{-S}$. For physically reasonable interactions such as the screened Coulomb interaction, the $s$-wave pairing interaction $g_{s}$ is typically the strongest while $g_{p}$ is the leading subdominant one. We may write the mean field Hamiltonian as $\int dr\psi^{\dagger}H_{m}\psi=\sum_{k}\psi_{k}^{\dagger}H_{m}^{k}\psi_{k}$ with

where $\sigma_{i}$ are the Pauli matrices acting in the c/v band space. The vector potential $A$ enters Eq. (2) through the minimal coupling $k\rightarrow k-A$ required by local gauge invariance (electric field is $E=-\partial_{t}A$) and we set electron charge $e$, speed of light and the Planck constant $\hbar$ to be one. Interband dipolar couplings could also occur Golež et al. (2016); Murakami et al. (2020) but do not affect our results. Since the global phase is not important, we choose the $s$-wave order parameter to be real. As we will show, the system develops an electrical polarization as a $p$-wave component $\pi/2$ out of phase with the equilibrium $\Delta_{s}$ is introduced. Due to an emergent ‘particle hole’ symmetry in the BCS weak coupling case defined as $|\Delta_{s}|,|\Delta_{p}|\ll G$ which we focus on, applied electric fields create $\Delta_{p}$ primarily in this channel (see Sec. VI A of SI for a rigorous proof), so we write $p$-wave pairing in the $\sigma_{2}$ channel Sun and Millis (2020b). The quasiparticle spectrum is $E_{k}=\pm\sqrt{\xi_{k}^{2}+\Delta_{s}^{2}+\Delta_{p}^{2}f_{k}^{2}}$. As shown by Fig. 1(c), the spectrum will have gapless points (nodes) at $(k_{x},k_{y})=(0,\pm k_{F})$ in the pure $p$-wave state ($\Delta_{s}=0$) in two dimensions (2D).

Charge pump—Spatially uniform changes in $\Delta_{s,p}$ produce uniform currents $J=\langle\sum_{k}\partial_{k}H_{m}^{k}\rangle$ (see SI Sec. II), whose time integral from the initial $(\Delta_{s},\Delta_{p})=(\Delta,0)$ to the final point then gives the pumped charge $P$ (difference of polarization between the final state and the initial state). In a one dimensional (1D) system, $P$ has a geometrical meaning King-Smith and Vanderbilt (1993); Resta (1994) in the limit of slow order parameter dynamics. It is the flux of the Berry curvature 2-form $B$ through the 2D surface $S$ spanned by the occupied 1D crystal momentum $k$ and the time varying trajectory of $\Delta_{s,p}$, or alternatively by the line integral of the Berry connection $\mathcal{A}_{\mu}=i\langle\psi|\partial_{\mu}|\psi\rangle$ around its boundary:

where $\mu=(k,\Delta_{s},\Delta_{p})$ (see Fig. 2).

The Berry curvature $B$ from the valence band of Eq. (2) is sourced by monopoles at the points $\xi_{k}=\Delta_{s}=\Delta_{p}=0$, i.e., the points $(k,\Delta_{s},\Delta_{p})=(\pm k_{F},0,0)$ each of which has monopole charge $1$. If the order parameter evolution completes a full cycle on the $s+ip$ plane, $S$ becomes the surface of the 2-torus shown in Fig. 2(a) and the net charge pumped is the total flux from the enclosed monopoles which is an integer $N=2$, the Chern number of the process. This quantized change in the polarization is known as the Thouless pump Thouless (1983), a topological phenomenon immune to disorder. Note that the monopoles exist only for the ‘BCS’ ($G>0$, band inversion) case where the excitons strongly overlap such that charge can jump between them. In the ‘BEC’ case $\xi_{k}\neq 0$ for all $k$ and there are no monopoles enclosed in $S$ (see SI Sec. II C).

To compute the polarization for the case the order parameter does not complete a full cycle, we use the line integral approach. For notational simplicity, we suppress the subscripts ‘$k$’ without causing ambiguities. An explicit expression for the valence band wave function from (2) at $(k,\Delta_{s},\Delta_{p})$ is

where $\Delta=\Delta_{s}+i\Delta_{p}f_{k}\equiv|\Delta|e^{i\phi}$ and $|u|^{2}(|v^{2}|)=\frac{1}{2}\left(1\pm\frac{\xi}{E}\right)$. The Berry connection $\mathcal{A}_{\mu}=|u|^{2}\partial_{\mu}\phi$ has singularities associated with the Dirac strings, the intersections of which with $S$ (marked by crosses in Fig. 2(b)) must be correctly treated in the evaluation of the line integral. Noting that $|u|^{2}\rightarrow 0$ when $\xi\ll-|\Delta|$ and $|u|^{2}\rightarrow 1$ when $\xi\gg|\Delta|$, we see that in the weak coupling BCS limit the contour can be collapsed to the red rectangles in Fig. 2(b). Parameterizing $S$ using $k$ and the angle $\theta$ defined by $\Delta_{s}+i\Delta_{p}=Re^{i\theta}$ in Fig. 1(a), one observes that the polarization of an state on the $s+ip$ plane depends only on the angle $\theta$. Specifically, we found

for a 1D EI (see SI Sec. II). This may be understood by noting that the low energy physics around $\pm k_{F}$ is of two massive Dirac models, each of which realizes a Goldstone-Wilczek Goldstone and Wilczek (1981) mechanism of charge pumping.

Higher dimensional systems can be viewed as 1D chains along $x$ direction stacked in momentum space. For a 2D circular fermi surface one finds

A full cycle pumps exactly two electrons along each 1D momentum chain that crosses the fermi surface, giving

for 1D, 2D and three dimensional (3D) isotropic systems respectively.

Although the charge pump is a bulk property carried by all valence band electrons, it is also revealed by the evolution of edge states Yang et al. (2020) as $\Delta_{s}$ and $\Delta_{p}$ are varied, as shown in Fig. 3 for a 1D wire connected with reservoirs. In the BCS limit, with open boundary conditions $\psi(0)=\psi(L)=0$, there are two edge states

where $C_{\pm}$ is a normalization constant. We suppose $\Delta_{s}+i\Delta_{p}=Re^{i\theta}$ and follow the evolution of $\psi_{+}$ as $\theta$ is varied (see Fig.3). At $\theta=0$ the state is delocalized and unoccupied with energy $R$. As $\theta$ is increased the state becomes localized near $x=0$ and decreases in energy. When $\theta$ passes through $\pi/2$, the state becomes maximally localized and becomes occupied by an electron from the left reservoir since its energy crosses the chemical potential. As $\theta$ further increases the state becomes delocalized and then localized at the right edge, delivering its electron to the right reservoir when $\theta$ crosses $3\pi/2$. Considering the $\psi_{-}$ state during the same cycle, two electrons in total are pumped. In higher dimensions, each 1D $k_{x}$ chain crossing the fermi surface has a similar edge state evolution (see SI Sec. III).

Dynamics—The coupled dynamics of electrons and the order parameters in the presence of an applied electric field is described by the action Eq. (1). To understand the qualitative dynamics, we use a low energy effective Ginzburg-Landau Lagrangian

for fields $\Delta_{s},\,\Delta_{p}$ obtained by integrating out the Fermions (see SI Sec. IV). The dynamics is given by the standard Euler-Lagrange equation $\frac{d}{dt}\frac{\delta L}{\delta\dot{\Delta}_{i}}=\frac{\delta L}{\delta\Delta_{i}}$ and is that of a point particle moving in the landscape defined by $F$, with kinetic energy $K$ and driven by an electric field through $L_{\text{drive}}$. We find

where $P$ is the adiabatic polarization in Eqs. (5) or (9), $s=\lim_{\omega\rightarrow 0}\sigma(\omega)/(2i\omega)$ and $\sigma(\omega)$ is the optical conductivity from virtual interband excitations (see SI Sec. IV). It is natural that electric field couples linearly to the polarization and therefore provides a ‘force’ $E\partial_{\theta}P$ to rotate the order parameter in the $\Delta_{s},\Delta_{p}$ plane.

$F(\Delta_{s},\Delta_{p})$ gives the potential landscape in which the dynamics takes place; it has the anisotropic ‘Mexican hat’ form shown in Fig. 4(a). For (quasi) 1D systems in the weak coupling BCS limit Altland and Simons (2010); Sun et al. (2020):

where $\nu$ is density of states in the metallic phase and $\Lambda\gg\sqrt{\Delta_{s}^{2}+\Delta_{p}^{2}}$ is a high energy cutoff Kozlov and Maksimov (1965). The first term becomes $-\nu\int\frac{d\theta_{k}}{2\pi}\left(\Delta_{s}^{2}+\Delta_{p}^{2}\cos^{2}\theta_{k}\right)\ln\left(2\Lambda/\sqrt{\Delta_{s}^{2}+\Delta_{p}^{2}\cos^{2}\theta_{k}}\right)$ for a 2D isotropic Fermi surface and $\frac{d\theta_{k}}{2\pi}\rightarrow\frac{\sin\theta_{k}d\theta_{k}d\phi_{k}}{4\pi}$ for 3D where $\theta_{k}$ and $\phi_{k}$ are angular variables on the Fermi surface. The landscape has a local maximum at $R=0$ surrounded by a trough at $R(\theta)$ of lower values of $F$. The ground state minima are at $(\pm\Delta,0)$ and the pure p-wave phases at $(0,\pm\Delta_{p0})$ are saddle points with energy higher by $F_{b}=\nu(\Delta^{2}-c\Delta_{p0}^{2})/2$ where $c$ is a constant depending on the space dimension.

We may estimate the minimal electric field required to drive the system from the minimum through the $p$-wave saddle point by equating the potential energy barrier $F_{b}$ to the work $EP(\theta=\pi/2)+\mathcal{O}(E^{2})$ done by the electric field, obtaining

where $\xi^{0}=v_{F}/\Delta$ is the coherence length (exciton size), $\kappa=\frac{1}{\pi}(1-\Delta_{p0}^{2}/\Delta^{2})$ in 1D and $\kappa=\frac{1}{2}-\frac{1}{4}\frac{\Delta_{p0}^{2}}{\Delta^{2}}$ in 2D, and $E_{0}$ is at the order of the dielectric breakdown field. For $v_{F}=10^{6}\,\mathrm{m/s}$, $\Delta=10\,\mathrm{meV}$ and $\Delta_{p0}\ll\Delta$, such as the case of electron hole bilayers, the threshold field is $E_{c}\sim 10^{3}\,\mathrm{V/cm}$ which can be easily achieved by modern optical technique. For a $100\,\mathrm{meV}$ gap such as that in Ta₂NiSe₅ Lu et al. (2017); Werdehausen et al. (2018) (assuming it is in the BCS regime), the threshold field is about $10^{5}\,\mathrm{V/cm}$. At such large field, $O(E^{2})$ terms in the Lagrangian will be important, which pushes the order parameter closer to zero but does not destroy the qualitative dynamics in the transient regime (See SI Sec. IV D).

The dynamical term $K$ has a relatively simple form if the gap never closes on the Fermi surface and the order parameter variation timescale is long compared to the inverse of the gap. For example for (quasi) 1D

to lowest order in time derivatives. For higher dimensions with closed Fermi surfaces, there are $O(1)$ changes to the coefficients and, crucially, dissipation and time non-locality arises from quasiparticle excitations near the nodes of the p-wave gap when $\Delta_{s}$ passes zero. This dissipation also brings a correction to the pumped charge: $P\rightarrow P_{2D}+P_{\text{dis}}$. To estimate $P_{\text{dis}}$, we observe that as the order parameter passes this gapless regime with a velocity $\dot{\Delta}_{s}$, the probability for exciting a particle-hole pair at $k$ is given by the Landau-Zener formula Wittig (2005): $P_{k}=e^{-2\pi\delta_{k}^{2}/|\partial_{t}2\Delta_{s}|}$ where $\delta_{k}=\sqrt{\xi_{k}^{2}+\Delta_{p}^{2}f_{k}^{2}}$ is half of its minimal excitation energy during the dynamics. In 2D, summing over momenta, one obtains the number of excited quasi particles $N=\frac{k_{F}}{2\pi^{2}v_{F}}|\frac{\dot{\Delta}_{s}}{\Delta_{p}}|$ and the non-adiabatic correction to the pumped charge

valid if $\sqrt{|\dot{\Delta}_{s}|}\ll|\Delta_{p}|$ (see SI Sec. VI C). Therefore, if the sub-dominant $p$-wave coupling constant is too small such that $\Delta_{p0}\sim\Lambda e^{-\frac{1}{g_{p}\nu}}<\sqrt{|\dot{\Delta}_{s}|/(8\pi^{2})}$, this dissipative correction will dominate over the adiabatic charge.

Numerics and Experiment—We numerically solved the mean field dynamics implied by Eq. (1) for a BCS weak coupling EI in 2D, driven by a train of widely separated electric field pulses (Fig. 4(b)). Mean field dynamics Barankov et al. (2004) means that each momentum state evolves in the time dependent mean field $(\Delta_{s},\,\Delta_{p}f_{k-A},\,\xi_{k-A})$ with $\Delta_{s,p}$ determined self consistently by the gap equation, neglecting any spatial fluctuations. We include a weak phenomenological damping $\gamma$ to represent energy loss caused by, e.g., a phonon bath (see SI Sec. VI). Each pulse drives the order parameter along the trajectory shown as the black dashed line in Fig. 4(a), advancing it by $\theta=\pi$ to stabilize the system in the other $s$-wave ground state. The total duration of the evolution from one minimum to the next is $T_{s}\approx 20\pi/\Delta$ and the amount of charge pumped is $WP_{2D}/2$ where $W$ is the width of the sample, as shown by Fig. 4(a). In a train of pulses with inter pulse separation $T_{0}\gg T_{s}$, each pulse advances the order parameter from one minimum to the next and allows it to stabilize before next pulse arrives, leading to a time-averaged current $I_{0}=eWk_{F}/(\pi T_{0})$. For a $10\,\mathrm{\mu m}$ wide sample with normal state carrier density of $10^{12}\,\mathrm{cm^{-2}}$ and inter pulse time $T_{0}=1\,\mathrm{ns}$, the current is $I_{0}=255\,\mathrm{nA}$ considering spin degeneracy.

A minimum field strength $\sim E_{c}$ (15) is required: as the maximum electric field $E_{\text{max}}$ of the pulse is increased beyond the threshold, the charge pumping (DC current) will onset sharply, as shown in Fig. 4(b). As $E_{\text{max}}$ further increases, each pulse induces a rotation of more cycles which pumps more charge, giving rise to the step structure. Deviations from perfect quantization arise from fast order parameter dynamics caused by the short duration pulse. A precisely engineered long duration pulse can substantially reduce these deviations; see SI Sec. V.

A static electric field in the DC transport regime could also drive such an order parameter rotation and charge pumping. However, unlike the case of well separated pulses, there is no time break to dump the generated heat into the environment which might destroy the system.

Discussion—We have shown theoretically that a Thouless charge pump may be realized as a collective many-body effect arising from order parameter steering in BCS type excitonic insulators. Similar dynamics and charge pumping can occur in general when the ground state order parameter and the sub dominant one have different parities under inversion. Its observation would provide both a verification of order parameter steering and a probe of the excitonic insulating state, in particular, distinguishing BCS and BEC states. It is interesting to study the dynamics in the vicinity of the BCS-BEC crossover and effects beyond mean field.

Supplemental Material for ‘Topological charge pumping in excitonic insulators’