Coherent, time-correlated tunneling of density wave electrons

A growing body of evidence reveals that charge density wave (CDW) transport is a high-temperature cooperative quantum phenomenon. According to the time-correlated soliton tunneling (ST) model, quantum solitons, or electron-phonon correlates within the CDW condensate, act much like electrons tunneling through a Coulomb-blockade tunnel junction. Pair creation of charged fluidic soliton droplets is prevented by their electrostatic energy below a Coulomb-blockade threshold electric field. Above threshold, the quantum fluid flows in a periodic fashion, via a hybrid between Zener-like and coherent Josephson-like tunneling. We summarize the time-correlated ST model and compare model simulations with experiment. The ST model shows excellent agreement with coherent voltage oscillations, and with CDW current-voltage characteristics. Finally, we discuss implications for physics and potential applications.


Introduction
Charge density waves (CDWs) exhibit correlated flow of electrons at the highest known temperatures of any macroscopic electron condensate [1][2][3][4]. CDW dynamical behavior has been observed above room temperature [5], in some cases even above the boiling point of water [6][7][8]. In linear chain compounds, the CDW condensate modulates the charge along each of N parallel chains, ( , ) = 0 ( , ) + 1 cos[2 − ( , )], where kinks in carry charge and can transport electric current. Spin density waves are similar and can be viewed as two out-of-phase CDWs for the spin-up and -down sub-bands.
Considerable evidence shows that CDWs do not classically "slide," but instead transport electric current as a fundamentally quantum process. In the classical picture, the CDW is expected to slide when the applied electric field tilts the washboard pinning potential enough for classical depinning. Just below this classical depinning field, the CDW should displace by at least a quarter wavelength, equivalent to a phase displacement of π/2, or 90°. The restoring force near threshold, moreover, should become vanishingly small, leading to a divergent dielectric response [3], as shown in figure 1. These classical predictions are refuted by: 1) NMR experiments [9] showing only 2° CDW phase displacement just below threshold; and flat 2) dielectric (figure 1 and ref. [3]), and 3) harmonic mixing [10] responses vs. bias field below threshold. These experiments reveal that the CDW remains near the bottom of the pinning potential well, and that the threshold field for nonlinear transport is often much smaller than the classical depinning field.
The Coulomb blockade effect in single electron tunneling shows how a threshold voltage or field can arise from a purely quantum process [11]. A similar Coulomb blockade mechanism was found to yield a threshold field for charged soliton pair creation [12]. This model, based on the (1+1)-D massive Schwinger model [13], was extended to a description of time-correlated soliton tunneling (ST) in a 1-D  [14]. Oscillations of period ℎ 2 ⁄ , in CDW conductance vs. magnetic flux of TaS3 rings, provided clear evidence for the quantum nature of CDW transport [15,16]. This Aharonov-Bohm-like behavior, first reported in 2009 [15], motivated further development of the time-correlated ST model. Later results, reported in 2012 [16], included telegraphic switching effects suggesting transitions between macroscopically distinct states. Additional evidence for quantum behavior includes linear ac admittance and mixing experiments showing agreement with photon-assisted tunneling theory [3,10]. Classical models include classical sine-Gordon (s-G); random pinning (RP); renormalization group (NM); and incommensurate harmonic chain (CF) models. See [3] for details.

Time-correlated soliton tunneling model
We treat CDW transport as the periodic flow of a quantum fluid of electron-phonon correlates, or quantum solitons, within the condensate [2][3][4]. Pair creation of charged soliton droplets by an applied field is prevented by their electrostatic energy below a Coulomb blockade threshold field, = * 2 ⁄ , where * is the internal field produced by a soliton-antisoliton pair. This threshold corresponds to a "vacuum angle" = 2 * ⁄ = , and is often much smaller than the classical depinning field.
This describes Josephson-like coupling between successive branches via the tunneling matrix element T, which has a Zener-like force dependence. Here Ψ depicts the macrostate amplitude for the system to be on branch n [figure 2(a)]. We interpret Ψ to be classically robust order parameters, whose magnitudes grow and diminish when the system evolves between successive branches [2][3][4]. The quantum fluid thus flows in drip-like fashion as microscopic entities tunnel coherently from one charging energy macrostate to the next. Using this model, we have performed simulations of coherent