Phase Slips, Dislocations, Half-Integer Vortices, Two-Fluid Hydrodynamics, and the Chiral Anomaly in Charge and Spin Density Waves

Abstract

This brief review recalls some chapters in theory of sliding incommensurate density waves which may have appeared after inspirations from studies of Dzyaloshinskii and collaborations with him. First we address the spin density waves which rich order parameter allows for an unusual object of a complex topological nature: a half-integer dislocation combined with a semi-vortex of the staggered magnetization. It becomes energetically preferable with respect to an ordinary dislocation due to the high Coulomb energy at low concentration of carriers. Generation of these objects should form a sequence of π-phase slips in accordance with experimental doubling of the phase-slips rate. Next, we revise the commonly employed TDGL approach which is shown to suffer from a violation of the charge conservation law resulting in nonphysical generation of particles which is particularly pronounced for electronic vortices in the course of their nucleation or motion. The suggested consistent theory exploits the chiral transformations taking into account the principle contribution of the fermionic chiral anomaly to the effective action. The derived equations clarify partitions of charges, currents, and rigidity among subsystems of condensed and normal carriers and the gluing electric field. Being non-analytical with respect to the order parameter, contrarily the conventional TDGL type, the resulting equations still allow for a numerical modeling of transient processes related to space- and spatiotemporal vorticity in DWs.

Appendices

APPENDICES

APPENDIX A

ANOMALOUS ENERGY AND SHAPE OF THE CHARGED DISLOCATION IN A DW

In the Fourier representation, the energy of phase deformations looks like

$$\begin{gathered} W\{ \varphi \} = \frac{{\hbar {{{v}}_{F}}}}{{4\pi }} \\ \times \sum {{\text{|}}{{\varphi }_{k}}{{{\text{|}}}^{2}}\left[ {C_{{||}}^{2}k_{{||}}^{2} + {{C}_{ \bot }}k_{ \bot }^{2} + \frac{{r_{0}^{{ - 2}}k_{{||}}^{2}}}{{(k_{{||}}^{2} + k_{ \bot }^{2} + r_{{{\text{scr}}}}^{{ - 2}})}}} \right]} \,, \\ \end{gathered} $$

(A.1)

where r₀ = $\sqrt {\hbar {{{v}}_{F}}s\epsilon {\text{/}}(8{{e}^{2}})}$ is the Thomas–Fermi screening length in the parent metal which is the atomic-scale length, and r_scr = r₀/$\sqrt {{{\rho }_{n}}}$ is the actual screening length by remnant charge carriers in the DW state. Here the normalized normal density ρ_n → 1 at T → $T_{c}^{0}$ and ρ_n ~ exp(–Δ/T) at low T being activated across the DW gap 2Δ. ρ_c = 1 – ρ_n = $C_{{||}}^{0}$ is the condensate density. Because of the very large Coulomb energy, the longitudinal gradients are strongly suppressed and we can neglect in (A.1) all appearances of k_|| except for the combination $r_{0}^{{ - 2}}k_{{||}}^{2}$. At large transverse distances beyond the screening length r_⊥ ≫ r_scr we can neglect also $k_{ \bot }^{2}$ in the denominator in (A.1); then the effective elastic theory is restored, while in stretched coordinates (x$\sqrt {{{C}_{ \bot }}{{\rho }_{n}}{\text{/}}{{\rho }_{c}}}$, r_⊥):

$$\begin{gathered} W\{ \varphi \} \approx \frac{{\hbar {{{v}}_{F}}}}{{4\pi }}\sum {{\text{|}}{{\varphi }_{k}}{{{\text{|}}}^{2}}[C_{{||}}^{2}k_{{||}}^{2} + {{C}_{ \bot }}k_{ \bot }^{2}],} \\ {{C}_{{||}}} = {{\rho }_{c}}{\text{/}}{{\rho }_{n}},\quad {{k}_{ \bot }} \ll {{r}_{{{\text{scr}}}}}\,. \\ \end{gathered} $$

(A.2)

Within the screening distance r < r_scr, the expression (A.1) describes a nonanalytic elastic theory, already known for uniaxial ferroelectrics, with the energy dependent on ratio of gradients rather on their values. In this nonscreened regime the energy dependence on the transverse length L_⊥ = $k_{ \bot }^{{ - 1}}$ at any given longitudinal scale L_|| = $k_{{||}}^{{ - 1}}$ is not monotonous, showing a minimum at $L_{ \bot }^{2}$ ~ L_||r₀$C_{ \bot }^{{1/2}}$. Multiplying the minimal energy density ~C_⊥$L_{ \bot }^{{ - 2}}$ by the characteristic volume L_||$L_{ \bot }^{2}$/$a_{ \bot }^{2}$ we estimate the energy as a function of R = L_⊥ as

$$\begin{gathered} C_{ \bot }^{{1/2}}{{R}^{2}}\hbar {{{v}}_{F}}{\text{/}}{{r}_{0}} = {{w}_{C}}N\quad {\text{with}} \\ {{w}_{C}}\sim {{T}_{c}}{{a}_{ \bot }}{\text{/}}{{r}_{0}}\quad {\text{at}}\quad N = \pi {{R}^{2}}{\text{/}}a_{ \bot }^{2}. \\ \end{gathered} $$

Consider a D-loop of a radius R in the (y, z) plane embracing a number N = πR²/$a_{ \bot }^{2}$ chains or a D-line stretched in z direction at a distance Y = a_⊥N from its counterpart or from the surface. For the DL energy W_D(N) the above estimations, with precise values found in [34], can be summarized as

(1) W_D(N) ~ T_c$\sqrt N$ln(N), valid within the length r₀—actually only at N ~ 1 i.e. for a elementary loop embracing only one chain, known as 2π soliton. A similar expression is valid at all distances for the spin vortex.

(2) W_D(N) ~ Nw_C, w_C ~ T_ca_⊥/r₀, valid at r₀ < R < r_scr.

(3) W_D(N) ~ T_c(r_scr/r₀)$\sqrt N$ln(Na_⊥/r_scr) at R > r_scr.

Within the wide (at low T) regime 2 the dislocation energy follows the confinement-type area law W ~ N rather than the usual perimetrical-logarithmic law ~$\sqrt N$lnN.

APPENDIX B

THE MICROSCOPIC ORIGIN OF THE CHIRAL ANOMALY

Technically (see [58]), the anomaly appears from uncertainty in calculations of fermionic determinants needing to be regularized, which procedure is usually treated in a relativistically invariant way by traditions of the field theory. It is instructive to see the origin of the anomaly in a transparent way related to rules of many-electronic systems, namely to the Tomas–Fermi procedure. Let the electrons in the parent metal occupy a parabolic spectrum p²/2m up to a Fermi energy E_F. In a smooth potential V(x) the electronic wave functions Ψ_E and their density distributions for an eigenenergy E are legitimately given by the WKB expressions

$$\begin{gathered} {{\Psi }_{E}} \propto (E - V{{(x)}^{{ - 1/4}}}\exp (i\int {dx{\kern 1pt} '(E - V(x{\kern 1pt} {{'}^{{1/2}}}),} \\ {{\rho }_{E}} = {\text{|}}\Psi (x){{{\text{|}}}^{2}} = (E - V{{(x)}^{{ - 1/2}}}). \\ \end{gathered} $$

Summing up ρ_E, the total density can be obtained and expanded in V/E_F as ρ(x) – 〈ρ〉 ≈ –V(x)N_F which yields the most important characterization of a metal: the linear response of the density to the applied potential. But if the spectrum linearization is done prematurely, then the Schrödinger equation becomes the first order one, the wave function Ψ_E looses the prefactor, ρ(x) becomes just constant, and the reaction to the local potential disappears. The role of the chiral anomaly action (W_CA in Eq. (10)) is just to restore this missing contribution.