Quantum Kinetic Theory

[Gabelo Camphire]

Thisbook presents quantum kinetic theory in a comprehensive way. The focus is on density operator methods and on non-equilibrium Green functions. The theory allows to rigorously treat nonequilibrium dynamics in quantum many-body systems. Of particular interest are ultrafast processes in plasmas, condensed matter and trapped atoms that are stimulated by rapidly developing experiments with short pulse lasers and free electron lasers. To describe these experiments theoretically, the most powerful approach is given by non-Markovian quantum kinetic equations that are discussed in detail, including computational aspects.

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We investigate the second-order nonlinear electronic thermal transport induced by the temperature gradient. We develop the quantum kinetic theory framework to describe thermal transport in the presence of a temperature gradient. Using this, we predict an intrinsic scattering time-independent nonlinear thermal current in addition to the known extrinsic nonlinear Drude and Berry curvature dipole contributions. We show that the intrinsic thermal current is determined by the band geometric quantities and is nonzero only in systems in which both the space inversion and time-reversal symmetries are broken. We employ the developed theory to study the thermal response in tilted massive Dirac systems. We show that besides the different scattering time dependencies, the various current contributions have distinct temperature dependencies in the low-temperature limit. Our systematic and comprehensive theory for nonlinear thermal transport paves the way for future theoretical and experimental studies on intrinsic thermal responses.

(a) Variation of the nonlinear anomalous Hall conductivity with the chemical potential μ, where solid line represents the numerically obtained result. In (b), we have shown the variation of the longitudinal (left) and transverse (right) total nonlinear intrinsic conductivity with chemical potential (μ). The total nonlinear intrinsic conductivity κtot is the sum of the intrinsic contributions arising from κdo, κod, and κoo. Therefore, in (b), the solid line represents the total intrinsic conductivity, while the dashed lines are the different thermal current contributions giving rise to intrinsic conductivity. Additionally, in this figure, the circles denote the analytical result calculated up to linear order in tilt velocity vt. The model parameters, scattering time, and temperature considered here are the same as in Fig. 3. For the numerical and analytical calculation of the nonlinear intrinsic thermal current, we considered the cutoff energy of the valence band to be 2eV.

Motivated by recent experimental studies of thin-film devices containing a single ferromagnetic layer, we develop a quantum kinetic theory of current-induced magnetic torques in Rashba-model ferromagnets. We find that the current-induced spin densities, responsible for the switching behavior, are due most essentially to spin-dependent quasiparticle lifetimes and derive analytic expressions for relevant limits of a simple model. Quantitative model parameter estimates suggest that spin-orbit coupling in the adjacent metal normal magnetic layer must play an essential role in the strength of the switching effect.

The quantum kinetic framework provides a versatile method for investigating the dynamical optical and transport currents of crystalline solids. In this paper, starting from the density-matrix equations of motion, we present a general theoretical path to obtain the nonlinear optical response in an elegant and transparent manner. We devise an extensive kinetic theory that can be applied to materials with arbitrary band structures and captures intraband and interband coherence effects, finite Fermi surfaces, and disorder effects. We present a classification of the nonlinear optical currents arising from the interference of the interband and intraband components of the density matrix with distinct symmetry and quantum geometrical origin for each contribution. In this context, we report the following four primary findings: (i) The Fermi golden rule approach is insufficient to derive the correct expression for the injection current, a shortcoming that we remedy in our theory while associating the injection current with the intraband-interband contribution to the second-order density matrix. (ii) The interband-intraband contribution yields a resonant current that survives irrespective of any symmetry constraint in addition to the well-known anomalous nonlinear current (nonresonant), which requires time-reversal symmetry. (iii) Quite generally, the nonlinear current is significantly enhanced by contributions arising from the finite Fermi surface. (iv) The finite Fermi surface and Fermi sea additionally lead to sizable novel nonlinear effects via contributions we term double resonant and higher-order pole. We investigate such effects in sum frequency and difference frequency generation. As an illustration, we compute the nonlinear response of the topological antiferromagnet CuMnAs and thin film tilted Weyl semimetals as model systems dominated by interband coherence contributions. We find that the nonlinear response of CuMnAs is responsive to the direction of the finite magnetization field and the response of Weyl semimetal to the tilt. In addition, the choice of the polarization angle of the beam is crucial to have a nonlinear current in CuMnAs, while it is not the case for Weyl semimetals.

A schematic tree for the generation of different contributions of the first-order and second-order density matrix ρ that leads to distinct forms of nonlinear currents. Here the subscripts d and o stand for the diagonal and off-diagonal parts of the density matrix. In the double subscripts such as dd, do, od, and oo, the first letter indicates the diagonal and off-diagonal part of the second-order density matrix and later letter corresponds to the dependence of the relevant part of the first-order density matrix on the second order.

Distribution of the geometric quantities in the momentum space. Top panel: For Topological antiferromagnetic CuMnAs where (a) corresponds to the quantum metric, (b) and (c) to metric connection. Bottom panel: For thin film tilted Weyl semimetal where (d) refers to the Berry curvature, (e) and (f) to the symplectic connection.

A kinetic theory for quantum many-particle systems in time-dependent electromagnetic fields is developed based on a gauge-invariant formulation. The resulting kinetic equation generalizes previous results to quantum systems and includes many-body effects. It is, in particular, applicable to the interaction of strong laser fields with dense correlated plasmas.

We present a general quantum kinetic theory of low-field magnetotransport in weakly disordered crystals that accounts fully for the interplay between electric-field-induced interband coherence, Bloch-state scattering, and an external magnetic field. The quantum kinetic equation we derive for the Bloch-state density matrix naturally incorporates the momentum-space Berry phase effects whose influence on Bloch-state wave-packet dynamics is normally incorporated into transport theory in an ad hoc manner. The Berry phase correction to the momentum-space density of states in the presence of an external magnetic field implied by semiclassical wave-packet dynamics is captured by our theory as an intrinsic density-matrix response to a magnetic field. We propose a simple and general procedure for expanding the linear response of the Bloch-state density matrix to an electric field in powers of magnetic field. As an illustration, we apply our theory to magnetotransport in Weyl semimetals. We show that the chiral anomaly (positive magnetoconductivity quadratic in magnetic field) that appears when separate Fermi surface pockets surround distinct Weyl points survives only when intervalley scattering is very weak compared to intravalley scattering.

(a) Pumping of valley population in a Weyl semimetal by parallel electric and magnetic fields oriented along the z direction in the absence of disorder. When the electronic structure of a Weyl semimetal is approximated by Dirac cones with masses that depend on kz, a magnetic field induces an anomalous (N=0) Landau level branch that has only one sign of velocity vz in a given valley. It follows that in each valley the density of states is increased for states with one sign of velocity and decreased for states with the other sign of velocity, and that the total current summed over a valley is already nonzero in equilibrium. When an electric field drives states through momentum space (diagonal red arrows), the total number of states in a valley varies. (b) Scattering within valleys (curved red arrows) can relax the current in each valley to its equilibrium value, but cannot establish a steady state because the number of states in each valley still changes at a constant rate. A steady state can be established only when intervalley scattering processes are present.

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