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Physical review. B, Solid state

Localized Dynamic Perturbations in Metals

E. Müller‐Hartmann, T. V. Ramakrishnan, G. Toulouse

Published Feb 15, 1971• 182 citations
Publisher

Abstract

The effect of a localized time-dependent potential on a Fermi gas is discussed. A simple bosonlike model for the particle-hole excitations is developed. Its use appears justified for the dissipative part of the response of the Fermi gas to localized time-dependent potentials. General expressions are obtained for the probability that the Fermi gas remains in the ground state and for the energy spectrum of excited states. Several illustrative examples are discussed. We consider the effect of an exponentially switched-on localized potential. It is shown that the state reached when the potential attains its full strength $V$ is orthogonal to the ground state of the Fermi gas with the potential $V$, no matter how slow the switching. It consists of a spectrum of excited states of the full Hamiltonian with a width proportional to the switching rate. Other aspects of this failure of adiabaticity such as nonreversibility and path dependence are investigated. Some possible applications are pointed out. The potential source may have a finite mass. The effect of its recoil is calculated for the special case of a suddenly switched-on potential. The probability of finding the Fermi gas in its final ground state after the perturbation is switched on is found to be ${P}_{G}=\mathrm{exp}[\frac{{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}}^{2}\mathrm{ln}\ensuremath{\gamma}}{(1\ensuremath{-}{\ensuremath{\gamma}}^{2})}]$, where $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}=V{\ensuremath{\rho}}_{\ensuremath{\epsilon}F}$ and is a measure of the coupling of the localized perturbation to the electron gas. $\ensuremath{\gamma}$ is the mass ratio $\frac{m}{M}$. The potential source is initially at rest. The above expression is critically discussed and corrections to it are calculated. They are found to be small. The spectrum of excitations produced is also calculated. In conclusion, connections are pointed out with other particle field problems exhibiting a similar infrared singularity.

DOI
10.1103/physrevb.3.1102

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Physical review. B, Condensed matter•Feb 15, 1982•19 citations
The effect of a localized time-dependent potential on a Fermi gas is discussed. A simple bosonlike model for the particle-hole excitations is developed. Its use appears justified for the dissipative part of the response of the Fermi gas to localized time-dependent potentials. General expressions are obtained for the probability that the Fermi gas remains in the ground state and for the energy spectrum of excited states. Several illustrative examples are discussed. We consider the effect of an exponentially switched-on localized potential. It is shown that the state reached when the potential attains its full strength VVV is orthogonal to the ground state of the Fermi gas with the potential VVV, no matter how slow the switching. It consists of a spectrum of excited states of the full Hamiltonian with a width proportional to the switching rate. Other aspects of this failure of adiabaticity such as nonreversibility and path dependence are investigated. Some possible applications are pointed out. The potential source may have a finite mass. The effect of its recoil is calculated for the special case of a suddenly switched-on potential. The probability of finding the Fermi gas in its final ground state after the perturbation is switched on is found to be PG=exp[V\ifmmode~\else˜\fi2ln\ensuremathγ(1\ensuremath−\ensuremathγ2)]{P}_{G}=\mathrm{exp}[\frac{{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}}^{2}\mathrm{ln}\ensuremath{\gamma}}{(1\ensuremath{-}{\ensuremath{\gamma}}^{2})}]PG​=exp[(1\ensuremath−\ensuremathγ2)V\ifmmode~\else˜\fi​2ln\ensuremathγ​], where V\ifmmode~\else˜\fi=V\ensuremathρ\ensuremathϵF\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}=V{\ensuremath{\rho}}_{\ensuremath{\epsilon}F}V\ifmmode~\else˜\fi​=V\ensuremathρ\ensuremathϵF​ and is a measure of the coupling of the localized perturbation to the electron gas. \ensuremathγ\ensuremath{\gamma}\ensuremathγ is the mass ratio mM\frac{m}{M}Mm​. The potential source is initially at rest. The above expression is critically discussed and corrections to it are calculated. They are found to be small. The spectrum of excitations produced is also calculated. In conclusion, connections are pointed out with other particle field problems exhibiting a similar infrared singularity.
We describe the Riemann-Hilbert (RH) approach to computing the long-time response of a Fermi gas to a time-dependent perturbation. The approach maps the problem onto a noncommuting RH problem. The method is nonperturbative, quite general, and can be used to compute the Fermi gas response in driven (out of equilibrium) as well as equilibrium systems. It has the appealing feature of working directly with scattering amplitudes defined at the Fermi surface rather than with the bare Hamiltonian. We illustrate the power of the method by rederiving standard results for the core-hole and open-line Green's functions for the equilibrium Fermi edge singularity (FES) problem. We then show that the case of the nonseparable potential can be solved nonperturbatively with no more effort than for the separable case. We compute the corresponding results for a biased (nonequilibrium) model tunneling device, similar to those used in single-photon detectors, in which a photon absorption process can significantly change the conductance of the barrier. For times much larger than the inverse bias across the device, the response of the Fermi gases in the two electrodes shows that the equilibrium Fermi edge singularity is smoothed, shifted in frequency, and becomes polarity dependent. These results have a simple interpretation in terms of known results for the equilibrium case but with (in general complex-valued) combinations of elements of the scattering matrix replacing the equilibrium phase shifts. We also consider the shot noise spectrum of a tunnel junction subject to a time-dependent bias and demonstrate that the calculation is essentially the same as that for the FES problem. For the case of a periodically driven device we show that the noise spectrum for coherent states of alternating current can be easily obtained using this approach.
A sudden change in a short-range single-particle potential excites an infinite number of low-energy particlehole pairs in a degenerate Fermion gas. Although each order of perturbation theory diverges, the perturbation series can be summed formally. The sum agrees in the limit of zero temperature with a result due to Nozi\`eres and De Dominicis. At finite temperature, it confirms a conjecture of Anderson's. The excitation spectrum at energies that are small compared to the temperature can be interpreted in terms of a relaxation frequency, which is proportional to the temperature times the square of the phase shift.
We discuss the quench dynamics near a quantum critical point focusing on the sine-Gordon model as a primary example. We suggest a unified approach to sudden and slow quenches, where the tuning parameter $\ensuremath{\lambda}(t)$ changes in time as $\ensuremath{\lambda}(t)\ensuremath{\sim}\ensuremath{\upsilon}{t}^{r}$, based on the adiabatic expansion of the excitation probability in powers of $\ensuremath{\upsilon}$. We show that the universal scaling of the excitation probability can be understood through the singularity of the generalized adiabatic susceptibility ${\ensuremath{\chi}}_{2r+2}(\ensuremath{\lambda})$, which for sudden quenches (r=0)(r=0)(r=0) reduces to the fidelity susceptibility. In turn this class of susceptibilities is expressed through the moments of the connected correlation function of the quench operator. We analyze the excitations created after a sudden quench of the cosine potential using a combined approach of form-factors expansion and conformal perturbation theory for the low-energy and high-energy sector, respectively. We find the general scaling laws for the probability of exciting the system, the density of excited quasiparticles, the entropy and the heat generated after the quench. In the two limits where the sine-Gordon model maps to hard-core bosons and free massive fermions we provide the exact solutions for the quench dynamics and discuss the finite temperature generalizations.
The ground-state energy and low-energy excitations of single-particle character of an infinite medium of fermions are discussed with the aid of time-dependent Green's functions, which are convenient generalizations of the exact particle correlation functions in the ground state of NNN particles. The power series development for the one- and two-particle functions, under the restriction to two-body forces, is derived and described by means of Feynman diagrams. The derivation of the linked-cluster expansion for the energy then follows immediately. The equivalence to previous versions is established. The one-particle function is examined in particular detail, and it is shown that the poles of its space-time Fourier transform studied as a function of the energy variable, for fixed momentum, determined the (N+1N+1N+1)-particle and ($N\ensuremath{-}1$)-particle excited states which have single-particle character. For a reasonable assumption about the full spectrum of excited states, it is found that for the interacting system, single-particle excitations with a real energy occur only at the Fermi momentum. It is pointed out that the corresponding energy, termed the perturbed Fermi energy, equals the binding energy per particle in the ground state of NNN particles for a saturating system at equilibrium density.It is shown, finally, that the entire structure of the theory may be carried over to the case of finite temperature, requiring only a redefinition of the Green's functions. The analogy is constructed from a discussion of the internal energy.
This chapter discusses the time-dependent perturbation theory applicable for small time-dependent perturbation. The incident electromagnetic wave makes the electric and magnetic potentials to oscillate and the perturbation is usually very small compared with the static Hamiltonian. The theory of time-dependent perturbation was developed by Dirac in 1927. Approximation methods are developed for systems where the time dependence of the Hamiltonian is small, slowly varying or rapidly varying. These correspond to perturbation, adiabatic and sudden approximations. An interesting consequence of sudden approximation is that the system can undergo an adiabatic change due to the sudden approximation. The rate of spontaneous emission is proportional to the number of atoms in the excited state. In the stimulated emission, an incident radiation of appropriate frequency induces an atom in an excited state to emit radiation. In the case of time-dependent perturbations, the system undergoes transition from one state to another during the action of the perturbation.