Dissipative processes in nonequilibrium many-body systems are fundamentally different than their equilibrium counterparts. Such processes are of great importance for the understanding of relaxation in single-molecule devices. As a detailed case study, we investigate here a generic spin-fermion model, where a two-level system couples to two metallic leads with different chemical potentials. We present results for the spin relaxation rate in the nonadiabatic limit for an arbitrary coupling to the leads using both analytical and exact numerical methods. The nonequilibrium dynamics is reflected by an exponential relaxation at long times and via complex phase shifts, leading in some cases to an ``antiorthogonality'' effect. In the limit of strong system-lead coupling at zero temperature we demonstrate the onset of a Marcus-like Gaussian decay with voltage difference activation. This is analogous to the equilibrium spin-boson model, where at strong coupling and high temperatures, the spin excitation rate manifests temperature activated Gaussian behavior. We find that there is no simple linear relationship between the role of the temperature in the bosonic system and a voltage drop in a nonequilibrium electronic case. The two models also differ by the orthogonality-catastrophe factor existing in a fermionic system, which modifies the resulting line shapes. Implications for current characteristics are discussed. We demonstrate the violation of pairwise Coulomb gas behavior for strong coupling to the leads. The results presented in this paper form the basis of an exact, nonperturbative description of steady-state quantum dissipative systems.
Dissipative processes in nonequilibrium many-body systems are fundamentally different than their equilibrium counterparts. Such processes are of great importance for the understanding of relaxation in single-molecule devices. As a detailed case study, we investigate here a generic spin-fermion model, where a two-level system couples to two metallic leads with different chemical potentials. We present results for the spin relaxation rate in the nonadiabatic limit for an arbitrary coupling to the leads using both analytical and exact numerical methods. The nonequilibrium dynamics is reflected by an exponential relaxation at long times and via complex phase shifts, leading in some cases to an ``antiorthogonality'' effect. In the limit of strong system-lead coupling at zero temperature we demonstrate the onset of a Marcus-like Gaussian decay with voltage difference activation. This is analogous to the equilibrium spin-boson model, where at strong coupling and high temperatures, the spin excitation rate manifests temperature activated Gaussian behavior. We find that there is no simple linear relationship between the role of the temperature in the bosonic system and a voltage drop in a nonequilibrium electronic case. The two models also differ by the orthogonality-catastrophe factor existing in a fermionic system, which modifies the resulting line shapes. Implications for current characteristics are discussed. We demonstrate the violation of pairwise Coulomb gas behavior for strong coupling to the leads. The results presented in this paper form the basis of an exact, nonperturbative description of steady-state quantum dissipative systems.
We theoretically study steady-state nonequilibrium properties of various open electronic systems subject to time-independent external bias. A charge current is established across each system by\nits coupling to two external particle reservoirs maintained at different chemical potentials. We discuss the impact of intra-reservoir electron correlations on transport, and examine how reservoir-generated dissipation and nonequilibrium-induced decoherence influence these systems.\n\n\nThe effect of intra-lead electron interactions on transport is investigated in the context of a phonon-coupled single molecule transistor driven by Luttinger-liquid source and drain leads. The semi-classical master equation approach is used to compute current and noise characteristics of the device for various interaction strengths in the leads. The results suggest the possibility of tuning the Fano factor of the device using intra-lead electron interactions.\n\n\nThe Keldysh path integral formalism is used to theoretically formulate models that describe the remaining open nonequilibrium systems. We consider voltage-induced electron-phonon scattering and electron mass enhancement due to phonons in a model metallic system. The possibility of adjusting the acoustic phonon velocity\nand the Thomas-Fermi screening length with external voltage is discussed. The effects of dissipation is investigated in an open BCS superconducting graphene, where the dissipation-induced rearrangement of its ground state from the BCS superconductor to the Fermi liquid is examined. The results theoretically infer prospects for a voltage-tuned metal-to-BCS quantum phase transition in graphene. Lastly, we develop a theory of nonequilibrium quantum criticality in open itinerant Ising and Heisenberg magnets. Both departures from equilibrium at conventional quantum critical points and the physics of phase transitions induced by the nonequilibrium drive are analyzed.
We summarize in this work the relations of the spin-boson Hamiltonian, which was recently studied in connection with the phenomenon of quantum coherence in the presence of dissipation, to three different fermionic Hamiltonians. These relations were obtained through well-known equivalences between Fermi and Bose operators in one dimension. The fermionic Hamiltonians correspond to (a) a two-level system coupled linearly with a fermionic bath, (b) the resonant-level model, and (c) the anisotropic Kondo model. For the first Hamiltonian we reobtain and discuss the relationship between the dimensionless dissipation coefficient \ensuremath{\alpha} and the phase shift of the fermions. The resonant-level model allows us to study the properties of the two-level system for values of \ensuremath{\alpha} around (1/2). At \ensuremath{\alpha}=(1/2) the model reduces to the Toulouse limit, where the Hamiltonian is exactly soluble. A pure exponential decay is obtained for the relaxation of P(t)=〈${\ensuremath{\sigma}}_{z}$(t)〉 for t>0, given that for t<0 the system is known to be localized in one of the two states. The comparison with the Kondo model gives the long-time-limit behavior of the system for (1/2)1 and provides a connection between universal numbers obtained previously for the spin-boson Hamiltonian and the Kondo model.
We study a generic spin-fermion model, where a two-level system (spin) is coupled to two metallic leads with different chemical potentials, in the presence of monochromatic driving fields. The real-time dynamics of the system is simulated beyond the Markovian limit by an iterative numerically exact influence functional path integral method. Our results show that although both system-bath coupling and chemical potential difference contribute to dissipation, their effects are distinct. In particular, under certain drivings the asymptotic Floquet states of the system exhibit robustness against a range of system-bath coupling strength: the asymptotic behaviors of the system are insensitive to different system-bath coupling strength, while they are highly tunable by the chemical potential difference of baths. Further simulations show that such robustness may be essentially a result of the interplay between driving, bath electronic structure, and system-bath coupling. Therefore, the robustness could break down depending on the characteristics of the interplay. In addition, under fast linearly polarized driving the quantum stochastic resonance is demonstrated that stronger system-bath coupling (stronger dissipation) enhances rather than suppresses the amplitude of coherent oscillations of the system.
Using detailed balance and scaling properties of integrals that appear in the Coulomb gas reformulation of quantum impurity problems, we establish exact relations between the nonequilibrium quantum decay rates of the boundary sine-Gordon and the anisotropic Kondo model at zero temperature. Combining these results with findings from the thermodynamic Bethe ansatz, we derive exact closed form expressions for the quantum decay rate of the dissipative two-state system in the scaling limit. These expressions illustrate how the crossover from weak to strong tunneling takes place. We trace out the regimes in which the usually applied Golden Rule (nonadiabatic) rate expression fails. Using a conjectured correspondence between the relaxation and dephasing rate, we obtain the exact lower bound of the dephasing rate as a function of bias and dissipation strength.
We study the dissipative dynamics of a class of interacting “gamma-matrix” spin models coupled to a Markovian environment. For spins on an arbitrary graph, we construct a Lindbladian that maps to a non-Hermitian model of free Majorana fermions hopping on the graph with a background classical Z 2 gauge field. We show, analytically and numerically, that the steady states and relaxation dynamics are qualitatively independent of the choice of the underlying graph, in stark contrast to the Hamiltonian case. We also show that the exponentially many steady states provide a concrete example of mixed-state topological order, in the sense of strong-to-weak spontaneous symmetry breaking of a one-form symmetry. While encoding only classical information, the steady states still exhibit long-range quantum correlations. Afterward, we examine the relaxation processes toward the steady state by numerically computing decay rates, which we generically find to be finite, even in the dissipationless limit. However, we identify symmetry sectors where fermion-parity conservation is enhanced to fermion-number conservation, where we can analytically bound the decay rates and prove that they vanish in the limits of both infinitely weak and infinitely strong dissipation. Finally, we show that while the choice of coherent dynamics is very flexible, exact solvability strongly constrains the allowed form of dissipation. Our work establishes an analytically tractable framework to explore nonequilibrium quantum phases of matter and the relaxation mechanisms toward them.
In this Thesis we discussed several issues related to the dynamics of isolated quantum many-body systems. The brief overview of Chapter 1 highlighted a series of open fundamental theoretical questions which are crucial for the investigation of nonequilibrium phenomena. Among them, we mainly focused on the emergence of thermal behaviors and on the characterization of dynamical phase transitions and of the associated universal dynamics. To this end, we analyzed some paradigmatic models, such as the Fermi-Hubbard model, the quantum Ising chain, and the O(N) model. -Chapter 2: Absence of thermalization in a Fermi liquid. In this Chapter we report two results that question the common expectation of thermalization in a Landau-Fermi liquid after an interaction quench. We first consider the perturbative expansion in the interaction strength of the long-wavelength structure factor S(q) and show that it does not satisfy the hypothesis that steady-state averages correspond to thermal ones. In particular, S(q) has an analytical component ∼ const. + O(q^2) for q → 0, compatible with thermalization, but it retains also a nonanalytic term ∼ |q| characteristic of a Fermi liquid at zero-temperature. In real space, this nonanalyticity turns in a power-law decay of the density-density correlations, in contrast with the exponential decay associated to thermalization. We next analyze the low-density case, where one can obtain results valid at any order in interaction but at leading in the density, and find that in the steady state the momentum distribution jump at the Fermi surface is strictly finite, though smaller than at equilibrium. -Chapter 3: Nonadiabatic stationary behavior in a driven Ising chain. In this Chapter we discuss the emergence of a nonadiabatic behavior in the dynamics of the order parameter in a low-dimensional system subject to a linear ramp of one of its parameters within a gapped phase, which should be the most favorable situation for an adiabatic evolution. We study in details the dynamics of the spontaneous magnetization m_x(t) in a quantum Ising chain after a linear variation in time of the transverse field within the ordered phase. In particular, focusing on the value of the order parameter at the end of the ramp m_x(τ), we show that the smaller the switching rate of the transverse field is the closer m_x(τ) gets to its ground state value. Nonetheless, this apparently small correction to adiabaticity eventually leads to the disruption of the order exponentially fast in the subsequent time evolution, no matter how slowly the ramp is performed. -Chapter 4: Aging and coarsening in isolated quantum systems after a quench. This Chapter analyzes the nonequilibrium dynamics of an isolated quantum system after a sudden quench to the dynamical critical point, where the emergence of scaling and universal exponents is expected, due to the absence of time and length scales. We explore these features for a bosonic interacting scalar field theory with O(N) symmetry in the large-N limit, where the model is exactly solvable and the exponents and scaling forms of the relevant two-time correlation functions can be analytically calculated. Moreover, we provide evidence of the emergence of a dynamical scaling behavior also for quenches below the dynamical critical point, associated with coarsening. We find that the latter case is characterized by the same scaling functions as those describing the critical case, yet with different exponents. -Chapter 5: Dynamical transitions and statistics of excitations. In this Chapter we study the dynamics of the O(N) model (introduced in the previous Chapter) resulting from a different protocol: a linear ramp of one of its parameters. We find that the presence of a dynamical phase transition, as well as its critical properties, are robust against the change of the protocol. We show that a characterization based on the critical dimensions and exponents would suggest that the dynamical phase transition is analogous to the equilibrium thermal one. However, its nonequilibrium nature becomes evident analyzing the statistics of excitations produced in the ramp process. In particular, the critical properties are encoded in the fluctuations in the number of excitations, which display qualitatively different behaviors depending on the ramp being performed above, at, or below the dynamical critical point. These behaviors bear no dependence on the duration of the protocol.