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Keldysh approach to dissipative dynamics of interacting spins with long range interactions

Muzaffar Qadir Lone, Mehboob Rashid Bhat, Junaid Majeed Bhat, Ahmed Farouk

Published Mar 3, 2018• 0 citations
OpenAlex

Abstract

The interplay of coherent dynamics and dissipation in many body sysems give rise to a rich class of non-equilibrium phenomena with the emergence of non-trivial phases. In this paper, we investigate this interplay of dynamics in a model of interacting spins with infinite range interactions described by Heisenberg interaction. Using Holestien-Primarkoff transformations, the spin model is bosonized to a collective mode with self interaction at $\frac{1}{N}$ order. Employing Keldysh field theoretic technique with saddle point approximation, we see that the system breaks $\mathcal{Z}_2$-symmetry at the transition point. An effective temperature arises due to dissipation which depends linearly on effective system-bath coupling, and is independent of the dissipation rate and cutoff frequency of the bath with Drude-Lorentz cutoff in wide class of bath spectral densities. Furthermore, fluctuations over mean field are studied and is shown that the dissipative spectrum is modified by ${\rm O}(\frac{1}{N})$ correction term which results change in various physically measurable quantities.

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The competition between Hamiltonian and Lindblad dynamics in quantum systems give rise to non-equilibrium phenomena with no counterpart in conventional condensed matter physics. In this paper, we investigate this interplay of dynamics in infinite range Heisenberg model coupled to a non-Markovian bath and subjected to Lindblad dynamics due to spin flipping at a given site. The spin model is bosonized via Holestien-Primakoff transformations and is shown to be valid for narrow range of parameters in the thermodynamic limit. Using Schwinger-Keldyshtechnique, we derive mean field solution of the model and observe that the system breaks z2 - symmetry at the transition point. We calculate effective temperature that has linear dependence on the effective system-bath coupling, and is independent of the dissipation rate and cutoff frequency of the bath spectral density. Furthermore, we study the fluctuations over mean field and show that the dissipative spectrum is modified by O(1/N) correction term which results change in various physically measurable quantities.
We investigate the fate of dissipative phase transitions in quantum many-body systems when the individual constituents are qudits (ddd-level systems) instead of qubits. As an example system, we employ a permutation-invariant XYXYXY model of NNN infinite-range interacting ddd-level spins undergoing individual and collective dissipation. In the mean-field limit, we identify a dissipative phase transition, whose critical point is independent of ddd after a suitable rescaling of parameters. When the decay rates between all adjacent levels are identical and $d\ensuremath{\ge}4$, the critical point expands, in terms of the ratio between dissipation and interaction strengths, to a critical region in which two phases coexist and which increases as ddd grows. In addition, a larger ddd leads to a more pronounced change in spin expectation values at the critical point. Numerical investigations for finite NNN reveal symmetry-breaking signatures in the Liouvillian spectrum at the phase transition. The phase transition is furthermore marked by maximum entanglement negativity and a significant purity change of the steady state, which become more pronounced as ddd increases. Considering qudits instead of qubits thus opens new perspectives on accessing rich phase diagrams in open many-body systems.
Many-body systems with both coherent dynamics and dissipation constitute a rich class of models which are nevertheless much less explored than their dissipationless counterparts. The advent of numerous experimental platforms that simulate such dynamics poses an immediate challenge to systematically understand and classify these models. In particular, nontrivial many-body states emerge as steady states under nonequilibrium dynamics. While these states and their phase transitions have been studied extensively with mean-field theory, the validity of the mean-field approximation has not been systematically investigated. In this paper, we employ a field-theoretic approach based on the Keldysh formalism to study nonequilibrium phases and phase transitions in a variety of models. In all cases, a complete description via the Keldysh formalism indicates a partial or complete failure of the mean-field analysis. Furthermore, we find that an effective temperature emerges as a result of dissipation, and the universal behavior including the dynamics near the steady state is generically described by a thermodynamic universality class.
We investigate the fate of dissipative phase transitions in quantum many-body systems when the individual constituents are qudits (d-level systems) instead of qubits. As an example system, we employ a permutation- invariant XY model of N infinite-range interacting d-level spins (Lipkin-Meshkov-Glick model) undergoing individual and collective dissipation. In the mean-field limit, we identify a dissipative phase transition, whose critical point is independent of d after a suitable rescaling of parameters. When the decay rates between all adjacent levels are identical and d ≥ 4, the critical point expands, in terms of the ratio between dissipation and interaction strengths, to a critical region in which two phases coexist and which increases as d grows. In addition, a larger d leads to a more pronounced change in spin expectation values at the critical point. Numerical investigations for finite N reveal symmetry breaking signatures in the Liouvillian spectrum at the phase transition. The phase transition is furthermore marked by maximum entanglement negativity and a significant purity change of the steady state, which become more pronounced as d increases. Considering qudits instead of qubits thus opens new perspectives on accessing rich phase diagrams in open many-body systems.
We consider a prototypical system of an infinite range transverse field Ising model coupled to a bosonic bath. By integrating out the bosonic degrees, an effective anisotropic Heisenberg model is obtained for the spin system. The phase diagram of the latter is calculated as a function of coupling to the heat bath and the transverse magnetic field. Collective excitations at low temeratures are assessed within a spin-wave like analysis that exhibits a vanishing energy gap at the quantum critical point. We also consider another limit where the system reduces to a generalized spin-boson model of two interacting spins. By increasing the coupling strength with the heat bath, the two-spin wavefunction changes from an entangled state to a factorized state of two spins which are aligned along the transverse field. We also discuss the possible realization and application of the model to different physical systems.
Long-range interacting models exhibit rich dynamical phenomena driven by deviations from collective descriptions. This work extends the study of these models to open quantum systems, focusing on the complex interplay between nonlocal interactions and local dissipation. The results demonstrate that the impact of long-range interactions on relaxation is twofold: they mitigate the impact of noises by stabilizing ferromagnetic domains and reducing local fluctuation, yet simultaneously accelerate memory loss by indirectly exposing each spin to a broader environment. While strong dissipation drives the system into glassy, range-insensitive relaxation, the results in weak dissipation regime reveal rich state-dependent dynamics. Specifically, long-range collective behaviors protect the early-stage coherence of systems quenched from polarized states, whereas they actively accelerate the relaxation of staggered configurations. Furthermore, nonlocal interactions continue to drive memory decorrelation even in the near-steady stages of a dissipative quench. These findings offer timely theoretical insights for the design, control, and calibration of analog quantum simulators operating with long-range interactions.