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arXiv:1311.1926 · Physical Review B

Finite-time interaction quench in a Luttinger liquid

Rashi Sachdeva, Tanay Nag, Amit Agarwal, Amit Dutta

Published Jul 23, 2014• 33 citations
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Abstract

We analyze the dynamics of a Luttinger model following a quench in the electron-electron interaction strength, where the change in the interaction strength occurs over a finite time scale $\ensuremath{\tau}$. We study the Loschmidt echo (the overlap between the initial and final state) as a function of time, both numerically and within a perturbation scheme, treating the change in the interaction strength as a small parameter, for all $\ensuremath{\tau}$. We derive the corrections appearing in (a) the Loschmidt echo for a finite quench duration $\ensuremath{\tau}$, (b) the scaling of the echo following a sudden $(\ensuremath{\tau}\ensuremath{\rightarrow}0)$ quench, and (c) the scaling of the echo after an adiabatic $(\ensuremath{\tau}\ensuremath{\rightarrow}\ensuremath{\infty})$ quench. We study in detail the limiting cases of the echo in the early-time and infinite-time limits, and provide scaling arguments to understand these in a general context. We also show that our perturbative results are in good agreement with the exact numerical ones.

DOI
10.1103/physrevb.90.045421

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We analyze the dynamics of a Luttinger model following a quench in the electron-electron interaction strength, where the change in the interaction strength occurs over a finite time scale \ensuremathτ\ensuremath{\tau}\ensuremathτ. We study the Loschmidt echo (the overlap between the initial and final state) as a function of time, both numerically and within a perturbation scheme, treating the change in the interaction strength as a small parameter, for all \ensuremathτ\ensuremath{\tau}\ensuremathτ. We derive the corrections appearing in (a) the Loschmidt echo for a finite quench duration \ensuremathτ\ensuremath{\tau}\ensuremathτ, (b) the scaling of the echo following a sudden (\ensuremathτ\ensuremath→0)(\ensuremath{\tau}\ensuremath{\rightarrow}0)(\ensuremathτ\ensuremath→0) quench, and (c) the scaling of the echo after an adiabatic (\ensuremathτ\ensuremath→\ensuremath∞)(\ensuremath{\tau}\ensuremath{\rightarrow}\ensuremath{\infty})(\ensuremathτ\ensuremath→\ensuremath∞) quench. We study in detail the limiting cases of the echo in the early-time and infinite-time limits, and provide scaling arguments to understand these in a general context. We also show that our perturbative results are in good agreement with the exact numerical ones.
We investigate the Loschmidt echo, the overlap of the initial and final wave functions of Luttinger liquids after a spatially inhomogeneous interaction quench. In studying the Luttinger model, we obtain an analytic solution of the bosonic Bogoliubov-de Gennes equations after quenching the interactions within a finite spatial region. As opposed to the power-law temporal decay following a potential quench, the interaction quench in the Luttinger model leads to a finite, hardly time-dependent overlap; therefore, no orthogonality catastrophe occurs. The steady state value of the Loschmidt echo after a sudden inhomogeneous quench is the square of the respective adiabatic overlaps. Our results are checked and validated numerically on the XXZ Heisenberg chain.
We study the response to sudden local perturbations of highly excited quantum Ising spin chains. The key quantity encoding this response is the overlap between time-dependent wave functions, which we write as a transient Loschmidt Echo. Its asymptotics at long time differences contain crucial information about the structure of the highly excited nonequilibrium environment induced by the quench. We compute the echo perturbatively for a weak local quench but for arbitrarily large global quench, using a cumulant expansion. Our perturbative results suggest that the echo decays exponentially, rather than power law as in the low-energy orthogonality catastrophe, a further example of quench-induced decoherence already found in the case of quenched Luttinger liquids. The emerging decoherence scale is set by the strength of the local potential and the bulk excitation energy.
We investigate the many-body generalization of the orthogonality catastrophe by studying the generalized Loschmidt echo of Luttinger liquids (LLs) after a global change of interaction. It decays exponentially with system size and exhibits universal behavior: the steady state exponent after quenching back and forth n times between 2 LLs (bang-bang protocol) is 2n times bigger than that of the adiabatic overlap and depends only on the initial and final LL parameters. These are corroborated numerically by matrix-product state based methods of the XXZ Heisenberg model. An experimental setup consisting of a hybrid system containing cold atoms and a flux qubit coupled to a Feshbach resonance is proposed to measure the Loschmidt echo using rf spectroscopy or Ramsey interferometry.
The equilibrium physics of quantum impurities frequently involves a universal crossover from weak to strong reservoir-impurity coupling, characterized by single-parameter scaling and an energy scale TK{T}_{K}TK​ (Kondo temperature) that breaks scale invariance. For the noninteracting resonant level model, the nonequilibrium time evolution of the Loschmidt echo after a local quantum quench was recently computed explicitly [R. Vasseur, K. Trinh, S. Haas, and H. Saleur, Phys. Rev. Lett. 110, 240601 (2013)]. It shows single-parameter scaling with variable TKt{T}_{K}tTK​t. Here, we scrutinize whether similar universal dynamics can be observed in various interacting quantum impurity systems. Using density matrix and functional renormalization group approaches, we analyze the time evolution resulting from abruptly coupling two noninteracting Fermi or interacting Luttinger liquid leads via a quantum dot or a direct link. We also consider the case of a single Luttinger liquid lead suddenly coupled to a quantum dot. We investigate whether the field-theory predictions for the universal scaling as well as for the large-time behavior successfully describe the time evolution of the Loschmidt echo and the entanglement entropy of microscopic models. Our study shows that for the considered local quench protocols the above quantum impurity models fall into a class of problems for which the nonequilibrium dynamics can largely be understood based on the knowledge of the corresponding equilibrium physics.
Luttinger liquids (LLs) arise by coupling left- and right-moving particles through interactions in one dimension. This most natural partitioning of LLs is investigated by the momentum-space entanglement after a quantum quench using analytical and numerical methods. We show that the momentum-space entanglement spectrum of a LL possesses many universal features both in equilibrium and after a quantum quench. The largest entanglement eigenvalue is identical to the Loschmidt echo, i.e., the overlap of the disentangled and final wave functions of the system. The second largest eigenvalue is the overlap of the first excited state of the disentangled system with zero total momentum and the final wave function. The entanglement gap is universal both in equilibrium and after a quantum quench. The momentum-space entanglement entropy is always extensive and saturates fast to a time independent value after the quench, in sharp contrast to a spatial bipartitioning.
We present a comprehensive analysis of the relaxation dynamics of a Luttinger liquid subject to a sequence of sudden interaction quenches. The critical exponent β governing the decay of the steady-state propagator is expressed as an explicit functional of the switching protocol. At long distances β depends only on the initial state while at short distances it is also history dependent. Continuous protocols of arbitrary complexity can be realized with infinitely long sequences. For quenches of finite duration we prove that there exists no protocol to bring the initial non-interacting system in the ground state of the Luttinger liquid, albeit thermalization occurs at short distances. The adiabatic theorem is then investigated with ramp switchings of increasing duration and several analytic results for both the propagator and the excitation energy are derived.