Understanding the non-equilibrium dynamics of extended quantum systems after the trigger of a sudden, global perturbation (quench) represents a daunting challenge, especially in the presence of interactions. The main difficulties stem from both the vanishing time scale of the quench event, which can thus create arbitrarily high energy modes, and its non-local nature, which curtails the utility of local excitation bases. We here show that nonperturbative methods based on integrability can prove sufficiently powerful to completely characterize quantum quenches: we illustrate this using a model of fermions with pairing interactions (Richardson's model). The effects of simple (and multiple) quenches on the dynamics of various important observables are discussed. Many of the features we find are expected to be universal to all kinds of quench situations in atomic physics and condensed matter.
The generic non-equilibrium evolution of a strongly interacting fermionic system is studied. For strong quenches, a collective collapse-and-revival phenomenon is found extending over the whole Brillouin zone. A qualitatively distinct behavior occurs for weak quenches where only weak wiggling occurs. Surprisingly, no evidence for prethermalization is found in the weak coupling regime. In both regimes, indications for relaxation beyond oscillatory or power law behavior are found and used to estimate relaxation rates without resorting to a probabilistic ansatz. The relaxation appears to be fastes…
We propose a new type of the Yang-Baxter equation (YBE) and the decorated Yang-Baxter equation (DYBE). Those relations for the fermionic R-operator were introduced recently as a tool to treat the integrability of the fermion models. Using the YBE and the DYBE for the XX fermion model, we construct the fermionic R-operator for the one-dimensional (1D) Hubbard model. It gives another proof of the integrability of the 1D Hubbard model. Furthermore a new approach to the SO(4) symmetry of the 1D Hubbard model is discussed.
We show that the dynamics of quenches in one dimension far off equilibrium can be described by power laws, but with exponents differing from the fully renormalized ones at lowest energies. Instead they depend on the initial state and its excitation energy. Furthermore, we found that for quenches to strong interactions unexpected similarities between systems in one and in infinite dimensions occur, indicating the dominance of local processes.
Using a Luttinger liquid theory we investigate the time evolution of the particle density of a one-dimensional fermionic system with open boundaries and subject to a finite duration quench of the inter-particle interaction. We provide analytical and asymptotic solutions to the unitary time evolution of the system, showing that both switching on and switching off the quench ramp create light-cone perturbations in the density. The post-quench dynamics is strongly affected by the interference between these two perturbations. In particular, we find that the discrepancy between the time-dependent d…
In this article, we derive the fermionic formalism of Hamiltonians as well as corresponding excitation spectrums and states of Calogero-Sutherland(CS), Laughlin and Halperin systems, respectively. In addition, we study the triangular property of these Hamiltonians and prove the integrability in these three cases.
We study the dynamics and thermalization of strongly correlated fermions in finite one-dimensional lattices after a quantum quench. Our calculations are performed using exact diagonalization. We focus on one- and two-body observables such as the momentum distribution function [n(k)] and the density-density structure factor [N(k)], respectively, and study the effects of approaching an integrable point. We show that while the relaxation dynamics and thermalization of N(k) for fermions is very similar to the one of hardcore bosons, the behavior of n(k) is distinctively different. The latter obser…
By means of full exact diagonalization, we study level statistics and the structure of the eigenvectors of one-dimensional gapless bosonic and fermionic systems across the transition from integrability to quantum chaos. These systems are integrable in the presence of only nearest-neighbor terms, whereas the addition of next-nearest neighbor hopping and interaction may lead to the onset of chaos. We show that the strength of the next-nearest neighbor terms required to observe clear signatures of nonintegrability is inversely proportional to the system size. Interestingly, the transition to chao…
Exact integrability and algebraic Bethe ansatz of the small-polaron model with the open boundary condition are discussed in the framework of the quantum inverse scattering method (QISM). We employ a new approach where the fermionic R-operator which consists of fermion operators is a key object. It satisfies the Yang-Baxter equation and the reflection equation with its corresponding K-operator. Two kinds of 'super-transposition' for the fermion operators are defined and the dual reflection equation is obtained. These equations prove the integrability and the Bethe ansatz equation which agrees w…
We consider the non-equilibrium physics induced by joining together two tight binding fermionic chains to form a single chain. Before being joined, each chain is in a many-fermion ground state. The fillings (densities) in the two chains might be the same or different. We present a number of exact results for the correlation functions in the non-interacting case. We present a short-time expansion, which can sometimes be fully resummed, and which reproduces the so-called `light cone' effect or wavefront behavior of the correlators. For large times, we show how all interesting physical regimes ma…
We implement numerical linked cluster expansions (NLCEs) to study dynamics of lattice systems following quantum quenches, and focus on a hard-core boson model in one-dimensional lattices. We find that, in the nonintegrable regime and within the accessible times, local observables exhibit exponential relaxation. We determine the relaxation rate as one departs from the integrable point and show that it scales quadratically with the strength of the integrability breaking perturbation. We compare the NLCE results with those from exact diagonalization calculations on finite chains with periodic bou…
The Sachdev-Ye-Kitaev (SYK) model attracts attention in the context of information scrambling, which represents delocalization of quantum information and is quantified by the out-of-time-ordered correlators (OTOC). The SYK model contains $N$ fermions with disordered and four-body interactions. Here, we introduce a variant of the SYK model, which we refer to as the Wishart SYK model. We investigate the Wishart SYK model for complex fermions and that for hard-core bosons. We show that the ground state of the Wishart SYK model is massively degenerate and the residual entropy is extensive, and tha…
Quantum quenches display universal scaling in several regimes. For quenches which start from a gapped phase and cross a critical point, with a rate slow compared to the initial gap, many systems obey Kibble-Zurek scaling. More recently, a different scaling behaviour has been shown to occur when the quench rate is fast compared to all other physical scales, but still slow compared to the UV cutoff. We investigate the passage from fast to slow quenches in scalar and fermionic free field theories with time dependent masses for which the dynamics can be solved exactly for all quench rates. We find…
We study the off-equilibrium dynamics of the infinite dimensional Bose Hubbard Model after a quantum quench. The dynamics can be analyzed exactly by mapping it to an effective Newtonian evolution. For integer filling, we find a dynamical transition separating regimes of small and large quantum quenches starting from the superfluid state. This transition is very similar to the one found for the fermionic Hubbard model by mean field approximations.
We consider the non-equilibrium evolution in the spin-1/2 XXZ Heisenberg chain for fixed magnetization after a local quantum quench. This model is equivalent to interacting spinless fermions. Initially an infinite magnetic field is applied to n consecutive sites and the ground state is calculated. At time t=0 the field is switched off and the time evolution of observables such as the z-component of spin is computed using the Time Evolving Block Decimation (TEBD) algorithm. We find that the observables exhibit strong signatures of linearly propagating spinon and bound state excitations. These p…
We investigate the dynamics following sudden quenches across quantum critical points belonging to different universality classes. Specifically, we use matrix product state methods to study the quantum Ising chain in the presence of two additional terms which break integrability. We find that in all models the rate function for the return probability to the initial state becomes a non-analytic function of time in the thermodynamic limit. This so-called `dynamical phase transition' was first observed in a recent work by Heyl, Polkovnikov, and Kehrein [Phys. Rev. Lett. 110, 135704 (2013)] for the…
The transport dynamics of a quenched Luttinger liquid tunnel-coupled to a fermionic reservoir is investigated. In the transient dynamics, we show that for a sudden quench of the electron interaction universal power-law decay in time of the tunneling current occurs, ascribed to the presence of entangled compound excitations created by the quench. In sharp contrast to the usual non universal power-law behavior of a zero-temperature non-quenched Luttinger liquid, the steady state tunneling current is ohmic and can be explained in terms of an effective quench-activated heating of the system. Our s…
We explore supersymmetric quantum quenches of the mass and coupling constants in the $\mathcal{N}=1$ supersymmetric vector model using Hartee-Fock approximation. We find that in the case of the free fermionic field, quench of the mass generates singularity localized at the instant of sudden change and we point out the essential role of the parity of spacetime. Focusing on a supersymmetric generalization of the $ϕ^6$ model and using stationary phase approximation, we demonstrate that supersymmetry is broken in the asymptotic state that emerges at late times after the quench. Finally, we confirm…
We study the non equilibrium dynamics in the fermionic Hubbard model after a sudden change of the interaction strength. To this scope, we introduce a time dependent variational approach in the spirit of the Gutzwiller ansatz. At the saddle-point approximation, we find at half filling a sharp transition between two different regimes of small and large coherent oscillations, separated by a critical line of quenches where the system is found to relax. Any finite doping washes out the transition, leaving aside just a sharp crossover. In order to investigate the role of quantum fluctuations, we map…
We use the variational quantum eigensolver (VQE) to simulate Kitaev spin models with and without integrability breaking perturbations, focusing in particular on the honeycomb and square-octagon lattices. These models are well known for being exactly solvable in a certain parameter regime via a mapping to free fermions. We use classical simulations to explore a novel variational ansatz that takes advantage of this fermionic representation and is capable of expressing the exact ground state in the solvable limit. We also demonstrate that this ansatz can be extended beyond this limit to provide e…
We consider a problem of non-adiabatic dynamics of a 2D fermionic system with $d+id$-wave symmetry of paring amplitude. Under the mean-field approximation, we determine the asymptotic behavior of the pairing amplitude following a sudden change of coupling strength. We also study an extended $d+id$ pairing system for which the long-time asymptotic states of the pairing amplitude in the collisionless regime can be determined exactly. By using numerical methods, we have identified three non-equilibrium steady states described by different long-time asymptotes of the pairing amplitude for both the…
We study inhomogeneous quantum quenches in the attractive regime of the sine-Gordon model. In our protocol, the system is prepared in an inhomogeneous initial state in finite volume by coupling the topological charge density operator to a Gaussian external field. After switching off the external field, the subsequent time evolution is governed by the homogeneous sine-Gordon Hamiltonian. Varying either the interaction strength of the sine-Gordon model or the amplitude of the external source field, an interesting transition is observed in the expectation value of the soliton density. This affect…
We investigate a class of exactly solvable quantum quench protocols with a finite quench rate in systems of one dimensional non-relativistic fermions in external harmonic oscillator or inverted harmonic oscillator potentials, with time dependent masses and frequencies. These hamiltonians arise, respectively, in harmonic traps, and the $c=1$ Matrix Model description of two dimensional string theory with time dependent string coupling. We show how the dynamics is determined by a single function of time which satisfies a generalized Ermakov-Pinney equation. The quench protocols we consider asympt…
We present a comparison between the bosonization results for quantum quenches and exact diagonalizations in microscopic models of interacting spinless fermions in a one-dimensional lattice. We show that important features are missed by the bosonization technique, which predicts the persistence of long-wavelength critical properties in the long-time evolution. Instead, numerical analysis provides puzzling evidences: while the momentum distribution appears to be consistent with the presence of a singularity at $k_F$, density-density correlations at small momenta clearly display a thermal-like be…
We investigate the growth of the entanglement entropy $S_{\textrm{ent}}$ following global quenches in two-dimensional free fermion models with potential and bond disorder. For the potential disorder case we show that an intermediate weak localization regime exists in which $S_{\textrm{ent}}(t)$ grows logarithmically in time $t$ before Anderson localization sets in. For the case of binary bond disorder near the percolation transition we find additive logarithmic corrections to area and volume laws as well as a scaling at long times which is consistent with an infinite randomness fixed point.
Real-time dynamics in a quantum many-body system are inherently complicated and hence difficult to predict. There are, however, a special set of systems where these dynamics are theoretically tractable: integrable models. Such models possess non-trivial conserved quantities beyond energy and momentum. These quantities are believed to control dynamics and thermalization in low dimensional atomic gases as well as in quantum spin chains. But what happens when the special symmetries leading to the existence of the extra conserved quantities are broken? Is there any memory of the quantities if the …
We develop a generic method to compute the dynamics induced by quenches in completely connected quantum systems. These models are expected to provide a mean-field description at least of the short time dynamics of finite dimensional system. We apply our method to the Bose-Hubbard model, to a generalized Jaynes-Cummings model, and to the Ising model in a transverse field. We find that the quantum evolution can be mapped onto a classical effective dynamics, which involves only a few intensive observables. For some special parameters of the quench, peculiar dynamical transitions occur. They resul…
We employ the mean-field approach in the fermionic picture of the spin-1/2 XXZ chain to investigate the dynamics of bipartite quantum discord and concurrence under sudden quenching. In the case, when quenching is performed in the anisotropy from an initial value to the critical point, the quantum correlations show periodic cusps in time. Moreover, the first suppression (cusp) in quantum correlations can be explained in terms of the semi-classical picture of quasiparticle propagation. On the other hand, quenching to, as well as away from the criticality point shows that the longtime pairwise qu…
The time evolution of the entanglement entropy in non-equilibrium quantum systems provides crucial information about the structure of the time-dependent state. For quantum quench protocols, by combining a quasiparticle picture for the entanglement spreading with the exact knowledge of the stationary state provided by Bethe ansatz, it is possible to obtain an exact and analytic description of the evolution of the entanglement entropy. Here we discuss the application of these ideas to several integrable models. First we show that for non-interacting systems, both bosonic and fermionic, the exact…
We study the effect of quenched disorder on the semimetal-superconductor quantum phase transition in a model of two-dimensional Dirac semimetal with $N$ flavors of two-component Dirac fermions, using perturbative renormalization group methods at one-loop order in a double epsilon expansion. For $N\geq 2$ we find that the Harris-stable clean critical behavior gives way, past a certain critical disorder strength, to a finite-disorder critical point characterized by non-Gaussian critical exponents, a noninteger dynamic critical exponent $z>1$, and a finite Yukawa coupling between Dirac fermions a…
Prethermal cooling with many-body quantum quenches
Institute for Quantum Information and Matter, California Institute of Technology（加州理工学院量子信息物质研究所）；Joint Quantum Institute, University of Maryland（马里兰大学联合量子研究所）；Institute for Theoretical Physics, Georg-August-Universität Göttingen（哥廷根大学理论物理研究所）
Non-analyticities in the logarithm of the Loschmidt echo, known as dynamical quantum phase transitions [DQPTs], are a recently introduced attempt to classify the myriad of possible phenomena which can occur in far from equilibrium closed quantum systems. In this work, we analytically investigate the Loschmidt echo in nonequilibrium $s$-wave and topological $p_x+ip_y$ fermionic superfluids. We find that the presence of non-analyticities in the echo is not invariant under global rotations of the superfluid phase. We remedy this deficiency by introducing a more general notion of a grand canonical…
Entanglement in a pure state of a many-body system can be characterized by the Rényi entropies $S^{(α)}=\ln\textrm{tr}(ρ^α)/(1-α)$ of the reduced density matrix $ρ$ of a subsystem. These entropies are, however, difficult to access experimentally and can typically be determined for small systems only. Here we show that for free fermionic systems in a Gaussian state and with particle number conservation, $\ln S^{(2)}$ can be tightly bound by the much easier accessible Rényi number entropy $S^{(2)}_N=-\ln \sum_n p^2(n)$ which is a function of the probability distribution $p(n)$ of the total parti…
The Lieb-Robinson bound implies that the unitary time evolution of an operator can be restricted to an effective light cone for any Hamiltonian with short-range interactions. Here we present a very efficient renormalization group algorithm based on this light cone structure to study the time evolution of prepared initial states in the thermodynamic limit in one-dimensional quantum systems. The algorithm does not require translational invariance and allows for an easy implementation of local conservation laws. We use the algorithm to investigate the relaxation dynamics of double occupancies in …
Motivated by recent experiments in ultracold atomic gases that explore the nonequilibrium dynamics of interacting quantum many-body systems, we investigate the nonequilibrium properties of a Fermi liquid. We apply an interaction quench within the Fermi liquid phase of the Hubbard model by switching on a weak interaction suddenly; then we follow the real-time dynamics of the momentum distribution by a systematic expansion in the interaction strength based on the flow equation method. In this paper we derive our main results, namely the applicability of a quasiparticle description, the observati…
