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arXiv:cond-mat/0003193 · Physical Review Letters

Long-Range Correlations in the Nonequilibrium Quantum Relaxation of a Spin Chain

Ferenc Iglói, Heiko Rieger

Published Oct 9, 2000• 162 citations
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Abstract

We consider the nonstationary quantum relaxation of the Ising spin chain in a transverse field of strength h. Starting from a homogeneously magnetized initial state the system approaches a stationary state by a process possessing quasi-long-range correlations in time and space, independent of the value of h. In particular, the system exhibits aging (or lack of time-translational invariance on intermediate time scales) although no indications of coarsening are present.

DOI
10.1103/physrevlett.85.3233

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We consider the nonstationary quantum relaxation of the Ising spin chain in a transverse field of strength h. Starting from a homogeneously magnetized initial state the system approaches a stationary state by a process possessing quasi-long-range correlations in time and space, independent of the value of h. In particular, the system exhibits aging (or lack of time-translational invariance on intermediate time scales) although no indications of coarsening are present.
The nonthermal quantum relaxation of the magnetization under nonequilibrium initial conditions is studied for the transverse Ising chain and for the one-dimensional isotropic XY model. In the absence of a heat bath the transverse Ising system exhibits an oscillating relaxation with a decaying amplitude. At the critical transverse field some stationary properties become independent of the initial state. The isotropic XY chain with a spatially inhomogeneous initial state shows a slow, modulated algebraic decay of the magnetization. Analysis of the two-time autocorrelation function reveals the phenomenon of aging.
We consider the ddd-dimensional transverse-field Ising model with power-law interactions $J/{r}^{d+\ensuremath{\sigma}}$ in the presence of a noisy longitudinal field with zero average. We study the longitudinal-magnetization dynamics of an initial paramagnetic state after a sudden switch-on of both the interactions and the noisy field. While the system eventually relaxes to an infinite-temperature state with vanishing magnetization correlations, we find that two-time correlation functions show aging at intermediate times. Moreover, for times shorter than the inverse noise strength $\ensuremath{\kappa}$ and distances longer than $a{(J/\ensuremath{\kappa})}^{2/\ensuremath{\sigma}}$ with aaa being the lattice spacing, we find a critical scaling regime of correlation and response functions consistent with the model A dynamical universality class with an initial-slip exponent $\ensuremath{\theta}=1$ and dynamical critical exponent $z=\ensuremath{\sigma}/2$. We obtain our results analytically by deriving an effective action for the magnetization field including the noise in a nonperturbative way. The above scaling regime is governed by a nonequilibrium fixed point dominated by the noise fluctuations.
Abstract Nonequilibrium dynamics in an isolated quantum spin chain with long-range Ising interactions that decay as ( ) with the distance r is studied. It turns out that long-range interactions give rise to a big timescale separation, which causes prethermalization for all . This conclusion is deduced by comparing two important timescales relevant for relaxation dynamics; one is the relaxation time of local permutation operators, which are quasi-conserved quantities in this system, and the other is the timescale of the initial relaxation of local quantities due to the growth of quantum fluctuations. We also explore the entire nonequilibrium dynamics by using the discrete truncated Wigner approximation, which is consistent with the result mentioned above.
We study the relaxation behaviour of the quantum Ising chain, focusing our attention onto the non-equilibrium dynamics of the transverse magnetization. The initial states, from which the magnetization relaxes, are product states such as those generated by setting in contact several systems, each initially equilibrated at a given temperature. Due to the free fermionic structure of the chain, the dynamics of the transverse magnetization is easily expressed in a compact form. In the completely factorized initial state, corresponding to a situation where all the spins are thermalized independently, we obtain in the scaling limit the Green function associated to the transverse magnetization. The relaxation behaviour is also considered in the system-bath case, where part of the chain called the system is thermalized at a temperature Ts and the remaining part is at a temperature Tb. The magnetization profiles show a scaling behaviour. Moreover, in the extreme case Tb=∞ and Ts=0, it is shown that the magnetization relaxes in quantized steps in the strong transverse field region.
Although statistical mechanics describes thermal equilibrium states, these states may or may not emerge dynamically for a subsystem of an isolated quantum many-body system. For instance, quantum systems that are near-integrable usually fail to thermalize in an experimentally realistic time scale, and instead relax to quasi-stationary prethermal states that can be described by statistical mechanics, when approximately conserved quantities are included in a generalized Gibbs ensemble (GGE). We experimentally study the relaxation dynamics of a chain of up to 22 spins evolving under a long-range transverse-field Ising Hamiltonian following a sudden quench. For sufficiently long-range interactions, the system relaxes to a new type of prethermal state that retains a strong memory of the initial conditions. However, the prethermal state in this case cannot be described by a standard GGE; it rather arises from an emergent double-well potential felt by the spin excitations. This result shows that prethermalization occurs in a broader context than previously thought, and reveals new challenges for a generic understanding of the thermalization of quantum systems, particularly in the presence of long-range interactions.
We study the dynamics caused by transport of transverse magnetization in one dimensional transverse Ising chain at zero temperature. We observe that a class of initial states in fermionic momentum-space produce spatial variation in transverse magnetization. Starting from such a state, we obtain the transverse magnetization analytically and then observe its dynamics in presence of a homogeneous constant field Γ\GammaΓ. In contradiction with general expectation, whatever be the strength of the field, the magnetization of the system does not become homogeneous even after infinite time. In each site, the dynamics is associated with oscillations having two different timescales. The envelope of the larger timescale oscillation decays algebraically having an exponent which is universal for the entire class of such initial states. Signature of quantum criticality is also embedded in the dynamics in two aspects : Behavioral change shown by the frequency of large timescale oscillation at any site and the corresponding magnetization at infinity when they cross the quantum critical point.