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arXiv:0901.1384 · Journal of Statistical Mechanics Theory and Experiment

Adiabatic dynamics in a spin-1 chain with uniaxial single-spin anisotropy

Elena Canovi, Davide Rossini, Rosario Fazio, Giuseppe E. Santoro

Published Mar 30, 2009• 12 citations
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Abstract

We study the adiabatic quantum dynamics of an anisotropic spin-1 XY chain across a second order quantum phase transition. The system is driven out of equilibrium by performing a quench on the uniaxial single-spin anisotropy, that is supposed to vary linearly in time. We show that, for sufficiently large system sizes, the excess energy after the quench admits a non trivial scaling behavior that is not predictable by standard Kibble-Zurek arguments for isolated critical points or extended critical regions. This emerges from a competing effect of many accessible low-lying excited states, inside the whole continuous line of critical points.

DOI
10.1088/1742-5468/2009/03/p03038

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We study the adiabatic quantum dynamics of an anisotropic spin-1 XY chain across a second order quantum phase transition. The system is driven out of equilibrium by performing a quench on the uniaxial single-spin anisotropy, that is supposed to vary linearly in time. We show that, for sufficiently large system sizes, the excess energy after the quench admits a non trivial scaling behavior that is not predictable by standard Kibble-Zurek arguments for isolated critical points or extended critical regions. This emerges from a competing effect of many accessible low-lying excited states, inside the whole continuous line of critical points.
It is well known that the dynamics of a quantum system is always nonadiabatic in passage through a quantum critical point and the defect density in the final state following a quench shows a power-law scaling with the rate of quenching. However, we propose here a possible situation where the dynamics of a quantum system in passage across quantum critical regions is adiabatic and the defect density decays exponentially. This is achieved by incorporating additional interactions which lead to quantum critical behavior and gapless phases but do not participate in the time evolution of the system. To illustrate the general argument, we study the defect generation in the quantum critical dynamics of a spin-1/2 anisotropic quantum XY spin chain with three spin interactions and a linearly driven staggered magnetic field.
We investigate the emergence of universal dynamical scaling in quantum critical spin systems adiabatically driven out of equilibrium, with emphasis on quench dynamics which involves non-isolated critical points ( i.e. , critical regions) and cannot be a priori described through standard scaling arguments nor time-dependent perturbative approaches. Comparing to the case of an isolated quantum critical point, we find that non-equilibrium scaling behavior of a large class of physical observables may still be explained in terms of equilibrium critical exponents. However, the latter are in general non-trivially path-dependent, and detailed knowledge about the time-dependent excitation process becomes essential. In particular, we show how multiple level crossings within a gapless phase may completely suppress excitation depending on the control path. Our results typify non-ergodic scaling in continuous finite-order quantum phase transitions.
We explore the robustness of universal dynamical scaling behavior in a quantum system near criticality with respect to initialization in a large class of states with finite energy. By focusing on a homogeneous XY\mathit{XY}XY quantum spin chain in a transverse field, we characterize the nonequilibrium response under adiabatic and sudden quench processes originating from a pure as well as a mixed excited initial state, and involving either a regular quantum critical or a multicritical point. We find that the critical exponents of the ground-state quantum phase transition can be encoded in the dynamical scaling exponents despite the finite energy of the initial state. In particular, we identify conditions on the initial distribution of quasiparticle excitation that ensure that Kibble-Zurek scaling persists. The emergence of effective thermal equilibrium behavior following a sudden quench toward criticality is also investigated, with focus on the long-time expectation value of the quasiparticle number operator. Despite the integrability of the XY\mathit{XY}XY model, this observable is found to behave thermally in quenches to a regular quantum critical point, provided that the system is initially prepared at sufficiently high temperature. However, a similar thermalization behavior fails to occur in quenches toward a multicritical point. We argue that the observed lack of thermalization originates in this case in the asymmetry of the impulse region that is also responsible for anomalous multicritical dynamical scaling.
By gradually changing the degree of the anisotropy in an XXZXXZXXZ chain, we study the defect formation in a quantum system that crosses an extended critical region. We discuss two qualitatively different cases of quenches, from the antiferromagnetic to the ferromagnetic phase and from the critical to the antiferromagnetic phase. By means of time-dependent density matrix renormalization group simulations, we calculate the residual energy at the end of the quench as a characteristic quantity gauging the loss of adiabaticity. We find the dynamical scalings of the residual energy for both types of quenches, and compare them to the predictions of the Kibble-Zurek and Landau-Zener theories.
We consider an inhomogeneous quantum phase transition across a multicritical point of the XY quantum spin chain. This is an example of a Lifshitz transition with a dynamical exponent z =2. Just like in the case z =1 considered by Dziarmaga and Rams (2010 New J. Phys. 12 055007), when a critical front propagates much faster than the maximal group velocity of quasiparticles v q , then the transition is effectively homogeneous: the density of excitations obeys a generalized Kibble–Zurek mechanism and scales with the sixth root of the transition rate. However, unlike for the case z =1, the inhomogeneous transition becomes adiabatic not below v q but at a lower threshold velocity , proportional to the inhomogeneity of the transition, where the excitations are suppressed exponentially. Interestingly, the adiabatic threshold is nonzero despite the vanishing minimal group velocity of low-energy quasiparticles. In the adiabatic regime below , the inhomogeneous transition can be used for efficient adiabatic quantum state preparation in a quantum simulator: the time required for the critical front to sweep across a chain of N spins adiabatically is merely linear in N , while the corresponding time for a homogeneous transition across the multicritical point scales with the sixth power of N . What is more, excitations after the adiabatic inhomogeneous transition, if any, are brushed away by the critical front to the end of the spin chain.
We analyze mechanisms for universal out-of-equilibrium dynamics near criticality by exploring the effect of randomized quantum resetting (QR) under a finite-time quench across a quantum phase transition. Using the transverse-field Ising chain as a generic model and exploiting its exact solution, QR is found to cause a crossover of the scaling of the topological defect density with the time scale τττ of the quench, from Kibble-Zurek to anti-Kibble-Zurek scaling as τττ increases. This reflects a competition between non-adiabatic quench-driven excitations and QR, giving rise to local minima of the defect densities at optimal annealing times. These times and the corresponding local minima are shown to scale as universal power laws with the rate of QR. Additional results for the scaling of the mean excess energy suggest that a system driven across a quantum critical point exhibits the same scaling behavior under a linear quench with QR as with uncorrelated noise.