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Register a new userQuantum Quenches to a Critical Point in One Dimension: some further results
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John Cardy
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We describe several results concerning global quantum quenches from states with short-range correlations to quantum critical points whose low-energy properties are described by a 1+1-dimensional conformal field theory (CFT), extending the work of Calabrese and Cardy (2006): (a) for the special class of initial states discussed in that paper we show that, once a finite region falls inside the horizon, its reduced density matrix is exponentially close in $L_2$ norm to that of a thermal Gibbs state; (b) small deformations of this initial state in general lead to a (non-Abelian) generalized Gibbs distribution (GGE) with, however, the possibility of parafermionic conserved charges; (c) small deformations of the CFT, corresponding to curvature of the dispersion relation and (non-integrable) left-right scattering, lead to a dependence of the speed of propagation on the initial state, as well as diffusive broadening of the horizon.
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In a recent letter, J. Cardy, Phys. Rev. Lett. textbf{112}, 220401 (2014), the author made a very interesting observation that complete revivals of quantum states after quantum quench can happen in a period which is a fraction of the system size. This is possible for critical systems that can be described by minimal conformal field theories (CFT) with central charge $c<1$. In this article, we show that these complete revivals are impossible in microscopic realizations of those minimal models. We will prove the absence of the mentioned complete revivals for the critical transverse field Ising chain analytically, and present numerical results for the critical line of the XY chain. In particular, for the considered initial states, we will show that criticality has no significant effect in partial revivals. We also comment on the applicability of quasi-particle picture to determine the period of the partial revivals qualitatively. In particular, we detect a regime in the phase diagram of the XY chain which one can not determine the period of the partial revivals using the quasi-particle picture.
We perform a numerical study of a spin-1/2 model with $mathbb{Z}_2 times mathbb{Z}_2$ symmetry in one dimension which demonstrates an interesting similarity to the physics of two-dimensional deconfined quantum critical points (DQCP). Specifically, we investigate the quantum phase transition between Ising ferromagnetic and valence bond solid (VBS) symmetry-breaking phases. Working directly in the thermodynamic limit using uniform matrix product states, we find evidence for a direct continuous phase transition that lies outside of the Landau-Ginzburg-Wilson paradigm. In our model, the continuous transition is found everywhere on the phase boundary. We find that the magnetic and VBS correlations show very close power law exponents, which is expected from the self-duality of the parton description of this DQCP. Critical exponents vary continuously along the phase boundary in a manner consistent with the predictions of the field theory for this transition. We also find a regime where the phase boundary splits, as suggested by the theory, introducing an intermediate phase of coexisting ferromagnetic and VBS order parameters. Interestingly, we discover a transition involving this coexistence phase which is similar to the DQCP, being also disallowed by Landau-Ginzburg-Wilson symmetry-breaking theory.
We study in general the time-evolution of correlation functions in a extended quantum system after the quench of a parameter in the hamiltonian. We show that correlation functions in d dimensions can be extracted using methods of boundary critical phenomena in d+1 dimensions. For d=1 this allows to use the powerful tools of conformal field theory in the case of critical evolution. Several results are obtained in generic dimension in the gaussian (mean-field) approximation. These predictions are checked against the real-time evolution of some solvable models that allows also to understand which features are valid beyond the critical evolution. All our findings may be explained in terms of a picture generally valid, whereby quasiparticles, entangled over regions of the order of the correlation length in the initial state, then propagate with a finite speed through the system. Furthermore we show that the long-time results can be interpreted in terms of a generalized Gibbs ensemble. We discuss some open questions and possible future developments.
Ferromagnetism in one dimension is a novel observation which has been reported in a recent work (P. Gambardella et.al., Nature {bf 416}, 301 (2002)), anisotropies are responsibles in that relevant effect. In the present work, another approach is used to obtain transition between two different magnetic ordering phases. Critical temperature has been estimated by Binder method. Ferromagnetic long range interactions have been included in a special Hamiltonian through a power law that decays at large inter-particle distance $r$ as $r^{-alpha}$, where $alphageq0$. For the present model, we have found that the trend of the critical temperature vanishes when the range of interactions decreases ($alphatoinfty$) and close to mean field approximation when the range of interactions increases ($alphato0$). The crossover between two these limit situations is discussed
We study the unitary time evolution of the order parameter of a quantum system after a sudden quench in the parameter driving the transition. By mapping the dynamics onto the imaginary time path-integral in a film geometry we derive the full mean-field non-equilibrium phase diagram for a one-component order parameter. The recently discovered non-equilibrium transition is identified with the shifted critical point in films and therefore it is generally expected to occur in more than one spatial dimension. We also find that anharmonic oscillations of the order parameter are a general feature of the mean-field quench dynamics.
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