Dynamics and Phases of Strongly Correlated Systems

Abstract

Non-equilibrium dynamics of an isolated quantum system driven through a quantum critical point shows Kibble-Zurek scaling. This scaling form is controlled by the critical exponents of the universality class of the quantum phase transition. We develop a projection operator formalism for studying both the zero temperature equilibrium phase diagram and the non-equilibrium dynamics of the Bose-Hubbard model. Our work shows that the method provides an accurate description of the equilibrium zero temperature phase diagram of the Bose-Hubbard model for several lattices in two- and three-dimensions (2D and 3D).We show that the accuracy of this method increases with the coordination number z0 of the lattice and reaches to within 0:5% of quantum Monte Carlo data for lattices with z0 = 6. We compute the excitation spectra of the bosons using this method in the Mott and the superfluid phases and compare our results with mean-field theory. We also show that the same method may be used to analyze the non-equilibrium dynamics of the model both in the Mott phase and near the superfluid-insulator quantum critical point where the hopping amplitude J and the on-site interaction U satisfy z0J=U 1. In particular, we study the non-equilibrium dynamics of the model, both subsequent to a sudden quench of the hopping amplitude J and during a ramp from Ji to Jf characterized by a ramp time t and exponent a: J(t) = Ji +(Jf Ji)(t=t)a. We compute the wave function overlap F, the residual energy Q, the superfluid order parameter D(t), the equal-time order parameter correlation function C(t), and the defect formation probability P for the above-mentioned protocols and provide a comparison of our results to their mean-field counterparts. We find that Q, F, and P do not exhibit the expected universal scaling. We explain this absence of universality and show that our results for linear ramps compare well with the recent experimental observations. We have generalized our above mentioned work to develop a time-dependent hopping expansion technique for studying the non-equilibrium dynamics of strongly interacting bosons in an optical lattice in the presence of a harmonic trap characterized by a force constant K. We show that after a sudden quench of the hopping amplitude J across the superfluid (SF)-Mott insulator(MI) transition, the SF order parameter jDr(t)j and the local density fluctuation dnr(t) exhibit sudden decoherence beyond a trap-induced time scale T0 K 1=2. We also show that after a slow linear ramp down of J, jDrj and the boson defect density Pr display a novel non-monotonic spatial profile. Both these phenomena can be explained as consequences of trap-induced time and length scales affecting the dynamics and can be tested by concrete experiments. We also study the statistics of the work distribution P(w) in a d dimensional closed quantum system with linear dimension L subjected to a periodic drive with frequency w0. We show that the corresponding rate function I(w) = ln[P(w)=L0]=Ld after a drive period satisfies an universal lower bound I(0) nd and has a zero at w = QLd=N, where nd and Q are the excitation and the residual energy densities generated during the drive, L0 is a constant fixed by the normalization of P(w), and N is the total number of constituent particles/spins in the system. We supplement our results by calculating I(w) for a class of d-dimensional integrable models and show that I(w) has oscillatory dependence on w0 originating from Stuckelberg interference generated due to double passage through critical point/region during the drive. We suggest experiments to test our theory.Research was conducted under the supervision of Prof. Krishnendu Sengupta of the Theoretical Physics division under the SPS [School of Physical Sciences]Research was carried out under IACS fellowship and gran