Non-equilibrium dynamics of the Bose-Hubbard model: A projection operator approach

We study the phase diagram and non-equilibrium dynamics, both subsequent to a sudden quench of the hopping amplitude $J$ and during a ramp $J(t)=Jt/\tau$ with ramp time $\tau$, of the Bose-Hubbard model at zero temperature using a projection operator formalism which allows us to incorporate the effects of quantum fluctuations beyond mean-field approximations in the strong coupling regime. Our formalism yields a phase diagram which provides a near exact match with quantum Monte Carlo results in three dimensions. We also compute the residual energy $Q$, the superfluid order parameter $\Delta(t)$, the equal-time order parameter correlation function $C(t)$, and the wavefunction overlap $F$ which yields the defect formation probability $P$ during non-equilibrium dynamics of the model. 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 compare well with recent experiments.Comment: Replaced with the accepted version, added one figure. 4 pages, 4 figures, to appear in Phys. Rev. Let

A projection operator approach to the Bose-Hubbard model

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, which constitutes an extension of Phys. Rev. Lett. {\bf 106}, 095702 (2011), 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 $z_0$ of the lattice and reaches to within 0.5% of quantum Monte Carlo data for lattices with $z_0=6$. 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 $z_0J/U \ll 1$. 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 $J_i$ to $J_f$ characterized by a ramp time $\tau$ and exponent $\alpha$: $J(t)=J_i +(J_f-J_i) (t/\tau)^{\alpha}$. We compute the wavefunction overlap $F$, the residual energy $Q$, the superfluid order parameter $\Delta(t)$, 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.Comment: v2; new references and new sections adde

Slow quench dynamics of the Kitaev model: anisotropic critical point and effect of disorder

We study the non-equilibrium slow dynamics for the Kitaev model both in the presence and the absence of disorder. For the case without disorder, we demonstrate, via an exact solution, that the model provides an example of a system with an anisotropic critical point and exhibits unusual scaling of defect density $n$ and residual energy $Q$ for a slow linear quench. We provide a general expression for the scaling of $n$ ($Q$) generated during a slow power-law dynamics, characterized by a rate $\tau^{-1}$ and exponent $\alpha$, from a gapped phase to an anisotropic quantum critical point in $d$ dimensions, for which the energy gap $\Delta_{\vec k} \sim k_i^z$ for $m$ momentum components ($i=1..m$) and $\sim k_i^{z'}$ for the rest $d-m$ components ($i=m+1..d$) with $z\le z'$: $n \sim \tau^{-[m + (d-m)z/z']\nu \alpha/(z\nu \alpha +1)}$ ($Q \sim \tau^{-[(m+z)+ (d-m)z/z']\nu \alpha/(z\nu \alpha +1)}$). These general expressions reproduce both the corresponding results for the Kitaev model as a special case for $d=z'=2$ and $m=z=\nu=1$ and the well-known scaling laws of $n$ and $Q$ for isotropic critical points for $z=z'$. We also present an exact computation of all non-zero, independent, multispin correlation functions of the Kitaev model for such a quench and discuss their spatial dependence. For the disordered Kitaev model, where the disorder is introduced via random choice of the link variables $D_n$ in the model's Fermionic representation, we find that $n \sim \tau^{-1/2}$ and $Q\sim \tau^{-1}$ ($Q\sim \tau^{-1/2}$) for a slow linear quench ending in the gapless (gapped) phase. We provide a qualitative explanation of such scaling.Comment: 10 pages, 11 Figs. v

Semiclassical Spectrum of Small Bose-Hubbard Chains: A Normal Form Approach

We analyze the spectrum of the 3-site Bose-Hubbard model with periodic boundary conditions using a semiclassical method. The Bohr-Sommerfeld quantization is applied to an effective classical Hamiltonian which we derive using resonance normal form theory. The derivation takes into account the 1:1 resonance between frequencies of a linearized classical system, and brings nonlinear terms into a corresponding normal form. The obtained expressions reproduce the exact low-energy spectrum of the system remarkably well even for a small number of particles N corresponding to fillings of just two particles per site. Such small fillings are often used in current experiments, and it is inspiring to get insight into this quantum regime using essentially classical calculations.Comment: Minor corrections to the coefficients of the effective Hamiltonian in Eqs 14,15,18,19. Figs 1,2 are slightly modified, correspondingl

Where does the gas fueling star formation in BCGs originate?

We investigate the relationship between X-ray cooling and star formation in brightest cluster galaxies (BCGs). We present an X-ray spectral analysis of the inner regions, 10-40 kpc, of six nearby cool core clusters (z<0.35) observed with Chandra ACIS. This sample is selected on the basis of the high star formation rate (SFR) observed in the BCGs. We restrict our search for cooling gas to regions that are roughly cospatial with the starburst. We fit single- and multi-temperature mkcflow models to constrain the amount of isobarically cooling intracluster medium (ICM). We find that in all clusters, below a threshold temperature ranging between 0.9 and 3 keV, only upper limits can be obtained. In four out of six objects, the upper limits are significantly below the SFR and in two, namely A1835 and A1068, they are less than a tenth of the SFR. Our results suggests that a number of mechanisms conspire to hide the cooling signature in our spectra. In a few systems the lack of a cooling signature may be attributed to a relatively long delay time between the X-ray cooling and the star burst. However, for A1835 and A1068, where the X-ray cooling time is shorter than the timescale of the starburst, a possible explanation is that the region where gas cools out of the X-ray phase extends to very large radii, likely beyond the core of these systems.Comment: to appear in A&