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arXiv:2106.12648 · Journal of Physics A Mathematical and Theoretical

Circuit complexity near critical points

Uday Sood, Martı́n Kruczenski

Published Mar 8, 2022• 10 citations
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Abstract

Abstract We consider the Bose–Hubbard model in two and three spatial dimensions and numerically compute the quantum circuit complexity of the ground state in the Mott insulator and superfluid phases using a mean field approximation with additional quadratic fluctuations. After mapping to a qubit system, the result is given by the complexity associated with a Bogoliubov transformation applied to the reference state taken to be the mean field ground state. In particular, the complexity has peaks at the O (2) critical points where the system can be described by a relativistic quantum field theory. Given that we use a Gaussian approximation, near criticality the numerical results agree with a free field theory calculation. To go beyond the Gaussian approximation we use general scaling arguments that imply that, as we approach the critical point t → t c , there is a non-analytic behavior in the complexity c 2 ( t ) of the form | c 2 ( t ) − c 2 ( t c )| ∼ | t − t c | νd , up to possible logarithmic corrections. Here d is the number of spatial dimensions and ν is the usual critical exponent for the correlation length ξ ∼ | t − t c | − ν . As a check, for d = 2 this agrees with the numerical computation if we use the Gaussian critical exponent ν = 1 2 . Finally, using AdS/CFT methods, we study higher dimensional examples and confirm this scaling argument with non-Gaussian exponent ν for strongly interacting theories that have a gravity dual.

DOI
10.1088/1751-8121/ac5b8f

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Abstract We consider the Bose–Hubbard model in two and three spatial dimensions and numerically compute the quantum circuit complexity of the ground state in the Mott insulator and superfluid phases using a mean field approximation with additional quadratic fluctuations. After mapping to a qubit system, the result is given by the complexity associated with a Bogoliubov transformation applied to the reference state taken to be the mean field ground state. In particular, the complexity has peaks at the O (2) critical points where the system can be described by a relativistic quantum field theory. Given that we use a Gaussian approximation, near criticality the numerical results agree with a free field theory calculation. To go beyond the Gaussian approximation we use general scaling arguments that imply that, as we approach the critical point t → t c , there is a non-analytic behavior in the complexity c 2 ( t ) of the form | c 2 ( t ) − c 2 ( t c )| ∼ | t − t c | νd , up to possible logarithmic corrections. Here d is the number of spatial dimensions and ν is the usual critical exponent for the correlation length ξ ∼ | t − t c | − ν . As a check, for d = 2 this agrees with the numerical computation if we use the Gaussian critical exponent ν = 1 2 . Finally, using AdS/CFT methods, we study higher dimensional examples and confirm this scaling argument with non-Gaussian exponent ν for strongly interacting theories that have a gravity dual.
Abstract The region near a critical point is studied using holographic models of second-order phase transitions. In a previous paper, we argued that the quantum circuit complexity of the vacuum (C 0) is the largest at the critical point. When deforming away from the critical point by a term the complexity C ( τ ) has a piece non-analytic in τ, namely C 0 − C ( τ ) ∼ | τ − τ c | ν ( d − 1 ) + a n a l y t i c . Here, as usual, ν = 1 d − Δ and ξ is the correlation length ξ ∼ | τ − τ c | − ν and there are possible logarithmic corrections to this expression. That was derived using numerical results for the Bose–Hubbard model and general scaling considerations. In this paper, we show that the same is valid in the case of holographic complexity providing evidence that the results are universal, and at the same time providing evidence for holographic computations of complexity.
We investigate the temperature-dependent behavior emerging in the vicinity of the superfluid (SF) to Mott insulator (MI) transition of interacting bosons in a two-dimensional optical lattice, described by the Bose-Hubbard model. The equilibrium phase diagram at finite temperatures is computed by means of the cluster mean-field theory (CMF) where the effect of non-local correlations is analyzed systematically by finite-size scaling of the cluster size. The phase diagram exhibits a rich structure including a transition and a crossover of the SF and MI phases respectively to a normal fluid (NF) state at finite temperature. In order to characterize these phases, and the NF transition and crossover scales, we calculate, in addition to the condensate amplitude, the superfluid fraction, sound velocity and compressibility. The phase boundaries obtained by CMF with finite-size scaling agree quantitatively with quantum Monte Carlo (QMC) results as well as with experiments. The von Neumann entanglement entropy of a cluster exhibits critical enhancement near the SF-MI quantum critical point (QCP). We also discuss the behavior of the transition lines near this QCP at the particle-hole symmetric point located at the tip of a Mott lobe as well as away from particle-hole symmetry.
We present a nonperturbative renormalization-group approach to the Bose-Hubbard model. By taking as initial condition of the renormalization-group flow the (local) limit of decoupled sites, we take into account both local and long-distance fluctuations in a nontrivial way. This approach yields a phase diagram in very good quantitative agreement with quantum Monte Carlo simulations, and reproduces the two universality classes of the superfluid-Mott-insulator transition. The critical behavior near the multicritical points, where the transition takes place at constant density, agrees with the original predictions of Fisher et al. [Phys. Rev. B 40, 546 (1989)] based on simple scaling arguments. At a generic transition point, the critical behavior is mean-field like with logarithmic corrections in two dimensions. In the weakly correlated superfluid phase (far away from the Mott insulating phase), the renormalization-group flow is controlled by the Bogoliubov fixed point down to a characteristic (Ginzburg) momentum scale kG{k}_{G}kG​, which is much smaller than the inverse healing length kh{k}_{h}kh​. In the vicinity of the multicritical points, when the density is commensurate, we identify a sharp crossover from a weakly to a strongly correlated superfluid phase where the condensate density and the superfluid stiffness are strongly suppressed and both kG{k}_{G}kG​ and kh{k}_{h}kh​ are of the order of the inverse lattice spacing.
We present a strong-coupling expansion of the Bose-Hubbard model which describes both the superfluid and the Mott phases of ultracold bosonic atoms in an optical lattice. By performing two successive Hubbard-Stratonovich transformations of the intersite hopping term, we derive an effective action which provides a suitable starting point to study the strong-coupling limit of the Bose-Hubbard model. This action can be analyzed by taking into account Gaussian fluctuations about the mean-field approximation as in the Bogoliubov theory of the weakly interacting Bose gas. In the Mott phase, we reproduce results of previous mean-field theories and also calculate the momentum distribution function. In the superfluid phase, we find a gapless spectrum and compare our results with the Bogoliubov theory.
We discuss how positions of critical points of the three-dimensional Bose-Hubbard model can be accurately obtained from variance of the on-site atom number operator, which can be experimentally measured. The idea that we explore is that the derivative of the variance, with respect to the parameter driving the transition, has a pronounced maximum close to critical points. We show that Quantum Monte Carlo studies of this maximum lead to precise determination of critical points for the superfluid-Mott insulator transition in systems with mean number of atoms per lattice site equal to one, two, and three. We also extract from such data the correlation-length critical exponent through the finite-size scaling analysis and discuss how the derivative of the variance can be reliably computed from numerical data for the variance. The same conclusions apply to the derivative of the nearest-neighbor correlation function, which can be obtained from routinely measured time-of-flight images.
I study the zero-temperature phase transition between the superfluid and the insulating ground states of the Bose-Hubbard model in a random chemical potential and at large integer average number of particles per site. Duality transformation maps the pure Bose-Hubbard model onto the sine-Gordon theory in one dimension (1D), and onto the three dimensional Higgs electrodynamics in two dimensions (2D). In 1D the random chemical potential in the dual theory couples to the space derivative of the dual field, and appears as a random magnetic field along the imaginary time direction in 2D. I show that the transition from the superfluid state both in 1D and 2D is always controlled by the random critical point. This arises due to a coupling constant in the dual theory with replicas that becomes generated at large distances by the random chemical potential, and represents a relevant perturbation at the pure superfluid-Mott insulator fixed point. At large distances the dual theory in 1D becomes equivalent to Haldane's macroscopic representation of a disordered quantum fluid, where the generated term is identified with the random backscattering. In 2D the generated coupling corresponds to the random mass of the complex field that represents vortex loops. I calculate the critical exponents at the superfluid-Bose-glass fixed point in 2D to be $\ensuremath{\nu}=1.38$ and z=1.93z=1.93z=1.93, and the universal conductivity at the transition ${\ensuremath{\sigma}}_{c}{=0.25e}_{*}^{2}/h$, using the one-loop field-theoretic renormalization group in fixed dimension.