Circuit complexity near critical points

Uday Sood; Martin Kruczenski

doi:10.1088/1751-8121/ac5b8f

paper

Circuit complexity near critical points

arXiv:2106.12648 · doi:10.1088/1751-8121/ac5b8f

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 $t \to t_{c}$ , there is a non-analytic behavior in the complexity $c_{2} (t)$ of the form $∣ c_{2} (t) - c_{2} (t_{c}) ∣ \sim ∣ t - t_{c} ∣^{ν d}$ , up to possible logarithmic corrections. Here $d$ is the number of spatial dimensions and $ν$ is the usual critical exponent for the correlation length $ξ \sim ∣ t - t_{c} ∣^{- ν}$ . As a check, for $d = 2$ this agrees with the numerical computation if we use the gaussian critical exponent $ν = \frac{1}{2}$ . 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.

arXiv comment36 pages, 11 figures, Latex ; added additional references

Circuit complexity near critical points

Abstract

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