Complexity and Criticality

Juan Claudio Toledo-Roy, Ana Leonor Rivera, Alejandro Frank

Uday Sood, Martı́n Kruczenski

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.

Luisa Eck

Kazuki Yamamoto, Kohei Kawabata

Motivated by the recent interest in the criticality of open quantum many-body systems, we study nonlinear sigma models with complexified couplings as a general framework for nonunitary field theory. Applying the perturbative renormalization-group analysis to the tenfold symmetric spaces, we demonstrate that fixed points with complex scaling dimensions and critical exponents arise generically, without counterparts in conventional nonlinear sigma models with real couplings. We further clarify the global phase diagrams in the complex-coupling plane and identify both continuous and discontinuous phase transitions. Our work thus identifies nonlinear sigma models as a representative setting for studying complex critical points and elucidates universal aspects of critical phenomena in complexified field theory.

Marc Baus, Carlos F. Tejero

Nitesh Jaiswal, Mamta Gautam, Tapobrata Sarkar

We study Nielsen complexity and Fubini-Study complexity for the transverse XY spin chain and establish their nonanalyticity at the critical lines. We obtain the scaling behavior of both complexities near quantum phase transitions in the thermodynamic limit, as a function of the system parameters. The scaling of the Nielsen complexity with the system size is also established, close to criticality. We also obtain analytic expressions for the Fubini-Study complexity in some special cases for this model, while a numerical analysis in more generic situations is carried out. Our study clearly demonstrates the differences in the two notions of complexity in the transverse XY model.