Complexity of Holographic Superconductors

Abstract

We study the complexity of holographic superconductors (Einstein-Maxwell-complex scalar actions in $d+1$ dimension) by the `complexity = volume' (CV) conjecture. First, it seems that there is a universal property: the superconducting phase always has a smaller complexity than the unstable normal phase below the critical temperature, which is similar to a free energy. We investigate the temperature dependence of the complexity. In the low temperature limit, the complexity (of formation) scales as $T^{\alpha }$, where $\alpha$ is a function of the complex scalar mass $m^2$, the $U(1)$ charge $q$, and dimension $d$. In particular, for $m^2=0$, we find $\alpha =d-1$, independent of $q$, which can be explained by the near horizon geometry of the low temperature holographic superconductor. Next, we develop a general numerical method to compute the time-dependent complexity by the CV conjecture. By this method, we compute the time-dependent complexity of holographic superconductors. In both normal and superconducting phase, the complexity increases as time goes on and the growth rate saturates to a temperature dependent constant. The higher the temperature is, the bigger the growth rate is. However, the growth rates do not violate the Lloyd's bound in all cases and saturate the Lloyd's bound in the high temperature limit at a late time.

a minor modification on the discussions of mass without changing the main results; references added

References in corpus (25)

• Building an AdS/CFT superconductor
• Complexity and Shock Wave Geometries
• Quantum Computation as Geometry
• Limits on Fundamental Limits to Computation
• Gravitational action with null boundaries
• Liouville Action as Path-Integral Complexity: From Continuous Tensor Networks to AdS/CFT
• Complexity and entanglement for thermofield double states
• Circuit complexity in interacting QFTs and RG flows
• Quantum Complexity and Negative Curvature
• Einstein's Equations from Varying Complexity
• On Volumes of Subregions in Holography and Complexity
• Time Evolution of Complexity: A Critique of Three Methods
• Switchbacks and the Bridge to Nowhere
• Complexity and the bulk volume, a new York time story
• Holographic Spacetimes as Quantum Circuits of Path-Integrations
• Tomography from Entanglement
• A geometric approach to quantum circuit lower bounds
• The Complexity of Quantum States and Transformations: From Quantum Money to Black Holes
• WdW-patches in AdS${}_3$ and complexity change under conformal transformations II
• Holographic complexity of anisotropic black branes
• More on complexity of operators in quantum field theory
• Circuit complexity for free Fermion with a mass quench
• Holographic complexity of local quench at finite temperature
• Low temperature properties of holographic condensates
• Holographic complexity under a global quantum quench

Cited by in corpus (32)

• Complexity growth of operators in the SYK model and in JT gravity
• Diving into a holographic superconductor
• Time evolution of the complexity in chaotic systems: concrete examples
• Circuit complexity for generalised coherent states in thermal field dynamics
• Interior Structure and Complexity Growth Rate of Holographic Superconductor from M-Theory
• Mixed State Entanglement for Holographic Axion Model
• Holographic entanglement entropy and complexity in St$\ddot{u}$ckelberg superconductor
• Holographic entanglement density for spontaneous symmetry breaking
• Homes' law in holographic superconductor with linear-$T$ resistivity
• Holographic complexity and thermodynamics of AdS black holes
• Entanglement entropy analysis of dyonic black holes using doubly holographic theory
• On the time dependence of holographic complexity for charged AdS black holes with scalar hair
• Volume complexity of dS bubbles
• Magnetotransport and Complexity of Holographic Metal-Insulator Transitions
• Mixed State Entanglement and Thermal Phase Transitions
• Time dependence of complexity for Lovelock black holes
• Generalized Volume-Complexity For Two-Sided Hyperscaling Violating Black Branes
• To be unitary-invariant or not?: a simple but non-trivial proposal for the complexity between states in quantum mechanics/field theory
• On the Complexity of a $2+1$--dimensional Holographic Superconductor
• Mixed-state Entanglement for AdS Born-Infeld Theory
• Holographic Complexity and Phase Transition for AdS Black Holes
• Complexity growth for topological black holes by holographic method
• Holographic Entanglement Entropy and Complexity for D-Wave Superconductors
• New Topological Gauss-Bonnet Black Holes in Five Dimensions
• Weighting gates in circuit complexity and holography
• Information theoretic measures for Lifshitz system
• Krylov complexity of purification
• Generalized volume-complexity for Lovelock black holes
• What kind of "complexity" is dual to holographic complexity?
• Complexity Growth of Dyonic Black holes with Quartic Field Strength Corrections
• Switchback effect of holographic complexity in multiple-horizon black holes
• Circuit Complexity for Coherent-Thermal States in Bosonic String Theory