Complexity for Charged Thermofield Double States

TL;DR

This work constructs and analyzes the charged thermofield double (cTFD) state for a free complex scalar field in a background electric field, using a lattice implementation and covariance-matrix methods to compute Nielsen circuit complexity. It shows that the complexity of formation, when evaluated with the LR basis and the F1 cost function, yields finite results that align with holographic expectations for charged black holes, including infrared behavior consistent with the third law, provided appropriate boundary-term choices are made. Time evolution of the complexity in the free theory saturates on a timescale set by the inverse temperature, in contrast to the linear late-time growth predicted holographically for strongly coupled theories; this discrepancy underscores the role of interactions in accessing the full chaotic Hilbert space and the corresponding complexity growth. The paper also demonstrates how a cMERA-inspired UV regulator enables a clean continuum limit and highlights the importance of basis and reference-scale choices in matching holographic results, suggesting that the F1 LR construction is a robust QFT counterpart to holographic complexity for both neutral and charged setups while pointing to boundary-term refinements as potential reconciliations for remaining gaps.

Abstract

We study Nielsen's circuit complexity for a charged thermofield double state (cTFD) of free complex scalar quantum field theory in the presence of background electric field. We show that the ratio of the complexity of formation for cTFD state to the thermodynamic entropy is finite and it depends just on the temperature and chemical potential. Moreover, this ratio smoothly approaches the value for real scalar theory. We compare our field theory calculations with holographic complexity of charged black holes and confirm that the same cost function which is used for neutral case continues to work in presence of $U(1)$ background field. For $t>0$, the complexity of cTFD state evolves in time and contrasts with holographic results, it saturates after a time of the order of inverse temperature. This discrepancy can be understood by the fact that holographic QFTs are actually strong interacting theories, not free ones.

Figures (33)

• Figure 1: All mode contribution to time dependence of ${C}_{\kappa=2}$ with the initial value subtracted for the neutral TFD at zero time on a circle with circumference $L$ with $\omega_{R} = 1/L$, $m = 10^{-5}/L$, $\mu q=10^{-5}/L$ and increasing temperatures $T = 1/\beta$. We use 1501 lattice sites on each side.
• Figure 2: All mode contribution to time dependence of ${C}_{\kappa=2}$ with the initial value subtracted for the neutral TFD at zero time on a circle with circumference $L$ with $\omega_{R} = 1/L$, $m = 10^{-5}/L$, $\mu q=10^{-1}/L$ and increasing temperatures $T = 1/\beta$. We use 1501 lattice sites on each side.
• Figure 3: Time dependence of $\kappa=2$ complexity with the initial value subtracted for TFD state on a circle with circumference $L$ with $\omega_{R} = 1/L$, $m=10^{-4}/L$, $\lambda_{R}=1$, $\beta = 10 L$ (left), $\beta = 10^{-1}L$ (right) and different increasing the chemical potential from $10^{-3}/L$ (blue) to $10^{-1}/L$ (green). We use 1501 lattice sites on each side. We see that complexity grows as $\log^2(t/L)$ up to times of the order of $1/(m+\mu q)$ when it starts oscillating around the saturated value. The zero mode is the source for this logarithmic growth but by increasing the temperature this behavior is lost.
• Figure 4: Time dependence of $\kappa=2$ complexity with the initial value subtracted for TFD state on a circle with circumference $L$ with $\omega_{R} = 1/L$, $\mu q = 10^{-1}/L$, $\beta = 10 L$ (left), $\beta = 10^{-1}/L$ (right) and different increasing the masses, from $10^{-4}/L$ (blue) to $10^{-2}/L$ (green). We use 1501 lattice sites on each side. We see that complexity grows as $\log^{2}(t/L)$ up to times of the order of $1/(m+\mu q)$ when it starts oscillating around the saturated value. The zero mode is the source for this logarithmic growth but by increasing the temperature this behavior is lost.
• Figure 5: All mode contribution to time dependence of ${C}_{\kappa=2}$ with the initial value subtracted for the neutral TFD at zero time on a circle with circumference $L$ with $\omega_{R} = 1/L$, $m = 10^{-5}/L$, $\mu q= 10^{-5}/L$. We use 1501 lattice sites on each side.