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arXiv:2404.08055

Complexity enriched dynamical phases for fermions on graphs

Xia, Wei, Jie Zou, Xiaopeng Li

Published Apr 11, 2024• 2 citations
arXivHTMLPDF

Abstract

Dynamical quantum phase transitions, encompassing phenomena like many-body localization transitions and measurement-induced phase transitions, are often characterized and identified through the analysis of quantum entanglement. Here, we highlight that the dynamical phases defined by entanglement are further enriched by complexity. We investigate both the entanglement and Krylov complexity for fermions on regular graphs, which can be implemented by systems like $^6$Li atoms confined by optical tweezers. Our investigations unveil that while entanglement follows volume laws on both types of regular graphs with degree $d = 2$ and $d = 3$, the Krylov complexity exhibits distinctive behaviors. We analyze both free fermions and interacting fermions models. In the absence of interaction, both numerical results and theoretical analysis confirm that the dimension of the Krylov space scales as $D\sim N$ for regular graphs of degree $d = 2$ with $N$ sites, and we have $D\sim N^2$ for $d = 3$. The qualitative distinction also persists in interacting fermions on regular graphs. For interacting fermions, our theoretical analyses find the dimension scales as $D\sim 4^{N^α}$ for regular graphs of $d = 2$ with $0.38\leqα\leq0.59$, whereas it scales as $D\sim 4^N$ for $d = 3$. The distinction in the complexity of quantum dynamics for fermions on graphs with different connectivity can be probed in experiments by measuring the out-of-time-order correlators.

DOI
10.48550/arxiv.2404.08055

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Dynamical quantum phase transitions, encompassing phenomena like many-body localization transitions and measurement-induced phase transitions, are often characterized and identified through the analysis of quantum entanglement. Here, we highlight that the dynamical phases defined by entanglement are further enriched by complexity. We investigate both the entanglement and Krylov complexity for fermions on regular graphs, which can be implemented by systems like 6^66Li atoms confined by optical tweezers. Our investigations unveil that while entanglement follows volume laws on both types of regular graphs with degree d=2d = 2d=2 and d=3d = 3d=3, the Krylov complexity exhibits distinctive behaviors. We analyze both free fermions and interacting fermions models. In the absence of interaction, both numerical results and theoretical analysis confirm that the dimension of the Krylov space scales as D∼ND\sim ND∼N for regular graphs of degree d=2d = 2d=2 with NNN sites, and we have D∼N2D\sim N^2D∼N2 for d=3d = 3d=3. The qualitative distinction also persists in interacting fermions on regular graphs. For interacting fermions, our theoretical analyses find the dimension scales as D∼4NαD\sim 4^{N^α}D∼4Nα for regular graphs of d=2d = 2d=2 with 0.38≤α≤0.590.38\leqα\leq0.590.38≤α≤0.59, whereas it scales as D∼4ND\sim 4^ND∼4N for d=3d = 3d=3. The distinction in the complexity of quantum dynamics for fermions on graphs with different connectivity can be probed in experiments by measuring the out-of-time-order correlators.
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We numerically investigate measurement-induced phase transitions in monitored free fermions through the spectral and eigenstate properties of the entanglement Hamiltonian. By analyzing entanglement scaling, we identify three nontrivial fixed points, for which the entanglement follows exact analytical scaling forms: chaotic unitary dynamics at infinitesimal monitoring, characterized by a Gaussian Page law; a Fermi-liquid fixed point at moderate monitoring, defining a metallic phase with logarithmic entanglement growth and emergent space-time invariance; and a quantum Lifshitz fixed point marking the measurement-induced phase transition into a localized area-law phase. Adopting a random-matrix perspective on the entanglement Hamiltonian, we show that short-range spectral correlations, such as the adjacent gap ratio $\ensuremath{\langle}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{r}\ensuremath{\rangle}$ and the Kullback-Leibler divergence KL1{\text{KL}}_{1}KL1​, sharply detect an ergodic to nonergodic phase transition at the quantum Lifshitz fixed point, and yield precise estimates of the critical point and correlation length exponent. Long-range probes, including the spectral form factor and the associated Thouless time, corroborate this picture, while the variant KL2{\text{KL}}_{2}KL2​ uncovers signatures of a possible nonergodic extended (multifractal) regime at intermediate monitoring strengths. Together, these results establish the entanglement Hamiltonian as a powerful framework for diagnosing metallic, localized, and multifractal regimes in monitored quantum dynamics, and highlight unexpected scaling structures, Fermi liquid, and Lifshitz criticality, that lie beyond current field-theoretic approaches.
Anyons obey fractional statistics that lie between bosonic and fermionic statistics, giving rise to a broad range of intriguing phenomena. However, how anyonic statistics govern quantum-state complexity is still largely unexplored. In this work, we investigate the interplay between the statistical phase and on-site interactions in the anyon-Hubbard model, identifying exact quantum many-body scar eigenstates and novel quench dynamics. The Krylov complexity exhibits perfect periodic revivals independent of the statistical phase in the scarred dynamics, whereas after a quench it depends on both the statistical phase and the interaction strength. In the strong-interaction regime, we find approximate scarred dynamics, while in the weak-interaction regime the state spreads over Krylov space and the complexity ultimately saturates. Moreover, for the bosonic initial state, the complexity of fermions exhibits the lowest saturation value, and vice versa. For fractional statistics, the saturation plateau is minimized when the post-quench statistical phase is close to that of the initial state. Our results demonstrate the central role of the statistical phase in governing many-body dynamics and provide new insights into Krylov complexity and quantum many-body scars.
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A bstract Quantum complexity, suitably defined, has been suggested as an important probe of late-time dynamics of black holes, particularly in the context of AdS/CFT. A notion of quantum complexity can be effectively captured by quantifying the spread of an operator in Krylov space as a consequence of time evolution. Complexity is expected to behave differently in chaotic many-body systems, as compared to integrable ones. In this paper we investigate Krylov complexity for the case of interacting integrable models at finite size and find that complexity saturation is suppressed as compared to chaotic systems. We associate this behavior with a novel localization phenomenon on the Krylov chain by mapping the theory of complexity growth and spread to an Anderson localization hopping model with off-diagonal disorder, and find that localization is enhanced in the integrable case due to a stronger disorder in the hopping amplitudes, inducing an effective suppression of Krylov complexity. We demonstrate this behavior for an interacting integrable model, the XXZ spin chain, and show that the same behavior results from a phenomenological model that we define: this model captures the essential features of our analysis and is able to reproduce the behaviors we observe for chaotic and integrable systems via an adjustable disorder parameter.