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arXiv:2505.02234 · Physical review. B./Physical review. B

Many-body critical phase in a quasiperiodic chain and dynamical Widom lines in Fock-space properties

Nilanjan Roy, Subroto Mukerjee, Sumilan Banerjee

Published Sep 22, 2025• 2 citations
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Abstract

We study a quasiperiodic model in one dimension, namely the extended Aubry-Andr\'e-Harper (EAAH) chain, that realizes a critical phase comprising entirely single-particle critical states in the noninteracting limit. In the presence of short-range interactions, the noninteracting critical phase transforms to a many-body critical (MBC) phase, separated by lines of MBC-ergodic, MBC-many-body localized (MBL) and ergodic-MBL phase transitions that meet at a triple point. We elucidate the unusual characteristics of the MBC phase compared to the ergodic and MBL phases through the localization properties of the excitations in real space and Fock space (FS), and the eigenstate inverse participation ratio (IPR). We show that the MBC phase, like the MBL phase, is well described by a multifractal scaling of the IPR and a linear finite-size scaling ansatz near the transition to the ergodic and MBL phases. However, the MBC phase, at the same time, exhibits delocalization of all single-particle excitations and a system-size-dependent Fock-space localization length, analogous to the ergodic phase. Remarkably, we find evidence of unusual Widom lines on the phase diagram in the form of lines of pronounced peaks or dips in the FS localization properties inside the MBC and MBL phases. These Widom lines either emerge as a continuation of the precursor phase transition line terminating at the triple point, or originate from a phase boundary.

DOI
10.1103/xc8b-yzx8

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We study a quasiperiodic model in one dimension, namely the extended Aubry-Andr'e-Harper (EAAH) chain, that realizes a critical phase comprising entirely single-particle critical states in the noninteracting limit. In the presence of short-range interactions, the noninteracting critical phase transforms to a many-body critical (MBC) phase, separated by lines of MBC-ergodic, MBC-many-body localized (MBL) and ergodic-MBL phase transitions that meet at a triple point. We elucidate the unusual characteristics of the MBC phase compared to the ergodic and MBL phases through the localization properties of the excitations in real space and Fock space (FS), and the eigenstate inverse participation ratio (IPR). We show that the MBC phase, like the MBL phase, is well described by a multifractal scaling of the IPR and a linear finite-size scaling ansatz near the transition to the ergodic and MBL phases. However, the MBC phase, at the same time, exhibits delocalization of all single-particle excitations and a system-size-dependent Fock-space localization length, analogous to the ergodic phase. Remarkably, we find evidence of unusual Widom lines on the phase diagram in the form of lines of pronounced peaks or dips in the FS localization properties inside the MBC and MBL phases. These Widom lines either emerge as a continuation of the precursor phase transition line terminating at the triple point, or originate from a phase boundary.
We provide real-space and Fock-space (FS) characterizations of ergodic, nonergodic extended (NEE) and many-body localized (MBL) phases in an interacting quasiperiodic system, namely, the generalized Aubry-Andr\'e-Harper model, which possesses a mobility edge in the noninteracting limit. We show that a mobility edge in the single-particle (SP) excitations survives even in the presence of interaction in the NEE phase. In contrast, all single-particle excitations get localized in the MBL phase due to the MBL proximity effect. We give complementary insights into the distinction of the NEE states from the ergodic and MBL states by computing local FS self-energies and decay length associated, respectively, with the local and the nonlocal FS propagators. Based on a finite-size scaling analysis of the typical local self-energy across the NEE to ergodic transition, we show that MBL and NEE states exhibit qualitatively similar multifractal character. However, we find that the NEE and MBL states can be distinguished in terms of the distribution of local self-energy and the decay of the nonlocal propagator in the FS, whereas the typical local FS self-energy cannot tell them apart.
The transition between ergodic and many-body localization (MBL) phases lies at the heart of understanding quantum thermalization of many-body systems. Here, we predict a many-body critical (MBC) phase with finite-size scaling analysis in the one-dimensional extended Aubry-André-Harper-Hubbard model, which is different from both the ergodic phase and MBL phase, implying that the quantum system hosts three different fundamental phases in the thermodynamic limit. The level statistics in the MBC phase are well characterized by the so-called critical statistics, and the wave functions exhibit deep multifractal behavior only in the critical region. We further study the half-chain entanglement entropy and thermalization properties and show that the former, in the MBC phase, manifest a volume law scaling, while the many-body states violate the eigenstate thermalization hypothesis. The results are confirmed by the state-of-the-art numerical calculations with system size up to L=22. This work unveils a novel many-body phase which is extended but nonthermal.
We study localization and many-body localization transition in one-dimensional systems in the presence of deterministic quasiperiodic potential. We use single-particle excitations obtained through single-particle Green's function in real space to characterize the localization to delocalization transition. A single parameter scaling analysis of the ratio of the typical to average value of the local density of states (LDOS) of single-particle excitations shows that the critical exponent with which the correlation length $\ensuremath{\xi}$ diverges at the transition point $\ensuremath{\xi}\ensuremath{\sim}|h\ensuremath{-}{h}_{c}{|}^{\ensuremath{-}\ensuremath{\nu}}$, coming from the localized side, satisfies the inequality $\ensuremath{\nu}\ensuremath{\ge}1$ for the noninteracting Aubry-Andre (AA) model. For the interacting system with AA potential, we study single-particle excitations produced in highly excited many-body eigenstates across the MBL transition and found that the critical exponent obtained from finite-size scaling of the ratio of the typical to average value of the LDOS satisfies $\ensuremath{\nu}\ensuremath{\ge}1$ here as well. This analysis of the local density of states shows that the localization and MBL transition in systems with quasiperiodic potential belong to a different universality class than the localization and MBL transition in systems with random disorder where $\ensuremath{\nu}\ensuremath{\ge}2$. In complete contrast to this, finite-size scaling of the level spacing ratio is known to support the same universality class for MBL transitions in systems with quasiperiodic as well as random disorder potentials. For the interacting systems with quasiperiodic potentials, though finite-size scaling of the level spacing ratio shows a transition at hclsr{h}_{c}^{\mathrm{lsr}}hclsr​ which is close to the transition point obtained from LDOS within numerical precision, the critical exponent obtained from finite-size scaling of level spacing ratio is $\ensuremath{\nu}\ensuremath{\sim}0.54$ in close similarity to the MBL systems with random disorder.
We study quench dynamics in an interacting spin chain with a quasiperiodic on-site field, known as the interacting Aubry-Andr\'e model of many-body localization. Using the time-dependent variational principle, we assess the late-time behavior for chains up to L=50L=50L=50. We find that the choice of periodicity $\mathrm{\ensuremath{\Phi}}$ of the quasiperiodic field influences the dynamics. For $\mathrm{\ensuremath{\Phi}}=(\sqrt{5}\ensuremath{-}1)/2$ (the inverse golden ratio) and interaction $\mathrm{\ensuremath{\Delta}}=1$, the model most frequently considered in the literature, we obtain the critical disorder ${W}_{c}=4.8\ifmmode\pm\else\textpm\fi{}0.5$ in units where the noninteracting transition is at W=2W=2W=2. At the same time, for periodicity $\mathrm{\ensuremath{\Phi}}=\sqrt{2}/2$ we obtain a considerably higher critical value, ${W}_{c}=7.8\ifmmode\pm\else\textpm\fi{}0.5$. Finite-size effects on the critical disorder Wc{W}_{c}Wc​ are much weaker than in the purely random case. This supports the enhancement of Wc{W}_{c}Wc​ in the case of a purely random potential by rare ``ergodic spots,'' which do not occur in the quasiperiodic case. Further, the data suggest that the decay of the antiferromagnetic order in the delocalized phase is faster than a power law.
We explore the Fock-space structure of eigenstates across the many-body localization (MBL) transition in a disordered, interacting quantum spin-12\frac{1}{2}21​ chain. Eigenstate expectation values of spatially local observables, which distinguish an MBL phase from an ergodic one, can be represented in terms of eigenstate amplitudes on the Fock space. Motivated by this, we introduce and study spatial correlations on the Fock space. From these, a correlation length emerges, which is found to vary discontinuously across the MBL transition, and is intimately connected to the discontinuous jump in the multifractal exponents characterizing the Fock-space wave functions. Exploiting the direct connection between the local observables and Fock-space correlations, we show that the discontinuity in the length scale also implies discontinuous behavior of the local observables across the transition. A scaling theory based on these Fock-space correlations is constructed, which is closely connected to that for the inverse participation ratio. It yields a volume scale in the ergodic phase and a length scale in the MBL phase, whose critical properties suggest a Kosterlitz-Thouless--type scenario for the MBL transition, as is predicted by recent phenomenological theories. Finally, we also show how correlation functions on the Fock space reveal the inhomogeneities in eigenstate amplitudes on the Fock space in the MBL phase.
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