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

TL;DR

This work demonstrates the existence of a many-body critical (MBC) phase in a 1D extended Aubry-André-Harper chain under short-range interactions, distinct from both ergodic and many-body localized phases. By combining real-space LDOS analyses with Fock-space diagnostics such as the inverse participation ratio and Feenberg self energy, it shows that the MBC phase features delocalized real-space single-pparticle excitations and multifractal FS statistics (0<D_2<1) with a FS localization length that scales with system size. The study uncovers Widom-like lines within the dynamical phase diagram, manifesting as peaks or dips in FS quantities (IPR and Delta_t) inside both MBC and MBL phases, and demonstrates KT-like and linear finite-size scaling behaviors near phase transitions. These findings provide a unified FS and real-space perspective on criticality and localization in quasiperiodic many-body systems and point to experimental routes for observing these phenomena in engineered quantum simulators.

Abstract

We study a quasiperiodic model in one dimension, namely the extended Aubry-André-Harper (EAAH) chain, that realizes a critical phase comprising entirely single-particle critical states in the non-interacting limit. In the presence of short-range interactions, the non-interacting 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 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.

Figures (19)

• Figure 1: Phase diagrams: (a) Phase diagram of the noninteracting $(V=0)$ EAAH model. The phases are delocalized, localized and critical separated by the solid lines. The vertical dashed line ($\lambda=2$) in the critical phase and horizontal dashed line ($\mu=1$) in the localized phase indicate the non-monotonic behavior ('Widom line') in IPR and $\Delta_t$ which are also used to obtain the phase boundaries. (b) Similar phase diagram for interacting $(V=1)$ EAAH model with ergodic, MBL and MBC phases separated by solid lines. The dashed lines in MBC and MBL phases denote similar Widom lines, implying non-monotonic behavior in IPR and $\Delta_t$ inside the phases, respectively. Here the strengths of the onsite quasi-periodic potential and quasi-periodic hoppings are denoted by $\lambda$ and $\mu$, respectively.
• Figure 2: Real-space single-particle excitations in noninteracting system: (a-b) Typical values of LDOS $\rho_t(\omega)$ for increasing $L$ in the delocalized and localized phases, for $\mu=0.5,~\lambda=1.0$ and $\mu=0.5,~\lambda=5.0$, respectively. The dashed curves in the figure represent the gapped excitations, where both $\rho_t(\omega)$ and $\rho_a(\omega)$ (not shown) decrease with $L$. (c-d) $\rho_t(\omega)$ for increasing $L$ at two different points, $(\mu=1.5,\lambda=1.0)$ and $(\mu=2.0,\lambda=1.0)$, in the critical phase on the phase diagram of Fig.\ref{['phase']}(a).
• Figure 3: Real-space single-particle excitations in interacting system: (a-b) Typical values of LDOS $\rho_t(\omega)$ as a function of system size $L$ in ergodic and MBL phases, for $\mu=0.5,~\lambda=2.0$ and $\mu=0.5,~\lambda=5.0$, respectively. (c-d) $\rho_t(\omega)$ for increasing $L$ at two different points, $(\mu=1.5,\lambda=1.0)$ and $(\mu=2.0,\lambda=1.0)$, in the MBC phase on the phase diagram of Fig.\ref{['phase']}(b).
• Figure 4: Fock-space (FS) lattice: FS lattice constructed out of real-space occupation-number basis states (black circles), illustrated for $L = 8$ at half filling, starting at the top with $\ket{11110000}$, i.e., all particles on the left side, and ending at the bottom with all particles on the right. The hoppings (black lines) and the slices (in lightblue) are indicated.
• Figure 5: Fock-space self-energy for noninteracting systems: Typical values of self-energy $\Delta_t$ across (a) the delocalization-localization transition as a function of $\lambda$ for $\mu=0.5$, (b) delocalization-critical transition as a function of $\mu$ for $\lambda=1.0$, (c) critical-localization transitions as a function of $\lambda$ for $\mu=1.5$, and (d) localization-critical transition as a function of $\mu$ for $\lambda=3.5$. The phase transition points are denoted by the vertical solid lines. In figures (c) and (d), the vertical dashed lines denote the peak/dip in $\Delta_t$ across the Widom-like lines in Fig.\ref{['phase']}(a) inside the critical and localized phases, respectively.