We characterize the Many-Body Localization (MBL) phase transition using the dynamics of spread complexity and inverse participation ratio in the Krylov space starting from different initial states. Our analysis of the disordered Heisenberg spin-1/2 chain unravels that the ergodic-to-MBL transition can be determined from the transition of the pre-saturation peak in the thermofield double state (TFD) spread complexity. On the other hand, if an initially ordered state or a superposition of a small number of such states is chosen, then the saturation value of spread complexity and Krylov inverse participation ratio (KIPR) can distinguish the ergodic phase from the integrable phases, with no sharp difference between the integrable phases. Interestingly, the distinction between the disorder-free integrable and the MBL integrable phase is established by the spread complexity study of random states chosen from unitary and orthogonal Haar ensembles. We also study the complexity dynamics by coupling the system to a bath, which shows distinctive profiles in different phases. A stretched exponential decay of KIPR is observed when the MBL system is connected to the bath, with the decay starting at an earlier time for a greater value of environmental dephasing. Our work sheds light on the efficacy of Krylov space dynamics in understanding phase transitions in quantum many-body systems.
We characterize the Many-Body Localization (MBL) phase transition using the dynamics of spread complexity and inverse participation ratio in the Krylov space starting from different initial states. Our analysis of the disordered Heisenberg spin-1/2 chain unravels that the ergodic-to-MBL transition can be determined from the transition of the pre-saturation peak in the thermofield double state (TFD) spread complexity. On the other hand, if an initially ordered state or a superposition of a small number of such states is chosen, then the saturation value of spread complexity and Krylov inverse participation ratio (KIPR) can distinguish the ergodic phase from the integrable phases, with no sharp difference between the integrable phases. Interestingly, the distinction between the disorder-free integrable and the MBL integrable phase is established by the spread complexity study of random states chosen from unitary and orthogonal Haar ensembles. We also study the complexity dynamics by coupling the system to a bath, which shows distinctive profiles in different phases. A stretched exponential decay of KIPR is observed when the MBL system is connected to the bath, with the decay starting at an earlier time for a greater value of environmental dephasing. Our work sheds light on the efficacy of Krylov space dynamics in understanding phase transitions in quantum many-body systems.
We study the behavior of spread complexity in the context of non-Hermitian many-body localization Transition (MBLT). Our analysis has shown that the singular value spread complexity is capable of distinguishing the ergodic and many-body localization (MBL) phase from the presaturation peak height for the non-hermitian models having time-reversal symmetry (TRS) and without TRS. On the other hand, the saturation value of the thermofield double (TFD) state complexity can detect the real-complex transition of the eigenvalues on increasing disorder strength. From the saturation value, we also distinguish the model with TRS and without TRS. The charge density wave complexity shows lower saturation values in the MBL phase for the model with TRS. However, the model without TRS shows a completely different behavior, which is also captivated by our analysis. So, our investigation unravels the real-complex transition in the eigenvalues, the difference between the model having TRS and without TRS, and the effect of boundary conditions for the non-hermitian models having MBL transitions, from the Krylov spread complexity perspective.
We investigate the anatomy and complexity of quantum states in Krylov space, in the ergodic and many-body localised (MBL) phases of a disordered, interacting spin chain. The Krylov basis generated by the Hamiltonian from an initial state provides a representation in which the spread of the time-evolving state constitutes a basis-optimised measure of complexity. We show that the long-time Krylov spread complexity sharply distinguishes the two phases. In the ergodic regime, the infinite-time complexity scales linearly with the Fock-space dimension, indicating that the state spreads over a finite fraction of the Krylov chain. By contrast, it grows sublinearly in the MBL regime, implying that the long-time state occupies only a vanishing fraction of the chain. Further, the profile of the infinite-time state along the Krylov chain exhibits a stretched-exponential decay in the MBL regime. This behaviour reflects a broad distribution of decay lengthscales, associated with different eigenstates contributing to the long-time state. Consistently, a large-deviation analysis of the statistics of eigenstate spread complexities shows that while the ergodic regime receives contributions from almost all eigenstates, the complexity in the MBL regime is dominated by a vanishing fraction of eigenstates, which have anomalously large complexity relative to the typical ones.
A bstract We explore spread and spectral complexity in quantum systems that exhibit a transition from integrability to chaos, namely the mixed-field Ising model and the next-to-nearest-neighbor deformation of the Heisenberg XXZ spin chain. We corroborate the observation that the presence of a peak in spread complexity before its saturation, is a characteristic feature in chaotic systems. We find that, in general, the saturation value of spread complexity post-peak depends not only on the spectral statistics of the Hamiltonian, but also on the specific state. However, there appears to be a maximal universal bound determined by the symmetries and dimension of the Hamiltonian, which is realized by the thermofield double state (TFD) at infinite temperature. We also find that the time scales at which the spread complexity and spectral form factor change their behaviour agree with each other and are independent of the chaotic properties of the systems. In the case of spectral complexity, we identify that the key factor determining its saturation value and timescale in chaotic systems is given by minimum energy difference in the theory’s spectrum. This explains observations made in the literature regarding its earlier saturation in chaotic systems compared to their integrable counterparts. We conclude by discussing the properties of the TFD which, we conjecture, make it suitable for probing signatures of chaos in quantum many-body systems.
Krylov complexity, a quantum complexity measure which uniquely characterizes the spread of a quantum state or an operator, has recently been studied in the context of quantum chaos. However, the definitiveness of this measure as a chaos quantifier is in question in light of its strong dependence on the initial condition. This article clarifies the connection between the Krylov complexity dynamics and the initial operator or state. We find that the saturation value of Krylov complexity depends monotonically on the inverse participation ratio (IPR) of the initial condition in the eigenbasis of the Hamiltonian. We explain the reversal of the complexity saturation levels observed in [Phys. Rev. E 107, 024217 (2023)2470-004510.1103/PhysRevE.107.024217] using the initial spread of the operator in the Hamiltonian eigenbasis. IPR dependence is present even in the fully chaotic regime, where popular quantifiers of chaos, such as out-of-time-ordered correlators and entanglement generation, show similar behavior regardless of the initial condition. Krylov complexity averaged over many initial conditions still does not characterize chaos.
We study many-body localization (MBL) in a pair-hopping model exhibiting strong fragmentation of the Hilbert space. We show that several Krylov subspaces have both ergodic statistics in the thermodynamic limit and a dimension that scales much slower than the full Hilbert space but still exponentially. Such a property allows us to study the MBL phase transition in systems including up to 64 spins. The different Krylov spaces that we consider show clear signatures of a many-body localization transition, both in the Kullback-Leibler divergence of the distribution of their level spacing ratio and their entanglement properties. However, they also present distinct scalings with the system size. Depending on the subspace, the critical disorder strength can be nearly independent of the system size or conversely show an approximately linear increase with the number of spins.
We investigate the phase transition between an ergodic and a many-body localized phase in infinite anisotropic spin-1/2 Heisenberg chains with binary disorder. Starting from the N\'eel state, we analyze the decay of antiferromagnetic order ms(t) and the growth of entanglement entropy Sent(t) during unitary time evolution. Near the phase transition we find that ms(t) decays exponentially to its asymptotic value ${m}_{s}(\ensuremath{\infty})\ensuremath{\ne}0$ in the localized phase while the data are consistent with a power-law decay at long times in the ergodic phase. In the localized phase, ${m}_{s}(\ensuremath{\infty})$ shows an exponential sensitivity on disorder with a critical exponent $\ensuremath{\nu}\ensuremath{\sim}0.9$. The entanglement entropy in the ergodic phase grows subballistically, ${S}_{\mathrm{ent}}(t)\ensuremath{\sim}{t}^{\ensuremath{\alpha}}$, $\ensuremath{\alpha}\ensuremath{\le}1$, with $\ensuremath{\alpha}$ varying continuously as a function of disorder. Exact diagonalizations for small systems, on the other hand, do not show a clear scaling with system size and attempts to determine the phase boundary from these data seem to overestimate the extent of the ergodic phase.