We study the entanglement in momentum space of a disordered one-dimensional fermion lattice model with attractive interaction. We observe that the many-body localization transition can be characterized by behaviors of two components in entanglement spectrum. One of the components is related to paired-fermion entanglement which contributes to the long-range correlation in position space, and the vanishing of it indicates the emerged many-body localized phase. Based on this understanding, we identify the critical point between delocalized and localized phases for interaction of different strength. Additionally by method of entanglement spectrum, we provide a new evidence to show the transition of two phases induced by interaction, and find this phase transition is not influenced by the disorder. Our result shows key characteristics in entanglement for different phases in the system, and provides a novel perspective to understand many-body localization phenomenon.
We study the entanglement in momentum space of the ground state of a disordered one-dimensional fermion lattice model with attractive interaction. We observe two components in the entanglement spectrum, one of which is related to paired-fermion entanglement and contributes to the long-range correlation in position space. The vanishing point of it indicates the localization phenomenon in the ground state of this model. Additionally, by method of entanglement spectrum, we provide a new evidence to show the transition of two phases induced by interaction, and find that this phase transition is not influenced by the disorder. Our result show key characteristics in entanglement for different phases in the system, and provide a novel perspective to understand localization phenomena.
We study the entanglement in momentum space of a disordered one-dimentional
fermion lattice model with attractive interaction and distinguish two
components in it. One of the components is related to paired-fermion
entanglement and thus the long-range correlation in position space, and the
vanishing of it indicates many-body localization phenomenon. Based on this
picture, we investigate the critical point between delocalized and localized
phases for different interaction. Also, we find a new signal to distinguish two
phases originating from the system only with the interaction, and show this
phase transition is not influenced by the disorder. This letter shows key
characteristics of this kind of systems, and provides a novel perspective to
understand many-body localization phenomenon.
The use of quantum entanglement to study condensed matter systems has been flourishing in critical systems and topological phases. Additionally, using real-space entanglement one can characterize localized and delocalized phases of disordered fermion systems. Here we instead propose the momentum-space entanglement spectrum as a means of characterizing disordered models. We show that localization in one dimension can be characterized by the momentum space entanglement between left and right movers and illustrate our methods using explicit models with spatially correlated disorder that exhibit phases which avoid complete Anderson localization. The momentum space entanglement spectrum clearly reveals the location of delocalized states in the energy spectrum, can be used as a signature of the phase transition between a delocalized and localized phase, and only requires a single numerical diagonalization to yield clear results.
We consider the one-dimensional lattice model of interacting fermions with disorder studied previously by Oganesyan and Huse [Phys. Rev. B 75, 155111 (2007)]. To characterize a possible many-body localization transition as a function of the disorder strength
W, we use an exact renormalization procedure in configuration space that generalizes the Aoki real-space renormalization procedure for Anderson localization one-particle models [H. Aoki, J. Phys. C 13, 3369 (1980)]. We focus on the statistical properties of the renormalized hopping
VL between two configurations separated by a distance
L in configuration space (distance being defined as the minimal number of elementary moves to go from one configuration to the other). Our numerical results point toward the existence of a many-body localization transition at a finite disorder strength
Wc. In the localized phase
W>Wc, the typical renormalized hopping ${V}_{L}^{typ}\ensuremath{\equiv}{e}^{\overline{\text{ln}\text{ }{V}_{L}}}$ decays exponentially in
L as $(\text{ln}\text{ }{V}_{L}^{typ})\ensuremath{\simeq}\ensuremath{-}\frac{L}{{\ensuremath{\xi}}_{loc}}$ and the localization length diverges as ${\ensuremath{\xi}}_{loc}(W)\ensuremath{\sim}{(W\ensuremath{-}{W}_{c})}^{\ensuremath{-}{\ensuremath{\nu}}_{loc}}$ with a critical exponent of order ${\ensuremath{\nu}}_{loc}\ensuremath{\simeq}0.45$. In the delocalized phase
W<Wc, the renormalized hopping remains a finite random variable as $L\ensuremath{\rightarrow}\ensuremath{\infty}$ and the typical asymptotic value ${V}_{\ensuremath{\infty}}^{typ}\ensuremath{\equiv}{e}^{\overline{\text{ln}\text{ }{V}_{\ensuremath{\infty}}}}$ presents an essential singularity $(\text{ln}\text{ }{V}_{\ensuremath{\infty}}^{typ})\ensuremath{\sim}\ensuremath{-}{({W}_{c}\ensuremath{-}W)}^{\ensuremath{-}\ensuremath{\kappa}}$ with an exponent of order $\ensuremath{\kappa}\ensuremath{\sim}1.4$. Finally, we show that this analysis in configuration space is compatible with the localization properties of the simplest two-point correlation function in real space.
Characterizing the delocalization transition in closed quantum systems with a many-body localized phase is a key open question in the field of nonequilibrium physics. We exploit the fact that localization of particles as realized in Anderson localization and standard many-body localization (MBL) implies Fock-space localization in single-particle basis sets characterized by a real-space index. Using a recently introduced quantitative measure for Fock-space localization computed from the density distributions, the occupation distance, we systematically study its scaling behavior across delocalization transitions and identify critical points from scaling collapses of numerical data. Excellent agreement with literature results is found for the critical disorder strengths of noninteracting fermions, such as the one-dimensional Aubry-Andr\'e model and the three-dimensional Anderson model. We observe a distinctively different scaling behavior in the case of interacting fermions with random disorder consistent with a Kosterlitz-Thouless transition. Finally, we use our measure to extract the transition point as a function of filling for interacting fermions.
In the presence of strong enough disorder one-dimensional systems of interacting spinless fermions at nonzero filling factor are known to be in a many-body localized phase. When represented in `Fock space,' the Hamiltonian of such a system looks like that of a single `particle' hopping on a Fock lattice in the presence of a random disordered potential. The coordination number of the Fock lattice increases linearly with the system size
L in one dimension. Thus in the thermodynamic limit $L\ensuremath{\rightarrow}\ensuremath{\infty}$, the disordered interacting problem in one dimension maps on to an Anderson model with infinite coordination number. Despite this, this system displays localization which appears counterintuitive. A close observation of the on-site disorder potentials on the Fock lattice reveals a large degree of correlation among them as they are derived from an exponentially smaller number of on-site disorder potentials in real space. This indicates that the correlations between the on-site disorder potentials on a Fock lattice has a strong effect on the localization properties of the corresponding many-body system. This intuition is also consistent with studies of quantum random energy model where the typical mid-spectrum states are ergodic and the on-site potentials in Fock space are completely uncorrelated. In this work we perform a systematic quantitative exploration of the nature of correlations of the Fock space potential required for localization. We study different functional variations of the disorder correlation in Fock lattice by analyzing the eigenspectrum obtained through exact diagonalization. Without changing the typical strength of the on-site disorder potential in Fock lattice we show that changing the correlation strength can induce thermalization or localization in systems. From among the various forms of correlations we study, we find that only the linear variation of correlations with Hamming distance in Fock space is able to drive a thermal-MBL phase transition where the transition is driven by the correlation strength. Systems with the other forms of correlations we study are found to be ergodic.