Entanglement entropy of low-lying excitation in localized interacting system: Signature of Fock space delocalization

Abstract

The properties of the entanglement entropy (EE) of low-lying excitations in one-dimensional disordered interacting systems are studied. The ground state EE shows a clear signature of localization, while low-lying excitation shows a crossover from metallic behavior at short sample sizes to localized at longer length. The dependence of the crossover as function of interaction strength and sample length is studied using the density matrix renormalization group (DMRG). This behavior corresponds to the presence of the predicted many particle critical energy in the vicinity of the Fermi energy. Implications of these results to experiments are discussed. PACS numbers: 73.20.Fz,03.65.Ud,71.10.Pm,73.21.Hb Many body localization (MBL) has drawn growing interest in recent years [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. The Anderson localization transition [21,22] is a zero-temperature quantum phase transition, which for the non-interacting case is manifested in the properties of the single-particle eigenstates and eigenvalues [23]. For one and two dimensional systems all the single-particle state are localized for any amount of disorder, while for three dimensional systems at a given disorder there exists a critical energy (mobility edge) below which all states are localized. For a many-particle system, as long as no particleparticle interactions are present, the properties of a many-particle excited state are determined by the properties of the single-particle states. Thus, as long as all the particles occupy localized single-particle states the manyparticle states should exhibit localized behavior. For example, the entanglement entropy (EE) between any subregion A of the system and the rest of it should saturate (i.e., S A ∝ ξ d−1 , where d is the dimensionality of the system and ξ is the single-electron localization length). While if some of the extended single-particle states are occupied the usual volume law for the EE (S A ∝ L d A , where L A is the length of region A) is observed [17]. Physically, the difference between the localized and extended many-particle states is manifested in the dynamics of a particle tunneling into some particular region of the system, or a particle excited by a confined external perturbation in the region. In the localized phase it will remain there (i.e., will have a very long lifetime -sharp level), while in the extended case it will leave the region (short lifetime -broad level).What is the influence of particle-particle interaction on the above picture? A cursory consideration may lead to the conclusion that since particles now interact with each other, any localized excitation will eventually spread all over the system. Thus, the rest of the system acts as a thermal bath for any excited sub-system [24][25][26]. Surprisingly, Basko, Aleiner and Altshuler [2] have shown that if all the single-electron states are localized, an excitation will remain localized even in the presence of particleparticle interactions up to a critical temperature or excitatio...

Universal Dynamics and Renormalization in Many-Body-Localized Systems

“…The studies mentioned above, together with many others [35][36][37][38][39][40][41][42][43], have also shown that, far from being a boring dead state, the localized phase can display remarkably rich dynamical behavior. In particular, it has been argued, that a number of distinct localized phases can exist [44], distinguished, for example, by presence of a broken symmetry or even quantum topological order in high energy eigenstates.…”

Entanglement entropy on the Cayley tree

“…Thus, for the EE of a branch of the CT there is a huge change between the ground-state EE and the excited-state EE not seen in more standard models. This may lead to a huge influence of localization on low-lying excitations for interacting systems [20]. (For noninteracting systems the localization is studied below in section 5.…”

“…It has been shown that EE is a very useful measure for locating and analyzing quantum phase transitions [14][15][16][17][18][19][20]. Ground-state EE typically scales like the boundary area of the region [14,15].…”