## Non-Ergodic Phases of Matter

Random matrix theory (RMT) has tremendous range of applications in physics spanning from the spectra of heavy nuclei to physics of glassy matter. This mathematical concept uses the symmetries and the topological structure of the underlying system in order to describe it statistically (taking account of the effects of impurities on average) in terms of the ensemble of matrices with random entries. The standard RMT ensembles are given by the matrices with identically independently distributed (i.i.d.) Gaussian random elements. These are usually called Wigner-Dyson (WD) ensembles. Random matrix theory has been successfully used in physics for describing various universal phenomena.

Indeed, RMT provides a very powerful analytical approach to describe the thermalizing many-body quantum systems and sets the basis for the eigenstate thermalization hypothesis (ETH). It provides an excellent statistical description of eigenvalues and eigenstate structure in the configuration (Hilbert) space in terms of the matrix ensemble with random entries. Given by the matrices with i.i.d. Gaussian random elements, RMT takes account of only universal symmetries and the topological structure of underlying systems, diminishing all system-specific details.

In applications, as opposed to thermalization, out-of-equilibrium and ergodicity-breaking quantum phenomena inevitably appear and play a crucial role. Thus, we are now at a stage when essentially quantum properties are used in proposed microscopic devices, in what has been called the “second quantum revolution”, involving devices that intrinsically rely on creating and manipulating quantum states, while unlike the equilibrium case, there is no framework of comparable generality to ETH to describe it.

The main difference from the ergodic (Wigner-Dyson) random-matrix ensembles and the ones describing ergodicity breaking and localization is the basis preference due to the enhanced amplitude of the potential energy fluctuations across the sites. The simplest modification of ergodic ensembles is the Rosenzweig-Porter RMT. It is also a complete graph of the Gaussian i.i.d. entries (like the WD), but with the special, stronger fluctuating diagonal matrix elements setting the above basis preference. Recently, my coauthors and I have found that the above model contains an additional phase beyond the Anderson localized and ergodic ones. We discovered that this model shows an entire phase of robust fractal states which was later confirmed with the mathematical rigor.

The presence of the non-ergodic delocalized states both in the MBL problems (in the Hilbert space) and in the Rosenzweig-Porter random matrix ensemble provide the link between these systems and hints that one can develop even some kind of more generic statistical mapping between the structures of their states. The same is also true for the disordered models on graphs being the prototypes of the MBL. This constitutes a prerequisite for the current research. If possible, this statistical mapping will open a simple way to describe the non-ergodic physics of the disordered many-body models in the Hilbert space and on the disordered hierarchical graphs, similar in spirit of how the Wigner-Dyson RMT describes the ergodic/chaotic behavior.

## Key Papers

**A random matrix model with localization and ergodic transitions** - V. E. Kravtsov, I. M. Khaymovich, E. Cuevas, M. Amini

**Survival probability in Generalized Rosenzweig-Porter random matrix ensemble** - G. De Tomasi, M. Amini, S. Bera, I. M. Khaymovich, V. E. Kravtsov

**Fragile ergodic phases in logarithmically-normal Rosenzweig-Porter model** - I. M. Khaymovich, V. E. Kravtsov, B. L. Altshuler, L. Ioffe

**Dynamical phases in a "multifractal" Rosenzweig-Porter model** - I. M. Khaymovich, V. E. Kravtsov

**Localization transition on the Random Regular Graph as an unstable tricritical point in a log-normal Rosenzweig-Porter random matrix ensemble** - V. E. Kravtsov, I. M. Khaymovich, B. L. Altshuler, L. B. Ioffe