Many-body localisation
Based on the eigenstate thermalisation hypothesis (ETH), we understand that ergodic closed quantum many-body systems generally thermalise and admit a thermodynamic description. More recently, however, it has become evident that there exists a large family of systems called many-body localised systems (MBL) which robustly violate the ETH and hence, evade thermalisation. Besides addressing the fate of Anderson localisation in the presence of interactions, MBL systems raise fundamental questions such as their statistical mechanics desccription, dynamics of quantum information in them, and possibilities of novel out-of-equilibrium phases of quantum matter.
My research in this direction has focussed on understanding the mechanisms that can stabilise an MBL phase, classical proxies for the MBL transition, and a statistical mechanics framework for the dynamical phases stabilised by MBL.
Fock-space correlations and MBL
The Hamiltonian of a disordered interacting system can be mapped onto a disordered tight-binding problem on the Fock-space graph of the system. However, the problem of MBL does not simply map onto a conventional Anderson localisation problem on the Fock-space graph. Complications arise due to the non-trivial topology and strong correlations on the Fock-space graph. Using a self-consistent theory on Fock space, we showed that the effective disorder correlations on Fock space play a crucial role in stabilising an MBL phase. In fact, a necessary requirement is that the correlations at finite distances on the graph should be maximal [1]. Indeed, local disordered Hamiltonians known to host an MBL phase can be analytically showed to satisfy this condition.
In another work [2], to analyse the effect of such strong disorder correlations in a more controlled setting, we studied an Anderson localisation problem on trees and random regular graphs where the disorder correlation mimicks that of the Fock space. We found that, despite the correlations, there exists a localised phase akin to the MBL phase with a qualitatively different scaling of the critical disorder with the graph connectivity.
Sthitadhi Roy and David E. Logan, "Fock-space correlations and the origins of many-body localisation" [Phys. Rev. B 101, 134202 (2020) (Editors' Suggestion)], [arXiv:1911.12370]
Sthitadhi Roy and D. E. Logan, "Localisation on certain graphs with strongly correlated disorder", [Phys. Rev. Lett., 125, 250402 (2020)] [arXiv:2007.10357]
MBL induced by local kinetic constraints
We showed that local kinetic constraints serve as a complementary route toward stabilising MBL, even in the presence of uncorrelated disorder on Fock space. Taking inspiration from classical glasses, we imposed an East-glass type constraint on the quantum random energy model (QREM) and found that it induces a fully MBL phase absent in the QREM. On the Fock-space graph, we find that potentially nonresonant bottlenecks in the Fock-space dynamics, caused by spatially local segments of frozen spins due to the constraints, lie at the root of localization.
Sthitadhi Roy and Achilleas Lazarides, "Strong ergodicity breaking due to local constraints in a quantum system", [Phys. Rev. Research 2, 023159 (2020)] [arXiv:1912.06660]
The broken lines denote the links on the Fock-space graph removed by the constraints. The blue and green paths show a representative shortest path between two configurations in the presence and absence of constraints.
The Fock space of a disordered interacting spin chain in the percolating phase (above) and the non-percolating phase below where each colour represents a distinct cluster.
Classical percolation on Fock space and MBL
Taking inspiration from the difference in behaviour of many-body eigenstates in Fock space across the many-body localisation transition, we ask the question that can a classical percolation problem be formulated in Fock space with percolation rules obtained from the microscopic quantum Hamiltonian, wherein a percolation transition and the two phases, percolating and otherwise, capture certain aspects of the many-body localisation problem?
We answer the question in the affirmative and show that the Fock-space cluster sizes behave quite analogously to the quantum eigenstates' participation entropies as well as local observables averaged over the clusters also behave similar to the eigenstate expectation values in the quantum problem.
For a certain disordered spin chain, the percolation problem allows for an exact solution for the critical proprties, and in general mapping the problem of cluster growth to kinetically constrained dynamics using Monte Carlo dynamics allows access to system sizes a couple of orders of magnitude larger than standard numerical techniques.
Sthitadhi Roy, David E. Logan, J. T. Chalker, "Exact solution of a percolation analogue for the many-body localisation transition" [Phys. Rev. B 99, 220201(R) (2019)] [arXiv:1812.05115]
Sthitadhi Roy, J. T. Chalker, David E. Logan, "Percolation in Fock space as a proxy for many-body localisation" [Phys. Rev. B 99, 104206 (2019) (Editors' Suggestion)] [arXiv:1812.06101]
MBL and long-range interactions
We considered the issue of MBL in models with long-ranged interaction, specifically, power-law interactions . On the Fock-space graph, the power law interactions non-trivially modify the scaling of connectivities and disorder correlations with system size, leading to a reach phase diagram. We obtained this phase doagram using an analytical self-consistent theory on Fock space and also using numerical diagnostics. We found that increasing the range of interactions between trans- verse spin components hinders localisation and enhances the critical disorder strength. In marked contrast, increasing the interaction range between longitudinal spin compo- nents is found to enhance localisation and lower the critical disorder.
Sthitadhi Roy and David E. Logan, "Self-consistent theory of many-body localisation in a quantum spin chain with long-range interactions"[SciPost Phys. 7, 042 (2019)] [arXiv:1903.04851]
MBL phase diagram in the space of disorder and power-law exponent of transverse interactions
Frequency dependent potential for a time crystal showing a response at half the frequency of the driving.
MBL and quantum order
Dynamical potentials
Localisation-protected quantum order extends the idea of symmetry breaking and order in ground states to individual eigenstates at arbitrary energy. Examples include many-body localised spin glasses and discrete time crystals. One of my current interests lies in developing a general framework, along the lines of equilibrium statistical mechanics, to study such out-of-equilibrium phases and transitions between them. As a first step, we introduced the notion of dynamical potentials which act as generating functions for such correlations and capture eigenstate phases and order. These potentials show formal similarities to their equilibrium counterparts, namely thermodynamic potentials.
Quantum order at infinite temperature
We further identified that spatiotemporal correlations are rather robust towards studying such phases as they betray eigenstate order even when the system lies in a featureless infinite temperature state. To incorporate such scenarios within the framework dynamical potentials we extended the framework for arbitrary mixed states and show that the potentials show features mirroring those of equilibrium statistical mechanics such as bimodal potentials in the symmetry-broken phase.
The next step in this direction is to come up with approximate but analytical methods of calculating these potentials so as to have a rigorous understanding of the phase transitions in the thermodynamic limit.
Sthitadhi Roy and Achilleas Lazarides, "Nonequilibrium quantum order at infinite temperature: spatiotemporal correlations and their generating functions" [Phys. Rev. B 98, 064208 (2018)] [arXiv:1804.08638]
Sthitadhi Roy, Achilleas Lazarides, Markus Heyl, Roderich Moessner, "Dynamical potentials for non-equilibrium quantum many-body phases"[Phys. Rev. B 97, 205143 (2018)][arXiv:1710.09388]