Shivaji Sondhi

Abstract:

We initiate the study of the classical mechanics of nonrelativistic fractons in its simplest setting—that of identical one-dimensional particles with local Hamiltonians characterized by a conserved dipole moment in addition to the usual symmetries of space and time translation invariance. We introduce a family of models and study the N -body problem for them. We find that locality leads to a “Machian” dynamics in which a given particle exhibits finite inertia only if within a specified distance of another particle. For well-separated particles, this dynamics leads to immobility, much as for quantum models of fractons discussed before. For two or more particles within inertial reach of each other at the start of motion, we obtain an interesting interplay of inertia and interactions. Specifically, for a solvable “inertia only” model of fractons, we find that two particles always become immobile at long times. Remarkably, three particles generically evolve to a late time state with one immobile particle and two oscillating about a common center of mass with generalizations of such “Machian clusters” for N>3. Interestingly, these Machian clusters exhibit physical limit cycles in a Hamiltonian system even though mathematical limit cycles are forbidden by Liouville's theorem.

Abstract:

The Luttinger liquid (LL) model of one-dimensional (1D) electronic systems provides a powerful tool for understanding strongly correlated physics, including phenomena such as spin–charge separation1. Substantial theoretical efforts have attempted to extend the LL phenomenology to two dimensions, especially in models of closely packed arrays of 1D quantum wires2,3,4,5,6,7,8,9,10,11,12,13, each being described as a LL. Such coupled-wire models have been successfully used to construct two-dimensional (2D) anisotropic non-Fermi liquids2,3,4,5,6, quantum Hall states7,8,9, topological phases10,11 and quantum spin liquids12,13. However, an experimental demonstration of high-quality arrays of 1D LLs suitable for realizing these models remains absent. Here we report the experimental realization of 2D arrays of 1D LLs with crystalline quality in a moiré superlattice made of twisted bilayer tungsten ditelluride (tWTe2). Originating from the anisotropic lattice of the monolayer, the moiré pattern of tWTe2 hosts identical, parallel 1D electronic channels, separated by a fixed nanoscale distance, which is tuneable by the interlayer twist angle. At a twist angle of approximately 5 degrees, we find that hole-doped tWTe2 exhibits exceptionally large transport anisotropy with a resistance ratio of around 1,000 between two orthogonal in-plane directions. The across-wire conductance exhibits power-law scaling behaviours, consistent with the formation of a 2D anisotropic phase that resembles an array of LLs. Our results open the door for realizing a variety of correlated and topological quantum phases based on coupled-wire models and LL physics.

Abstract:

Quantum many-body systems display rich phase structure in their low-temperature equilibrium states¹. However, much of nature is not in thermal equilibrium. Remarkably, it was recently predicted that out-of-equilibrium systems can exhibit novel dynamical phases^2-8 that may otherwise be forbidden by equilibrium thermodynamics, a paradigmatic example being the discrete time crystal (DTC)^7,9-15. Concretely, dynamical phases can be defined in periodically driven many-body-localized (MBL) systems via the concept of eigenstate order^7,16,17. In eigenstate-ordered MBL phases, the entire many-body spectrum exhibits quantum correlations and long-range order, with characteristic signatures in late-time dynamics from all initial states. It is, however, challenging to experimentally distinguish such stable phases from transient phenomena, or from regimes in which the dynamics of a few select states can mask typical behaviour. Here we implement tunable controlled-phase (CPHASE) gates on an array of superconducting qubits to experimentally observe an MBL-DTC and demonstrate its characteristic spatiotemporal response for generic initial states^7,9,10. Our work employs a time-reversal protocol to quantify the impact of external decoherence, and leverages quantum typicality to circumvent the exponential cost of densely sampling the eigenspectrum. Furthermore, we locate the phase transition out of the DTC with an experimental finite-size analysis. These results establish a scalable approach to studying non-equilibrium phases of matter on quantum processors.

Abstract:

The “quantum complexity” of a unitary operator measures the difficulty of its construction from a set of elementary quantum gates. While the notion of quantum complexity was first introduced as a quantum generalization of the classical computational complexity, it has since been argued to hold a fundamental significance in its own right, as a physical quantity analogous to the thermodynamic entropy. In this paper, we present a unified perspective on various notions of quantum complexity, viewed as functions on the space of unitary operators. One striking feature of these functions is that they can exhibit non-smooth and even fractal behaviour. We use ideas from Diophantine approximation theory and sub-Riemannian geometry to rigorously quantify this lack of smoothness. Implications for the physical meaning of quantum complexity are discussed.

Abstract:

We develop a theory of the excitonic phase recently proposed as the zero-field insulating state observed near charge neutrality in monolayer WTe$_2$. Using a Hartree-Fock approximation, we numerically identify two distinct gapped excitonic phases: a spin density wave state for weak but non-zero interaction strength $U_0$, and spin spiral order at larger $U_0$, separated by a narrow window of trivial insulator. We introduce a simplified model capturing essential features of the WTe$_2$ band structure, in which the two phases may be viewed as distinct valley ferromagnetic orders. We link the competition between the two phases to the orbital structure of the electronic wavefunctions at the Fermi surface and hence its proximity to the underlying gapped Dirac point in WTe$_2$. We briefly discuss collective modes of the two excitonic states, and comment on implications for experiments.