Monte Carlo Study of a U(1)×U(1)U(1)\times U(1) Loop Model with Modular Invariance

We study a $U(1)\times U(1)$ system in (2+1)-dimensions with long-range interactions and mutual statistics. The model has the same form after the application of operations from the modular group, a property which we call modular invariance. Using the modular invariance of the model, we propose a possible phase diagram. We obtain a sign-free reformulation of the model and study it in Monte Carlo. This study confirms our proposed phase diagram. We use the modular invariance to analytically determine the current-current correlation functions and conductivities in all the phases in the diagram, as well as at special "fixed" points which are unchanged by an operation from the modular group. We numerically determine the order of the phase transitions, and find segments of second-order transitions. For the statistical interaction parameter $\theta=\pi$, these second-order transitions are evidence of a critical loop phase obtained when both loops are trying to condense simulataneously. We also measure the critical exponents of the second-order transitions.Comment: 14 pages, 13 figure

Many body localization and thermalization: insights from the entanglement spectrum

We study the entanglement spectrum in the many body localizing and thermalizing phases of one and two dimensional Hamiltonian systems, and periodically driven `Floquet' systems. We focus on the level statistics of the entanglement spectrum as obtained through numerical diagonalization, finding structure beyond that revealed by more limited measures such as entanglement entropy. In the thermalizing phase the entanglement spectrum obeys level statistics governed by an appropriate random matrix ensemble. For Hamiltonian systems this can be viewed as evidence in favor of a strong version of the eigenstate thermalization hypothesis (ETH). Similar results are also obtained for Floquet systems, where they constitute a result `beyond ETH', and show that the corrections to ETH governing the Floquet entanglement spectrum have statistical properties governed by a random matrix ensemble. The particular random matrix ensemble governing the Floquet entanglement spectrum depends on the symmetries of the Floquet drive, and therefore can depend on the choice of origin of time. In the many body localized phase the entanglement spectrum is also found to show level repulsion, following a semi-Poisson distribution (in contrast to the energy spectrum, which follows a Poisson distribution). This semi-Poisson distribution is found to come mainly from states at high entanglement energies. The observed level repulsion only occurs for interacting localized phases. We also demonstrate that equivalent results can be obtained by calculating with a single typical eigenstate, or by averaging over a microcanonical energy window - a surprising result in the localized phase. This discovery of new structure in the pattern of entanglement of localized and thermalizing phases may open up new lines of attack on many body localization, thermalization, and the localization transition.Comment: 17 pages, 20 figure

Characterizing the many-body localization transition through the entanglement spectrum

We numerically explore the many body localization (MBL) transition through the lens of the {\it entanglement spectrum}. While a direct transition from localization to thermalization is believed to obtain in the thermodynamic limit (the exact details of which remain an open problem), in finite system sizes there exists an intermediate `quantum critical' regime. Previous numerical investigations have explored the crossover from thermalization to criticality, and have used this to place a numerical {\it lower} bound on the critical disorder strength for MBL. A careful analysis of the {\it high energy} part of the entanglement spectrum (which contains universal information about the critical point) allows us to make the first ever observation in exact numerics of the crossover from criticality to MBL and hence to place a numerical {\it upper bound} on the critical disorder strength for MBL.Comment: 4 pages+appendi

Phases and phase transitions in a U(1) × U(1) system with θ = 2π/3 mutual statistics

We study a U(1) × U(1) system with short-range interactions and mutual θ = 2π/3 statistics in (2+1) dimensions. We are able to reformulate the model to eliminate the sign problem and perform a Monte Carlo study. We find a phase diagram containing a phase with only small loops and two phases with one species of proliferated loop. We also find a phase where both species of loop condense, but without any gapless modes. Lastly, when the energy cost of loops becomes small, we find a phase that is a condensate of bound states, each made up of three particles of one species and a vortex of the other. We define several exact reformulations of the model that allow us to precisely describe each phase in terms of gapped excitations. We propose field-theoretic descriptions of the phases and phase transitions, which are particularly interesting on the “self-dual” line where both species have identical interactions. We also define irreducible responses useful for describing the phases

Composite fermions in bands with N-fold rotational symmetry

We study the effect of band anisotropy with discrete rotational symmetry $C_N$ (where $N\ge 2$) in the quantum Hall regime of two-dimensional electron systems. We focus on the composite Fermi liquid (CFL) at half filling of the lowest Landau level. We find that the magnitude of anisotropy transferred to the composite fermions decreases very rapidly with $N$. We demonstrate this by performing density matrix normalization group calculations on the CFL, and comparing the anisotropy of the composite fermion Fermi contour with that of the (non-interacting) electron Fermi contour at zero magnetic field. We also show that the effective interaction between the electrons after projecting into a single Landau level is much less anisotropic than the band, a fact which does not depend on filling and thus has implications for other quantum Hall states as well. Our results confirm experimental observations on anisotropic bands with warped Fermi contours, where the only detectable effect on the composite Fermi contour is an elliptical distortion ($N = 2$).Comment: 6 pages + bibliography, 5 figure

Monte Carlo Study of a U(1)xU(1) system with \pi-statistical Interaction

We study a $U(1)\times U(1)$ system with two species of loops with mutual $\pi$-statistics in (2+1) dimensions. We are able to reformulate the model in a way that can be studied by Monte Carlo and we determine the phase diagram. In addition to a phase with no loops, we find two phases with only one species of loop proliferated. The model has a self-dual line, a segment of which separates these two phases. Everywhere on the segment, we find the transition to be first-order, signifying that the two loop systems behave as immiscible fluids when they are both trying to condense. Moving further along the self-dual line, we find a phase where both loops proliferate, but they are only of even strength, and therefore avoid the statistical interactions. We study another model which does not have this phase, and also find first-order behavior on the self-dual segment.Comment: 4 pages, 5 figure

Exact realization of Integer and Fractional Quantum Hall Phases in U(1) × U(1) models in (2 + 1)d

In this work we present a set of microscopic U(1) × U(1) models which realize insulating phases with a quantized Hall conductivity σ_(xy). The models are defined in terms of physical degrees of freedom, and can be realized by local Hamiltonians. For one set of these models, we find that σ_(xy) is quantized to be an even integer. The origin of this effect is a condensation of objects made up of bosons of one species bound to a single vortex of the other species. For other models, the Hall conductivity can be quantized as a rational number times two. For these systems, the condensed objects contain bosons of one species bound to multiple vortices of the other species. These systems have excitations carrying fractional charges and non-trivial mutual statistics. We present sign-free reformulations of these models which can be studied in Monte Carlo, and we use such reformulations to numerically detect a gapless boundary between the quantum Hall and trivial insulator states. We also present the broader phase diagrams of the models