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We investigate quantum order by disorder in a frustrated spin nanotube formed by wrapping a $J_1$-$J_2$ Heisenberg model at 45$^\circ$ around a cylinder. Using Schwinger boson theory and Density Matrix Renormalization Group (DMRG), we have computed the ground-state phase diagram to reveal a $\mathbb{Z}_2$ phase in which collinear spin stripes form a right or left handed helix around the nanotube. We have derived an analytic estimate for the critical $\eta_c=J_1/2J_2$ of the $\mathbb{Z}_2$ helical phase transition, which is in agreement with the DMRG results. By evaluating the entanglement spectrum and nonlocal string order parameters we discuss the topology of the $\mathbb{Z}_2$-helical phase., Comment: 16 pages, 13 figures

The idea of breaking time-translation symmetry has fascinated humanity at least since ancient proposals of the perpetuum mobile. Unlike the breaking of other symmetries, such as spatial translation in a crystal or spin rotation in a magnet, time translation symmetry breaking (TTSB) has been tantalisingly elusive. We review this history up to recent developments which have shown that discrete TTSB does takes place in periodically driven (Floquet) systems in the presence of many-body localization. Such Floquet time-crystals represent a new paradigm in quantum statistical mechanics --- that of an intrinsically out-of-equilibrium many-body phase of matter. We include a compendium of necessary background, before specializing to a detailed discussion of the nature, and diagnostics, of TTSB. We formalize the notion of a time-crystal as a stable, macroscopic, conservative clock --- explaining both the need for a many-body system in the infinite volume limit, and for a lack of net energy absorption or dissipation. We also cover a range of related phenomena, including various types of long-lived prethermal time-crystals, and expose the roles played by symmetries -- exact and (emergent) approximate -- and their breaking. We clarify the distinctions between many-body time-crystals and other ostensibly similar phenomena dating as far back as the works of Faraday and Mathieu. En route, we encounter Wilczek's suggestion that macroscopic systems should exhibit TTSB in their ground states, together with a theorem ruling this out. We also analyze pioneering recent experiments detecting signatures of time crystallinity in a variety of different platforms, and provide a detailed theoretical explanation of the physics in each case. In all existing experiments, the system does not realize a `true' time-crystal phase, and we identify necessary ingredients for improvements in future experiments., Invited Review article; 70 pages

International audience

Recent work suggests that a sharp definition of `phase of matter' can be given for periodically driven `Floquet' quantum systems exhibiting many-body localization. In this work we propose a classification of the phases of interacting Floquet localized systems with (completely) spontaneously broken symmetries -- we focus on the one dimensional case, but our results appear to generalize to higher dimensions. We find that the different Floquet phases correspond to elements of $Z(G)$, the centre of the symmetry group in question. In a previous paper we offered a companion classification of unbroken, i.e., paramagnetic phases., Published version

Fractionalisation -- the breaking up of an apparently indivisible microscopic degree of freedom -- is one of the most counterintuitive phenomena in many-body physics. Here we study its most fundamental manifestation in spin ice, the only known fractionalised magnetic compound in 3D: we directly visualise the $1/r^2$ magnetic Coulomb field of monopoles which emerge as the atomic magnetic dipoles fractionalise. We analyse the internal magnetic field distribution, relevant for local experimental probes. In particular, we present new zero-field NMR measurements which exhibit excellent agreement with the calculated lineshapes, noting that this experimental technique can in principle measure directly the monopole density in spin ice. The distribution of field strengths is captured by a simple analytical form which exhibits a low density of low-field sites---in apparent disagreement with reported $\mu$SR results. Counterintuitively, the density of low-field locations decreases as the local ferromagnetic correlations imposed by the ice rules weaken., Comment: (7 pages; 6 figures; v2 revised version for publication)

A set of recent results indicates that fractionally filled bands of Chern insulators in two dimensions support fractional quantum Hall states analogous to those found in fractionally filled Landau levels. We provide an understanding of these results by examining the algebra of Chern band projected density operators. We find that this algebra closes at long wavelengths and for constant Berry curvature, whereupon it is isomorphic to the W-Infinity algebra of lowest Landau level projected densities first identified by Girvin, MacDonald and Platzman [Phys. Rev. B 33, 2481 (1986).] For Hamiltonians projected to the Chern band this provides a route to replicating lowest Landau level physics on the lattice., 4 pages; v2 contains minor changes, including brief discussion of disorder, version published in Phys. Rev. B (Rapid Communications)

The quest for quantum computers is motivated by their potential for solving problems that defy existing, classical, computers. The theory of computational complexity, one of the crown jewels of computer science, provides a rigorous framework for classifying the hardness of problems according to the computational resources, most notably time, needed to solve them. Its extension to quantum computers allows the relative power of quantum computers to be analyzed. This framework identifies families of problems which are likely hard for classical computers (``NP-complete'') and those which are likely hard for quantum computers (``QMA-complete'') by indirect methods. That is, they identify problems of comparable worst-case difficulty without directly determining the individual hardness of any given instance. Statistical mechanical methods can be used to complement this classification by directly extracting information about particular families of instances---typically those that involve optimization---by studying random ensembles of them. These pose unusual and interesting (quantum) statistical mechanical questions and the results shed light on the difficulty of problems for large classes of algorithms as well as providing a window on the contrast between typical and worst case complexity. In these lecture notes we present an introduction to this set of ideas with older work on classical satisfiability and recent work on quantum satisfiability as primary examples. We also touch on the connection of computational hardness with the physical notion of glassiness., 25 pages, 8 figures. Lecture notes for lectures given by R. Moessner at the Les Houches School on "Modern theories of correlated electron systems", May 11-29, 2009

Alongside the effort underway to build quantum computers, it is important to better understand which classes of problems they will find easy and which others even they will find intractable. We study random ensembles of the QMA$_1$-complete quantum satisfiability (QSAT) problem introduced by Bravyi. QSAT appropriately generalizes the NP-complete classical satisfiability (SAT) problem. We show that, as the density of clauses/projectors is varied, the ensembles exhibit quantum phase transitions between phases that are satisfiable and unsatisfiable. Remarkably, almost all instances of QSAT for any hypergraph exhibit the same dimension of the satisfying manifold. This establishes the QSAT decision problem as equivalent to a, potentially new, graph theoretic problem and that the hardest typical instances are likely to be localized in a bounded range of clause density., 9 pages, 3 figures

We investigate the finite temperature magnetothermoelectric response in the vicinity of a superfluid--Mott insulator quantum phase transition. We focus on the particle-hole symmetric transitions of the Bose--Hubbard model, and combine Lorentz invariance arguments with quantum Boltzmann calculations. By means of an epsilon expansion, we find that a non-vanishing thermoelectric tensor and a finite thermal conductivity are supported in this quantum critical regime. We comment on the singular Nernst effect in this problem., 5 pages

We calculate the magnetization of a two-dimensional superconductor in a perpendicular magnetic field near its Kosterlitz-Thouless transition and at lower temperatures. We find that the critical behavior is more complex than assumed in the literature and that, in particular, the critical magnetization is {\it not} field independent as naive scaling predicts. In the low temperature phase we find a substantial fluctuation renormalization of the mean-field result. We compare our analysis with the data on the cuprates., 8 pages, 3 figures

Scaling arguments imply that quantum critical points exhibit universal non-linear responses to external probes. We investigate the origins of such non-linearities in transport, which is especially problematic since the system is necessarily driven far from equilibrium. We argue that for a wide class of systems the new ingredient that enters is the Schwinger mechanism--the production of carriers from the vacuum by the applied field-- which is then balanced against a scattering rate which is itself set by the field. We show by explicit computation how this works for the case of the symmetric superfluid-Mott insulator transition of bosons.

We construct a family of rotationally invariant, local, S=1/2 Klein Hamiltonians on various lattices that exhibit ground state manifolds spanned by nearest-neighbor valence bond states. We show that with selected perturbations such models can be driven into phases modeled by well understood quantum dimer models on the corresponding lattices. Specifically, we show that the perturbation procedure is arbitrarily well controlled by a new parameter which is the extent of decoration of the reference lattice. This strategy leads to Hamiltonians that exhibit i) $Z_2$ RVB phases in two dimensions, ii) U(1) RVB phases with a gapless ``photon'' in three dimensions, and iii) a Cantor deconfined region in two dimensions. We also construct two models on the pyrochlore lattice, one model exhibiting a $Z_2$ RVB phase and the other a U(1) RVB phase., 16 pages, 15 figures; 1 figure and some references added; some minor typos fixed

Spin ice in a magnetic field in the [111] direction displays two magnetization plateaux, one at saturation and an intermediate one with finite entropy. We study the crossovers between the different regimes from a point of view of (entropically) interacting defects. We develop an analytical theory for the nearest-neighbor spin ice model, which covers most of the magnetization curve. We find that the entropy is non-monotonic, exhibiting a giant spike between the two plateaux. This regime is described by a monomer-dimer model with tunable fugacities. At low fields, we develop an RG treatment for the extended string defects, and we compare our results to extensive Monte Carlo simulations. We address the implications of our results for cooling by adiabatic (de)magnetization., 13 pages, 13 figures

Physical Review Letters

We study the quantum dimer model on the triangular lattice, which is expected to describe the singlet dynamics of frustrated Heisenberg models in phases where valence bond configurations dominate their physics. We find, in contrast to the square lattice, that there is a truly short ranged resonating valence bond (RVB) phase with no gapless collective excitations and with deconfined, gapped, spinons for a {\it finite} range of parameters. We also establish the presence of three crystalline phases in this system., 4 pages, 2 figures, Revtex 3.0

We propose a device, consisting of a Hall bar with two weak barriers, that can be used to study quantum interference effects in a strongly correlated system. We show how the device provides a way of measuring the fractional charge and fractional statistics of quasiparticles in the quantum Hall effect through an anomalous Aharanov-Bohm period. We discuss how this disentangling of the charge and statistics can be accomplished by measurements at fixed filling factor and at fixed density. We also discuss a another type of interference effect that occurs in the nonlinear regime as the source-drain voltage is varied. The period of these oscillations can also be used to measure the fractional charge, and details of the oscillations patterns, in particular the position of the nodes, can be used to distinguish between Fermi-liquid and Luttinger-liquid behavior. We illustrate these ideas by computing the conductance of the device in the framework of edge state theory and use it to estimate parameters for the experimental realization of this device., 17 pages, 4 embedded eps figures

We have studied the effect of including nearest-neighbor, electron-electron interactions, in particular the off-diagonal (non density-density) terms, on the spectra of truncated tetrahedral and icosahedral ``Hubbard molecules,'' focusing on the relevance of these systems to the physics of doped C$_{60}$. Our perturbation theoretic and exact diagonalization results agree with previous work in that the density-density term suppresses pair-binding. However, we find that for the parameter values of interest for $C_{60}$ the off-diagonal terms {\em enhance} pair-binding, though not enough to offset the suppression due to the density-density term. We also find that the critical interaction strengths for the Hund's rules violating level crossings in C$_{60}^{-2}$, C$_{60}^{-3}$ and C$_{60}^{-4}$ are quite insensitive to the inclusion of these additional interactions., 20p + 5figs, Revtex 3.0, UIUC preprint P-94-10-085