We consider various two-leg ladder models exhibiting gapped phases. All of these phases have short-ranged valence bond ground states, and they all exhibit string order. However, we show that short-ranged valence bond ground states divide into two topologically distinct classes, and as a consequence, there exist two topologically distinct types of string order. Therefore, not all gapped phases belong to the same universality class. We show that phase transitions occur when we interpolate between models belonging to different topological classes, and we study the nature of these transitions.
In this work, we propose a testing procedure to distinguish between the different approaches for computing complexity. Our test does not require a direct comparison between the approaches and thus avoids the issue of choice of gates, basis, etc. The proposed testing procedure employs the informationtheoretic measures Loschmidt echo and Fidelity; the idea is to investigate the sensitivity of the complexity (derived from the different approaches) to the evolution of states. We discover that only circuit complexity obtained directly from the wave function is sensitive to time evolution, leaving us to claim that it surpasses the other approaches. We also demonstrate that circuit complexity displays a universal behaviour-the complexity is proportional to the number of distinct Hamiltonian evolutions that act on a reference state. Due to this fact, for a given number of Hamiltonians, we can always find the combination of states that provides the maximum complexity; consequently, other combinations involving a smaller number of evolutions will have less than maximum complexity and, hence, will have resources. Finally, we explore the evolution of complexity in non-local theories; we demonstrate the growth of complexity is sustained over a longer period of time as compared to a local theory.
We propose a new diagnostic for quantum chaos. We show that time evolution of complexity for a particular type of target state can provide equivalent information about the classical Lyapunov exponent and scrambling time as out-of-time-order correlators. Moreover, for systems that can be switched from a regular to unstable (chaotic) regime by a tuning of the coupling constant of the interaction Hamiltonian, we find that the complexity defines a new time scale. We interpret this time scale as recording when the system makes the transition from regular to chaotic behaviour.
The realization of the Hofstadter model in a strongly anisotropic ladder geometry has now become possible in one-dimensional optical lattices with a synthetic dimension. In this work, we show how the Hofstadter Hamiltonian in such ladder configurations hosts a topological phase of matter which is radically different from its two-dimensional counterpart. This topological phase stems directly from the hybrid nature of the ladder geometry, and is protected by a properly defined inversion symmetry. We start our analysis considering the paradigmatic case of a three-leg ladder which supports a topological phase exhibiting the typical features of topological states in one dimension: robust fermionic edge modes, a degenerate entanglement spectrum and a non-zero Zak phase; then, we generalize our findings-addressable in the state-of-the-art cold atom experiments-to ladders with an higher number of legs. PACS numbers: 67.85-d, 03.65.Vf The Hofstadter problem [1] describing a particle hopping on a two-dimensional lattice pierced by a magnetic field, is a paradigm of quantum mechanics. Formulated more then forty years ago, it embeds a multitude of semi-nal notions in modern condensed matter physics [2]: topo-logical bands, edge excitations, fractal properties of the spectrum, just to mention some of them. Despite its apparent simplicity and the enormous body of investigation both theoretical and experimental [3-7], the Hofstadter problem still hides some surprises, as we are going to discuss in the following. The motivation of our work stems from the recent realization [8-10] of the Hofstadter Hamiltonian in optical lattices with a synthetic dimension. The possibility of engineering an additional (synthetic) few sites long dimension [11, 12] with non-trivial boundary conditions by using some internal degrees of freedom of the atoms has encouraged the study of the Hofstadter Hamiltonian in a strongly anisotropic geometry [13]. (a) m = 0 m = 1 m = +1 t j = 1 j = 2 j = 3 j = L x j = L x 1 te i te +i ⌦ ⌦ ˜ ⌦ j (b) ⌦/t ˜ ⌦/t t o p o l o g i c a l ⌦ + ⌦ 1 2 3 4 0 1 2 3 4 5 FIG. 1: (a) Schematic representation of the Hofstadter Hamil-tonian on a three-leg ladder; in the presence of a non-vanishing coupling between the extremal states m = +1 and m = −1 (˜ Ωj = 0), the ladder is equivalent to a cylinder. (b) Phase diagram for a three leg-ladder. * simone.barbarino@sns.it Is this a new territory for the Hofstadter problem or are we bound to detect a smooth crossover from a two-to a one-dimensional behavior? A na¨ıvena¨ıve expectation would induce to think that synthetic lattices can simulate a two-dimensional geometry when the transverse dimension is much longer than the correlation length of the system along the transverse dimension itself, and they turn out to be effectively one-dimensional when the number of sites in the transverse direction is small, such as in synthetic ladders. In this work, we show that in the presence of a selected applied magnetic field and for an odd number of legs (L y), this simple picture fails and the H...
The electronic nematic phase occurs when the point-group symmetry of the lattice structure is broken, due to electron-electron interactions. We study a model for the nematic phase on a square lattice with emphasis on the phase transition between isotropic and nematic phases within mean field theory. We find the transition to be first order, with dramatic changes in the Fermi surface topology accompanying the transition. Furthermore, we study the conductivity tensor and Hall constant as probes of the nematic phase and its transition. The relevance of our findings to Hall resistivity experiments in the high-Tc cuprates is discussed.
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