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arXiv:1701.00121 · Physical Review Letters

Nonlocal Parity Order in the Two-Dimensional Mott Insulator

Serena Fazzini, Federico Becca, Arianna Montorsi

Published Apr 12, 2017• 18 citations
arXivHTMLPDF

Abstract

The Mott insulator is characterized by having small deviations around the (integer) average particle density n, with pairs with n-1 and n+1 particles forming bound states. In one dimension, the effect is captured by a nonzero value of a nonlocal "string" of parities, which instead vanishes in the superfluid phase where density fluctuations are large. Here, we investigate the interaction induced transition from the superfluid to the Mott insulator, in the paradigmatic Bose Hubbard model at n=1. By means of quantum Monte Carlo simulations and finite size scaling analysis on L×M ladders, we explore the behavior of "brane" parity operators from one dimension (i.e., M=1 and L→∞) to two dimensions (i.e., M→∞ and L→∞). We confirm the conjecture that, adopting a standard definition, their average value decays to zero in two dimensions also in the insulating phase, evaluating the scaling factor of the "perimeter law" [S. P. Rath et al., Ann. Phys. (Berlin) 334, 256 (2013)ANPYA20003-380410.1016/j.aop.2013.04.006]. Upon introducing a further phase in the brane parity, we show that its expectation value becomes nonzero in the insulator, while still vanishing at the transition to the superfluid phase. These quantities are directly accessible to experimental measures, thus providing an insightful signature of the Mott insulator.

DOI
10.1103/physrevlett.118.157602

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The Mott insulator is characterized by having small deviations around the (integer) average particle density n, with pairs with n-1 and n+1 particles forming bound states. In one dimension, the effect is captured by a nonzero value of a nonlocal "string" of parities, which instead vanishes in the superfluid phase where density fluctuations are large. Here, we investigate the interaction induced transition from the superfluid to the Mott insulator, in the paradigmatic Bose Hubbard model at n=1. By means of quantum Monte Carlo simulations and finite size scaling analysis on L×M ladders, we explore the behavior of "brane" parity operators from one dimension (i.e., M=1 and L→∞) to two dimensions (i.e., M→∞ and L→∞). We confirm the conjecture that, adopting a standard definition, their average value decays to zero in two dimensions also in the insulating phase, evaluating the scaling factor of the "perimeter law" [S. P. Rath et al., Ann. Phys. (Berlin) 334, 256 (2013)ANPYA20003-380410.1016/j.aop.2013.04.006]. Upon introducing a further phase in the brane parity, we show that its expectation value becomes nonzero in the insulator, while still vanishing at the transition to the superfluid phase. These quantities are directly accessible to experimental measures, thus providing an insightful signature of the Mott insulator.
The Mott insulating state of the Hubbard model at half filling could be depicted as a spin liquid of singly occupied sites with holon-doublon quantum fluctuations localized in pairs. In one dimension the behavior is captured by a finite value of the charge parity string correlator, which fails to remain finite when generalized to higher dimensions. We recover a definition of parity brane correlator which may remain nonvanishing in the presence of interchain coupling, by assigning an appropriate fractional phase to the parity breaking fluctuations. In the case of Hubbard ladders at half filling, we find that the charge parity brane is nonzero at any repulsive value of interaction. The spin-parity brane instead becomes nonvanishing in the even-leg case, in correspondence to the onset of the spin gapped D-Mott phase, which is absent in the odd-leg case. The behavior of the parity correlators is also analyzed by means of a numerical DMRG analysis of the one- and two-leg ladder.
The Mott-insulating phase of the two-dimensional (2D) Bose-Hubbard model is expected to be characterized by a nonlocal brane parity order.Parity order captures the presence of microscopic particlehole fluctuations and entanglement, whose properties depend on the underlying lattice geometry.We realize 2D Bose-Hubbard models in dynamically tunable lattice geometries, using neutral atoms in a passively phase-stable tunable optical lattice in combination with programmable site-blocking potentials.We benchmark the performance of our system by single-particle quantum walks in the square, triangular, kagome, and Lieb lattices.In the strongly correlated regime, we microscopically characterize the geometry dependence of the quantum fluctuations and experimentally validate brane parity as a proxy for the nonlocal order parameter signaling the superfluid-to-Mott-insulating phase transition.
Nonlocal order parameters capture the presence of correlated fluctuations between specific degrees of freedom, in otherwise disordered quantum matter. Here, we provide a further example of their fundamental role, deriving the ground state phase diagram of the filling one extended Bose-Hubbard model, exclusively in terms of their ordering. By means of a density matrix renormalization group numerical analysis, we show that in addition to the (even) parity order characteristic of the Mott insulating phase and the string order nonvanishing in the Haldane insulator, the recently proposed odd parity order completes the picture, becoming nonvanishing at the transition from the normal superfluid to the paired superfluid phase. The above three nonlocal parameters capture all the distinct phases, including the density wave phase, in which the local order is seen as the simultaneous presence of correlated fluctuations in different channels. They provide a unique tool for the experimental observation of the full phase diagram of strongly correlated quantum matter, by means of local density measurements.
We characterize the Mott-insulator and Luther-Emery phases of the 1D Hubbard model through correlators that measure the parity of spin and charge strings along the chain. These nonlocal quantities order in the corresponding gapped phases and vanish at the critical point U(c)=0, thus configuring as hidden order parameters. The Mott insulator consists of bound doublon-holon pairs, which in the Luther-Emery phase turn into electron pairs with opposite spins, both unbinding at U(c). The behavior of the parity correlators is captured by an effective free spinless fermion model.
The phase diagrams and phase transitions of bosons with short-ranged repulsive interactions moving in periodic and/or random external potentials at zero temperature are investigated with emphasis on the superfluid-insulator transition induced by varying a parameter such as the density. Bosons in periodic potentials (e.g., on a lattice) at T=0 exhibit two types of phases: a superfluid phase and Mott insulating phases characterized by integer (or commensurate) boson densities, by the existence of a gap for particle-hole excitations, and by zero compressibility. Generically, the superfluid onset transition in d dimensions from a Mott insulator to superfluidity is ``ideal,'' or mean field in character, but at special multicritical points with particle-hole symmetry it is in the universality class of the (d+1)-dimensional XY model. In the presence of disorder, a third, ``Bose glass'' phase exists. This phase is insulating because of the localization effects of the randomness and analogous to the Fermi glass phase of interacting fermions in a strongly disordered potential.The Bose glass phase is characterized by a finite compressibility, no gap, but an infinite superfluid susceptibility. In the presence of disorder the transition to superfluidity is argued to occur only from the Bose glass phase, and never directly from the Mott insulator. This zero-temperature superfluid-insulator transition is studied via generalizations of the Josephson scaling relation for the superfluid density at the ordinary \ensuremath{\lambda} transition, highlighting the crucial role of quantum fluctuations. The transition is found to have a dynamic critical exponent z exactly equal to d and correlation length and order-parameter correlation exponents \ensuremath{\nu} and \ensuremath{\eta} which satisfy the bounds \ensuremath{\nu}\ensuremath{\ge}2/d and \ensuremath{\eta}\ensuremath{\le}2-d, respectively. It is argued that the superfluid-insulator transition in the presence of disorder may have an upper critical dimension dc{d}_{c}dc​ which is infinite, but a perturbative renormalization-group calculation wherein the critical exponents have mean-field values for weak disorder above d=4 is also discussed. Many of these conclusions are verified by explicit calculations on a model of one-dimensional bosons in the presence of both random and periodic potentials. The general results are applied to experiments on 4He^{4}\mathrm{He}4He absorbed in porous media such as Vycor. Some measurable properties of the superfluid onset are predicted exactly [e.g., the exponent x relating the \ensuremath{\lambda} transition temperature to the zero-temperature superfluid density is found to be d/2(d-1)], while stringent bounds are placed on others. Analysis of preliminary data is consistent with these predictions.
Order parameters represent a fundamental resource to characterize quantum matter. We show that pair superfluids can be rigorously defined in terms of a nonlocal order parameter, named odd parity, which derivation is experimentally accessible by local density measurements. As a case of study, we first investigate a constrained Bose-Hubbard model at different densities, both in one and two spatial dimensions. Here, our analysis finds pair superfluidity for relatively strong attractive interactions. The odd parity operator acts as the unique order parameter for such phase irrespectively to the density of the system and its dimensionality in regimes of total particle number conservation. In order to enforce our finding, we confirm the generality of our approach also on a two-component Bose-Hubbard Hamiltonian, which experimental realization represents a timely topic in ultracold atomic systems. Our results shed new light on the role of correlated density fluctuations in pair superfluids. In addition, they provide a powerful tool for the experimental detection of such exotic phases and the characterization of their transition to the atomic superfluid phase.