A short introduction to numerical linked-cluster expansions

Tang, Baoming; Khatami, Ehsan; Rigol, Marcos

doi:10.1016/j.cpc.2012.10.008

Cited by 109 publications

(141 citation statements)

References 19 publications

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“…20,21,[29][30][31][32] In its original formulation, which includes all connected clusters up to a given size, one encounters a bottleneck arising from the task of computing every possible way these clusters can be embedded on a lattice; this severely limits the maximum cluster size (to about 16 sites). 32 Here we follow the approach of Ref. 20 and consider only rectangular clusters, relevant for the calculation of quantities on square-lattice systems.…”

Section: Methods For Computing the Corner Coefficientmentioning

confidence: 99%

Corner contribution to the entanglement entropy of strongly interacting O(2) quantum critical systems in 2+1 dimensions

et al. 2014

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In a D = 2+1 quantum critical system, the entanglement entropy across a boundary with a corner contains a subleading logarithmic scaling term with a universal coefficient. It has been conjectured that this coefficient is, to leading order, proportional to the number of field components N in the associated O(N ) continuum φ 4 field theory. Using density matrix renormalization group calculations combined with the powerful numerical linked cluster expansion technique, we confirm this scenario for the O(2) Wilson-Fisher fixed point in a striking way, through direct calculation at the quantum critical points of two very different microscopic models. The value of this corner coefficient is, to within our numerical precision, twice the coefficient of the Ising fixed point. Our results add to the growing body of evidence that this universal term in the Rényi entanglement entropy reflects the number of low-energy degrees of freedom in a system, even for strongly interacting theories.

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Section: Methods For Computing the Corner Coefficientmentioning

confidence: 99%

Corner contribution to the entanglement entropy of strongly interacting O(2) quantum critical systems in 2+1 dimensions

et al. 2014

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“…32). These considerations are closely related to the existence of linked-cluster-theorems for degenerate systems where non-local terms corresponding to un-linked diagrams cancel [33][34][35] . In section III A 4 we provide numerical evidence that this is indeed the case by excluding disconnected terms from the pertubative expansions.…”

Section: Third and Higher Order Correctionsmentioning

confidence: 99%

Parafermionic clock models and quantum resonance

et al. 2017

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We explore the ZN parafermionic clock-model generalisations of the p-wave Majorana wire model. In particular we examine whether zero-mode operators analogous to Majorana zero-modes can be found in these models when one introduces chiral parameters to break time reversal symmetry. The existence of such zero-modes implies N -fold degeneracies throughout the energy spectrum. We address the question directly through these degeneracies by characterising the entire energy spectrum using perturbation theory and exact diagonalisation. We find that when N is prime, and the length L of the wire is finite, the spectrum exhibits degeneracies up to a splitting that decays exponentially with system size, for generic values of the chiral parameters. However, at particular parameter values (resonance points), band crossings appear in the unperturbed spectrum that typically result in a splitting of the degeneracy at finite order. We find strong evidence that these preclude the existence of strong zero-modes for generic values of the chiral parameters. In particular we show that in the thermodynamic limit, the resonance points become dense in the chiral parameter space. When N is not prime, the situation is qualitatively different, and degeneracies in the energy spectrum are split at finite order in perturbation theory for generic parameter values, even when the length of the wire L is finite. Exceptions to these general findings can occur at special "anti-resonant" points. Here the evidence points to the existence of strong zero modes and, in the case of the achiral point of the the N = 4 model, we are able to construct these modes exactly. There has recently been growing interest in a class of Z N symmetric one dimensional lattice models known as parafermion chain or quantum clock models 1-6 . These models generalise the Kitaev wire model 7 which exhibits localised unpaired Majorana zero-modes at each end. The recent surge of interest is inspired in part by proposals for their physical realisation and their potential application to universal topological quantum computation 8-10 , something which is not possible with Majorana zero-modes.Clock-like systems of this type have been studied earlier 11,12 , and much is known about the exactly solvable chiral Potts models that occur at special values of the coupling constants (see e.g. Refs 13 and 14). From the perspective of topological quantum computation however, there are a number of interesting open problems surrounding the topological degeneracies and potential zero-modes of the models. In particular, one may ask whether zero-modes exist which result in topological degeneracies throughout the energy spectrum 3,5 . In this context one speaks of strong vs. weak zero-modes (see e.g. Ref. 15) and the question is important because degeneracies that exist at energies above the ground-state potentially allow for topologically fault-tolerant quantum devices at higher temperatures 16 . This area is of course also interesting on a fundamental level, as it addresses if and when decoupled/fr...

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“…NLCEs use the same basis as hightemperature expansions, however, properties of clusters are calculated via exact diagonalisation, as opposed to a perturbative expansion in powers of the inverse temperature [8,33]. The site-based NLCE for the Hubbard model [34] is implemented here for a three-dimensional lattice and carried out to the eighth order for all thermodynamic quantities, except for S θ , where due to the reduced symmetry, only seven orders were obtained.…”

Section: Numerical Calculationsmentioning

confidence: 99%

“…The convergence region extends to significantly lower T /t at n = 1 and generally improves by increasing the interaction strength. At lower T /t, we take advantage of numerical resummations, such as Euler and Wynn transformations [33], to obtain an estimate. The NLCE provides a fast tool, which, given the value of U/t, generates results on a dense temperature and chemical potential grid in a single run.…”

Section: Numerical Calculationsmentioning

confidence: 99%

Observation of antiferromagnetic correlations in the Hubbard model with ultracold atoms

Hart

Duarte

Yang

et al. 2015

Nature

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408

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Ultracold atoms in optical lattices have great potential to contribute to a better understanding of some of the most important issues in many-body physics, such as high-temperature (high-Tc) superconductivity [1]. Thirty years ago, Anderson suggested that the Hubbard model, a simplified representation of fermions moving on a periodic lattice, may contain the essence of copper oxide superconductivity [2]. The Hubbard model describes many of the features shared by the copper oxides, including an interaction-driven Mott insulating state and an antiferromagnetic (AFM) state. Optical lattices filled with a two-spin-component Fermi gas of ultracold atoms can faithfully realise the Hubbard model with readily tunable parameters, and thus provide a platform for the systematic exploration of its phase diagram [3,4]. Realisation of strongly correlated phases, however, has been hindered by the need to cool the atoms to temperatures as low as the magnetic exchange energy, and also by the lack of reliable thermometry [5]. Here we demonstrate spin-sensitive Bragg scattering of light to measure AFM spin correlations in a realisation of the three-dimensional (3D) Hubbard model at temperatures down to 1.4 times that of the AFM phase transition. This temperature regime is beyond the range of validity of a simple high-temperature series expansion, which brings our experiment close to the limit of the capabilities of current numerical techniques. We reach these low temperatures using a unique compensated optical lattice technique [6], in which the confinement of each lattice beam is compensated by a blue-detuned laser beam. The temperature of the atoms in the lattice is deduced by comparing the light scattering to determinantal quantum Monte Carlo [7] (DQMC) and numerical linked-cluster expansion [8] (NLCE) calculations. Further refinement of the compensated lattice may produce even lower temperatures which, along with light scattering thermometry, would open avenues for achieving and characterising other novel quantum states of matter, such as the pseudogap regime of the 2D Hubbard model.A two-spin-component Fermi gas in a simple cubic optical lattice may be described by a single-band Hubbard model with nearest-neighbour tunnelling t and on-site interaction U > 0. At a density n of one atom per site, and for sufficiently large U/t there is a crossover from a 'metallic' state to a Mott insulating regime [9] as the temperature T is reduced below U . The Mott regime has been demonstrated with ultracold atoms in an optical lattice by observing the reduction of doubly occupied sites [10] and the related reduction of the global compressibility [11]. For T below the Néel ordering temperature T N , which for U t is approximately equal to the exchange energy J = 4t 2 /U , the system undergoes a phase transition to an AFM state [12]. In the context of quantum simulations, AFM phases of Ising spins have been previously engineered with bosonic atoms in an optical lattice [13] and with spin-1 2 ions [14,15]. Also, nearest-neighbour AFM correlat...

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A short introduction to numerical linked-cluster expansions

Cited by 109 publications

References 19 publications

Corner contribution to the entanglement entropy of strongly interacting O(2) quantum critical systems in 2+1 dimensions

Corner contribution to the entanglement entropy of strongly interacting O(2) quantum critical systems in 2+1 dimensions

Parafermionic clock models and quantum resonance

Observation of antiferromagnetic correlations in the Hubbard model with ultracold atoms

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