† These authors contributed equally to the calculations in this work.Competing inhomogeneous orders are a central feature of correlated electron materials including the high-temperature superconductors. The twodimensional Hubbard model serves as the canonical microscopic physical model for such systems. Multiple orders have been proposed in the underdoped part of the phase diagram, which corresponds to a regime of maximum numerical difficulty. By combining the latest numerical methods in exhaustive simulations, we uncover the ordering in the underdoped ground state. We find 1 arXiv:1701.00054v3 [cond-mat.str-el]

Numerical results for ground state and excited state properties (energies, double occupancies, and Matsubara-axis self energies) of the single-orbital Hubbard model on a two-dimensional square lattice are presented, in order to provide an assessment of our ability to compute accurate results in the thermodynamic limit. Many methods are employed, including auxiliary field quantum Monte Carlo, bare and bold-line diagrammatic Monte Carlo, method of dual fermions, density matrix embedding theory, density matrix renormalization group, dynamical cluster approximation, diffusion Monte Carlo within a fixed node approximation, unrestricted coupled cluster theory, and multireference projected Hartree-Fock. Comparison of results obtained by different methods allows for the identification of uncertainties and systematic errors. The importance of extrapolation to converged thermodynamic limit values is emphasized. Cases where agreement between different methods is obtained establish benchmark results that may be useful in the validation of new approaches and the improvement of existing methods. arXiv:1505.02290v2 [cond-mat.str-el] 15

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Motivated by recent developments in conformal field theory (CFT), we devise a Quantum Monte Carlo (QMC) method to calculate the moments of the partially transposed reduced density matrix at finite temperature. These are used to construct scale invariant combinations that are related to the negativity, a true measure of entanglement for two intervals embedded in a chain. These quantities can serve as witnesses of criticality. In particular, we study several scale invariant combinations of the moments for the 1D hard-core boson model. For two adjacent intervals unusual finite size corrections are present, showing parity effects that oscillate with a filling dependent period. These are more pronounced in the presence of boundaries. For large chains we find perfect agreement with CFT. Oppositely, for disjoint intervals corrections are more severe and CFT is recovered only asymptotically. Furthermore, we provide evidence that their exponent is the same as that governing the corrections of the mutual information. Additionally we study the 1D Bose-Hubbard model in the superfluid phase. Remarkably, the finite-size effects are smaller and QMC data are already in impressive agreement with CFT at moderate large sizes.

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We perform large-scale quantum Monte Carlo (QMC) simulations for strongly interacting bosons in a twodimensional optical lattice trap and confirm an excellent agreement with the benchmarking in situ density measurements by the Chicago group [N. Gemelke et al., Nature (London) 460, 995 (2009)]. We further present a general finite-temperature phase diagram for both the uniform and the trapped systems, demonstrating how the universal scaling properties near the superfluid-to-Mott insulator transition can be observed from the in situ density profile. The characteristic temperature to find such quantum criticality is estimated to be of the order of the single-particle bandwidth, which should be achievable in the present experiments. Finally, we examine the validity regime of the local fluctuation-dissipation theorem, which can be a used as a thermometry in the strongly interacting regime.In this recent decade, systems of ultracold atoms have become the most promising candidate for the realization of quantum simulators, which are designed to explore various challenging and exotic many-body physics. One of the most well-understood ultracold systems is made of bosonic atoms loaded in an optical lattice, whereby the three-dimensional (3D) superfluid-to-Mott insulator (SF-MI) phase transition had been observed in time-of-flight (TOF) experiments [1] and then directly compared to theory with great accuracy [2]. However, the TOF images do not directly represent physical quantities of atoms inside the optical lattice and thus subject the quantitative study of critical phenomena to question, especially given the complexity of the expansion dynamics [3,4]. As a result, several schemes have been proposed to determine the critical properties based on density-related quantities [5-8], which have been successfully measured in a bosonic system loaded into a 2D optical lattice [9,10]. Before these theoretical schemes could be trusted and applied, an essential step is to validate them from the first-principles calculation in the parameter regime of realistic experiments.In this paper, we perform an ab initio quantum Monte Carlo (QMC) simulation [11] to have a direct comparison with the in situ data of Chicago group [9] in a 2D optical lattice with a harmonic trap and find excellent agreement. We then calculate results for both trapped and uniform systems to confirm the general validity of the local density approximation (LDA) in a 2D system, except near the phase boundary. We further quantitatively demonstrate the universal scaling properties near the SF-MI quantum phase transition point through the finite temperature density profile in a trapped system. The characteristic temperature, T * , to find such scaling is about the single particle bandwidth. Finally, we show the regime where the local fluctuation-dissipation theorem (FDT) is valid, providing useful thermometry in the strongly interacting regime. Our work therefore paves the way for the future experimental realization of quantum simulators.Our numerical simulation starts fro...

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We present a numerical scheme to reconstruct a subset of the entanglement spectrum of quantum many body systems using quantum Monte Carlo. The approach builds on the replica trick to evaluate particle number resolved traces of the first n of powers of a reduced density matrix. From this information we reconstruct n entanglement spectrum levels using a polynomial root solver. We illustrate the power and limitations of the method by an application to the extended Bose-Hubbard model in one dimension where we are able to resolve the quasidegeneracy of the entanglement spectrum in the Haldane-insulator phase. In general, the method is able to reconstruct the largest few eigenvalues in each symmetry sector and typically performs better when the eigenvalues are not too different.