Showing papers by "Matthias Troyer published in 2011"

Continuous-time Monte Carlo methods for quantum impurity models

Abstract: Quantum impurity models describe an atom or molecule embedded in a host material with which it can exchange electrons. They are basic to nanoscience as representations of quantum dots and molecular conductors and play an increasingly important role in the theory of "correlated electron" materials as auxiliary problems whose solution gives the "dynamical mean field" approximation to the self energy and local correlation functions. These applications require a method of solution which provides access to both high and low energy scales and is effective for wide classes of physically realistic models. The continuous-time quantum Monte Carlo algorithms reviewed in this article meet this challenge. We present derivations and descriptions of the algorithms in enough detail to allow other workers to write their own implementations, discuss the strengths and weaknesses of the methods, summarize the problems to which the new methods have been successfully applied and outline prospects for future applications.

The ALPS project release 2.0: open source software for strongly correlated systems

Abstract: We present release 2.0 of the ALPS (Algorithms and Libraries for Physics Simulations) project, an open source software project to develop libraries and application programs for the simulation of strongly correlated quantum lattice models such as quantum magnets, lattice bosons, and strongly correlated fermion systems. The code development is centered on common XML and HDF5 data formats, libraries to simplify and speed up code development, common evaluation and plotting tools, and simulation programs. The programs enable non-experts to start carrying out serial or parallel numerical simulations by providing basic implementations of the important algorithms for quantum lattice models: classical and quantum Monte Carlo (QMC) using non-local updates, extended ensemble simulations, exact and full diagonalization (ED), the density matrix renormalization group (DMRG) both in a static version and a dynamic time-evolving block decimation (TEBD) code, and quantum Monte Carlo solvers for dynamical mean field theory (DMFT). The ALPS libraries provide a powerful framework for programmers to develop their own applications, which, for instance, greatly simplify the steps of porting a serial code onto a parallel, distributed memory machine. Major changes in release 2.0 include the use of HDF5 for binary data, evaluation tools in Python, support for the Windows operating system, the use of CMake as build system and binary installation packages for Mac OS X and Windows, and integration with the VisTrails workflow provenance tool. The software is available from our web server at http://alps.comp-phys.org/.

Stripes in the two-dimensional t-J model with infinite projected entangled-pair states

Abstract: We simulate the t-J model in two dimensions by means of infinite projected entangled-pair states (iPEPS) generalized to arbitrary unit cells, finding results similar to those previously obtained by the density-matrix renormalization group (DMRG) for wide ladders. In particular, we show that states exhibiting stripes, that is, a unidirectional modulation of hole-density and antiferromagnetic order with a p-phase shift between adjacent stripes, have a lower variational energy than uniform phases predicted by variational and fixed-node Monte Carlo simulations. For a fixed unit-cell size the energy per hole is minimized for a hole density rho(l) similar to 0.5 per unit length of a stripe. The superconducting order parameter is maximal around rho(l) similar to 0.75-0.8.

Implementing global Abelian symmetries in projected entangled-pair state algorithms

Abstract: Due to the unfavorable scaling of tensor-network methods with the refinement parameter M, new approaches are necessary to improve the efficiency of numerical simulations based on such states, in particular for gapless, strongly entangled systems. In one-dimensional density matrix renormalization group methods, the use of Abelian symmetries has led to large computational gain. In higher-dimensional tensor networks, this is associated with significant technical efforts and additional approximations. We explain a formalism to implement such symmetries in two-dimensional tensor-network states and present benchmark results that confirm the validity of these approximations in the context of projected entangled-pair state algorithms.

Thermodynamics of the 3D Hubbard model on approaching the Néel transition.

Abstract: We study the thermodynamic properties of the 3D Hubbard model for temperatures down to the Neel temperature by using cluster dynamical mean-field theory. In particular, we calculate the energy, entropy, density, double occupancy, and nearest-neighbor spin correlations as a function of chemical potential, temperature, and repulsion strength. To make contact with cold-gas experiments, we also compute properties of the system subject to an external trap in the local density approximation. We find that an entropy per particle S/N approximate to 0.65(6) at U/t = 8 is sufficient to achieve a Neel state in the center of the trap, substantially higher than the entropy required in a homogeneous system. Precursors to antiferromagnetism can clearly be observed in nearest-neighbor spin correlators.

Simultaneous Dimerization and SU(4) Symmetry Breaking of 4-Color Fermions on the Square Lattice

Abstract: Using infinite projected entangled-pair states, exact diagonalization, and flavor-wave theory, we show that the SU(4) Heisenberg model undergoes a spontaneous dimerization on the square lattice, in contrast with its SU(2) and SU(3) counterparts, which develop Neel and three-sublattice stripelike long-range order. Since the ground state of a dimer is not a singlet for SU(4) but a 6-dimensional irreducible representation, this leaves the door open for further symmetry breaking. We provide evidence that, unlike in SU(4) ladders, where dimers pair up to form singlet plaquettes, here the SU(4) symmetry is additionally broken, leading to a gapless spectrum in spite of the broken translational symmetry.

The ALPS project release 2.0: Open source software for strongly correlated systems

Abstract: We present release 2.0 of the ALPS (Algorithms and Libraries for Physics Simulations) project, an open source software project to develop libraries and application programs for the simulation of strongly correlated quantum lattice models such as quantum magnets, lattice bosons, and strongly correlated fermion systems. The code development is centered on common XML and HDF5 data formats, libraries to simplify and speed up code development, common evaluation and plotting tools, and simulation programs. The programs enable non-experts to start carrying out serial or parallel numerical simulations by providing basic implementations of the important algorithms for quantum lattice models: classical and quantum Monte Carlo (QMC) using non-local updates, extended ensemble simulations, exact and full diagonalization (ED), the density matrix renormalization group (DMRG) both in a static version and a dynamic time-evolving block decimation (TEBD) code, and quantum Monte Carlo solvers for dynamical mean field theory (DMFT). The ALPS libraries provide a powerful framework for programers to develop their own applications, which, for instance, greatly simplify the steps of porting a serial code onto a parallel, distributed memory machine. Major changes in release 2.0 include the use of HDF5 for binary data, evaluation tools in Python, support for the Windows operating system, the use of CMake as build system and binary installation packages for Mac OS X and Windows, and integration with the VisTrails workflow provenance tool. The software is available from our web server at this http URL.

A Provenance-Based Infrastructure to Support the Life Cycle of Executable Papers

Abstract: As publishers establish a greater online presence as well as infrastructure to support the distribution of more varied information, the idea of an executable paper that enables greater interaction has developed. An executable paper provides more information for computational experiments and results than the text, tables, and figures of standard papers. Executable papers can bundle computational content that allow readers and reviewers to interact, validate, and explore experiments. By including such content, authors facilitate future discoveries by lowering the barrier to reproducing and extending results. We present an infrastructure for creating, disseminating, and maintaining executable papers. Our approach is rooted in provenance, the documentation of exactly how data, experiments, and results were generated. We seek to improve the experience for everyone involved in the life cycle of an executable paper. The automated capture of provenance information allows authors to easily integrate and update results into papers as they write, and also helps reviewers better evaluate approaches by enabling them to explore experimental results by varying parameters or data. With a provenance-based system, readers are able to examine exactly how a result was developed to better understand and extend published findings.

Microscopic models of interacting Yang–Lee anyons

Abstract: Collective states of interacting non-Abelian anyons have recently been studied mostly in the context of certain fractional quantum Hall states, such as the Moore-Read state proposed to describe the physics of the quantum Hall plateau at filling fraction = 5/2. In this paper, we further expand this line of research and present non-unitary generalizations of interacting anyon models. In particular, we introduce the notion of Yang-Lee anyons, discuss their relation to the so-called 'Gaffnian' quantum Hall wave function and describe an elementary model for their interactions. A one-dimensional (1D) version of this model— a non-unitary generalization of the original golden chain model—can be fully understood in terms of an exact algebraic solution and numerical diagonalization. We discuss the gapless theories of these chain models for general su(2)k anyonic theories and their Galois conjugates. We further introduce and solve a 1D version of the Levin-Wen model for non-unitary Yang-Lee anyons.

Dynamical mean-field theory for bosons

Abstract: We discuss the recently developed bosonic dynamical mean-field (B-DMFT) framework, which maps a bosonic lattice model onto the selfconsistent solution of a bosonic impurity model with coupling to a reservoir of normal and condensed bosons. The effective impurity action is derived in several ways: (i) as an approximation to the kinetic energy functional of the lattice problem, (ii) using a cavity approach, and (iii) by using an effective medium approach based on adding a one-loop correction to the selfconsistently defined condensate. To solve the impurity problem, we use a continuous-time Monte Carlo algorithm based on a sampling of a perturbation expansion in the hybridization functions and the condensate wave function. As applications of the formalism we present finite temperature B-DMFT phase diagrams for the bosonic Hubbard model on a 3d cubic and 2d square lattice, the condensate order parameter as a function of chemical potential, critical exponents for the condensate, the approach to the weakly interacting Bose gas regime for weak repulsions, and the kinetic energy as a function of temperature.

Mutual information in classical spin models

Abstract: The total many-body correlations present in finite temperature classical spin systems are studied using the concept of mutual information. As opposed to zero-temperature quantum phase transitions, the total correlations are not maximal at the phase transition, but reach a maximum in the high-temperature paramagnetic phase. The Shannon mutual information and the Renyi mutual information in both Ising and Potts models in two dimensions are calculated numerically by combining matrix product state algorithms and Monte Carlo sampling techniques.

Dynamical mean-field theory for bosons

Abstract: We discuss the recently developed bosonic dynamical mean-field theory (B-DMFT) framework, which maps a bosonic lattice model onto the self-consistent solution of a bosonic impurity model with coupling to a reservoir of normal and condensed bosons. The effective impurity action is derived in several ways: (i) as an approximation to the kinetic energy functional of the lattice problem, (ii) using a cavity approach and (iii) using an effective medium approach based on adding a one-loop correction to the self-consistently defined condensate. To solve the impurity problem, we use a continuous-time Monte Carlo algorithm based on the sampling of a perturbation expansion in the hybridization functions and the condensate wave function. As applications of the formalism, we present finite-temperature B-DMFT phase diagrams for the bosonic Hubbard model on a three-dimensional (3D) cubic and a 2D square lattice, the condensate order parameter as a function of chemical potential, critical exponents for the condensate, the approach to the weakly interacting Bose gas regime for weak repulsions and the kinetic energy as a function of temperature.

Two-dimensional quantum liquids from interacting non-Abelian anyons

Abstract: A set of localized, non-Abelian anyons?such as vortices in a px+ipy superconductor or quasiholes in certain quantum Hall states?gives rise to a macroscopic degeneracy. Such a degeneracy is split in the presence of interactions between the anyons. Here, we show that in two spatial dimensions this splitting selects a unique collective state as ground state of the interacting many-body system. This collective state can be a novel gapped quantum liquid nucleated inside the original parent liquid (of which the anyons are excitations). This physics is of relevance for any quantum Hall plateau realizing a non-Abelian quantum Hall state when moving off the center of the plateau.

Spectral properties of the three-dimensional Hubbard model

Abstract: We present momentum-resolved single-particle spectra for the three-dimensional Hubbard model for the paramagnetic and antiferromagnetically ordered phase obtained within the dynamical cluster approximation. The effective cluster problem is solved by continuous-time quantum Monte Carlo simulations. The absence of a time discretization error and the ability to perform Monte Carlo measurements directly in Matsubara frequencies enable us to analytically continue the self-energies by maximum entropy, which is essential to obtaining momentum-resolved spectral functions for the N\'eel state. We investigate the dependence on temperature and interaction strength and the effect of magnetic frustration introduced by a next-nearest-neighbor hopping. One particular question we address here is the influence of the frustrating interaction on the metal-insulator transition of the three-dimensional Hubbard model.

Dynamics at and near conformal quantum critical points

Abstract: We explore the dynamical behavior at and near a special class of two-dimensional quantum critical points. Each is a conformal quantum critical point (CQCP), where in the scaling limit the equal-time correlators are those of a two-dimensional conformal field theory. The critical theories include the square-lattice quantum dimer model, the quantum Lifshitz theory, and a deformed toric code model. We show that under generic perturbation the latter flows toward the ordinary Lorentz-invariant ($2+1$)-dimensional Ising critical point, illustrating that CQCPs are generically unstable. We exploit a correspondence between the classical and quantum-dynamical behavior in such systems to perform an extensive numerical study of two lines of CQCPs in a quantum eight-vertex model or, equivalently, two coupled deformed toric codes. We find that the dynamical critical exponent $z$ remains 2 along the U(1)-symmetric quantum Lifshitz line, while it continuously varies along the line with only ${\mathbb{Z}}_{2}$ symmetry. This illustrates how two CQCPs can have very different dynamical properties, despite identical equal-time ground-state correlators. Our results equally apply to the dynamics of the corresponding purely classical models.

Quantum spin ladders of non-Abelian anyons

Abstract: Quantum ladder models, consisting of coupled chains, form intriguing systems bridging one and two dimensions and have been well studied in the context of quantum magnets and fermionic systems Here we consider ladder systems made of more exotic quantum mechanical degrees of freedom, so-called non-Abelian anyons, which can be thought of as certain quantum deformations of ordinary SU(2) spins Such non-Abelian anyons occur as quasiparticle excitations in topological quantum fluids, including ${p}_{x}+i{p}_{y}$ superconductors, certain fractional quantum Hall states, and rotating Bose-Einstein condensates Here we use a combination of exact diagonalization and conformal field theory to determine the phase diagrams of ladders with up to four chains We discuss how phenomena familiar from ordinary SU(2) spin ladders are generalized in their anyonic counterparts, such as gapless and gapped phases, odd and even effects with the ladder width, and elementary ``magnon'' excitations Other features are entirely due to the topological nature of the anyonic degrees of freedom In general, two-dimensional systems of interacting localized non-Abelian anyons are anyonic generalizations of two-dimensional quantum magnets

Diagrammatic quantum Monte Carlo solution of the two-dimensional cooperon-fermion model

Abstract: We investigate the two-dimensional cooperon-fermion model in the correlated regime with a continuous-time diagrammatic determinant quantum Monte Carlo algorithm. We estimate the transition temperature ${T}_{c}$, examine the effectively reduced band gap and cooperon mass, and find that delocalization of the cooperons enhances the diamagnetism. When applied to diamagnetism of the pseudogap phase in high-${T}_{c}$ cuprates, we obtain results in qualitative agreement with recent torque magnetization measurements.

Thermodynamics of the 3D Hubbard model on approach to the Neel transition

Abstract: We study the thermodynamic properties of the 3D Hubbard model for temperatures down to the Néel temperature by using cluster dynamical mean-field theory. In particular, we calculate the energy, entropy, density, double occupancy, and nearest-neighbor spin correlations as a function of chemical potential, temperature, and repulsion strength. To make contact with cold-gas experiments, we also compute properties of the system subject to an external trap in the local density approximation. We find that an entropy per particle S/N ≈ 0.65(6) at U/t = 8 is sufficient to achieve a Néel state in the center of the trap, substantially higher than the entropy required in a homogeneous system. Precursors to antiferromagnetism can clearly be observed in nearest-neighbor spin correlators.

Trapped ultracold bosons in periodically modulated lattices

Abstract: Motivated by the recent rapid development of the field of quantum gases in optical lattices, we present a comprehensive study of the spectrum of ultracold atoms in a one-dimensional optical lattice subjected to a periodic lattice modulation. Using the time-dependent density matrix renormalization group method, we study the dynamical response due to lattice modulations in different quantum phases of the system with varying density. For the Mott-insulating state, we identify several excitation processes, which provide important information about the density profile of the gases. For the superfluid, the dynamical response can be well described in a local density approximation. This simplification can be valuable in understanding the strong-correlated superfluid in a slow-varying harmonic potential. All these spectroscopic features of an inhomogeneous system can be used as a test for the validity of the Bose-Hubbard model in a parabolic trapping potential.

Identifying quantum topological phases through statistical correlation

Abstract: nature of the symmetries defining each sector. We then use the indistinguishability to underscore a key difference between topological order in the toric code and the FQH regime. By diagonalizing models of the FQH effect we show that distincttopologicalsectors(anddistinctFQHstatesingeneral) differ in that symmetry operators must span the entire system rather than just one-dimensional operators. The measure can be used to identify mechanisms of topological ordering in more nontrivial models where symmetries and a complete characterization of states have not been performed. In Sec. II we review the indistinguishability as a measure of distinct quantum orders. In Sec. III we examine the scaling behavior of the indistinguishability in the toric code. In Sec. IV we examine the scaling of the indistinguishability in FQH models of the Laughlin, charge density wave 16 (CDW), and Moore-Read 17 states. We summarize in Sec. V with a comparison of results for both sets of models.

Publishing Provenance-rich Scientific Papers.

Abstract: Complete documentation and reproducibility of results are important goals for scientific publications. Standard scientific papers, however, usually contain only final results and document only parameters and processing steps that the authors considered important enough. By recording the complete provenance history of the data leading to a publication one can overcome this limitation and allow reproducibility for reviewers, publishers and readers of scientific publications. While the process of capturing provenance information is a growing research subject, here we discuss usually overlooked challenges involved in publishing provenance-complete papers. We report on our experience in preparing and publishing two specific executable papers where we used the VisTrails workflow system to embed full provenance information of the paper and discuss open challenges and issues we encountered.

Topological Phases: An Expedition off Lattice

Abstract: Motivated by the goal to give the simplest possible microscopic foundation for a broad class of topological phases, we study quantum mechanical lattice models where the topology of the lattice is one of the dynamical variables. However, a fluctuating geometry can remove the separation between the system size and the range of local interactions, which is important for topological protection and ultimately the stability of a topological phase. In particular, it can open the door to a pathology, which has been studied in the context of quantum gravity and goes by the name of `baby universe', Here we discuss three distinct approaches to suppressing these pathological fluctuations. We complement this discussion by applying Cheeger's theory relating the geometry of manifolds to their vibrational modes to study the spectra of Hamiltonians. In particular, we present a detailed study of the statistical properties of loop gas and string net models on fluctuating lattices, both analytically and numerically.