DISS. ETH No. 18124 Continuous-Time Quantum Monte Carlo Algorithms for Fermions A dissertation submitted to ETH ZURICH for the degree of Doctor of Sciences presented by Emanuel Gull Dipl. Phys. ETH born 3. Aug. 1980 citizen of Schwerzenbach ZH accepted on the recommendation of Prof. Dr. M. Troyer, examiner Prof. Dr. A. Georges, co-examiner Prof. Dr. P. Werner, co-examiner 2008 Contents 1 Introduction 9 2 Dynamical Mean Field Theory 13 2.1 SingleSiteDMFT ............................ 14 2.2 Single Impurity Anderson Model (SIAM) – Hamiltonian . ... 14 2.2.1 ImpuritySolvers ........................ 15 2.2.2 SelfConsistencyEquations . 16 2.3 SelfConsistencyLoop.......................... 19 2.4 MultipleOrbitalExtensions. 20 2.5 ClusterExtensions ............................ 21 2.5.1 CDMFT ............................. 22 2.5.2 DCA ............................... 24 2.6 Alternative Methods for the Solution of the Impurity Model .... 25 2.6.1 Numerical Renormalization Group (NRG) Solvers . 25 2.6.2 ExactDiagonalization(ED)Solvers . 26 2.6.3 DMRG.............................. 26 2.6.4 Discrete Time QMC Solver – The Hirsch-Fye Algorithm . 26 2.7 InteractionRepresentation. 28 2.8 ActionFormulation ........................... 29 2.9 Continuous-TimeQMCAlgorithms . 30 3 Monte Carlo sampling of partition functions 32 3.1 MonteCarloIntegration.... .... .... .... .... .... 32 3.1.1 ImportanceSampling. 34 3.2 MarkovprocessandMetropolisAlgorithm . 35 3.3 Partition Function Expansions – the Sampling of Infinite Series . 36 3.4 TheSignProblem ............................ 40 4 Weak Coupling algorithm 43 4.1 Single Impurity Anderson Model – Partition Function Expansion . 43 4.1.1 EnsuringPositivityoftheWeights . 44 4.1.2 MultipleOrbitalProblems . 46 4.1.3 ClusterCalculations . 47 4.2 Updates .................................. 47 4.2.1 InsertionandRemovalUpdates . 48 CONTENTS 3 4.2.2 ShiftUpdates .......................... 48 4.3 Measurements .............................. 49 4.4 ImplementationNotes .... .... .... .... .... .... .. 50 4.4.1 FactorizationoftheMeasurement. 50 4.4.2 Frequency versus Imaginary Time Measurement . 51 4.4.3 SelfEnergyBinningMeasurement . 52 4.4.4 DiscretizationoftheExponentialFactors . 53 4.4.5 DCAMomentumMeasurements . 53 4.5 Green’sFunction(“Worm”)–Sampling . 53 4.5.1 DerivationoftheWormAlgorithm . 54 4.5.2 WormUpdates.......................... 55 4.5.3 Measurements.......................... 57 4.5.4 ImplementationNotes . 58 4.5.5 Normalization.......................... 59 4.5.6 Reweighing ........................... 59 4.5.7 Results .............................. 60 4.6 WangLandauSampling ......................... 60 4.6.1 TheClassicalWang-LandauAlgorithm. 61 4.6.2 QuantumWangLandau .. .... .... .... .... .. 63 5 Continuous-Time Auxiliary Field Algorithm 65 5.1 LatticeVersion–HamiltonianFormulation . ... 65 5.2 Updates .................................. 69 5.2.1 SpinflipUpdates......................... 70 5.2.2 InsertionandRemovalUpdates . 70 5.3 Measurements .............................. 71 5.3.1 MeasurementoftheGreen’sFunction . 71 5.3.2 FourPointFunctions . 72 5.3.3 Role of the Expansion Parameter K –PotentialEnergy . 73 5.4 SignProblem ............................... 74 5.5 ImpurityVersion–ActionFormulation . 75 6 Hybridization algorithm 77 6.1 PartitionFunctionExpansion . 77 6.1.1 NoninteractingCase . 79 6.1.2 Interactions–Density-DensityCase . 81 6.2 Updates .................................. 82 6.3 Measurements .............................. 84 6.4 ImplementationNotes .... .... .... .... .... .... .. 86 7 Hybridization Algorithm – General Interactions 88 7.1 PartitionFunctionExpansion . 88 7.1.1 ConfigurationSpace .... .... .... .... .... .. 89 7.1.2 Computationof theHybridization Determinant . 90 7.1.3 ComputationoftheTrace . 90 4 CONTENTS 7.2 Measurements .............................. 92 7.3 ImplementationNotes . .... .... .... .... .... .... 93 7.4 TreeAlgorithm.............................. 94 8 Performance Comparison 98 8.1 MatrixSize ................................ 98 8.2 AccuracyforConstantCPUTime . 99 8.2.1 KineticandPotentialEnergy . 100 8.2.2 Green’sFunctionandSelfEnergy . 101 8.3 AwayfromHalfFilling .... .... .... .... .... .... .103 8.4 ScalingConsiderations . .104 9 Local Order and the gapped phase of the Hubbard model 106 10 LDA + DMFT – Beyond Model Hamiltonians 115 10.1 LDA–SuccessandLimitation . .116 10.2 LDA+DMFT–theMethod .......................116 10.3 SelfConsistencyLoop. .118 10.4 DoubleCountingCorrections. 119 10.4.1 Double Counting Correction in the Hamiltonian . 119 10.4.2 HartreeTerm ..........................119 10.5 Hybridization...............................120 10.6 Approximations .............................121 10.7 SourcesofErrors.............................122 10.8 CeriumModelandHamiltonian . 122 10.8.1 ThePhysicsofCerium . .123 10.8.2 CeriumResults .... .... .... .... .... .... .124 11 Spin freezing transition in a 3-orbital model 125 A Inverse Matrix Formulas 132 A.1 InversionbyPartitioning . .132 A.2 Spinflips..................................134 B Fourier Transforms in Continuous-Time DMFT Algorithms 136 B.1 Green’sFunctioninFrequencySpace . 136 B.2 SplineInterpolation ...........................137 B.3 FourierTransformusingtheHighFrequencyTail . 138 B.4 Density-DensityMulti-OrbitalProblems . 139 B.4.1 Lattice Green’s Function and Self Energy Terms . 140 B.4.2 Bare Green’s Function and Hybridization Function . 140 B.4.3 SelfEnergyCalculation . .141 Abstract The numerical investigation of strongly correlated electron systems is a funda- mental goal of modern condensed-matter systems. Unfortunately, standard meth- ods to solve the many-body system have their limitations: straight-forward exact the problem is only possible for a small number of sites, as the effort grows ex- ponentially in the number of sites [1]. Standard lattice Monte Carlo methods fail, as the sign problem [2, 3] makes low temperatures and large systems inaccessi- ble. Other methods like the density-matrix renormalization group theory [4], are limited to the ground state of (quasi) one-dimensional systems. A useful and numerically feasible approach to treat fermionic systems in the thermodynamic limit is the so-called dynamical mean field theory or DMFT. De- velopment of this field started with the demonstration by M¨uller-Hartmann and by Metzner and Vollhardt [5, 6] that the diagrammatics of lattice models of in- teracting fermions simplifies dramatically in an appropriately chosen infinite di- mensional (or infinite coordination) limit. This insight was developed by Georges, Kotliar and co-workers [7, 8] who showed that if the momentum dependence of the electronic self-energy may be neglected (Σ p,ω Σ ω ), as occurs in the infinite coordination number limit, then the solution( ) of → the(lattice) model may be obtained from the solution of a quantum impurity model plus a self-consistency condition. In this thesis, we explain recent algorithmic improvements in the field of fermionic lattice Monte Carlo solvers and their application. These novel solvers, known as continuous-time solvers, are able to solve the impurity problems or- ders of magnitude more efficiently than previous attempts [9] and therefore open new horizons to the field. All impurity solvers described herein have been imple- mented and tested thoroughly as part of this thesis, and one algorithm has been newly developed [10]. We then apply these algorithms to physical problems: The four-site DCA method of including intersite correlations in the dynamical mean field theory is used to investigate the metal-insulator transition in the Hubbard model. At half filling a gap-opening transition is found to occur as the interaction strength is in- creased beyond a critical value. The gapped behavior found in the 4-site DCA approximation is shown to be associated with the onset of strong antiferromag- netic and singlet correlations and the transition is found to be potential energy driven. It is thus more accurately described as a Slater phenomenon (induced by strong short ranged order) than as a Mott phenomenon. Doping the gapped phase leads to a non-Fermi-liquid state with a Fermi surface only in the nodal regions 6 CONTENTS and a pseudogap in the antinodal regions at lower dopings x 0.15 and to a Fermi liquid phase at higher dopings. ≲ A single-site dynamical mean field study of a three band model with the ro- tationally invariant interactions appropriate to the t2g levels of a transition metal oxide reveals a quantum phase transition between a paramagnetic metallic phase and an incoherent metallic phase with frozen moments. The Mott transitions oc- curring at electron densities n=2,3 per site take place inside the frozen moment phase. The critical line separating the two phases is characterized by a self energy with the frequency dependence Σ ω √ω and a broad quantum critical regime. The findings are discussed in the( context) ∼ of the power law observed in the optical conductivity of SrRuO3. Finally, a simulation on Cerium using the realistic band structure and interac- tion matrix is used to revisit the properties of the α - and γ - phase of Cerium. Using LDA+DMFT-techniques we obtain spectra for the two phases and observe the development of a Kondo resonance as well as crystal field splitting effects. This thesis on continuous-time algorithms is arranged in two parts: the first part presents algorithms and their implementation as well as an introduction to the quantum mechanical framework needed to derive them. The second part uses these algorithms and applies