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Quantum Monte Carlo in Condensed Matter Physics:Past, Present, and Future0. Overview1. Past: Algorithms and Algorithm Development World-Line Methods Auxiliary Field Methods2. Present: Results- Where are we? World-Line Methods Auxiliary Field Methods3. Future Algorithmic Challenges Computational Physics Education

MahmudEliasAssmann

0: OVERVIEW: Progress in Superconductivity: Tc versus Time200150Rapid acceleration in field(Note change in scales!)Many new materialsHeavy fermionsCuprates (high Tc )Iron-Pnictides · · ·100504030New mechanismsElectron-phonon mediated Spin mediatedCharge inhomogeneities20101900 1940 19801985 1990 1995 2000 2005 2010

OVERVIEW: Progress in World Line Quantum Monte CarloDisorderedWorm2D Bose HubLoopSupersolidsHeliumSSEDisorderedSpinsAFLRO2D pinLiquidsSupersolidsMultispinTri Bose HubInteractionsBose Liquid DimerizedSpinsRing Exch1D Holstein1D FermionHubbard2D Bose HubCritical ptEntanglementEntropySupersolids2D Bose HubSF MI1D Bose Hub19902000Year2010Same time frame as ‘modern’ SC era:Algorithm development:Initial Formulations: 1982-84Advances SSE, Loop, Worm: 1989-96Rapidly Accelerating ApplicationsCritical Points (eg SF-MI) to 0.1%Settle Bose Glass controversyNew Paradigms for Phase TransitionsExcept for fermions in d 1 !!

OVERVIEW: Progress in Auxiliary Field Quantum Monte CarloAFQMCDFTMergerLGTmored wave2D HubAFLRO2D HubPAMAF2D RuO4LaSrCuOα δ Pu3D HubMnOEntropyn(k)2D HubAFQMC(q,t)d wave2D HubAFQMC(.,t)19902000Year2010Algorithm development102 103 sites/orbitalsMerger with DFTApps: materials on SC timeline!2D sign problem much better than WLQMCLattice Gauge Theory ‘Cousin’

Providing Context: The Hubbard HamiltonianH tX(c†iσ cjσ c†jσ ciσ ) UXni ni µihijiσXniσiσOperators c†iσ (ciσ ) create (destroy) an electron of spin σ on site i.Includes electron kinetic energy (t) and interaction energy (U ).tUU large favors local moments: Single or per site.What is optimal spin arrangement?Hopping of neighboring parallel spins forbidden by Pauli.Antiparallel arrangement lower in second order perturbation theory.xt E (2) 0t E (2) t2 /U JAntiferromagnetism is accompanied by insulating behavior.Transition Metal Oxides: AF insulator (Hubbard’s motivation).

Mott InsulatorU/t large and hni 1.All sites occupied by exactly one e .Hopping causes double occupancy, costs U .Chemical potential µ Cost to add particle Jumps at ρ 11.5ρ1.25U 18 tT 0.5 tN 8x81κ finite"Mott Plateau"κ dρ/dµ 00.75κ finite0.5-8-40µ48

Mott InsulatorU/t large and hni 1.All sites occupied by exactly one e .Hopping causes double occupancy, costs U .Chemical potential µ Cost to add particle Jumps at ρ 11.5ρ1.25U 18 tT 0.5 tN 8x81κ finite"Mott Plateau"κ dρ/dµ 00.75κ finite0.5-8-40µ48

Mott InsulatorU/t large and hni 1.All sites occupied by exactly one e .Hopping causes double occupancy, costs U .Chemical potential µ Cost to add particle Jumps at ρ 11.5ρ1.25U 18 tT 0.5 tN 8x81κ finite"Mott Plateau"κ dρ/dµ 00.75κ finite0.5-8-40µ48

Mott InsulatorU/t large and hni 1.All sites occupied by exactly one e .Hopping causes double occupancy, costs U .Chemical potential µ Cost to add particle Jumps at ρ 11.5ρ1.25U 18 tT 0.5 tN 8x81κ finiteSummary: Hubbard model ρ 1: AF-insulator ρ 6 1: d-wave SC ?"Mott Plateau"κ dρ/dµ 00.75κ finite0.5-8-40µ48

1. PAST: Path Integral Quantum Monte Carlo SimulationsPartition FunctionZ Tr e βHInverse Temperature discretized: β L τZ Tr [e τ H ]L Tr [e τ H e τ H e τ H · · · e τ H ]Trotter approximation H Ha Hb · · · Hme τ H e τ Ha e τ Hb · · · e τ HmIndividual e τ Hx calculable.Extrapolate τ 0 (but see later · · · )World line and auxiliary field methods differ in how individual e τ Hx treated.Here: Focus on lattice models. (Continuum, eg Helium, Ceperley)

World-Line Quantum Monte Carlo Simulationse τ Hx calculated by spatial separation:{Hx } contain independent small spatial clusters, e.g. 1D HubbardXXX ††n̂i n̂i µn̂iσH t(ciσ ci 1σ ci 1σ ciσ ) UiiσH a (1,2)Hb(3,4)iσ(5,6)1 2 3 4 5 6 7(2,3) (4,5) (6,7)Independent2 site clustersInsert complete sets of occupation number statesXhn0 e τ Ha n1 ihn1 e τ Hb n2 ihn2 e τ Ha n3 ihn3 e τ Hb n4 iZ nl. . .hn2L 2 e τ Ha n2L 1 ihn2L 1 e τ Hb n0 ihnl e τ Hx nl 1 i breaks into product of independent cluster problems.Sign problem: Are hnl e τ Hx nl 1 i positive?

State of system represented by occupation number paths ni (τ )Paths sampled stochastically (locally distort world lines).Weight: product of matrix )Zero WindingNon-Zero winding (SF density nonzero)World line topology linked to underlying physics (Ceperley, Pollock)

Key features of (original) World Line QMC: Linear scaling in particle number (system size). (Very) Long autocorrelation times. Cannot measure Green’s function Gσij hc†iσ cjσ i (would break world lines). Sign problem for fermions and frustrated quantum spins.Advances: Continuous time algorithms eliminate β discretization. No Trotter errors. Loop and worm algorithms improve sampling.Autocorrelation times decrease by 3-4 orders of magnitude.Sample nonzero winding sectors. Extend measurements.Superfluid density ρs .Greens function (‘worms’ allow ‘broken’ world lines); n(k).Bottom line: Can simulate 104 106 quantum bosons/spins. Very high precision on critical points, exponents. Address very subtle issues in nature of phase transitions. Sign problem remains for fermions.

Auxiliary Field Quantum Monte CarloSeparate H K P into kinetic and interaction pieces. (Discrete) Hubbard-Stratonovich Fields decouple interaction: τ U ni ni eX τ λS (n n )1 U τ(ni ni )iτi i 2ee e τ Pi (τ )2S iτwhere cosh( τ λ) eU τ2. Quadratic Form in fermion operators: Do trace analyticallyXZ Tr [e τ K e τ P(1) e τ K e τ P(2) e τ K . . . e τ P(L) ]{Siτ } XdetM ({Siτ })detM ({Siτ }){Siτ }dim(Mσ ) is the number of spatial sites. Sample HS field stochastically.Si0 τ0 Si0 τ0detMσ ({Siτ }) detMσ ({Siτ }′ )

Key features of (original) Determinant QMC: Algorithm is order N 3 L. (Computation of determinants.) N 102 lattice sites (fermions).L β/ τ 200 (to reach low temperatures).Ground state projection methods (Sorella, S-W. Zhang) Sign ProblemAt low temperature detMσ can go negative.However, typically occurs at T t/4.Order of magnitude lower T than WLQMC!T cold enough to characterize short range spin/charge correlations.Special symmetry cases (U 0, ρ 1; or any U 0, all ρ): sign detM , detM same.Constrained Path Approaches (S-W. Zhang)Advances: N 102 N 103 fermions (collaborate with CS researchers: D’Azevedo; Bai) Integration with Electronic Structure Codes via DMFT Better scaling in β (Khatami) Continuous time solvers for DMFT.Treat strongly correlated material rather than models.

2. Present WLQMC2D Boson Hubbard model ground state phase diagram (bosons- no sign problem)UtMISFWeakly interacting bosons (U small):3USuperfluid (SF) phase.tU

2. Present WLQMCBoson Hubbard model ground state phase diagram.UtMIXSFStrongly interacting bosons ( (U large, ρ 1): Mott Insulator (MI) phase.Locate position of SF-MI (U/t)crit to a small fraction of a percent.Capogrosso-Sansone etal.

Disordered Boson Hubbard model ground state phase diagram.Disordered site energies ( 6 0) induce additional Bose Glass phase.Both order parameters (Superfluid density and Mott gap) zero.Precision determination of crit .Bose glass phase always intervenes between MI and SF.Prokof’ev etal.

“Deconfined Quantum Criticality” (DQC)Transition between 2 competing phases doesn’t follow traditional Landau description.ML2 (1 ηs),D L2 (1 ηd)2.522 (1 ηd),D L2 (1 ηs)Spin1.5L 24L 32L 48L 641Hamiltonian with competing AF (spin)and valence bond (dimer) phases.Scaling collapse demonstrates DQCin appropriate spin n2L 24L 32L 48L 641.51Dimer0.5-200201/νL40(q-qc)/qcInterest: Could DQC underlie physics of high Tc and heavy fermion superconductors?!Competition between AF and pairing order parametersUnusual behavior observed in vicinity of transition.

2. Present AFQMCRecent Progress in Lattice Gauge Theory(Auxiliary Field Quantum Monte Carlo Cousin)Last five years: computation of parameters of Standard ModelCabibbo, Kobayashi, Maskawa (CKM) matrix elements.Examine effects of QCD on weak interactions.Close methodological connections to AFQMC.LGT: Quarks and gluons on a lattice.CM: Electrons and phonons (or Hubbard-Stratonovich field) on a lattice.LGT size now at 643 x 192 versus 322 x 192 for Condensed Matter AFQMC.Reason is that LGT has linear scaling algorithm!

Decay of D meson (charmed light quark)to K meson (strange light quark), lepton, and neutrino22qmax/mD*s2.5D Klν2lattice QCD [Fermilab/MILC, hep-ph/0408306]experiment [Belle, hep-ex/0510003]experiment [BaBar, 0704.0020 [hep-ex]]experiment [CLEO-c, 0712.0998 [hep-ex]]experiment [CLEO-c, 0810.3878 [hep-ex]]2f (q )1.510.5000.050.10.150.20.252/mD*s0.30.350.42q Overall normalization measures CKM matrix elements. Functional dependence on q 2 (outgoing lepton momentum)matches between LGT (2004) and experiment (2005-2008).0.45

Lattice (Hubbard) Models in Condensed MatterIncreased DQMC lattice size: resolution of Fermi surface (occupied/empty k)U 4 Fermi function:ρ 0.4 β 8ρ 0.6 β 6ρ 0.8 β 4ρ 1.0 β 8π10.80.60.40.20π/20-π/2-π-π -π/2 0 π/2 π-π/2 0 π/2 π-π/2 0 π/2 π-π/2 0 π/2 πn(k)ρ 0.2 β 8-π/2 0 π/2 πU 4 Gradient of Fermi function:ππ/2100.5-π/2-π0-π -π/2 0 π/2 π-π/2 0 π/2 π-π/2 0 π/2 π-π/2 0 π/2 π-π/2 0 π/2 π n(k)1.5

Antiferromagnetic spin correlations form at low temperature.βt 12, 20, 32T /t 0.083, 0.050, 0.033.20 x 20U 0,0)(10,0)(10,0)β 32β 20β 12(10,10)(0,0)

Dynamic Cluster Quantum Monte Carlo: superconductivity in the 2D Hubbard modelWhat is the interplay between stripes and pairing?Nonzero V0 (charge inhomogeneity scale) initially increases Tc .Dynamic Cluster Approximation: 100 site momentum space clusters.Can do 100 site momentum space clusters. Maier etal; other DCA: Jarrell etal.

Combining Density Functional Theory and AFQMCHkin Vion(e.g. in LDA basis)Wannier projectionHkin Vion H0 (k)VCoul. (r r ′ ) VijGloc. Xk1iωn H0 (k)Eq. of MotionDyson Σ(k, iω); G(k, iωn ) new AIMkXGloc. Parquet Eq.:with Vij Γir ViiG(k, iωn )AIM: Γir (ω, ν, ν ′ )Density Functional Theory: Compute band structure H0 (k) for given material.Dynamical Mean Field Theory: Computes additional self energy from strong correlations.Renaissance of diagrammatic methods to refine/extend DMFT:Dynamical Cluster ApproximationDynamical Vertex Approximation (shown) · · ·

DMFT Revealing Unexpected Physics in Multi-Orbital/Band ModelsTwo band Hubbard model with crystal field splitting (appropriate to LaCoO3 )2χ (µB /eV/atom)“Spin Disproportionation”: At low temperature T sites equivalent (same spin).As T increases (!) order arises: alternation of high spin (magnetic) and low spin.(nonmagnetic) sites at intermediate temperature.3050200Intermediate T hasEnhanced magnetic susceptibility050010and different LS and HS occupations0.1na cf 3.40, (3.42) for red(black)HS0.0500200400Kunes etal, arXiv:1103.2249LS600Temperature (K)800

“Linear Scaling Algorithms”Many computational problems involve linear algebraDense Matrix Multiplication, Matrix inversion o(N 3 )Can algorithms be formulated as o(N ) ?(Electronic Structure and Quantum Monte Carlo, for example)General Argument that such algorithms might exist in principle:“Nearsightedness principle” (Kohn 1996)Influence of degrees of freedom in problem fall off sufficiently rapidly:Partition problem into local spatial domains.Issues in Practical Implementation:How is domain size determined? Done for each separate problem?How robust? Do small errors associated with partitioning blow up in “time”?(eg as Molecular Dynamics or Monte Carlo simulation progresses)

Example from Quantum Monte Carlo for fermions:ZZZhiT 1Z Dx detM(x) Dx DΦ exp Φ(M(x) M(x)) Φx: Field coupled to fermions, eg phonons, gluons, Hubbard-Stratonovich.M is a sparse matrix.TˆTΦ update is trivial: Φ M R where P (R) exp R Rx update requires computation of (M T M ) 1 Φ. Do iteratively. Involves sparse matrix multiplication: o(N ) !Works well in Lattice Gauge Theory.Works poorly in simulations of the Hubbard Hamiltonian.Number of iterations grows slowly with linear lattice size.Grows very rapidly (even becoming unstable) in imaginary time.Molecular Dynamics in LGT!dx pdtidpd hT 1 Φ(M(x) M(x)) ΦdtdxWhat happens at zero eigenvalues of M?

Computational Physics EducationNSF, NAS emphasize the need for K-12 computing education:“It’s one of the most vexing paradoxes facing the U.S. today, even if most people are notaware of it. American IT and software companies dominate the world market place andthe vast majority of colleges and universities have excellent computer science programs,yet at the K-12 level, computer science education is almost nonexistent.”http://www.nsf.gov/news/news summ.jsp?cntn id 116059Yet, computer programming classes remain absent from secondary schools,or moving in the wrong direction.Percentages of high schools offering:Introductory Programming CourseAP Programming Course200578%40%201065%27%Meanwhile, other countries have implemented a comprehensive (required)secondary school computer science curriculum.

Undergraduate and graduate level: 5-6 undergraduate programs in computational physics 25 minors/concentrations/tracksBut, in many cases, computational physics is not being emphasized in university physicscurriculum despite its increasing pervasiveness in research and in industry.“we are teaching the same things we taught 50 years ago”.“Report of the Joint AAPT-APS Task Force of Graduate Education in Physics”, June2006.Rubin Landau/Steven Gottlieb(editors of new series of textbooks incorporating computational physics)

3. FutureAlgorithms Better treatments of dynamics e β Ĥ e iĤt (Rigol). Linear

Quantum Monte Carlo in Condensed Matter Physics: Past, Present, and Future 0. Overview 1. Past: Algorithms and Algorithm Development •World-Line Methods •Auxiliary Field Methods 2. Present: Results- Where are we? •World-Line Methods •Auxiliary Field Methods 3. Future •Algorithmic Challenges •Computational Physics Education 0-0