DPHYS Department of Physics Institute for Theoretical Physics. Simulating quantum many body systems. M. Troyer (ETH Zurich)

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1 Simulating quantum many body systems M. Troyer (ETH Zurich)

2 Supercomputing: petaflop and beyond Supercomputing technology has broken through the petaflop-barrier and Moore s law continues to hold 2

3 What can we do with such a fast computer? Look back 56 years 3

4 Complexity of a quantum many-body problem DPHYS Solving the one-body Schrödinger equation: i Ψ( x ) t = 1 2m ΔΨ( x ) + V ( x )Ψ( x ) a complex partial differential equation in 3 dimensions Solving the N-body Schrödinger equation requires solving a complex partial differential equation in 3 dimensions Discretizing each dimension with M points the complexity is exponential: M 3N We cannot do large N, not even on a petaflop computer 4

5 Density functional theory and quantum chemistry Reduces the N-body problem to a 1-body problem: computationally tractable Successful in calculating properties many metals, insulators, semiconductors Example: band structure of silicon 1998 Nobel prize in chemistry band gap Walter Kohn John A. Pople J.R.Chelikowsky and M.L.Cohen, PRB (1974) 5

6 Strong correlations and novel phases Cuprate high temperature superconductor Undoped material: half-filled band and metal according to DFT but antiferromagnetic insulator in experiment! Band structure calculation breaks down! Doped material: a high-temperature superconductor not yet fully theoretically understood after more than two decades Fermi level Band structure of La2CuO4 6

7 Effective model: the Hubbard model Simplify to an effective model capturing the relevant physics single-band model for the partially filled Cu dx2-y2 orbital H = t i,j,σ(c i,σ c j,σ + c j,σ c i,σ)+u n i, n i, i t hopping of an electron to nearest neighbors U repulsion between two electrons on the same site U t Strong on-site repulsion turns a half-filled metal into an insulator strong repulsion But even solving this simplified model is exponentially complex! 7

8 Solving quantum many body problem Mean-field approximations work well deep inside ordered phases but fail when fluctuations are strong, close to phase transitions Weak-coupling approximations work perfectly in weakly interacting systems but fail when interactions become strong Variational methods work well if a good approximation for the true wave function is known but can be very wrong if we guess incorrectly will fail to find novel unexpected phases We need unbiased accurate numerical methods 8

9 1. An overview of methods 9

10 Exact Diagonalization Construct a basis for the Hilbert space of N-site lattice 2 N states for spin-1/2 model, 4 N states for fermionic Hubbard possible to store up to 10 9 states Construct matrix representation of the Hamiltonian using all possible symmetries to reduce the size Obtain low-lying eigenvalues and eigenvectors numerically High accuracy, (15 digits) Static and dynamic properties but restricted to small systems of about sites 10

11 Example: spectrum of 1-d anyonic chain Feiguin, et al, PRL (2007) rescaled energy E(K) 4 4 1/ /5 + 2 L = 36 1/5 + 2 primary fields 7/ / / descendants 3/ / /5 1/ / / / / Neveu-Schwarz Ramond 1/5 sector sector 0 0 3/40 0 7/ momentum K [2π/L] 11

12 DMRG (Density Matrix Renormalization Group) S.R. White, PRL 1992 DMRG cleverly reduces the Hilbert space dimension DPHYS It is a variational method but so accurate in one dimension that it can essentially be viewed as an exact 3-10 digits accuracy on chains and ladders with many hundreds of sites 12

13 DMRG example: stripe formation Stripe (charge density wave) formation in 6-chain fermionic Hubbard ladder S.R. White and D.J. Scalapino (2003) 13

14 New developments in DMRG Time evolution in non-equilibrium quantum systems G. Vidal, PRL (2003) A. Daley et al, JSTAT (2004), S. White and A. Feiguin, PRL (2004) Extensions to two dimensions PEPS: F. Verstraete and I. Cirac (2004) MERA: G. Vidal (2007) and variants Still unclear whether these will be as accurate in 2D as the DMRG is in 1D 14

15 Time evolution in DMRG Example: evolution of two polarized domains in an antiferromagnetic Heisenberg chain D. Gobert et al, Phys. Rev. E (2005) 15

16 Series expansion methods Use the computer to calculate a high (10-20) order symbolic series expansion high-temperature series in β=1/t f = n g(n)β n /n! or perturbation series in a small coupling constant λ H = H 0 + λv A(λ) = n c A (n)λ n /n! Works for any lattice model, in any dimension on infinite systems but only high to intermediate temperature or weak to intermediate coupling 16

17 The Monte Carlo Method What is the probability to win in Solitaire? Ulam s answer: play it 100 times, count the number of wins and you have a pretty good estimate 17

18 What can we do with such a fast computer? Look back 56 years 18

19 Modern QMC algorithms System Temperature Modern algorithms Spins J spins Spins 0.1 J spins Spins J spins hard-core bosons 0.1 t bosons Bose-Hubbard t bosons on sites 19

20 Algorithmic advances as important as Moore s law Moore s law is impressive but algorithmic advances are even more impressive in many fields Example from D.P. Landau, simulating the Ising model Important algorithmic advances cluster updates histogram reweighting! 20

21 2. Some applications Magnetic properties of quantum magnets Supersolidity in Helium-4 Validation of experiments on ultracold atomic gases Critical temperature of the resonant Fermi gas 21

22 Quantum Monte Carlo simulations to fit experiments Spin-1 chains in NaVGe2O6 B. Pedrini et al. PRB (2004) a 0.012!"! 0, H = 2 T S=1 chain, quantum Monte Carlo simulation (a) b c! (emu/mol) J =18.9 ± 0.5 K J = 3.4 ± 0.2 K Molecular magnet Mn-84 V. Tangoulis T (K) (b)! (emu/mol) T N = 0.95 J / k B T max = 1.32 J / k B using open-source codes Matthias 22 Troyer T (K)

23 Supersolidity in Helium-4 A supersolid is simultaneously superfluid: broken gauge symmetry solid: broken translational symmetry Superflow due to motion of vacancies or interstitials Andreev and Lifshitz, Sov. Phys. JETP 29, 1107 (1969). Thouless, Ann. Phys. 52, 403 (1969). Chester, Phys. Rev. A 2, 256 (1970). But only recent experimental evidence Kim and Chan, Nature 427, 225 (2004); Science 305, 1941 (2004). 23

24 ave rity torant tas a 17 as by the he ow of han o a nd the C). A nd IF as lid ults od of the same Buniversal[ supersolid behavior, Kim-Chan A experiment ρ s /ρ Torsional oscillator containing 38 µm/s Helium-4 NCRIF frequency changes when 198 Helium-4 µm/s 1 µm/s becomes 2.5 µm/s µm/s 6.5 µm/s superfluid, since part of the mass then decouples 14 µm/s 42 µm/s 420 µm/s from the oscillator B NCRIF C T c 0.2K s (T ns, 0)/ρ 2886 ns, and 3143 ns for solid samples at ρ 26, 41, and 65 bars. NCRIF curves measured s /ρ with oscillation speed less than the critical RIF ρ s /ρ 26 bars 5 µm/s 7 µm/s 17 µm/s 65 µm/s NCRIF 41 bars 3 µm/s 6 µm/s 9 µm/s 22 µm/s 35 µm/s 85 µm/s 145 µm/s 205 µm/s 520 µm/s 65 bars DPHYS creasing D so /D with pressure. We speculate that the variation in D so /D is related to the less than ideal crystallinity of the solid helium samples. Given the temperature gradient and different wall materials that exist in the torsional cell and the capillary leading to the cell, solid helium grown by the blocked cap- ρ s /ρ 65 bars illary method is likely to be polycrystalline with grain boundaries that may affect the coherence of the superflow and possibly the magnitude of the supersolid fraction. The largest D so /D value, 0.017, is comparable to that found for the experiment with helium T (K) confined in Vycor (0.025) (15). The Vycor value includes a multiplicative factor of 5 to correct for the tortuous pore structure of the Vycor glass. The theoretical estimate of the 70 zero-temperature supersolid fraction varies 60 from 1 part per million to 40% (11, 23 26). In addition to the comparable ampli- 50 Normal Supersolid solid tude, the temperature40 dependence of the D s / Fig. 3. (A to C) NCRIF as a function of temperature for three solid samples at different maximum oscillation speeds v max. The observed period increases due to filling of the cell with 4 He at 300 mk are, respectively, 2785 velocity of superflow collapse into a single curve. These curves represent the supersolid fraction, D s /D, as a function of temperature. (D) NCRIF in the low-temperature limit as a function Matthias of v max Troyer. 1 µm/s 2.5 µm/s 6.5 µm/s (bar) D of 4 He in Vycor resembles 30 that found in bulk solid samples (15). These similarities 20 suggest that the observed superflow in Pressure Superfluid Normal liquid 10 these two systems is an intrinsic lowtemperature property of solid helium. One may argue that the decrease in the resonant T (K) Fig. 4. Phase diagram of liquid and solid helium. 24

25 Is Helium-4 really supersolid? Simulate solid Helium and check! Continuous space QMC worm algorithm Boninsegni, Prokof ev & Svistunov PRL (2006) Unbiased ab-initio method No approximation besides use of 2-body potential Tests of potential: superfluid Tc reproduced to 0.5% LJ Aziz 13K long-range attractive Perfect single crystal is a boring insulator Boninsegni, Prokof ev and Svistunov, PRL Clark and Ceperley, PRL Boninsegni, Kuklov, Pollet, Prokof ev, 30 Svistunov, Troyer, PRL Pollet, Boninsegni, Kuklov, Prokof ev, 10 Svistunov, Troyer, PRL Vacancies and interstitials are gapped µ(k) n c n f n m I V µ I µ solid µ V n(a -3 ) 25

26 But crystal defects can be superfluid! Pollet, Boninsegni, Kuklov, Prokof ev, Svistunov, Troyer, Phys. Rev. Lett. 98, (2007) Boninsegni, Kuklov, Pollet, Prokof ev, Svistunov, Troyer, Phys. Rev. Lett. 99, (2007) Pollet, Boninsegni, Kuklov, Prokof ev, Svistunov, Troyer, Phys. Rev. Lett (2008) Superflow along grain boundary between two crysallites pinned Lz grain boundary 3 layers, Tc = 0.5K pinned N = atoms 0 condensate map L y Superflow also seen along dislocations Kim-Chan experiment not finally explained yet, but crystal defects play the key role in superflow in solid Helium 26

27 Bose-Einstein condensation in cold atomic gases At close to zero temperatures, a macroscopic fraction of all atoms in a Bose gas occupy the same quantum state A diverging occupation of the zero momentum state Momentum distribution function 27

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29 How do we detect these quantum gases? release the atoms faster atoms fly farther the image reflects the momentum distribution

30 A quantum simulator for a quantum lattice model Load an ultracold quantum gas into an optical lattice I. Bloch P. Zoller Controlled and tunable experimental realization of the Hubbard model H = t ) (b i b j + h.c. + U n i (n i 1)/2 µ n i + V ij i i i r 2 i n i Can be used as quantum simulator to explore its phase diagram validate it for known phase diagram of bosonic model then use it for unknown phase diagram of fermionic models 30

31 Validation of experiment by QMC QMC simulation of experimental setup. Approximation-free with no no free parameter a b Experiment V 0 = 8E r, U/J = 8.11, T c = 26.5nK 13.6 nk 18.8 nk 26.5 nk 30.7 nk 43.6 nk 2hk 11.9 nk 19.1 nk 26.5 nk 31.8 nk 47.7 nk QMC c OD 2 1 Exp. QMC x TOF (2hk)

32 Critical temperature of the resonant Fermi gas Phase diagram of dilute Fermi gas BCS superfluid of Cooper pairs??? BEC Bose condensate of strongly bound pairs BCS Universal behavior at resonance (unitary point) 32

33 Previous results for the unitary point 0.5 M Holland, SJJMF Kokkelmans, ML Chiofalo, and R. Walser 0.4 J Kinhast, A Turlapov, JE Thomas, 0.3 Q Chen, J Stajic, K Levin V.K. Akkineni, N. Trivedi, D. Ceperley 0.2 P Nozieres, S Schmitt-Rink XJ Liu, H Ho A Bulgac, JE Drut, P Magierski BEC limit P. Nikolic, S Sachdev R. Haussmann, W. Rantner S. Cerrito, W. Zwerger M Wingate unbiased QMC result Burovski et al 33

34 3. Monte Carlo Integration 34

35 Integrating a function Convert the integral to a discrete sum b! f (x)dx = b " a N a N ' i =1 # f a + i b " a % $ N & + O(1/N) Higher order integrators: Trapezoidal rule: b! f (x)dx = b " a N a Simpson rule: # ( $ N "1 1 2 f (a) + f # a + i b " a % ' $ N & i = f (b) % ) + O(1/N 2 ) & b! f (x)dx = b " a N "1 # 3N f (a) + (3 " # ("1)i ) f a + i b " a % % ( ' + f (b) $ $ N & ) + O(1/N 4 ) & a i =1 35

36 High dimensional integrals Simpson rule with M points per dimension one dimension the error is O(M -4 ) d dimensions we need N = M d points the error is order O(M -4 ) = O(N -4/d ) An order - n scheme in 1 dimension is order - n/d d in d dimensions! In a statistical mechanics model with N particles we have 6N-dimensional integrals (3N positions and 3N momenta). Integration becomes extremely inefficient! 36

37 Ulam: the Monte Carlo Method What is the probability to win in Solitaire? Ulam s answer: play it 100 times, count the number of wins and you have a pretty good estimate Matthias 37 Troyer

38 Throwing stones into a pond How can we calculate π by throwing stones? Take a square surrounding the area we want to measure: π/4 Choose M pairs of random numbers ( x, y ) and count how many points ( x, y ) lie in the interesting area 38

39 Monte Carlo integration Consider an integral Instead of evaluating it at equally spaced points evaluate it at M points xi chosen randomly in Ω: The error is statistical: " x )d x! d x! f = f (! In d>8 dimensions Monte Carlo is better than Simpson! f!! 1 M M " i =1 f (! x i ) "!! = Var f M " M #1/ 2 Var f = f 2 # f 2 39

40 Sharply peaked functions wasted effort In many cases a function is large only in a tiny region Lots of time wasted in regions where the function is small The sampling error is large since the variance is large 40

41 Importance sampling f(x)/p(x) p(x) Choose points not uniformly but with probability p(x): f = f p p := f ( x! ) " p( x! ) p(! x )d x! d x!! The error is now determined by Var f/p Find p similar to f and such that p-distributed random numbers are easily available "! 41

42 4. Generating Random Numbers 42

43 Random numbers Real random numbers are hard to obtain classical chaos (atmospheric noise) quantum mechanics Commercial products: quantum random number generators based on photons and semi-transparent mirror 4 Mbit/s from a USB device, too slow for most MC simulations 43

44 Pseudo Random numbers Are generated by an algorithm Not random at all, but completely deterministic Look nearly random however when algorithm is not known and may be good enough for our purposes Never trust pseudo random numbers however! 44

45 Linear congruential generators are of the simple form xn+1=f(xn) A reasonably good choice is the GGL generator x n +1 = (ax n + c)mod m with a = 16807, c = 0, m = quality depends sensitively on a,c,m Periodicity is a problem with such 32-bit generators The sequence repeats identically after iterations With 500 million numbers per second that is just 4 seconds! Should not be used anymore! 45

46 Lagged Fibonacci generators x n = x n p x n q mod m Good choices are (2281,1252,+) (9689,5502,+) (44497,23463,+) Seed blocks usually generated by linear congruential Has very long periods since large block of seeds A very fast generator: vectorizes and pipelines very well 46

47 More advanced generators As well-established generators fail new tests, better and better generators get developed Mersenne twister (Matsumoto & Nishimura, 1997) Well generator (Panneton and L'Ecuyer, 2004) Number theory enters the generator design: predicting the next number is equivalent to solving a very hard mathematical problem 47

48 Are these numbers really random? No! Are they random enough? Maybe? Statistical tests for distribution and correlations Are these tests enough? No! Your calculation could depend in a subtle way on hidden correlations! What is the ultimate test? Run your simulation with various random number generators and compare the results 48

49 Non-uniform random numbers we found ways to generate pseudo random numbers u in the interval [0,1[ How do we get other uniform distributions? uniform x in [a,b[: x = a+(b-a) u Other distributions: Inversion of integrated distribution Rejection method 49

50 Non-uniform distributions How can we get a random number x distributed with f(x) in the interval [a,b[ from a uniform random number u? Look at probabilities: P[x < y] = y a x = F 1 (u) f (t) dt =: F(y) P[u < F(y)] This method is feasible if the integral can be inverted easily exponential distribution f(x)=λ exp(-λx) can be obtained from uniform by x=-1/λ ln(1-u) 50

51 Normally distributed numbers The normal distribution f (x) = 1 2π exp( 2 x ) cannot easily be integrated in one dimension but can be easily integrated in 2 dimensions! We can obtain two normally distributed numbers from two uniform ones (Box-Muller method) n 1 = 2 ln(1 u 1 ) sinu 2 n 2 = 2ln(1 u 1 ) cosu 2 51

52 Rejection method (von Neumann) f / h reject accept u u x Look for a simple distribution h that bounds f: f(x) < λh(x) Choose an h-distributed number x Choose a uniform random number number 0 u < 1 Accept x if u < f(x)/ λh(x), otherwise reject x and get a new pair (x,u) Needs a good guess h to be efficient, numerical inversion of integral might be faster if no suitable h can be found x 52

53 5. The Metropolis Algorithm 53

54 Monte Carlo for classical systems Evaluate phase space integral by importance sampling A = Ω A(c) p(c)dc A A = 1 M M A ci Ω p(c)dc i=1 Pick configurations with the correct Boltzmann weight P[c] = p(c) Z = exp( βe(c)) Z But how do we create configurations with that distribution? The key problem in statistical mechanics! 54

55 The top-10 Algorithms of the 20th Century G UES T E D I T O R S I N T R O D UCT I O N t he Top Metropolis Algorithm for Monte Carlo Simplex IMethod for Linear Programming Krylov Subspace Iteration Methods The Decompositional Approach to Matrix Computations The Fortran Optimizing Compiler QR Algorithm for Computing Eigenvalues Quicksort Algorithm for Sorting Fast Fourier Transform Integer Relation Detection Fast Multipole Method Matthias 55 Troyer

56 The Metropolis Algorithm (1953) Matthias 56 Troyer

57 Markov chain Monte Carlo Instead of drawing independent samples ci we build a Markov chain c 1 c 2... c i c i Transition probabilities W x,y for transition x y need to satisfy: Normalization: W x,y = 1 y Ergodicity: any configuration reachable from any other x,y n : ( W n ) x,y 0 Balance: the distribution should be stationary p(x)w x,y p(x) = p(y)w y,x 0 = d dt p(x) = p(y)w y,x y y Detailed balance is sufficient but not necessary for balance y W x,y W y,x = p(y) p(x) 57

58 The Metropolis algorithm Teller s proposal was to use rejection sampling: Propose a change with an a-priori proposal rate A x,y Accept the proposal with a probability P x,y The total transition rate is W x,y =A x,y P x,y The choice P x,y = min 1, A p(y) y,x A x,y p(x) satisfies detailed balance and was first proposed by Metropolis et al 58

59 Metropolis algorithm for the Ising model 1.Pick a random spin and propose to flip it 2.Accept the flip with probability P = min 1,e (E new E old )/T 3.Perform a measurement independent of whether the proposed flip was accepted or rejected! 59

60 Equilibration Starting from a random initial configuration it takes a while to reach the equilibrium distribution The desired equilibrium distribution is a left eigenvector with eigenvalue 1 (this is just the balance condition) p(x) = Convergence is controlled by the second largest eigenvalue y p(y)w y,x p(x,t) = p(x) + O(exp( λ 2 t)) We need to run the simulation for a while to equilibrate and only then start measuring 60

61 6. Monte Carlo Error Analysis 61

62 Dogs and fleas: direct sampling exact Monte Carlo 0.08 P[n] n 62

63 Dogs and fleas: naïve errors exact Monte Carlo Monte Carlo 0.08 P[n] n 63

64 Dogs and fleas: uncorrelated samples exact Monte Carlo 0.08 P[n] n 64

65 Monte Carlo error analysis The simple formula ΔA = Var A M is valid only for independent samples The Metropolis algorithm gives us correlated samples! The number of independent samples is reduced ΔA = Var A M ( 1 + 2τ ) A The autocorrelation time is defined by τ A = t =1 ( A i +t A i A 2 ) Var A 65

66 Binning analysis Take averages of consecutive measurements: averages become less correlated and naive error estimates converge to real error A 1 A 2 A 3 A 4 A 5 A 6 A 7 A 8 A 9 A 10 A 11 A 12 A 13 A 14 A 15 A 16 A 1 (1) A 2 (1) A 3 (1) A 4 (1) A 5 (1) A 6 (1) A 7 (1) A 8 (1) ( ) (l A ) i = 1 2 A (l 1) l 2i 1 + A 2i A 1 (2) A 2 (2) A 3 (2) A 4 (2) A 1 (3) A 2 (3) not converged Δ (l ) = Var A (l ) M 1 τ A = lim l 2 2 l Var A (l ) Var A (0) (l ) l 1 Δ = (1 + 2τ A )Var A M a smart implementation needs only O(log(N)) memory for N measurements Δ (l) L = 4 L = 48 converged binning level l 66

67 Dogs and fleas: binning analysis exact Monte Carlo 0.08 P[n] n 67

68 Correlated quantities How do we calculate the errors of functions of correlated measurements? specific heat Binder cumulant ratio c V = E 2 E 2 U = The naïve way of assuming uncorrelated errors is wrong! It is not even enough to calculate all crosscorrelations due to nonlinearities except if the errors are tiny! m4 m 2 2 T 2 68

69 Splitting the time series Simplest idea: split the time series and evaluate for each segment X Y U=f(X,Y) X1 X2 X3... XM Y1 Y2 Y3... YM U1 U2 U3... UM U U = 1 M U i i =1 ΔU M 1 M(M 1) ( U i U ) 2 Problem: can be unstable and noisy for nonlinear functions such as X/Y M i 1 69

70 Jackknife-analysis Evaluate the function on all and all but one segment U 0 = 1 M M i =1 f (X i,y i ) f(x1,y1) f(x2,y2) f(x3,y3)... f(xm,ym) U 1 = U j = 1 M 1. 1 M 1. M i =2 M i =1 i j f (X i,y i ) f (X i,y i ) f(x1,y1) f(x2,y2) f(x3,y3)... f(xm,ym). f(x1,y1) f(x2,y2) f(x3,y3)... f(xm,ym). U U 0 (M 1)(U U 0 ) U = 1 M U i M i =1 ΔU M 1 M M i 1 ( U i U ) 2 70

71 ALPS Alea library in C++ The ALPS class library implements reliable error analysis Adding a measurement: alps::realobservable mag; mag << new_value; Evaluating measurements std::cout << mag.mean() << +/- << mag.error(); std::cout Autocorrelation time: << mag.tau(); Correlated quantities? Such as in Binder cumulant ratios m 4 m 2 2 ALPS library uses jackknife analysis to get correct errors alps::realobsevaluator binder = mag4/(mag2*mag2); std::cout << binder.mean() << +/- << binder.error(); 71

72 ALPS Alea library in Python The ALPS class library implements reliable error analysis Adding a measurement: mag = pyalps.pyalea.realobservable( Magnetization ); mag << new_value; Evaluating measurements print mag.mean, +/-, mag.error; print Autocorrelation time:, mag.tau; Correlated quantities? Such as in Binder cumulant ratios m 4 m 2 2 ALPS library uses jackknife analysis to get correct errors of functions of data, after reading data from file print mag4/(mag2*mag2) 72

73 7. Quantum Monte Carlo 73

74 Quantum Monte Carlo Not as easy as classical Monte Carlo Z = c e E c / k B T Calculating the eigenvalues E c is equivalent to solving the problem Need to find a mapping of the quantum partition function to a classical problem Z = Tre βh c Negative sign problem if some p c < 0 p c 74

75 Quantum Monte Carlo Feynman (1953) lays foundation for quantum Monte Carlo Map quantum system to classical world lines classical imaginary time quantum mechanical particles space world lines Use Metropolis algorithm to update world lines Matthias 75 Troyer

76 The Suzuki-Trotter Decomposition Generic mapping of a quantum spin system to Ising model basis of most discrete time QMC algorithms not limited to special models Split Hamiltonian into two easily diagonalized pieces H = H 1 + H 2 e εh = e ε ( H 1 + H 2 ) = e εh 1 e εh 2 + O(ε2 ) Obtain the checkerboard decomposition H H 1 H 2 = + Z = Tr[ exp( βh) ] = Tr[ e β ( H 1 + H 2 ) ] = Tr[ e (β / M )H 1 e (β / M )H ] M 2 + O(β 3 / M 2 ) imaginary time space direction 76

77 Path integral QMC Use Trotter-Suzuki or a a simple low-order formula gives a mapping to a (d+1)-dimensional classical model imaginary time Z = Tre βh = Tr e MΔτH = Tr( e ΔτH ) M = Tr( 1 ΔτH) M + O(βΔτ) i 1 i 8 i 7 i 6 i 5 i 4 i 3 i 2 i 1 {(i 1...i M )} = i 1 1 ΔτH i 2 i 2 1 ΔτH i 3 i M 1 ΔτH i 1 space direction place particles (spins) for Hamiltonians conserving particle number (magnetization) we get world lines partition function of quantum system is sum over classical world lines Matthias 77 Troyer

78 Calculating configuration weights Z = Tre βh = Tr e MΔτH = Tr( e ΔτH ) M = Tr( 1 ΔτH) M + O(βΔτ) {(i 1...i M )} = i 1 1 ΔτH i 2 i 2 1 ΔτH i 3 i M 1 ΔτH i 1 Examples: particles with nearest neighbor repulsion H = t i,j (a i a j + a j a i)+v i,j n i n j 1 ( τt) 2 (1 τv) 3 ( τt) 2 Matthias 78 Troyer

79 Monte Carlo updates just move the world lines locally probabilities given by matrix element of Hamiltonian example: tight binding model H = t ( c i c i +1 + c i +1 c ) i i, j introduce or remove two kinks: shift a kink: P =1 P = ( Δτt) 2 P = Δτt P = min[ 1,(Δτt) 2 ] P = min[ 1,1/(Δτt) 2 ] P = P =1 Matthias 79 Troyer

80 The continuous time limit the limit Δτ 0 can be taken in the algorithm [Prokof'ev et al., Pis'ma v Zh.Eks. Teor. Fiz. 64, 853 (1996)] i 1 imaginary time i 8 i 7 i 6 i 5 i 4 i 3 imaginary time τ 5 τ 6 τ 4 τ 2 τ 3 i 2 τ 1 i 1 space direction space direction discrete time: store configuration at all time steps continuous time: store times at which configuration changes Matthias 80 Troyer

81 Diagrammatic QMC Split the Hamiltonian into diagonal term H0 and perturbation V Then perform time-dependent perturbation theory H = H 0 + V, H 0 = J z ij S z i S z j hs z i, V = J xy ij (S x x i S j + S y i S y j ) <i, j > i < i, j > Z = Tr(e βh ) = Tr(e βh 0 β dτv (τ ) 0 Te ) Z = Tr(e βh 0 β (1 dτv (τ ) + dτ 1 dτ 2 V (τ 1 ) V (τ 2 ) +...)) 0 β 0 β τ 1 Each term is represented by a diagram (world line configuration) τ τ 2 τ 1 81

82 Calculating configuration weights Continuous time algorithms just sample time-depedent perturbation expansion Z =Tr (e βh 0 T e β 0 dτv(τ) ) Examples: particles with nearest neighbor repulsion H 0 = V i,j n i n j V = t i,j (a i a j + a j a i) β τ 2 τ 1 0 β τ 2 τ t 2 dτ 1 dτ 2 e βv e τ 1V e (β τ 2)V t 2 dτ 1 dτ 2 Matthias 82 Troyer

83 Updates in continuous time Shift a kink to any new position: Insert a pair of kinks: vanishing P =1 acceptance rate P = Δτt ( ) 2 0 Λ τ 2 τ 1 0 P = min[ 1,(Δτt) 2 ] 0 solution: integrate over a$ possible insertions in an interval Λ Λ P = t 2 dτ 2 dτ 1 = Λ2 t τ 1 P = min[ 1,Λ 2 t 2 /2] 0 Matthias 83 Troyer

84 Advantages of continuous time No need to extrapolate in time step a single simulation is sufficient no additional errors from extrapolation Less memory and CPU time required Instead of a time step Δτ << t we only have to store changes in the configuration happening at mean distances t Speedup of 1 / Δτ 10 Conceptual advantage we directly sample a diagrammatic perturbation expansion 84

85 8. Cluster updates The Swendsen-Wang algorithm The loop algorithm 85

86 Autocorrelation effects The Metropolis algorithm creates a Markov chain c 1 c 2... c i c i+1... successive configurations are correlated, leading to an increased statistical error ΔA = ( A A ) 2 = Var A M (1+ 2τ ) A Critical slowing down at second order phase transition τ L 2 Exponential tunneling problem at first order phase transition τ exp(l d 1 ) Matthias 86 Troyer

87 Problems with local updates Local updates cannot change global topological properties number of world lines (particles, magnetization) conserved winding conserved braiding conserved cannot sample grand-canonical ensemble Critical slowing down at second order phase transitions solved by cluster updates (today) Tunneling problem at first order phase transitions solved by extended sampling techniques (later in the course) Matthias 87 Troyer

88 From local to cluster updates Energy of configurations in Ising model J if parallel: + J if anti-parallel: Probability for flip Anti-parallel: flipping lowers energy, always accepted ( ) =1 2ΔE /T ΔE = 2J P = min 1,e Parallel: 2ΔE /T ΔE = +2J P = min 1,e no change with probability 1 exp( 2βJ)!!! Alternative: flip both! ( ) = exp( 2βJ) DPHYS P = exp( 2J /T) P =1 exp( 2J /T) 88

89 Swendsen-Wang Cluster-Updates DPHYS No critical slowing down (Swendsen and Wang, 1987)!!! Ask for each spin: do we want to flip it against its neighbor? antiparallel: yes parallel: costs energy Accept with Otherwise: also flip neighbor! Repeat for all flipped spins => cluster updates P = exp( 2βJ) P =1 exp( 2βJ)???????? Shall we flip neighbor???? Done building cluster?? Flip all spins in cluster 89

90 Cluster algorithms: the formal explanation Extend the phase space to configurations + graphs (C,G) Choose graph weights independent of configuration Perform updates Z = W (C) = W (C,G) with W (C) = W (C,G) C C G W (C,G) = Δ(C,G)V (G) where Δ(C,G) = G 2. Discard configuration 1 graph G allowed for C 0 otherwise 4. Discard graph Detailed balance is satisfied C i (C i,g) G (C i +1,G) C i Pick a graph G P[G] = V (G) W (C) 3. Pick any allowed new configuration P[(C i,g) (C i +1,G)] P[(C i +1,G) (C i,g)] = 1/N C =1= Δ(C i +1,G)V (G) 1/N C Δ(C i,g)v (G) = P[(C i +1,G)] P[(C i,g)] 90

91 Cluster algorithms: Ising model We need to find Δ(C,G) and V(G) to fulfill W (C) = W (C,G) = Δ(C,G)V (G) G G Δ(C,G) o-o o o W(C), 1 1 e + βj, 0 1 e βj W(G) e + βj -e βj e βj This means for: C i (C i,g) G Parallel spins: pick connected graph o-o with Antiparallel spins: always pick open graph o o P( ) = e +βj + e βj o-o e +βj =1 e 2βJ And for: G (C i +1,G) C i +1 Configuration must be allowed connected spins must be parallel connected spins flipped as one cluster 91

92 92

93 The loop algorithm (Evertz et al) Swendsen-Wang cluster algorithm for the Ising model two choices on each bond: connected (flip both spins) or disconnected all connected spins are flipped together Loop algorithm is a generalization to quantum systems world lines may not be broken always 2 or 4 spins must be flipped together four different connection types Matthias 93 Troyer

94 Hamiltonian of spin-1/2 models Consider a 2-site quantum spin-1/2 model H XXZ = J xz (S x 1 S x 2 + S y 1 S y 2 ) + J z S z 1 S z 2 h( S z z 1 + S ) 2 ( ) = J xz 2 (S + 1 S 2 + S 1 S + 2 ) + J z S z 1 S z 2 h S z z 1 + S 2 Heisenberg model if J xy = J z = J H = J S 1 S 2 h S 1 z + S 2 z ( ) Hamiltonian matrix in 2-site basis,,, { } H XXZ = J z 4 + h J xy J z 4 J 0 xy 2 J z h J z 94

95 Cluster building rules: XY-like antiferromagnet DPHYS H XXZ = J xz 2 (S + i S j + S i S + j ) + J z S z z i S j i, j with 0 J z J xy i, j W (C) = W (C,G) = Δ(C,G)V (G) G G Δ(C,G) W(C) (J z /4) dτ (J z /4) dτ 1 1 (J xy /2) dτ V(G) 1 (J z /4) dτ (J z /2) dτ (J xy- J z )/2 dτ 95

96 How do we deal with infinitesimals? How do we deal with the vanishing dτ terms in continuous time? First example: the exchange process Possible graph connections: Graph weights: J z 2 dτ J xy J z 2 dτ Probability to pick graph: (divide weight by sum) J z J xy J xy J z J xy The infinitesimal dτ terms cancel out Randomly pick one of the graphs (with appropriate probabilities) 96

97 How to deal with infinitesimals? How do we deal with the vanishing dτ terms in continuous time? Second example: the decay process Possible graph connections: Graph weights: Probability to pick graph: (divide weight by sum) 1 J z 4 dτ J z 2 dτ 1 J z 2 dτ J z 2 dτ The infinitesimal dτ terms remain Infinitesimal acceptance rate at infinitely many time steps? 97

98 How to deal with infinitesimals? We have to tackle the problem of vanishing probabilities Example: Heisenberg antiferromagnet P =1 J Δτ 1 2 P = = J 2 Δτ 0 Interpret the connection as a decay process where the loop jumps P = = J 2 dτ decay constant λ = J 2 mean distance d = 2 J Matthias 98 Troyer

99 Loop-cluster updates 1. Connect spins according to loop-custer building rules 2. Build and flip loop-cluster 99

100 Heisenberg antiferromagnet H Heisenberg = J i, j S i S j W (C) = DPHYS W (C,G) = Δ(C,G)V (G) G G Δ(C,G) W(C) (J/4) dτ (J/4) dτ 0 1 (J/2) dτ V(G) 1 (J/4) dτ (J/2) dτ Connected spins form a cluster and have to be flipped together Very simple and deterministic for Heisenberg model 100

101 Ising-like ferromagnet H XXZ = J xz 2 (S + i S j + S i S + j ) J z S z z i S j i, j i, j W (C) = W (C,G) = Δ(C,G)V (G) G G with 0 J xy J z Δ(C,G) W(C) (J z /4) dτ (J z /4) dτ (J xy /2) dτ V(G) 1 (J z /4) dτ (J xy /2) dτ (J z -J xy )/2 dτ Now 4-spin freezing graph is needed: connects (freezes) loops 101

102 Ising ferromagnet H Ising = J S z z i S j = J 4 i, j i, j σ i σ j W (C) = W (C,G) = Δ(C,G)V (G) G G Δ(C,G) W(C) (J/4) dτ (J/4) dτ V(G) 1 (J/4) dτ (J/2) dτ Two spins are frozen if there is any freezing graph along the world line P no freezing = lim (1 (β / M)J /2) M = exp( βj /2) = exp( 2βJ classical ) M We recover the Swendsen Wang algorithm: probability for no freezing 102

103 9. Stochastic Series Expansion (SSE) 103

104 Path integral representation Split the Hamiltonian into diagonal term H0 and perturbation V Then perform time-dependent perturbation theory H = H 0 + V, H 0 = J z ij S z i S z j hs z i, V = J xy ij (S x x i S j + S y i S y j ) <i, j > i < i, j > Z = Tr(e βh ) = Tr(e βh 0 β dτv (τ ) 0 Te ) Z = Tr(e βh 0 β (1 dτv (τ ) + dτ 1 dτ 2 V (τ 1 ) V (τ 2 ) +...)) 0 β 0 β τ 1 Each term is represented by a diagram (world line configuration) τ τ 2 τ 1 104

105 Stochastic Series Expansion based on high temperature expansion, developed by Sandvik Z = Tr(e βh ) = = with H = β n n = 0 n! i H i ( β) n Tr(H n ) n = 0 α ( H bi ) α α ( ) b 1,...,b n n i=1 H b1 H b2 H b1 Similar world line representation but without times assigned 105

106 Splitting the Hamiltonian for SSE Break up the Hamiltonian into offdiagonal and diagonal bond terms i, j o H = H (i, j ) d + H (i, j ) Example: Heisenberg antiferromagnet H XXZ = J xz 2 (S + i S j + S i S + j ) + J z S z z z i S j h S 1 i, j i, j i convert site terms into bond terms = J xz 2 (S + i S j + S i S + j ) + J z S z z i S j i, j i, j h z i, j ( S z z i + S ) j split into diagonal and = o d H (i, j ) + H (i, j ) i, j i, j offdiagonal bond terms with o H (i, j ) d H (i, j ) = J xz 2 (S + i S j + S i S + j ) = J z S z i S z j h ( z S z z i + S ) j 106

107 Ensuring positivity of diagonal bond weights DPHYS Recall the SSE expansion: Z = β n n! n = 0 α ( H bi ) α α ( ) b 1,...,b n n i=1 Negative matrix elements of H are the weights Need to make all matrix elements non-positive Diagonal matrix elements: subtract an energy shift d H (i, j ) = d H (i, j ) = J z S z i S z j h ( z S z z i + S ) j C J z 4 + h C J z 4 C J z 4 C h C J z C J z 4 + h 107

108 Positivity of off-diagonal bond weights Energy shift will not help with off-diagonal matrix elements o H (i, j ) = J xz 2 (S + i S j + S i S + j ) Ferromagnet (J xy < 0) is no problem o H (i, j ) Antiferromagnet on bipartite lattice perform a gauge transformation on one sublattice J xy 2 S + i S j + S + i S j ( ) S ± i ( 1) i ± S i J xy 2 S + i S j + S + i S j ( ) Frustrated antiferromagnet: we have a sign problem J 0 0 xy 0 = 2 J 0 xy

109 Fixed length operator strings SSE sampling requires variable length n operator strings Z = β n n! n = 0 α ( H bi ) α α ( ) b 1,...,b n n i=1 Extend operator string to fixed length by adding extra identity operators: n number of non-unit operators i, j d H bi H (i, j ) { } o,h (i, j ) Z = Λ n = 0 (Λ n)!β n Λ α ( H bi ) α Λ! α ( ) b 1,...,b Λ i=1 H id = 1 H bi d { H id } H (i, j ) i, j { } o,h (i, j ) pick Λ large enough during thermalization And now just perform updates 109

110 Step 1: Local diagonal updates Recall the weight of a configuration: Z = Λ n = 0 (Λ n)!β n Λ α ( H bi ) α Λ! α ( ) b 1,...,b Λ i=1 Walk through operator string Propose to insert diagonal operators instead of unit operators d P[1 H (i, j ) ] = min 1, βn d bonds α H (i, j ) α Λ n configuration index operator Propose to remove diagonal operators d P[H (i, j ) 1] = min 1, Λ n +1 d βn bonds α H (i, j ) α H1 d (1,2) H o (3,4) 1 H1 o d (1,2) (3,4) 110

111 Step 2: Offdiagonal updates (local) Are very easy, can be done in any representation Problem 1: only local changes Nonergodic No change of magnetization, particle number, winding number Problem 2: Critical slowing down Solution: loop algorithm 111

112 Loop building rules in SSE Example: XY-like AFM: H XXZ = J xz 2 (S + i S j + S i S + j ) + J z S z z i S j i, j with 0 J z J xy i, j W (C) = W (C,G) = Δ(C,G)V (G) G G Δ(C,G) W(C) 1 0 J z / J xy /2 W(G) J z /2 (J xy- J z )/2 112

113 Loop updates in path integrals 1. Define breakups (graphs) for exchange processes 2. Insert decay graphs 3. Build and flip one or more loops 113

114 Loop updates in SSE 1. Insert/remove diagonal operators 2. Decide breakups (graphs) 3. Build and flip one or more loops Similarity is no chance: an exact mapping exists 114

115 Measurements All diagonal operators can be measured easily diagonal spin correlation functions magnetization diagonal components of the energy Green s function The S + - S - offdiagonal Green s function can be measured during loop construction. See the papers for details. Energy in path integrals: in SSE: H = H 0 n/β H = n/β 115

116 Extensions of the loop algorithm The loop algorithm can also extended to other models Hubbard model (Kawashima Gubernatis, Evertz, PRB 1994) higher spin S > 1/2 (Kawashima, J. Stat. Phys. 1996) t-j model (Ammon, Evertz, Kawashima, MT, PRB 1998) SU(N) models (Harada and Kawashima, JPSJ, 2001) dissipative quantum models (Werner and Troyer, PRL 2005) Josepshon junction arrays (Werner and Troyer, PRL 2005)

117 10. The worm algorithm 117

118 Loop algorithm in a magnetic field Loop cluster algorithm requires spin inversion symmetry Magnetic field implemented by a-posteriori acceptance rate Example: spin dimer at J = h =1 H = J S 1 S 2 h S 1 z + S 2 z ( ) DPHYS p = exp( βj /2) Triplet E = J/4 - h = -3/4 Matthias 118 Troyer Triplet E = -J/4 Singlet Loop algorithm must go through high energy intermediate state Exponential slowdown E = -3J/4 = -3/4

119 High-T expansion of the Ising model Z = e Ks is j = cosh(k) 1 + tanh(k)s i s j s 1...s N ij s 1...s N ij i.e. Z = cosh(k) 2N 1 tanh(k)] n b s n b i s n b j tanh(k) P n b {n b } s n b i s n b j s 1...s N bonds n b =0 s 1...s N bonds n b =0, 1: power associated to bond ij s n i s n j s pi i, p i total power associated to site i s 1...s N ij i s i For a spin-1/2 system one has s sp = 2 if p is even, zero otherwise Hence, Z =2 N {n b } tanh(k) P n b (closed loops) 119

120 Closed loops NO YES Nonzero weight only if total even number of bond powers, thus all labeled bonds form closed loops Open-ended loops require one additional spin operator for each end: give correlation function measurements 120

121 Classical worm algorithm Prokof ev and Svistunov, PRL (2001) Correlator sector Partition function sector No critical slowing down faster than cluster updates Correlator = distribution function for the ends Normal state: small loops, short distance between ends Ordered state: macroscopic entangled loops 121

122 The complete algorithm I=M I M M M M If I=M, select a new site for both at random Otherwise move I or M in a random direction, with acceptance rates min [1,tanh(J/T)] for n=0 -> n=1 min [1,1/tanh(J/T)] for n=0 -> n=1 Easier to implement than local updates but faster than cluster updates 122

123 Worm updates DPHYS Break a world line by inserting a pair of creation/annihilation operators ( ) H H + η c i + c i H H + η S + i + S i i ( ) move these operators ( Ira and Masha ) using local moves until Ira and Masha meet i insert worm move worm continue until head and tail meet shift remove jump insert jump Matthias 123 Troyer

124 Worm algorithm in a magnetic field Worm algorithm performs a random walk Change of configuration done in small steps Example: spin dimer at J = h =1 Triplet E = J/4 - h = -3/4 No high energy intermediate state Efficient update in presence of a magnetic field Singlet E = -3J/4 = -3/4 124

125 Directed loops in SSE Instead of following a pre-chosen path given by graphs we pick randomly E.g. using heat bath method weight probability J z 4 + C straight J z 4 + C 2C + J xy 2 + h J xy 2 0 jump turn J xy 2 2C + J xy 2 + h 0 J z 4 + h + C bounce J z 4 + h + C 2C + J xy 2 + h 125

126 Better directed loop schemes Bounces are bad since they undo the last change If bounce can be eliminated loop algorithm possible Directed loops give loop algorithm as a limit for some models Bounce path can be minimized In models where there is no loop algorithm Better choices for path selection O.F. Syljuåsen and A.W. Sandvik, PRE (2002) O.F. Syljuåsen, PRE (2003) F. Alet, S. Wessel and M. Troyer, PRE (2004) 126

127 An earlier attempt Prokof ev et al 98 detailed balance at each step of random walk Cullen and Landau 83 unbiased random walk less efficient since the physics does not enter worm construction Matthias 127 Troyer

128 When to use which algorithm? Stochastic Series Expansion (SSE) is simpler to implement Continuous-time path integrals needs lower orders Use SSE for local actions with not too large diagonal terms SSE Path Integrals Loop algorithm Spin models Spin models with dissipation Worm algorithm Spin models in magnetic field Bose-Hubbard models 128

129 First order phase transitions Tunneling out of meta-stable state is suppressed exponentially gas liquid gas liquid τ exp( cl d 1 /T) P Critical slowing down solved by cluster updates How can we tunnel out of metastable state?? liquid critical point gas T Matthias 129 Troyer

130 First order phase transitions DPHYS Tunneling problem at a first order phase transition is solved by changing the ensemble to create a flat energy landscape Multicanonical sampling (Berg and Neuhaus, Phys. Rev. Lett. 1992) Wang-Landau sampling (Wang and Landau, Phys. Rev. Lett. 2001) Quantum version (MT, Wessel and Alet, Phys. Rev. Lett. 2003) Optimized ensembles (Trebst, Huse and MT, Phys. Rev. E 2004)?? liquid solid Matthias 130 Troyer

131 Canonical sampling n w (E) = exp( βe) g(e) ~ 2 N histogram n w (E) canonical weight ln g(e) 2 2D ferromagnet ln( density of states ) energy -1 0 energy / 2N critical energy Matthias 131 Troyer E c = E(T c ) 0.74E 0

132 First-order phase transition DPHYS n w (E) = exp( βe) g(e) histogram T=T c canonical weight ln g(e) 10-state Potts model ln( density of states ) energy -1 0 energy / 2N Exponentially suppressed tunneling out of metastable states. Matthias 132 Troyer critical energies

133 Flat-histogram sampling n w (E) = 1/g(E) g(e) flat-histogram weight How do we obtain the weights? Flat-histogram MC algorithms Multicanonical recursions Wang-Landau B. A. Berg and algorithm T. Neuhaus (1992) energy F. Wang and D.P. Landau (2001) Quantum version M. Troyer, S. Wessel and F. Alet (2003) Matthias 133 Troyer

134 alculating the density of states Start with any ensemble w(e) = 1 g(e) Simulate using Metropolis algorithm The Wang-Landau algorithm g(e) = 1 estimated density of states p(e 1 E 2 ) = min ( 1, w(e ) 2) w(e 1 ) Iteratively improve ensemble during simulation = min ( 1, g(e ) 1) g(e 2 ) g(e) = g(e) f modification factor Reduce modification factor f when histogram is flat. Matthias 134 Troyer

135 Details of the Wang-Landau method Initially ρ(e) is unknown Start with ρ(e)=1 and adjust iteratively Only a few modifications to usual sampling needed Start with modification factor f=1 do { do { Metropolis updates with transition probability W[i->j] = min[1,ρ(e i ) / ρ(e j )] Adjust ρ(e) at each step: ρ(e) < ρ(e) x exp(f) } until histogram H(E) is flat decrease f < f / 2 } until f

136 Wang-Landau in action Movie by Emanuel Gull (2004) 136

137 Flat-histogram sampling Idea: sampling all energies increases equilibration. slow equilibration energy temperature fast round-trips! How can we quantify the performance? Matthias 137 Troyer

138 Typical round-trip time DPHYS -1 0 energy / 2N The energy range scales like N. τ N 2 The round-trip time should scale like N 2. τ Matthias 138 Troyer

139 Performance of flat-histogram sampling Phys. Rev. Lett. 92, (2004) τ / N z = 0.9 fully frustrated Ising model ferromagnetic Ising model z = L = N 1 / 2 Flat-histogram random walk in energy space: power law scaling O( N 2+z ) dynamical exponent Flat-histogram sampling suffers from slowing down. Matthias 139 Troyer

140 Where is the bottleneck? DPHYS local diffusivity in energy D(E, t D ) = [E(t) E(t + t D )] 2 /t D 5 local diffusivity E / 2N Matthias 140 Troyer

141 Labeled walkers DPHYS energy Matthias 141 Troyer

142 Labeled walkers DPHYS ferromagnetic Ising model fraction f(e) label down 0.2 label up E / 2N Steady-state data for current in energy space. Matthias 142 Troyer

143 Optimizing the ensemble DPHYS Measure the current in the energy interval S. Trebst, D. Huse and MT Phys. Rev. E 70, (2004) local diffusivity j = D(E) n w (E) df de current histogram derivative of fraction Determine the local diffusivity. Maximize current by varying histogram/ensemble. Matthias 143 Troyer

144 Feedback for speed Calculate the current Rearrange this j = D(E)n w (E) df de 1 j df de = 1 D(E)n w (E) and integrate 1 j = Emax E min de D(E)n w (E) Matthias 144 Troyer

145 Feedback for speed Normalize n (E) by introducing a Lagrange multiplier w Emax E min de ( ) 1 D(E)n w (E) + λn w(e) Extremize by simply varying n w (E) n (opt) w = 1 D(E)λ Matthias 145 Troyer

146 Optimizing the ensemble DPHYS Optimal histogram is found Phys. Rev. E 70, (2004) n (opt) w 1 D(E) Feedback of local diffusivity optimized ensemble w (E) w(e) df de 1 n w (E) Iterate feedback until convergence. original ensemble Matthias 146 Troyer

147 Optimized histogram ferromagnet DPHYS 3 2 histogram E / 2N critical energy Feedback reallocates resources towards the critical energy. Matthias 147 Troyer

148 erformance of optimal DPHYS nsemble τ / N flat-histogram ensemble speedup ~ optimized ensemble 5,000 cpu hours L = N 1 / 2 The round-trip times scale like O( [N log N] 2 ). Matthias 148 Troyer

149 Applications of optimized ensembles Nonfrustrated and frustrated magnets (PRE 2004) Dense Lennard-Jones liquids (Trebst, Gull, MT, JCP 2005 ) Parallel tempering (Katzgraber, Trebst, Huse & MT, JSTAT 2006) optimal temperature set more temperature points in critical region Protein folding (Trebst, MT, Hansmann, JCP 2006) Quantum systems (MT et al, PRL 2003, Wessel et al, JSTAT 2008) 149

150 uantum systems DPHYS Classical: c E Z = e E c / k B T = ρ(e)e E / k B T Quantum: ρ(e) is not accessible formulation in terms of high-temperature series Z = Tr(e βh ) = or perturbation series β n Tr( H) n = n! n= 0 n= 0 β n g(n) n! Z = Tre βh = Tr e β ( H 0 +λv ) = λ n g(n) Flat histogram, parallel tempering, histogram reweighting, n= 0 etc done in order n of series expansion instead of energy Matthias 150 Troyer

151 Stochastic Series Expansion (SSE) DPHYS based on high temperature expansion, (A. Sandvik, 1991) Z = Tr(e β H ) = = β n β n n=0 n! Tr ( H ) n α n! 1 H α 2 α 2 H α 3 α n H α 1 also has a graphical representation in terms of world lines n=0 α 1,...,α n is very similar to path integrals perturb in all terms of the Hamiltonian, not just off-diagonal terms H b1 H b2 H b1 Matthias 151 Troyer

152 Wang-Landau sampling for quantum SSE: Z = Tr(e βh ) = = β n β n n=0 n! Tr ( H ) n α 1 H α 2 α 2 H α 3 α n H α 1 n! n=0 α 1,...,α n compare to classical Monte Carlo: β n n=0 n! g(n) flat histogram obtained by changing the ensemble: Z = e E c / k B T = ρ(e)e E / k B T classically: c E quantum: e βe c 1 ρ(e) β n n! 1 g(n) 152

153 Wang-Landau updates in SSE We want flat histogram in order n Use the Wang-Landau algorithm to get Λ from Λ (Λ n)!β n Λ Z = β n g(n) Z = α ( H bi ) α ( Λ! n = 0 n = 0 α b Small change in acceptance rates 1,...,b for Λ ) i=1 diagonal updates d P[1 H (i, j ) ] = min 1, βn d bonds α H (i, j ) α Wang Landau min 1, N d bonds α H (i, j ) α Λ n Λ n Loop update does not change n and is thus unchanged! d Λ n +1 Wang-Landau Λ n +1 P[H Cutoff (i, j ) 1] = min 1, Λ limits temperatures βn bonds α H d (i, j ) α min 1, d to β < Λ / E 0 N bonds α H (i, j ) α g(n) g(n +1) g(n) g(n 1) 153

154 The first test L=10 site Heisenberg chain with Λ = 250 temperature cutoff due to finite L 154

155 Wang-Landau sampling for quantum Example: 3D quantum Heisenberg antiferromagnet T/J T/J 155

156 Speedup at first order phase DPHYS Greatly reduced tunneling times at free energy barriers Example: stripe rotation in 2D hard-core bosons ( ) H = t a i a j + h.c. + V 2 n i n j i, j i, j τ av conventional SSE Wang Landau sampling T / t 156

157 Perturbation expansion Instead of temperature a coupling constant can be changed Based on finite temperature perturbation expansion with H = H 0 + λv = Z = Tr(e βh ) = Λ n = 0 ( β) n n! n Flat histogram λ = 0 in order n λ α ( H bi ) α λ n λ (b 1,...,b n ) α ( ) b 1,...,b n (Λ n)!β n α ( H bi ) α λ n λ (b 1,...,b Λ ) Λ! n = 0 Λ α ( ) b 1,...,b Λ = λ n λ g(n λ ) i λ n λ (i) H i Λ i=1 n i=1 n λ (b 1,...,b n ) counts # of λ terms of perturbation expansion 157

158 Perturbation series by Wang-Landau We want flat histogram in order n λ Use Λthe Wang-Landau algorithm Λ to get (Λ n)!β n Λ from Z = α ( H bi ) α λ n λ (b 1,...,b Λ Z = λ n λ g(n ) λ ) Λ! n λ = 0 n = 0 Small change in acceptance rates for diagonal updates α ( ) b 1,...,b Λ i=1 d P[1 H (i, j ) ] = min 1, βn d bonds α H (i, j ) Λ n α Wang Landau min 1, βn d bonds α H (i, j ) Λ n α g(n λ ) g(n λ + Δn λ ) d P[H (i, j ) Λ n +1 1] = min 1, βn bonds α H (i, j ) Wang-Landau Λ n +1 min 1, βn bonds α H (i, j ) d d Loop update does not change α n and is thus unchanged! α Cutoff Λ limits value of λ for which the series converges g(n λ ) g(n λ Δn λ ) 158

159 The antiferromagnetic bilayer J J J << J : spin gap, no long range order J >> J : long range order Quantum phase transition at J / J 2.524(2) Spin gap vanishes Magnetic order vanishes Universal properties 159

160 Quantum phase transition Quantum phase transition in bilayer quantum Heisenberg antiferromagnet J/Jʼ 160

161 Summary Extension of Wang-Landau sampling to quantum systems Stochastically evaluate series expansion coefficients High-temperature series Perturbation series Features Flat histogram in the expansion order Allows calculation of free energy Like classical systems, allows tunneling through free energy barriers Z = Z = β n g(n) n = 0 n! λ n λ g(n λ ) n λ = 0 Optimized ensembles are also possible 161

162 Quantum Monte Carlo Not as easy as classical Monte Carlo Calculating the eigenvalues E c is equivalent to solving the problem Z = c e E c / k B T Need to find a mapping of the quantum partition function to a classical problem Negative sign problem if some p c < 0 Z = Tre βh c p c 162

163 The negative sign problem DPHYS In mapping of quantum to classical system Z =Tre βh = i p i there is a sign problem if some of the pi < 0 Appears e.g. in simulation of electrons when two electrons exchange places (Pauli principle) i 1 > i 4 > i 3 > i 2 > i 1 > Matthias 163 Troyer

164 The negative sign problem DPHYS Sample with respect to absolute values of the weights A = i A i p i p i = A i sgn p i p i i Exponentially growing cancellation in the sign i i sgn p i p i i i p i p i A sign p sign p sign = i p i Exponential growth of errors i p i = Z/Z p = e βv (f f p ) sign sign = sign2 sign 2 M sign M eβv (f f p ) NP-hard problem (no general solution) [Troyer and Wiese, PRL 2005] Matthias 164 Troyer

165 Is the sign problem exponentially The sign problem is basis-dependent Diagonalize the Hamiltonian matrix A = Tr[ Aexp(!"H) ] Tr[ exp(!"h) ] = i A i i exp(!"# i ) All weights are positive $ $ exp(!"# i ) But this is an exponentially hard problem since dim(h)=2 N! Good news: the sign problem is basis-dependent! But: the sign problem is still not solved Despite decades of attempts Reminiscent of the NP-hard problems No proof that they are exponentially hard No polynomial solution either i i 165

166 Complexity of decision problems Partial hierarchy of decision problems Undecidable ( This sentence is false ) Partially decidable (halting problem of Turing machines) EXPSPACE Exponential space and time complexity: diagonalization of Hamiltonian PSPACE Exponential time, polynomial space complexity: Monte Carlo NP Polynomial complexity on non-deterministic machine Traveling salesman problem 3D Ising spin glass P Polynomial complexity on Turing machine 166

167 Complexity of decision problems Some problems are harder than others: Complexity class P Can be solved in polynomial time on a Turing machine Eulerian circuit problem Minimum spanning Tree (decision version) Detecting primality Complexity class NP Polynomial complexity using non-deterministic algorithms Hamiltonian cirlce problem Traveling salesman problem (decision version) Factorization of integers 3D spin glasses 167

168 The complexity class P The Eulerian circuit problem Seven bridges in Königsberg (now Kaliningrad) crossed the river Pregel Can we do a roundtrip by crossing each bridge exactly once? Is there a closed walk on the graph going through each edge exactly once? Looks like an expensive task by testing all possible paths. Euler: Desired path exits only if the coordination of each edge is even. This is of order O(N 2 ) Concering Königsberg: NO! 168

169 The complexity class NP The Hamiltonian cycle problem Sir Hamilton's Icosian game: Is there a closed walk on going through each vertex exactly once? DPHYS Looks like an expensive task by testing all possible paths. No polynomial algorithm is known, nor a proof that it cannot be constructed 169

170 The complexity class NP Polynomial time complexity on a nondeterministic machine Can execute both branches of an if-statement, but branches cannot merge again Has exponential number of CPUs but no communication It can in polynomial time Test all possible paths on the graph to see whether there is a Hamiltonian cycle Test all possible configurations of a spin glass for a configuration smaller than a given energy It cannot c : E c < E Calculate a partition function since the sum over all states cannot be performed c Z = exp( βε c ) 170

171 NP-hardness and NP-completeness Polynomial reduction Two decision problems Q and P: : there is an polynomial algorithm for Q, provided there is one for P Typical proof: Use the algorithm for P as a subroutine in an algorithm for P Many problems have been reduced to other problems Q P NP-hardness A problem P is NP-hard if This means that solving it in polynomial time solves all problems in NP too NP-completeness A problem P is NP-complete, if P is NP-hard and Q NP : Q P Most Problems in NP were shown to be NP-complete DPHYS P NP 171

172 The P versus NP problem Hundreds of important NP-complete problems in computer science Despite decades of research no polynomial time algorithm was found Exponential complexity has not been proven either The P versus NP problem Is P=NP or is P NP? One of the millenium challenges of the Clay Math Foundation 1 million US$ for proving either P=NP or P NP? The situation is similar to the sign problem 172

173 The Ising spin glass: NP-complete 3D Ising spin glass The NP-complete question is: Is there a configuration with energy E 0? Solution by Monte Carlo: Perform a Monte Carlo simulation at H = J ij σ j σ j with J ij = 0,±1 i, j Measure the energy: β = N ln2 + ln N + ln E < E if there exists a state with energy E 0 A Monte Carlo simulation can decide the question E > E 0 +1 otherwise 173

174 The Ising spin glass: NP-complete 3D Ising spin glass is NP-complete H = J ij σ j σ j with J ij = 0,±1 i, j Frustration leads to NP-hardness of Monte Carlo? Exponentially long tunneling and autocorrelation times c 1! c 2!...! c i! c i +1!... ΔA = ( A A ) 2 = Var A M (1 + 2τ A ) 174

175 Frustration Antiferronmagnetic couplings on a triangle:? Leads to frustration, cannot have each bond in lowest energy state With random couplings finding the ground state is NP-hard Quantum mechanical: negative probabilities for a world line configuration Due to exchange of fermions Negative weight (-J) 3 -J! -J! -J! 175

176 What is a solution of the sign problem? Consider a fermionic quantum system with a sign problem (some p i < 0 A = Tr[ Aexp( βh )] Tr [ exp( βh )] = A i p i i i p i Where the sampling of the bosonic system with respect to p i scales polynomially T ε 2 N n β m A solution of the sign problem is defined as an algorithm that can calculate the average with respect to p i also in polynomial time Note that changing basis to make all p i 0 might not be enough: the algorithm might still exhibit exponential scaling 176

177 Solving an NP-hard problem by QMC Take 3D Ising spin glass View it as a quantum problem in basis where H it is not diagonal H = J ij σ j σ j with J ij = 0,±1 i, j H (SG ) = J ij σ x jσ x j with J ij = 0,±1 The randomness ends up in the sign of offdiagonal matrix elements i, j Ignoring the sign gives the ferromagnet and loop algorithm is in P The sign problem causes NP-hardness solving the sign problem solves H (FM ) = all the σ x jσnp-complete x j problems and prove NP=P i, j 177

178 Summary A solution to the sign problem solves all problems in NP Hence a general solution to the sign problem does not exist unless P=NP If you still find one and thus prove that NP=P you will get 1 million US $! A Nobel prize? A Fields medal? What does this imply? A general method cannot exist Look for specific solutions to the sign problem or model-specific methods 178

179 The origin of the sign problem We sample with the wrong distribution by ignoring the sign! We simulate bosons and expect to learn about fermions? will only work in insulators and superfluids We simulate a ferromagnet and expect to learn something useful about a frustrated antiferromagnet? We simulate a ferromagnet and expect to learn something about a spin glass? This is the idea behind the proof of NP-hardness 179

180 Working around the sign problem 1.Simulate bosonic systems Bosonic atoms in optical lattices Helium-4 supersolids Nonfrustrated magnets 2.Simulate sign-problem free fermionic systems Attractive on-site interactions Half-filled Mott insulators 3.Restriction to quasi-1d systems Use the density matrix renormalization group method (DMRG) 4.Use approximate methods Dynamical mean field theory (DMFT) 180

181 Diagrammatic expansion in DPHYS too many to draw all p! 2 possible topologies but easy to sum all 2 of them! (about 5000! ) 181

182 Diagrammatic expansion in A.N. Rubtsov & A.I. Lichtenstein, Pis'ma v JETP 80, 67 (2004) 1 A.N. Rubtsov, V.V. Savkin, A.I. Lichtenstein, Phys. Rev. B 72, (2005) The sum of all p! 2 diagrams for a given vertex configuration is a determinant squared Sign-problem free for attractive interactions U<0 and balanced population of up and down spins 182

183 Repulsive interactions Repulsive interactions: U>0 (-U) n can be negative: sign problem Perform particle-hole transformation on one spin-species U changes sign Un i, n i, Un i, (1 n i, )=Un i, Un i, n i, Equal population condition changes to a half-band filling condition n i, = n i, n i, =1 n i, n i, + n i, =1 We can simulate repulsive fermions (only) at half filling 183

184 The fermion worm algorithm E. Burovski, N. Prokof ev, B. Svistunov, M. Troyer Phys. Rev. Lett. 96, (2006) New J. Phys. 8, 153 (2006) DPHYS The worm algorithm can be adapted to the diagrammatic expansion for fermions of Rubtsov et al. create two half-vertices (Ira and Masha) move Ira around, creating or annihilating vertices remove Ira and Masha when they get close 184

185 Critical temperature of the resonant Fermi DPHYS Phase diagram of dilute Fermi gas ak attraction: S of Cooper pairs??? BEC strong attraction: BEC of molecules BCS Universal behavior at resonance (unitary point) where 2-particle bound state crosses lower band edge at about U = -8t in lattice model 185

186 Previous results for the unitary point 0.5 M Holland, SJJMF Kokkelmans, ML Chiofalo, and R. Walser J Kinhast, A Turlapov, JE Thomas, Q Chen, J Stajic, K Levin V.K. Akkineni, N. Trivedi, D. Ceperley P Nozieres, S Schmitt-Rink A Bulgac, JE Drut, P Magierski XJ Liu, H Ho P. Nikolic, S Sachdev R. Haussmann, W. Rantner S. cerrito, W. Zwerger M Wingate BEC limit 186

187 Previous results for the unitary point 0.5 M Holland, SJJMF Kokkelmans, ML Chiofalo, and R. Walser J Kinhast, A Turlapov, JE Thomas, Q Chen, J Stajic, K Levin V.K. Akkineni, N. Trivedi, D. Ceperley 0.2 P Nozieres, S Schmitt-Rink XJ Liu, H Ho A Bulgac, JE Drut, P Magierski BEC limit 0.1 P. Nikolic, S Sachdev R. Haussmann, W. Rantner S. cerrito, W. Zwerger M Wingate Our result 0 187

188 Dynamical Mean Field Theory DPHYS is an approximative but successful method for describing strongly interacting (repulsive) fermions in high dimensions E. Müller-Hartmann, Z. Phys. B (1989). M. Metzner and D. Vollhard, PRL 62, 324 (1989). A. Georges and G. Kotliar Phys. Rev. B 45, 6479 (1992). A. Georges et al., Rev. Mod. Phys. 68, 13 (1996). solves a few-site problem in the presence of a self-consistent bath provided by the rest of the system with an effective nonlocal action 188

189 Mean-field theory for Ising Model Lattice model (nearest neighbor coupling J, coordination number z) H latt = J i,j S is j J Single site model (m i = S i, h eff = J 0,i m i = zjm) H 0 = h eff S 0 h eff Self-consistency condition m = m 0 H0 ( = tanh(βh eff ) = tanh(βzjm) ) 189

190 Dynamical mean field theory Lattice model H latt = U i n i n i t i,j,σ c iσ c jσ Metzner & Vollhardt, PRL (1989) Georges & Kotliar, PRB (1992) t Quantum impurity model H imp = Un n k,σ (t kc σa bath k,σ + h.c.) + H bath t k 190

191 Dynamical mean field theory DPHYS Self-consistency loop lattice model Metzner & Vollhardt, PRL (1989) impurity model Georges & Kotliar, PRB (1992) t t k G latt dk 1 iω n +µ k Σ latt Σ latt self-consistency condition G latt G imp DMFT approximation H imp impurity solver G imp, Σ imp Σ latt Σ imp Computationally expensive step: solution of the impurity model 191

192 DMFT impurity solvers Analytical solvers: just perturbative approximations Numerical solvers: Exact Diagonalization (ED) Density Matrix Renormalization Group Methods (DMRG) Numerical Renormalization Group (NRG) Quantum Monte Carlo solvers Hirsch-Fye solver (Hirsch & Fye, PRL 1986) Weak coupling (Rubtsov et al, PRB 2005, Gull et al, EPL 2008) Hybridization expansion (Werner et al, PRL 2006) 192

193 Diagrammatic QMC General recipe: Split Hamiltonian into two parts: Use interaction representation in which DPHYS Write partition function as time-ordered exponential, expand in powers of Z = T r = k H 2 e βh 1 T e R β 0 dτh 2(τ) β 0 dτ 1... β 0 ( 1) k dτ k k! Weak-coupling expansion: Rombouts et al., (1999), Rubtsov et al. (2005), Gull et al. (2008) expand in interactions, treat quadratic terms exactly Hybridization expansion: Werner et al., (2006), Werner & Millis (2006), Haule (2007) expand in hybridizations, treat local terms exactly H = H 1 + H 2 O(τ) = e τh 1 Oe τh 1 T r e βh 1 T H 2 (τ 1 )... H 2 (τ k ) 193

194 Hirsch-Fye QMC solver J.E. Hirsch & R.M. Fye, Phys. Rev. Lett. 56, 2521 (1986) DPHYS Uses M discrete time steps τ = β/m Decouples quartic interaction using Hubbard-Stratonovich transformation and auxiliary fields si e τu(n n +1/2(n +n )) = 1 2 s=±1 Integration over fermionic Gaussian integrals gives eλ(u, τ)s(n +n ), determinants Z = s i det G 1 0, (s 1,..., s N )G 1 0, (s 1,..., s N ) Monte Carlo sampling of auxiliary fields si Was the standard QMC solver until recently, suffers from all disadvantages of discrete time solvers 194

195 Impurity model given by Expand partition function into powers of the interaction term Decouple the interaction terms using Rombouts et al., PRL (1999) DPHYS CT-auxiliary field QMC Rombouts et al., PRL (1999) H = H 0 + H U H 0 = K/β (µ U/2)(n + n ) + H hyb + H bath H U = U(n n (n + n )/2) K/β Z = k ( 1) k k! dτ 1... Gull et al., EPL (2008) dτ k T r T e βh 0 H U (τ 1 )... H U (τ k ) H U = K 2β then integrate over fermionic Gaussian integrals s=±1 e γs(n n ), cosh(γ) = 1 + βu 2K 195

196 CT-auxiliary field QMC DPHYS Rombouts et al., PRL (1999) Gull et al., EPL (2008) Configuration space: all possible time-ordered spin configurations Weight: w(τ 1, s 1 ;... ; τ k, s k ) = Kdτ 2β k N 1 σ = e Γ σ G 0σ e Γ σ 1 Monte Carlo updates: random insertion/removal of a spin Formally similar to discrete-time Hirsch-Fye method Closely related to weak-coupling solver by Rubtsvov (2005) σ det N 1 σ ({τ i, s i }) e Γ σ = diag(e γσs 1,..., e γσs k ) 196

197 Rubtsov et al., PRB (2005) DPHYS Weak-coupling CT-QMC Impurity model given by H = H 0 + H U H 0 = µ(n + n ) + α n + α n + H hyb + H bath H U = U(n α )(n α ) Expand partition function into powers of the interaction term Z = k ( 1) k k! dτ 1... dτ k T r T e βh 0 H U (τ 1 )... H U (τ k ) Wick s theorem yields weight of vertex configurations alpha-terms necessary to avoid sign problem 197

198 Weak-coupling CT-QMC Rubtsov et al., PRB (2005) DPHYS Configuration space: all possible time-ordered vertex configurations Weight: w(τ 1,..., τ k ) = ( Udτ) k σ det G (α,α ) 0σ ({τ i }) Monte Carlo updates: random insertion/removal of a vertex 198

199 M-I transition in the 2D Hubbard Hubbard model with nn hopping t, nnn hopping t =0 (bandwidth 8t) H = p,α p c p,αc p,α + i n i, n i, p = 2t(cos(p x ) + cos(p y )) DMFT: approximate momentum-dependence of the self-energy Σ(p, ω) = a DCA: ``tiling of the Brillouin zone φ a (p)σ a (ω) 199

200 M-I transition in the 2D Hubbard Doping the insulator produces electron/hole pockets 8-site cluster has a ``tile at the expected position of the pockets 8-site DCA-result at U/t=7: first 8% of dopants go into the B sector t=40, B t=40, C t=20, B t=20, C n B C B C !/t 200

201 M-I transition in the 2D Hubbard Doping the insulator produces electron/hole pockets 8-site cluster has a ``tile at the expected position of the pockets 8-site DCA-result at U/t=7: first 8% of dopants go into the B sector Assuming an ellipsoidal shape for the pocket, we can estimate the aspect ratio b a 1 10 a b 201

202 Hybridization expansion Impurity model given by DPHYS Werner et al., PRL (2006) Werner & Millis, PRB (2006) Haule, PRB (2007) H = H loc + H bath + H hyb H loc = Un n µ(n + n ) H hyb = t σ p c σa p,σ + h.c. p,σ Expand partition function into powers of the hybridization term Z = 1 dτ 1... dτ 2k T r T e β(h loc+h bath ) H hyb (τ 1 )... H hyb (τ 2k ) 2k! k Trace over bath degrees of freedom yields determinant of hybridization functions F T r bath [...] = σ det Mσ 1, Mσ 1 (i, j) = F σ (τ (c) i τ (c ) j ) t σ p 2 F σ ( iω n ) = p iω n p 202

203 Hybridization expansion DPHYS Werner et al., PRL (2006) Werner & Millis, PRB (2006) Haule, PRB (2007) Monte Carlo configurations consist of segments for spin up and down Monte Carlo updates: random insertion/removal of (anti-)segments Weight of a segment configuration: w τ σ(c) 1, τ σ(c ) 1 ;... ; τ σ(c) k σ, τ σ(c ) k σ = e Ul overlap +µ(l +l ) T r imp [...] Determinant of a k x k matrix resums k! diagrams F σ (τ (c) 1 τ (c ) 1 ) F σ (τ (c) 1 τ (c ) 2 ) det F σ (τ (c) 2 τ (c ) 1 ) F σ (τ (c) 2 τ (c ) Eliminates sign problem 2 ) σ det M 1 σ T r bath [...] dτ 2k σ 203

204 Generalizations Multi-orbital and cluster problems P. Werner and A.J. Millis, Phys. Rev. B 74, (2006) K. Haule, Phys. Rev. B 75, (2007) Phonons can be added at no cost P. Werner and A.J. Millis, Phys. Rev. Lett. 99, (2007) 204

205 Multi-orbital case Single site: treat interaction by looking at overlap of lines: 0 β Un n Un n Un n Un n Extension to density - density interactions for multiple orbitals is straightforward: 0 Un n Un n Un n Un n β 205

206 Spin freezing transition in a 3-orbital 1 site, 3 degenerate orbitals (semi-circular DOS, bandwidth 4t) H loc = α,σ µn α,σ + α Un α, n α, + α>β,σ U n α,σ n β, σ + (U J)n α,σ n β,σ α=β J(ψ α, ψ β, ψ β, ψ α, + ψ β, ψ β, ψ α, ψ α, + h.c.) Important for SrRuO3 other transition metal oxides, actinides, Fe based superconductors,

207 Spin freezing transition in a 3-orbital Phase diagram for U = U = 2J, J/U = 1/6, βt = U/t 8 U/t Mott insulator ( t=50) µ/t Mott insulating ``lobes with 1, 2, 3, (4, 5) electrons n 207

208 Spin freezing transition in a 3-orbital U/t Phase diagram for <n 0 (0)n 1 ( )>, <S z (0)S z ( )> µ/t glass transition Mott insulating ``lobes with 1, 2, 3, (4, 5) electrons -0.1 U = U = 2J, J/U = 1/6, βt = 50 In the metallic phase: transition from Fermi liquid to ``spin glass U/t Fermi liquid glass transition Mott insulator ( t=50) t n=1.21 n=1.75 n=2.23 n=2.62 n=2.97 n frozen moment 208

209 Spin freezing transition in a 3-orbital U/t Phase diagram 3 for Im /t U = U = 2J, J/U = 1/6, βt = 50 n=2.62 n=2.35 n=1.99 n=1.79 n=1.60 U/t Fermi liquid frozen moment µ/t glass transition Critical exponents 0 associated with the transition can be seen in a wide quantum 0 critical regime e. g. non Fermi-liquid self-energy( n /t) glass transition Mott insulator ( t=50) ImΣ/t (iω n /t) α, α 0.5 n 209

210 Spin freezing transition in a 3-orbital A self-energy with frequency dependence optical conductivity σ(ω) 1/ω 1/2 Σ(ω) ω 1/2 implies an 210

211 Comparing the QMC solvers All three solvers need to calculate determinants same scaling as O(β 3 ) per update sweep ratio of matrix sizes determines low-temperature CPU scaling Benchmarks at intermediate temperature compare error bars of a calculation using 140 hours of CPU time on a single 4-year old PC 211

212 Solver Comparison - Matrix Sizes Weak Coupling Algorithm Hybridization Expansion Hirsch Fye Matrix Size βt U/t=4 212

213 Matrix Sizes - coupling depenence 100 Weak Coupling Algorithm Hybridization Expansion Matrix Size 50 Typical region of interest: U/t βt=30 213

214 Summary and Conclusions Diagrammatic continuous-time QMC impurity solvers enable efficient DMFT simulations of fermionic lattice models Weak-coupling solver ideal for large impurity clusters Hybridization expansion allows to treat multi-orbital models with complicated interactions Applications Aspect ratio of electron/hole pockets in Hubbard model from 8-site DCA Spin freezing transition in a 3-orbital model with Hund coupling Ongoing and future projects LDA+DMFT studies of transition metal oxides, f-electron materials, superconductivity in multi-band models,

215 Divergence of the correlation Typical length scale ξ divegres at phase transition at Tc m (T c T) β ξ T T c ν To avoid system size effects we need to have L >> ξ T T c T >> T c L ξ L ξ 215