7. Overview of modern QMC algorithms Friday, 9 July 2010 Modern Monte Carlo algorithms • Which system sizes can be studied? temperature local updates modern algorithms 3D Tc 16’000 spins 16’000’000 spins 0.1 J 200 spins 1’000’000 spins 0.005 J ––– 50’000 spins 3D Tc 32 bosons 1’000’000 bosons 0.1 t 32 bosons 10’000 bosons Friday, 9 July 2010 When to use SSE? • For quantum magnets • loop cluster algorithm if there is spin inversion symmetry • directed loops if there is no spin inversion symmetry • For hardcore bosons: • loop cluster algorithm if there is particle-hole symmetry • directed loops if there is no particle-hole symmetry • Which models? • 2-site interactions are rather straightforward • multi-site interactions require more thought Friday, 9 July 2010 When to use path integrals? • For Bose-Hubbard models • Use the worm algorithm in continuous time path integrals • This expands only in the hopping t and not the much larger repulsion U • For non-local in time actions • Appear in dissipative (Caldeira-Legget type) models, coupling to phonons, DMFT, ... • Cluster algorithms are again possible in case of spin-inversion symmetry Friday, 9 July 2010 8. Wang-Landau sampling and optimized ensembles for quantum systems Friday, 9 July 2010 First order phase transitions Tunneling out of meta-stable state is suppressed exponentially liquid gas gas liquid τ ∝exp(−cLd −1 /T) P Critical slowing down solved by cluster updates liquid How can we tunnel critical ? point out of metastable state? gas T Friday, 9 July 2010 First order phase transitions • 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 Friday, 9 July 2010 Quantum systems Z e−Ec / kBT (E)e−E / kBT Classical: = ∑ = ∑ρ • c E • Quantum: ρ(E) is not accessible • formulation in terms of high-temperature series ∞ β n ∞ β n Z = Tr(e−βH ) = ∑ Tr(−H)n = ∑ g(n) n= 0 n! n= 0 n! • or perturbation series ∞ Z = Tre−βH = Tre−β ( H 0 +λV ) = ∑λn g(n) n= 0 • Flat histogram, parallel tempering, histogram reweighting, etc done in order n of series expansion instead of energy Friday, 9 July 2010 Stochastic Series Expansion (SSE) • based on high temperature expansion, (A. Sandvik, 1991) ∞ β n −β H ⎡ n ⎤ Z = Tr(e ) = ∑ Tr ⎣(−H) ⎦ n=0 n! ∞ β n = α − H α α − H α ⋅⋅⋅ α − H α ∑ n! ∑ 1 2 2 3 n 1 n=0 α1 ,...,αn • also has a graphical representation in terms of world lines H b2 Hb 1 H b1 • is very similar to path integrals • perturb in all terms of the Hamiltonian, not just off-diagonal terms Friday, 9 July 2010 Wang-Landau sampling for quantum systems ∞ n −βH β n Z = Tr(e ) = ∑ Tr ⎣⎡(−H ) ⎦⎤ • SSE: n=0 n! ∞ β n = ∑ ∑ α1 − H α 2 α 2 − H α 3 ⋅⋅⋅ α n − H α1 n=0 n! α1 ,...,αn ∞ β n ≡ ∑ g(n) n=0 n! • compare to classical Monte Carlo: Z = ∑e−Ec / kBT = ∑ρ(E)e−E / kBT c E • flat histogram obtained by changing the ensemble: • classically: E 1 e−β c → ρ(E) β n 1 • quantum: → n! g(n) Friday, 9 July 2010 Wang-Landau updates in SSE • We want flat histogram in order n • Use the Wang-Landau algorithm to get Λ Λ (Λ − n)!β n Λ Z = ,,,β n g(n) from Z = α (−H ) α ∑ ∑∑ ∑ Λ! ∏ bi n= 0 n= 0 α (b1 ,...,bΛ ) i=1 • Small change in acceptance rates for diagonal updates ⎛ d ⎞ ⎛ d ⎞ d βNbonds α H(i, j ) α Wang−Landau Nbonds α H(i, j ) α g(n) P[1→ H(i, j )] = min⎜1 , ⎟ ⎯ ⎯ ⎯ ⎯⎯ → min⎜1 , ⎟ ⎝ Λ − n ⎠ ⎝ Λ − n g(n +1)⎠ ⎛ Λ − n +1 ⎞ ⎛ Λ − n +1 g(n) ⎞ P[H d →1] = min⎜1 , ⎟ ⎯Wang ⎯ ⎯-Landau ⎯⎯ → min⎜1 , ⎟ (i, j ) ⎜ N H d ⎟ ⎜ N H d g(n 1)⎟ ⎝ β bonds α (i, j ) α ⎠ ⎝ bonds α (i, j ) α − ⎠ • Loop update does not change n and is thus unchanged! • Cuto' Λ limits temperatures to β < Λ / E0 Friday, 9 July 2010 The first test • L=10 site Heisenberg chain with Λ = 250 temperature cutoff due to finite L Friday, 9 July 2010 Wang-Landau sampling for quantum systems • Example: 3D quantum Heisenberg antiferromagnet 0.8 1.0 0.8 0.6 0.6 0.4 0.4 0.2 0.0 0.0 0.4 0.8 1.2 1.6 2.0 0.2 T/J 0.0 0.4 0.8 1.2 1.6 2.0 T/J Friday, 9 July 2010 Speedup at first order phase transition • Greatly reduced tunneling times at free energy barriers • Example: stripe rotation in 2D hard-core bosons 108 † H = −t∑(ai a j + h.c.) + V2 ∑nin j i, j i, j 106 conventional SSE 4 10 av 102 Wang−Landau sampling 100 0.0 0.5 1.0 1.5 2.0 T / t Friday, 9 July 2010 Perturbation expansion • Instead of temperature a coupling constant can be changed • Based on finite temperature perturbation expansion nλ (i) with H = H0 + λV = ∑λ Hi i ∞ (−β)n n Z = Tr(e−βH ) = α (−H ) α λnλ (b1 ,...,bn ) ∑ n! ∑ ∑ ∏ bi n= 0 α (b1 ,...,bn ) i=1 Λ (Λ − n)!β n Λ ≈ α (−H ) α λnλ (b1 ,...,bΛ ) ∑∑ ∑ Λ! ∏ bi n= 0 α (b1 ,...,bΛ ) i=1 Λ nλ = λ g(nλ ) ∑ nλ (b1,...,bn ) counts # of λ terms nλ = 0 Flat histogram in order n of perturbation expansion • λ Friday, 9 July 2010 Perturbation series by Wang-Landau • We want flat histogram in order nλ • Use the Wang-Landau algorithm to get Λ Λ n Λ n (Λ − n)!β Z = ,,,λ λ g(n ) from Z = α (−H ) α λnλ (b1 ,...,bΛ ) ∑ λ ∑∑ ∑ Λ! ∏ bi nλ = 0 n= 0 α (b1 ,...,bΛ ) i=1 • Small change in acceptance rates for diagonal updates ⎛ d ⎞ ⎛ d ⎞ d βNbonds α H(i, j ) α Wang−Landau βNbonds α H(i, j ) α g(nλ ) P[1→ H(i, j )] = min⎜1 , ⎟ ⎯ ⎯ ⎯ ⎯⎯ → min⎜1 , ⎟ ⎝ Λ − n ⎠ ⎝ Λ − n g(nλ + Δnλ )⎠ ⎛ Λ − n +1 ⎞ ⎛ Λ − n +1 g(n ) ⎞ P[H d →1] = min⎜1 , ⎟ ⎯Wang ⎯ ⎯-Landau ⎯⎯ → min⎜1 , λ ⎟ (i, j ) ⎜ N H d ⎟ ⎜ N H d g(n n )⎟ ⎝ β bonds α (i, j ) α ⎠ ⎝ β bonds α (i, j ) α λ − Δ λ ⎠ • Loop update does not change n and is thus unchanged! • Cuto' Λ limits value of λ for which the series converges Friday, 9 July 2010 The antiferromagnetic bilayer J⊥ J J >> J : long range order J << J⊥: spin gap, no long range order ⊥ Quantum phase transition at J⊥ / J ≈ 2.524(2) Spin gap vanishes Magnetic order vanishes Universal properties Friday, 9 July 2010 Quantum phase transition • Quantum phase transition in bilayer quantum Heisenberg antiferromagnet 2.0 1.5 1.0 0.5 0.0 0.0 0.2 0.4 0.6 0.8 1.0 J/Jʼ Friday, 9 July 2010 Summary • Extension of Wang-Landau sampling to quantum systems • Stochastically evaluate series expansion coefficients ∞ β n • High-temperature series Z = ∑ g(n) n= 0 n! ∞ Perturbation series nλ • Z = ∑λ g(nλ ) nλ = 0 • Features • Flat histogram in the expansion order • Allows calculation of free energy • Like classical systems, allows tunneling through free energy barriers • Optimized ensembles are also possible Friday, 9 July 2010 9. The negative sign problem in quantum Monte Carlo Friday, 9 July 2010 Quantum Monte Carlo • Not as easy as classical Monte Carlo Z = ∑e−Ec / kBT c • Calculating the eigenvalues Ec is equivalent to solving the problem • Need to find a mapping of the quantum partition function to a classical problem −βH Z = Tre ≡ ∑ pc c • “Negative sign” problem if some pc < 0 Friday, 9 July 2010 The negative sign problem • In mapping of quantum to classical system =Tr −βH = Z e ! pi 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) |i1> |i4> |i3> |i2> |i1> Friday, 9 July 2010 The negative sign problem • Sample with respect to absolute values of the weights A sgn p p p ∑ i i i ∑ i A ⋅ sign A = A p p = i i ≡ p ∑ i i ∑ i sign i i ∑sgn pi pi ∑ pi p i i • Exponentially growing cancellation in the sign ! pi = i = = −βV (f−f|p|) !sign" Z/Z|p| e !i |pi| • Exponential growth of errors − ∆sign ! sign2 sign 2 eβV (f f|p|) = ! "−! " sign √M sign ≈ √M ! " ! " • NP-hard problem (no general solution) [Troyer and Wiese, PRL 2005] Friday, 9 July 2010 Is the sign problem exponentially hard? • The sign problem is basis-dependent • Diagonalize the Hamiltonian matrix A = Tr[Aexp(!"H)] Tr[exp(!"H)] = $ i Ai i exp(!"#i ) $exp(!"#i ) i i • All weights are positive • But this is an exponentially hard problem since dim(H)=2N ! • 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 Friday, 9 July 2010 What is a solution of the sign problem? • Consider a fermionic quantum system with a sign problem (some pi < 0 A = Tr[Aexp(−βH )] Tr[exp(−βH )] = ∑Ai pi ∑ pi i i • Where the sampling of the bosonic system with respect to |pi| scales polynomially T ∝ε−2N nβ m • A solution of the sign problem is defined as an algorithm that can calculate the average with respect to pi also in polynomial time • Note that changing basis to make all pi ≥ 0 might not be enough: the algorithm might still exhibit exponential scaling Friday, 9 July 2010 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 Friday, 9 July 2010 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 Friday, 9 July 2010 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.