Stabilizing Canonical- Ensemble Calculations in the Auxiliary-Field Monte Carlo Method C. N. Gilbreth In collaboration with Y. Alhassid Yale University Advances in quantum Monte Carlo techniques for non-relativistic many-body systems Institute for Nuclear Theory July 5, 2013
Outline Auxiliary-field quantum Monte Carlo (AFMC) in the configuration-interaction framework Exact particle-number projection Numerical stabilization More efficient method for particle-number projection Application: Thermodynamic properties of a trapped finite cold atomic Fermi gas Heat capacity, condensate fraction, pairing gap Spatial density Density due to an extra particle 2
Auxiliary-field Quantum Monte Carlo (AFMC) A method for studying highly-correlated systems which is free of systematic errors Benefits: Permits finite-temperature calculations No fixed-node approximations (for good sign interactions) Useful in different contexts (electronic structure, nuclear physics, atomic physics, chemistry) Allows calculation of any one- or two-body observable Challenges: Sign problem -- for repulsive interactions and certain projections Scaling is Ns 3 N t or Ns 4 N t, depending on the application Numerical stability at low temperatures / large model spaces 3
AFMC in Configuration (Fock) Space Goal: compute the thermal expectation hôi = Tr(Ôe Ĥ ) Tr(e Ĥ ) for a collection of fermions. Steps to formulate AFMC: (a) Formulation of the Configuration-Interaction (CI) Hamiltonian as a sum of quadratic one-body operators (b) Trotter decomposition (imaginary time discretization) (c) Path-integral representation (d) Calculation of observables for a given set of fields (e) Monte Carlo evaluation 4
(a) Hamiltonian and (b)trotter decomposition Single-particle basis: ii,i=1,...,n s Model space: set of all Slater determinants from the s.p. basis Hamiltonian: Ĥ = X i,j Rewrite in terms of one-body densities a i a k, a j a l and diagonalize the matrix Ĥ = Ĥ0 + 1 2 h ij a i a j + 1 2 X Ô2 X i,j,k,l v ijkl v ijkl a i a j a la k to obtain where and are one-body operators. Ĥ 0 Ô Trotter decomposition: and e Ĥ = e Ĥ 0 Y e Ĥ =(e Ĥ ) N t, /N t e Ô2 /2 + O(( ) 2 ) 5
Discretization error in observables condensate fraction 0.24 0.2 0.16 0.12 T/T F = 0.125 N max =11 fit ( β ( >= apple 1/32) 0 0.02 0.04 0.06 β The error due to the Trotter decomposition scales as We perform a linear fit and extrapolate to! 0 6
{ { { (c) Path-Integral Formulation Hubbard-Stratonovich (HS) transformation: e Ô 2 /2 = Z 1 This linearizes the two-body part of the Hamiltonian into a one-body operator, with the addition of an auxiliary field. End result of the HS transformation: 1 d e 2 /2 e s Ô, s = ( 1, < 0 i, > 0 e Ĥ = Z Time-dependent auxiliary fields (many) D[ ]G Û( ) Integration measure Non-interacting propagator Gaussian weight A path integral of a non-interacting propagator with respect to fluctuating time-dependent auxiliary fields. 7
(d) Observables in the Canonical Ensemble Want to compute: Z D[ ]G Tr[Û( )Ô] Û( ) is a non-interacting propagator and Ô is a one-or two- body operator. To compute traces for fixed particle number, we use exact particle-number projection: Number of single-particle states Tr N [ÔÛ( )] = 1 XN s N s m=1 e i' mn Tr GC [ÔÛ( )ei' m ˆN ]. Canonical trace N = number of particles Can be derived by writing ' m 2 m/n s Tr GC Grand-canonical trace as a sum of canonical traces [W. E. Ormand, et al., Phys. Rev. C 49, 1422 (1994)] 8
(d) Canonical ensemble cont. To compute grand-canonical quantities, can use matrix algebra in the single particle space (of order ~100-1000): Tr GC (Û( )ei' m ˆN )=det(1+ue i' m ) Tr GC [a i ajû( )ei' m ˆN ] Tr GC [Û( )ei' m ˆN ] = 1 1+U 1 e i' m ji where U is the matrix representing space. (e) Monte Carlo evaluation Û in the single-particle Z Discretize the integrals over the auxiliary fields: D[ ]G Û( )! X w( )Û( ) Apply the Metropolis algorithm. [Koonin, et al., Phys. Rep. 278, 1 (1997)] 9
Accuracy of AFMC: Three cold atoms E 25 20 15 10 5 E 4.5 4.4 4.3 4.2 4.1 4 3.9 diagonalization 3 3.5 4 4.5 5 5.5 6 β 0 0 1 2 3 4 5 6 β Ground-state energy AFMC AFMC vs. diagonalization for a fixed interaction with three cold atoms (""#) in a harmonic trap 10
Numerical Stabilization
Numerical Stabilization Need to compute, for each m, Tr GC [Û( )ei' m ˆN ]=det(1+ue i' m ) where U = matrix representing Û in the single-particle space. At long imaginary times, the propagator U becomes unstable. (1) Compute U: (a) Unstabilized method (b) Stabilized method (2) Compute determinant: (i) Unstabilized method (ii) Standard stabilized method (iii) New stabilized method (faster) 12
Unstabilized calculation of U As N t grows large, the matrix product U = U( Nt ) U( 2 )U( 1 ) becomes ill-conditioned: ratio of largest to smallest eigenvalues becomes very large. The matrix elements of U, however, all become large, and it is impossible to extract smaller scales (those near the Fermi surface) accurately. E 120 100 80 60 40 Energy vs. inverse temperature for 20 cold atoms in the unitary limit. At low temperatures, calculation becomes unreliable. 1 2 3 4 5 6 7 8 β Unstabilized calculation is only good for high temperatures (small ) 13
Stabilized Calculation of U Stabilized calculation of U : Use method of [E. Y. Loh Jr. and J. E. Gubernatis, in Electronic Phase Transitions, 1992] Instead of computing U, compute a decomposition U = ADB of U which explicitly displays all numerical scales. A and B are well-conditioned, and D is diagonal with positive entries: U = ADB = x x x x x x x x x E.g., singular value decomposition or QR decomposition X X X x x x x x x x x x 14
To compute Stabilized Matrix Multiplication decomposition of each new factor:, begin with an initial and carefully update when multiplying in U = U Nt U 2 U 1 U 1 U 1 = A 1 D 1 B 1. U n+1 (A n D n B n )=((U n+1 A n )D n )B n =(A n+1 D n+1 B 0 )B n = A n+1 D n+1 (B 0 B n ) (decompose) (group terms) (decompose) (group terms) = A n+1 D n+1 B n+1. U = A Nt D Nt B Nt The intermediate matrix (U n+1 A n )D n therefore can be decomposed stably is column-stratified and 15
Stabilized Matrix Multiplication cont. A column-stratified matrix displays its scales in the columns: U n+1 A n D n = x x x x x x x x x x x x x x x x x x X X X = X X X X X X X X X Similarly, a row-stratified matrix displays its scales in the rows: DBA = 0 @ X X X 1 0 A @ x x x x x x x x x 1 0 A @ x x x x x x x x x Stratified matrices are much more stable for decomposition and diagonalization than general matrices with widely varying scales. 1 A = 0 @ X X X X X X X X X 1 A 16
(2) Calculation of the Determinant Wish to compute det(1 + Ue i' m ) (i) Unstabilized calculation: for each m = 1, 2,..., N s Compute U = U Nt U 1 Diagonalize U and compute Only for high temperatures. Scales as using standard matrix multiplication (diagonalization) (ii) Standard stabilized calculation: [Alhassid et al., 101, 082501 (2008)] Cannot multiply out ADB to diagonalize, as this would destroy information. Instead: 1+ADBe i' m = A(A 1 B 1 + De i' m )B Requires decomposition for each as O(Ns 4 ) det(1 + Ue i' m )= O(N 3 s ) YN s k=1 (1 + k e i' m ) = A(A 0 D 0 B 0 )B (decompose) m =1,...,N s, so scales 17
Stabilized Matrix Diagonalization [Gilbreth & Alhassid, arxiv:1210.4131] (iii) New method: Transform ADBx = x Let x = Ay, DBAy = y The matrix DBA is row-stratified and can be multiplied out & diagonalized stably Eigenvalues are identical to those of ADB. Reduces time back to O(Ns 3 ) Test 1: Compute eigenvectors and eigenvalues of is a random complex matrix, and n =1,2,3,... (e C ) n, where C Test 2: Compare to standard stabilization method 18
Accuracy of Diagonalizing an ill-conditioned Matrix max relative error 10 4 10 2 10 0 10-2 10-4 10-6 10-8 10-10 10-12 unstabilized eigenvalues unstabilized eigenvectors stabilized eigenvalues stabilized eigenvectors 10 0 10 50 10 100 10 150 10 200 10 250 10 300 condition number (e C ) n Error in stabilized and unstabilized eigenvalues and eigenvectors of the matrix (e C ) n of 19
Numerical Stabilization -- Accuracy and Timing E 120 100 80 60 40 unstabilized stabilized 1 2 3 4 5 6 7 8 time per sample (s) 1.5x10 4 1.0x10 4 5.0x10 3 0.0x10 0 New method Old Method 50 100 150 200 β Number of single-particle states Accuracy Timing 20
Summary AFMC method in the configuration-interaction framework Strongly-interacting systems with arbitrary goodsign interactions can be studied Calculations in the canonical ensemble A new stabilization method in the canonical ensemble Much faster, scales as O(N 3 s ) O(N 4 s ) instead of 21
Thermodynamics of a finite trapped cold atomic Fermi gas
Advances with Atomic Fermi Gases 1999: First realization of an ideal degenerate Fermi gas ( 40 K). Evaporative cooling to T = 0.5 TF (JILA, CO) 2002: Feshbach resonance allows tuning to stronglyinteracting regime ( 6 Li, Duke Univ.) 2005: Observation of vortex lattice after stirring confirms superfluidity (MIT) 2012: Measurement of lambda peak in heat capacity (MIT) [Zwierlein, M. W. et al., Nature 435, 1047 (2005)] [Ku, et al., Science 335, 563 (2012)] B C V /Nk B 2.5 2.0 1.5 1.0 0.5 0.2 C 0.0 0.5 T/TF 0.2 23
Clean, strongly interacting, tunable systems Two species ( " and # ) of Fermions interact at very short range in a harmonic trap. Zero-range (s-wave) interactions s-wave scattering length a controllable via external magnetic field Unitary limit saturates (a!1) s-wave scattering cross section. Strongly interacting, nonperturbative system when a is large [Zwierlein, M. W. et al., Nature 435, 1047 (2005)] 24
Unitary, Finite, Trapped Fermi Gas Goal: Describe thermodynamics of trapped finite-size (~20 particle) Fermi gas in the unitary limit. Heat capacity Pairing gap Condensate fraction. Particle density Questions: Is the superfluid phase transition visible in a system of this size? If so, is there a pseudogap effect? [Q. Chen et al., Physics Reports 412 (2005) 1-88] 25
AFMC for Trapped Cold Atoms Hamiltonian: H = X N 1 +N 2 i=1 Single-particle basis of harmonic-oscillator states: Interaction: 1 2 r2 i + 1 2 m!2 r 2 i V (r i r j )=V 0 (r i r j ) Shell-model decomposition: Our calculations are done in the canonical ensemble. + X i<j V (r i r j ) ii = nlmi = R nl (r)y m l (, ), 2n + l apple N max H = XN s i,j=1 h i,j a i a j + 1 2 X K,, Cutoff parameter (V0 renormalized for each N max to reproduce two-particle ground-state energy) K, ( ) KX M=0 ( ˆQ K M ( ) 2 + ˆP K M ( ) 2 ) ˆQ, ˆP one-body operators 26
Monte Carlo Sign of the Contact Interaction It is well-known that the contact interaction has good Monte Carlo sign in coordinate space. V (r) =V 0 (r) It is not as clear in the Configuration-Interaction formalism. The proof proceeds as follows: The Hamiltonian takes the form H = nlmσ ε nl a nlmσ a nlmσ + 1 2 (ab ˆV cd)a aσ a bσ a dσ a cσ, and in a time-reversed density decomposition : Ĥ = Ĥ1 + 1 2 (a b ˆV c d)( ) m b ( ) m d a aσa cσ ā bσ ā dσ, One can show that the interaction matrix ā a, = time reverse V ac,bd =(a b ˆV c d)( ) m a ( ) m b has only nonpositive eigenvalues when V 0 < 0. 27
Sign of the Contact Interaction (cont.) We show this as follows: ac,bd v ac(a b ˆV c d)( ) m b+m d v bd This implies e = V 0 = V 0 = V 0 Ô 2 /2 = s =1 ac,bd Z 1 d 3 r d 3 r v acϕ a(r)ϕ c (r)ϕ b(r)( ) m b ϕ d(r)( ) m d v bd ( ac ac v acϕ a(r)ϕ c (r) )( d 3 r vacϕ a(r)ϕ c (r) 2 0. in the Hubbard-Stratonovich transformation: 1 d e 2 /2 e Ô, bd ϕ b (r)ϕ d(r)v bd ) So the auxiliary-field Hamiltonian is complex-conjugation invariant. 28
Sign of the Contact Interaction (cont.) The propagator factorizes: Û = Û"Û# Û ", Û# are invariant under complex conjugation: K 0 Û ",# K 0 = Û",# Û ",# vi = vi K 0 Û ",# vi = Û",#K 0 vi = K 0 vi ) Therefore, complex eigenvalues come in complex-conjugate pairs So Tr N",# (Û",#) = X i (N ",# ) i = real When N " = N #, Tr N",N # (Û) =Tr N " (Û")Tr N# (Û#) =[Tr N" (Û")] 2 0 so the calculation has good sign. 29
Accuracy of Renormalized Contact Interaction 18 16 14 Exact 3-particle energy at unitarity 12 E 10 8 Fermi gas Contact interaction, Nmax = 8 6 4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 T/T F Tuning V0 to reproduce two-particle ground-state energy provides an accurate interaction for the three-particle system. [Exact results based on S. Tan and Werner & Castin] 30
C = de(t ) dt Heat capacity When computing the heat capacity, the statistical error in the energy can be greatly magnified. We use a method of correlated errors introduced in [S. Liu and Y. Alhassid, PRL 87, 022501 (2001)] to avoid this. Sample fields at a single temperature Compute energies at T + T and T T for each sample: E(T ± T )= Statistical error: Var(C) = [E(T + T ) E(T T )]/2 T R D[ ]G (T ± T )Tr[ Û(,T ± T )] R D[ ]G (T ± T )Tr Û(,T ± T ) T 1 h (2 T ) 2 Var(hĤ(T + T )i) + Var(hĤ(T T )i) i 2 Cov(hĤ(T + T )i, hĥ(t T )i) Covariance greatly reduces statistical error in heat capacity 31
15 10 C 5 54 Fe 54 Fe 0 0 0.5 1 1.5 2 T (MeV) 0 0.5 1 1.5 2 T (MeV) [S. Liu and Y. Alhassid, PRL 87, 022501 (2001)] 32
Condensate Fraction Condensate fraction does not have a standard meaning in finite-size systems We define a condensate fraction from the two-body density matrix (TBDM). C. N. Yang (1962): Off-diagonal long-range order (ODLRO) is equivalent to the existence of a large eigenvalue in the TBDM Calculation: 1. Compute C(ij, kl) ha i," a j,# a l,"a k,# i 2. Diagonalize to obtain a pair wavefunction corresponding to the largest eigenvalue of C. B X ij ' ij a i" a j# 3. The largest eigenvalue satisfies (for N " = N # = N/2) and = hb Bi 0 apple apple N/2 so n /(N/2) defines a condensate fraction. 33
Energy-Staggering Pairing gap Energy-staggering pairing gap where gap [E(N ",N # + 1) + E(N " +1,N # ) E(N " +1,N # + 1) E(N ",N # )]/2 E(N ",N # )= energy of a system with spin-up particles and spin-down particles. N # gap systems and a systems with an unpaired particle. measures the difference in energy between fully paired Calculation: (for N " = N # = N/2) 1. Sample auxiliary fields for the system. 2. Compute E(N ",N # ), E(N " +1,N # )=E(N ",N # + 1), and E(N " +1,N # + 1) for each sample 3. Compute variance of using correlated errors. gap N " (N ",N # + 1) 34
Signatures of the Phase Transition: 20 Atoms [Gilbreth and Alhassid, arxiv:1210.4131] gap /ε F condensate fraction C 0.3 0.2 0.1 0 0.4 0.3 0.2 0.1 0 35 30 25 20 Fermi gas Fermi gas 15 0.05 0.1 0.15 0.2 0.25 T/T F gap =[E(N ",N # + 1) + E(N " +1,N # ) E(N ",N # ) E(N " +1,N # + 1)]/2 C L (ab, cd) =ha LM (ab)a LM(cd)i n = max /(N/2) 35
Particle Density & Odd Particle Effect ρ(r) (units of a -3 osc ) 1.6 1.2 0.8 0.4 0 T/T F = 0.100 T/T F = 0.150 T/T F = 0.200 T/T F = 0.250 0 0.5 1 1.5 2 2.5 3 ρ (units of a -3 osc ) 0.01 0-0.01-0.02-0.03 T/T F = 0.075 0 1 2 3 4 r/a osc r/a osc (r) = P (r) (r) (r) =[ (r) (9,10) + (r) (10,9) ]/2 [ (r) (10,10) + (r) (9,9) ]/2 Cloud spreads out at higher temperatures Unpaired particle prefers the edge of the trap 36
Summary: AFMC in the CI Framework for Cold Atoms We devised a more efficient numerical stabilization method in the canonical ensemble which allows calculations in much larger model spaces. First ab initio calculations of heat capacity and energy-staggering pairing gap across the superfluid phase transition in any system of cold atoms Condensate fraction and particle density for a finite trapped system of cold atoms Clear signatures of a superfluid phase transition The addition of an extra particle to the spin-balanced system produces extra density at the edge of the trap. 37
Questions Questions: Does gap show a pseudogap effect in the trapped system for larger numbers of particles at unitarity? Does gap (as opposed to the spectral function) show a pseudogap effect in the uniform system? How can we extend AFMC in the CI framework to larger model spaces? - Represent the interaction in coordinate space? - Optimize matrix exponential, diagonalization, etc. methods? - Optimize single-particle basis functions? - Alternatives to Metropolis? 38
Acknowledgements -- Thank You! Yoram Alhassid Abhishek Mukherjee Cem Özen Yale HPC Department of Physics Konstantin Nesterov Marco Bonet-Matiz 39