Quantum Monte Carlo Methods at Work for Novel Phases of Matter Trieste, Italy, Jan 23 - Feb 3, 2012 Ground State Projector QMC in the valence-bond basis Anders. Sandvik, Boston University Outline: The valence-bond basis Projector QMC with valence bonds Amplitude-product states J-Q chain: 1D valence-bond solid 1
Common bases for quantum spin systems Lattice of S=1/2 spins, e.g., Heisenberg antiferromagnet H = J S i S j = J Si z Sj z + (S + i S j + S i S+ j )/2 i,j i,j The most common basis is that of up and down spins = = S z = +1/2 = = S z = 1/2 One can also use eigenstates of two or more spins dimer singlet-triplet basis = ( )/ 2 = = ( + )/ 2 = The hamiltonian is more complicated in this basis 2
The valence bond basis for S=1/2 spins Valence-bonds between sublattice A, B sites Basis states; singlet products V r = N/2 b=1 (i rb, j rb ), r = 1,... (N/2)! (i, j) = ( i j i j )/ 2 The valence bond basis is overcomplete and non-orthogonal expansion of arbitrary singlet state is not unique = r f r V r (all fr positive for non-frustrated system) All valence bond states overlap with each other V l V r = 2 N N/2 N = number of loops in overlap graph Spin correlations from loop structure V l S i S j V r V l V r = 3 4 ( 1)x i x j +y i y j 0 (i,j in different loops) (i,j in same loop) A B More complicated matrix elements (e.g., dimer correlations) are also related to the loop structure K.S.D. Beach and A.W.S., Nucl. Phys. B 750, 142 (2006) V l V r V l V r 3
Projector Monte Carlo in the valence-bond basis Liang, 1991; Sorella et al. (1998); AWS, Phys. Rev. Lett 95, 207203 (2005) (-H) n projects out the ground state from an arbitrary state ( H) n =( H) n i S=1/2 Heisenberg model c i i c 0 ( E 0 ) n 0 H = S i S j = H ij, H ij = ( 1 4 S i S j ) i,j i,j Project with string of bond operators n H i(p)j(p) r 0 (r = irrelevant) {H ij } p=1 (a,d) Action of bond operators (a,b) (c,b) (c,d) H ab...(a, b)...(c, d)... =...(a, b)...(c, d)... H bc...(a, b)...(c, d)... = 1...(c, b)...(a, d)... 2 A B A B (i, j) = ( i j i j )/ 2 Simple reconfiguration of bonds (or no change; diagonal) no minus signs for A B bond direction convetion sign problem does appear for frustrated systems 4
Sampling the wave function Simplified notation for operator strings n H i(p)j(p) = P k, k = 1,... Nb n {H ij } p=1 k 6-site chain Simplest trial wave function: a basis state V r P k V r = W kr V r (k) The weight Wkr of a path is given by the number of off-diagonal operations ( bond flips ) nflip W kr = 1 2 nflip n = n dia + n flip H ab...(a, b)...(c, d)... =...(a, b)...(c, d)... H bc...(a, b)...(c, d)... = 1...(c, b)...(a, d)... 2 Note: all paths contribute - no dead (W=0) paths Sampling: Trivial way: Replace m (m 2-4) operators at random P accept = 1 2 n new flip n old flip The state has to be re-propagated with the full operator string More efficient updating scheme exists (later...) 5
Calculating the energy Using a state which has equal overlap with all VB basis states e.g., the Neel state N N V r = ( 2) N/2 E 0 = N H 0 N 0 = k N HP k V r k N P k V r H acts on the projected state nf = number of bond flips nd = number of diagonal operations E 0 = n d + n f /2 H 6
General expectation values: A = 0 A 0 Strings of singlet projectors P k = n p=1 H ik (p)j k (p), k =1,...,N n b (N b = number of interaction bonds) We have to project bra and ket states P k V r = W kr V r (k) ( E 0 ) n c 0 0 k k V l P g = V l (g) W gl 0 c 0 ( g g 6-spin chain example: E 0 ) n A = g,k V l P g AP k V r g,k V l P g P k V r = g,k W glw kr V l (g) A V r (k) g,k W glw kr V l (g) V r (k) V l A V r Monte Carlo sampling of operator strings 7
Sampling an amplitude-product state A better trial state leads to faster n convergence bond-amplitude product state [Liang, Doucot, Anderson, 1990] 0 = k N/2 b=1 h(x rb, y rb ) V k Update state by reconfiguring two bonds a c b d P accept = h(x c, y c )h(x d, y d ) h(x a, y a )h(x b, y b ) If reconfiguration accepted calculate change in projection weight used for final accept/reject prob. S. Liang [PRB 42, 6555 (1990)] used parametrized state amplitudes determined parameters variationally improved state by projection V l A V r 8
Variational wave function (2D Heisenberg) All amplitudes h(x,y) can be optimized [J. Lou and A.W.S., PRB 2007, AWS and H.-G. Evertz, PRB 2010] variational energy error 50% smaller than previously best (<0.1%) spin correlations deviate by less than 1% from exact values amplitudes decay as 1/r 3 0.16 0.14 M s 2 0.12 0.10 0 20 40 60 80-0.669 E/N -0.670-0.671-0.672 variational projected -0.673-0.674 0 20 40 60 80 L 9
More efficient ground state QMC algorithm larger lattices Loop updates in the valence-bond basis AWS and H. G. Evertz, PRB 2010 Put the spins back in a way compatible with the valence bonds (a i,b i )=( i j i j )/ 2 and sample in a combined space of spins and bonds A Loop updates similar to those in finite-t methods (world-line and stochastic series expansion methods) good valence-bond trial wave functions can be used larger systems accessible sample spins, but measure using valence bonds (as before) 10
τ V hai = (graphs by Ying Tang) h m m ( H) A( H) 1 h 1 ( H)2m 2 i 2i m m V V power m should be large enough to obtain ground state V 11
τ V n oper(n) 7 1 1 01 67 6 1 01 32 5 1 0 2 4 0 0 3 1 1 01 76 2 1 0 0 4 1 1 1 01 76 V 0 0 1 2 3 4 5 1 0 2 use bit operation to flip operators 12
Convergence Trial state expanded in H-eigenstates 0 i = X c n ni n Projected state after m-th power m i = H m 0 i = X n Expectation value hai m = h0 A 0i +2h1 A 0i c 1 hai m = h0 A 0i + c exp c n E m n ni c 0 E1 E 0 m N e 0 e 0 = E 0 /M, =E 1 E 0 m +... Conclusion: m/n >> e0/δ in valence-bond basis Δ is the singlet-singlet gap M s 2 E/N 0.08 0 2 4 6 8 10 trial state also can have fixed momentum k=0 (e.g., ampl. product state) - only k=0 excited states (gap) 0.14 0.12 0.10-0.667-0.668-0.669 32 32 Heisenberg p=2 p=3 p=4 optimized (1) optimized (2) -0.670 0 1 2 3 4 m/n 13
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VBS states from multi-spin interactions Sandvik, Phys. Rev. Lett. 98, 227202 (2007) The Heisenberg interaction is equivalent to a singlet-projector C ij = 1 4 S i S j s s C ij ij = ij, C ij tm ij =0 (m = 1, 0, 1) we can construct models with products of singlet projectors no frustration in the conventional sense (QMC can be used) correlated singlet projection reduces antiferromagnetic order/correlations H = J Q 2 i j i j k l J Q 3 i j i j k l m n J X hiji C ij Q 2 X hijkli C ij C kl including all translations - H is translationally invariant The J-Q chains have the same critical-vbs transition as the J1-J2 Heisenberg chain! - Heisenberg SSE and projector codes can be easily adapted to Q-terms 15
S=1/2 Heisenberg chain with frustrated interactions (J1-J2 chain) = J 1 = J 2 Different types of ground states, depending on the ratio g=j2/j1 (both >0) Antiferromagnetic quasi order (critical state) for g<0.2411... - exact solution - Bethe Ansatz - for J2=0 - bosonization (continuum field theory) approach gives further insights - spin-spin correlations decay as 1/r C(r) = S i S i+r ( 1) r ln1/2 (r/r 0 ) r - gapless spin excitations ( spinons, not spin waves!) VBS order for g>0.2411... the ground state is doubly-degenerate state - gap to spin excitations; exponentially decaying spin correlations C(r) = S i S i+r ( 1) r e r/ - singlet-product state is exact for g=1/2 (Majumdar-Gosh point) 0 critical 0.241... VBS g 16
VBS state in J-Q chains (more in tutorial) Y. Tang and AWS, Phys. Rev. Lett. 107, 157201 (2011) S. Sanyal, A. Banerjee, and K. Damle, arxiv:1107.1493 J Q 2 i j i j k l critical VBS J Q 3 0 (Q/J)c Q/J i j i j k l m n dimer operator: B i = ~ S i ~S i+1 In a symmetry-broken VBS: hb i i = a + ( 1) i In a finite system in which the symmetry is not broken: <Bi>=0 detect VBS with dimer correlation function D(r) = 1 N NX hb i B i+r i i=1 This is a 4-spin correlation function can be evaluated using the transition graphs (1- and 2-loop contributions) expression in the afternoon tutorial 17
Animation of the projected states - transition graph Animations by Ying Tang J =0 18
J/Q =0.5 19
J/Q =(J/Q) c 6 20
Estimator for the singlet-triplet gap The original VB basis spans the singlet space with one triplet bond, one can obtain the lowest triplet state (i, j) = ( i j i j )/ 2 [i, j] = ( i j + i j )/ 2 Under propagation, the triplet flips like a singlet but a diagonal operation on a triplet kills it H bc...[a, b]...(c, d)... = 1...(c, b)...[a, d]... 2 H ab...[a, b]...(c, d)... = 0 The initial triplet can be placed anywhere N/2 different triplet propagations Those that survive contribute to E1 Partial error cancellations in the gap = E 1 E 0 The ability to generate singlet and triplet states in the same run is a unique feature of VB projector Monte Carlo 21
Singlet-triplet matrix elements It is also possible to project one singlet and one triplet matrix elements between the lowest singlet and triplet states e.g., magnon weight in dynamic structure factor T (q) Sq z S(0) 22