Quantum Monte Carlo Simulationsin the Valence Bond BasisAnders Sandvik, Boston University

Outline• The valence bond basis for S=1/2 spins• Projector QMC in the valence bond basis• Heisenberg model with 4-spin interactions• The valence bond basis for frustrated systems
Papers• A. S., Phys. Rev. Lett. 95, 207203 (2005)• K. Beach and A. S., Nucl. Phys. B 750, 142 (2006)
Funded by the NSF

The valence bond basis for S=1/2 spins• Dates back to the 1930s; Pauling, Romer, Hulthen...
• Spans the singlet space
AConsider N (even) spins• divide into 2 groups; A and B• e.g., sublattices (not necessarily)• bonds from A sites to B sites; singlets

BBasis states:
The valence bond basis is overcomplete, non-orthogonal• expansion of singlet sate not unique

Overlap of VB states• given by the number of loops Nl in superimposed graphs• all basis states overlap with each other (useful...)Matrix elements• spin correlations related to loop structure

More complicated matrix elements discussed inK. Beach, A.S., Nucl. Phys. B 750, 142 (2006)

Projector Monte Carlo in the valence bond basis(-H)n, or (C-H)n, projects out ground state from an arbitrary state
S=1/2 Heisenberg model

Project with strings of bond operators

Arrow - ”direction” of (i,j)

Action of bond operator on VB state• No branching when acting on a VB basis state• No minus signs for bipartite lattice if “direction” of singlet(i,j) is(i on sublattice A, j on sublattice B)Sampling the wave function
Simplest trial wave function; a single basis state |Vk>

The weight of the path is given by # of off-diagonal operations

How to sample? Trivial way:
• Randomly replace m operators at string positions p1,...,pm• Recalculate weight (noff) by propagating |Vk>
• Accept using Metropolis probability• Gives good acceptance probability for• Works because of non-orthogonality of basis• Faster way to calculate acceptance probability desired...

Calculating the energy
Consider the Neel state |N>• All VB basis states have equal overlap with |N>• The energy can be calculated using any equal-overlap stateOperation with1/2 (off-diagonal)

gives sum of 1(diagonal) andnd = number of diagonal bond operations Hij
no = number of off-diagonal operations Hij
Nb = number of bonds on the lattice (no+nd=Nb)

Direct improved estimator for the singlet-triplet gap
The valence bond basis spans the singlet sector• with one triplet bond, one can study the lowest triplet state

Hij propagates the triplet off-diagonally almost like a singlet
but a diagonal operation gives zero (kills the triplet state)
E1 can be calculated by propagating a state with one triplet• propagate for all initial triplet locations (N/2 different bonds)• surviving paths contribute to the triplet energy• subset of singlet and triplet configurations are the same
- error cancellations in, improved gap estimator

General operator expectation values
We have to project both a bra and a ket state:
Operator string notation:

Importance sampling ofEstimator to be averaged

Sampling by operator replacements as before• note that the weight includes an overlap

IIlustration: 6-site chain, n=3Note: in this basis the propagation is non-hermitean• the left state is propagated from the left
• the right state is propagated from the right• the propagated states always have some overlap

Sampling a bond-amplitude product state
Instead of using a single basis state as the trial state,
it is possible to sample a wave function

bond b of state kUpdate by reconfiguring two bonds[Liang, Doucot, Anderson, PRL 61, 365 (1988)]
acb

Acceptance probability

dIf reconfiguration accepted• calculate change in string-projection weight as before• final accept/reject based on projection weight only
Liang [PRB 42, 6555 (1990)] used parametrized h(x,y) for 2D Heisenberg• determined parameters variationally• improved the variational wave function by projection

Variational wave function
It is very time consuming to fully optimize all h(x,y) variationally• Newton/conjugate-gradient method [J. Lou and A. S., cond-mat/0605034]
EnergySpin correlations

Variational energy error• 25% smaller than Liang et al.’s
Asymptotic form h(x,y)=h(r)• h(r) decays as 1/r3
• previous work gave 1/rp, p = 2-5• mean-field theory [K. Beach]explains 1/r3 form

Self-optimized trial wave function
Projector method can access thebond-length probability P(x,y)• related to the amplitude h(x,y)• for wave function with h(x,y)
P(x,y) h(x,y)

Results for 2D Heisenberg

P(x,y) can be used to construct h(x,y)
almost as good as the variational hDefinitions• h(x,y) = bond amplitude of the trial state• P0(x,y) = bond probability of the trial state• Pn(x,y) = bond probability of the Hn projected state
For large enough n, Pn(x,y) is the exact ground-state distribution• if P0(x,y) > Pn(x,y), then reduce h(x,y)• if P0(x,y) < Pn(x,y), then increase h(x,y)• repeat until P0(x,y)=Pn(x,y) for all x,y• fast method to obtain almost optimal h(x,y)• can be generalized to include bond correlations

Energy convergence; 2D Heisenberg, 162 spins• comparison of different trial wave functions

Energy of self-optimized stateConvergence of long-distance correlation function

Correlation function of self-optimized state

2D Heisenberg model with 4-site interactions• Neel to VBS quantum phase transition• Candidate for “deconfined” quantum-criticality- contunuous transition; “Landau rules” say 1st order

VBS order; open boundaries break symmetry - unique VBS• order parameter; bond correlation

J=0 - columnar VBS

Q=0 - no VBS orderFinite-size scaling
periodic boundaries• dimer correlations• submattice magnetization• singlet-triplet gap• J/Q=0.0; VBS• J/Q=0.1; antiferromagnet

Singlet-triplet gap scaling; dynamic exponent z
z relates length and time scales

Critical gap:

at J/Q≈0.04

Consistent with deconfined quantum-criticality (z=1 theory)

Neel and VBS orders• finite-size scaling• order parameters
vanish at the same
coupling; (J/Q)c≈0.035• correlation function
exponent h is large;h ≈ 0.4 for spin
• smaller for dimers• more careful analysis
in progress

Frustrated systems
Consider the full valence-bond basis, including• normal bonds, connecting A,B spins (sublattices)• frustrated bonds, connecting A,A or B,B
For a non-frustrated system• projection eliminates frustarted bonds

For a frustrated system• frustrated bonds remain and cause a sign problem• frustrated bonds can be eliminated using over-completenessIn a simulation, one of the branches can be randomly chosen• but there is a sign problem

Summary
The valence bond basis can be used in projector QMC• some observables easier to calculated than in z-basis• easy to study certain types of multi-spin interactions- interesting phases/transitions; Neel-VBS• self-optimized trial wave functions- including bond-correlations; explored currently
Sign problems for frustrated systems• but freedom offered by overcompleteness should beexplored; potentially there are sign-problem-free
frustrated systems
More details for 2D J-Q model in symposium talk• evidence for deconfined quantum-criticality,
including emergent U(1) symmetry