vbs

Valence-bond methods and valence-bond solid (VBS) states Anders W. Sandvik, Boston University The valence-bond basis and resonating valence-bond states • Alternative to single-spin ↑,↓ basis • for qualitative insights and computational utility • Exact solution of the frustrated chain at the “Majumdar-Ghosh” point • Amplitude-product states XIV Training Course on Strongly Correlated Systems Vietri Sul Mare, Salerno, Italy, October 5-16, 2009 Neel to valence-bond solid transition (T=0) • sign-problem free models exhibiting Neel and VBS phases • evidence for “deconfined” quantum criticality • method for direct detection of spinon confinement/deconfinement Valence-bond basis and resonating valence-bond states As an alternative to single-spin ↑ and ↓ states, we can use singlets and triplet pairs static dimers (complete basis) arbitrary singlets (overcomplete in singlet subspace) one triplet in the singlet soup (overcomplete in triplet subspace) In the valence-bond basis (b,c) one normally includes pairs connecting two groups of spins - sublattices A and B (bipartite system, no frustration) (a, b) = (↑a↓b − ↓a↑b)/ √ 2 a ∈ A, b ∈ B arrows indicate the order of the spins in the singlet definition Superpositions, “resonating valence-bond” states |Ψs〉 = ∑ α fα|(aα1 , bα1 ) · · · (aαN/2, bαN/2)〉 = ∑ α fα|Vα〉 Marshallʼs sign rule for the ground-state wave function The (a,b) singlet definition corresponds to a particular choice of ↑↓ wave-function signs |Ψ〉 = ∑ σ Ψ(σ)|σ〉 |σ〉 = |Sz1 , Sz2 , . . . , SzN 〉 E = 〈Ψ|H|Ψ〉 = ∑ σ ∑ τ Ψ∗(τ)Ψ(σ)〈τ |H|σ〉 Consider this as a variational state; we want to minimize the energy E = ∑ σ |Ψ(σ)|2〈σ|Hdia|σ〉+ ∑ σ |Ψ(σ)|2 ∑ τ Ψ∗(τ) Ψ∗(σ) 〈τ |Hoff |σ〉 Let us consider a bipartite Heisenberg model H = J N∑ i=1 Si · Si+1 = J N∑ i=1 [Sxi S x i+1 + S y i S y i+1 + S z i S z i+1], = J N∑ i=1 [Szi S z i+1 + 1 2 (S + i S − i+1 + S − i S + i+1)] i ∈ A, i+ 1 ∈ B Diagonal and off-diagonal energy terms To minimize E, the wave-function ratio should be negative → sign[Ψ(Sz1 , . . . , S z N )] = (−1)nB This holds for the singlets (a, b) = (↑a↓b − ↓a↑b)/ √ 2 a ∈ A, b ∈ B Useful operator: singlet projector operator Cij = −(Si · Sj − 14 ) Creates a singlet (valence bond), if one is not present on (i,j); Graphical representation of the off-diagonal operation Cab(a, b) = (a, b) Cbc(a, b)(c, d) = 1 2 (c, b)(a, d) In (b) there are non-bipartite bonds; can be eliminated using The Heisenberg hamiltonian is a sum of singlet projectors Calculating with valence-bond states All valence-bond basis states are non-orthogonal • the overlaps are obtained using transposition graphs (loops) 〈Vβ | 〈Vβ |Vα〉 |Vα〉 Each loop has two compatible spin states → 〈Vβ |Vα〉 = 2Nloop−N/2 This replaces the standard overlap for an orthogonal basis; 〈β|α〉 = δαβ Many matrix elements can also be expressed using the loops, e.g., 〈Vβ |Si · Sj |Vα〉 〈Vβ |Vα〉 = { 0, for λi #= λj 3 4φij , for λi = λj λi is the loop index (each loop has a label), staggered phase factor φij = {−1, for i, j on different sublattices +1, for i, j on the same sublattice More complicated cases derived by K.S.D. Beach and A.W.S., Nucl. Phys. B 750, 142 (2006) Solution of the frustrated chain at the Majumdar-Ghosh point H = N∑ i=1 [ J1Si · Si+1 + J2Si · Si+2 ] We will show that this state is an eigenstate when J2/J1=1/2 |ΨA〉 = |(1, 2)(3, 4)(5, 6) · · · 〉 Act with one “segment” of these terms on the state; graphically Eigenstate for g=1/2; one can also show that itʼs the lowest eigenstate H = − N∑ i=1 (Ci,i+1 + gCi,i+2) +N 1 + g 4 , Write H in terms of singlet projectors Cij = −(Si · Sj − 14 ) Amplitude-product states Good variational ground state for bipartite models can be constructed fα = ∏ r h(r)nα(r), |Ψs〉 = ∑ α fα|(aα1 , bα1 ) · · · (aαN/2, bαN/2)〉 = ∑ α fα|Vα〉 Let the wave-function coefficients be products of “amplitudes” (real positive numbers) The amplitudes h(r), r=1,3,...,N/2 (in one dimension) are adjustable parameters What are the properties of such states (independently of any model H)? • given fixed h(r), one can study the state using Monte Carlo sampling of bonds • elementary move by reconfiguring two bonds • simple accept/reject probability (Metropolis algorithm) Test of two cases for the amplitudes in 1D: h(r) = e−r/κ h(r) = r−κ Liang, Doucot, Anderson (1990) Spin and dimer correlations obtained with MC sampling (N=256) exponentially-decaying amplitudes The state is always a valence-bond solid dimer correlation: D(rij) = 〈(si · si+xˆ)(sj · sj+xˆ)〉 Used to defined VBS order parameter • subtract average = C2(1), gives (-1)r sign oscillations in VBS state power-law decaying amplitudes Spin and dimer correlations obtained with MC sampling for different N at r=N/2 There is a “quantum phase transition” when κ≈1.52 • Long-range antiferromagnetic order for κ<1.52 • Valence-bond-solid order for κ>1.52 The critical state is similar to the ground state of the Heisenberg chain • but not quite C(r)∼1/r and D(r)∼1/r for Heisenberg chain • C(r)∼1/r and D(r)∼1/r1/2 for amplitude-product state • exponents depend on details of the amplitudes Neel to VBS Quantum Phase transitions Introduction to quantum phase transitions Finite-size scaling at critical points “J-Q” models exhibiting Neel - VBS transitions Simulation results • exponents, emergent U(1) symmetry Method for detecting spinon deconfinement 2D quantum-criticality (T=0 transition) “Manually” dimerized S=1/2 Heisenberg models Singlet formation on strong bonds ➙ Neel - disordered transition Examples: bilayer, dimerized single layer weak interactions strong interactions 2D quantum spins map onto (2+1)D classical spins (Haldane) • Continuum field theory: nonlinear σ-model (Chakravarty, Halperin, Nelson) ⇒3D classical Heisenberg (O3) universality class expected Ground state (T=0) phases Example: 2D Heisenberg model (T→0) simulations Finite-size scaling of the sublattice magnetization Simulations & theory agree: O(3) universality class (e.g., η≈0.03) for bilayer ordered near-critical 〈M2〉 ∼ L−(1+η)At critical point: !M = 1 N ∑ i (−1)xi+yi !Si In simulations we calculate many papers, e.g., L. Wang and A.W.S., Phys. Rev. B 73, 014431 (2006) Finite-size scaling, critical exponents Order parameter close to Tc in a classical system or at critical coupling gc at T=0 in a quantum system m ∼ { (Tc − T )β (gc − g)β At Tc or gc in finite system of length L: m ∼ L−β/ν A quantity A in the neighborhood of the critical point scales as A(L, T ) = L−κ/νf(tLν) t = T − Tc Tc Data collapse: plot ALκ/ν versus tLν What is the nature of the non-magnetic ground state for g=J2/J1≈1/2? H = ∑ 〈i,j〉 Jij !Si · !Sj= J1 = J2 A challenging problem: frustrated quantum spins Quantum phase transition between AF and VBS state expected at J2/J1≈0.45 • but difficult to study in this model • exact diagonalization only up to 6×6 • sign problems for QMC = 〈!Si · !Sj〉 Are there models with AF-VBS transitions that do not have QMC sign problems? = (| ↑1↓2〉 − | ↓1↑2〉)/ √ 2 • No spin (magnetic) order • Broken translational symmetry • most likely a Valence-bond solid (crystal) [Read & Sachdev (1989)] • no sign problems in QMC simulations • has an AF-VBS transition at J/Q≈0.04 • microscopic interaction not necessarily realistic for materials • macroscopic physics (AF-VBS transition) relevant for ‣ testing and stimulating theories (e.g., quantum phase transitions) ‣ there may already be an experimental realization of the critical point 2D S=1/2 Heisenberg model with 4-spin interactions H = J ∑ 〈ij〉 Si · Sj −Q ∑ 〈ijkl〉 (Si · Sj − 1 4 )(Sk · Sl − 1 4 ) = 〈!Si · !Sj〉 Questions • is the transition continuous? ‣ normally order-order transitions are first order (Landau-Ginzburg) ‣ theory of “deconfined” quantum critical points has continuous transition • nature of the VBS quantum fluctuations ‣ emergent U(1) symmetry predicted A.W.S, Phys. Rev. Lett. 98, 227202 (2007) = 〈!Si · !Sj〉 Deconfined quantum criticality weakly 1st order argued by Jiang et al., JSTAT, P02009 (2008) Kuklov et al., PRL 101, 050405 (2008) Senthil et al., Science 303, 1490 (2004) Generic continuous AF-VBS transition • beyond the Ginzburg-Landau paradigm (generically 1st order AF-VBS point) Spinon deconfinement at the critical point Confinement inside VBS phase associated with new length scale and emergent U(1) symmetry [Read & Sachdev (1989)] (C-H)n projects out the ground state from an arbitrary state (C −H)n|Ψ〉 = (C −H)n ∑ i ci|i〉 → c0(C − E0)n|0〉 H = ∑ 〈i,j〉 !Si · !Sj = − ∑ 〈i,j〉 Hij , Hij = (14 − !Si · !Sj) S=1/2 Heisenberg model Project with string of bond operators Hab|...(a, b)...(c, d)...〉 = |...(a, b)...(c, d)...〉 Hbc|...(a, b)...(c, d)...〉 = 12 |...(c, b)...(a, d)...〉 ∑ {Hij} n∏ p=1 Hi(p)j(p)|Ψ〉 → r|0〉 (r = irrelevant) Action of bond operators Simple reconfiguration of bonds (or no change; diagonal) • no minus signs for A→B bond ‘direction’ convetion • sign problem does appear for frustrated systems A BAB (a,b) (a,d) (c,d)(c,b) (i, j) = (| ↑i↓j〉 − | ↓i↑j〉)/ √ 2 Projector Monte Carlo in the valence-bond basis Liang, 1991; AWS, Phys. Rev. Lett 95, 207203 (2005) Expectation values We have to project bra and ket states 〈A〉 = 〈0|A|0〉 |Vr〉〈Vl| A 〈A〉 = ∑ g,k〈Vl|P ∗gAPk|Vr〉∑ g,k〈Vl|P ∗g Pk|Vr〉 = ∑ g,kWglWkr〈Vl(g)|A|Vr(k)〉∑ g,kWglWkr〈Vl(g)|Vr(k)〉 ∑ k Pk|Vr〉 = ∑ k Wkr|Vr(k)〉 → ( 14 − E0)nc0|0〉∑ g 〈Vl|P ∗g = ∑ g 〈Vl(g)|Wgl → 〈0|c0( 14 − E0)n Results: VBS phase in the J-Q model ➭ VBS order parameter columnar dimer-dimer correlations ➭ AF (Neel) order parameter sublattice magnetization ➤ J/Q=0.0 → VBS ➤ J/Q=0.1 → antiferromagnet D2 = 〈D2x +D2y〉, Dx = 1 N N∑ i=1 (−1)xiSi · Si+xˆ, Dy = 1 N N∑ i=1 (−1)yiSi · Si+yˆ !M = 1 N ∑ i (−1)xi+yi !SiM2 = 〈 !M · !M〉 Singlet-triplet gap scaling → Dynamic exponent z z relates length and time scales: ωq ∼ |q| z finite size → There is an improved estimatorfor the gap in the VB basis QMC∆ ∼ L−z Finite-size scaling of L∆ ∆(L) = a1 L + a2 L2 + · · ·Critical gap scaling: ⇒ z = 1 The gap at J=0 is small; Δ/Q=0.07 The VBS is near-critical Exponents; finite-size scaling Correlation lengths (spin, dimer): ξs,d Binder ratio (for spins): qs=/2 long-distance spin and dimer correlations: Cs,d(L/2,L/2) All scale with a single set of critical exponents at gc≈0.04 (with subleading corrections) ν = 0.78(3), η = 0.26(3) z=1,η≈0.3: consistent with deconfined quantum-criticality • z=1 field theory and ”large” η predicted (Senthil et al.) g = J Q T,L scaling properties R. G. Melko and R. Kaul, PRL 100, 017203 (2008) Additional confirmation of a critical point • finite-T stochastic series expansion • larger systems (because T>0) • good agreement on critcal Q/J z = 1, η ≈ 0.35 4 0.1 1 10 100 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 2 10 3 L=12 L=16 L=24 L=32 L=48 L=64 L=80 L=128 LzT Ld−2 T ρs TLdχu FIG. 6: Scaling of χu and ρs at J = 0.038 ≈ Jc, with z = 1 and d = 2. These plots are the universal functions Y(x, 0) and Z(x, 0) up to the non-universal scale factor c on the x-axis. The expected asymptotes (see text) are plotted as dashed lines Y(x → 0, 0) = Aρ/x and Z(x→ ∞, 0) = Aχx d/z. From fits to the data, we find Aχ/c2 = 0.041(4) and Aρc = 0.37(3), allowing us to estimate a universal model-independent num- ber associated with the QCP, Aρ p Aχ ≈ 0.075(4). support for a z = 1 QCP between 0.038 ≤ J ≤ 0.040. Finally, we hold the second argument of the scaling functions [Eqs. (2,3)] constant by tuning the system to g = 0. One then expects a data collapse for ρs/T and Lχu when they are plotted as a function of LzT (with z = 1). Fig. 6 shows this collapse for simulations carried out with extremely anisotropic arguments LT , varying over almost three orders of magnitude. There is an excellent data collapse over 8 orders of magnitude of the range of the universal functions, with no fit parameters. This data together with that in Fig. 4 provide our most striking evidence for the existence of a QCP with z = 1 in the proximity of J/Q ≈ 0.038. Discussion: In this paper we have presented exten- sive data for the SU(2) symmetric JQ model which in- dicates that the Ne´el order (present when J $ Q) is destroyed at a continuous quantum transition as Q is in- creased [6]. In the finite-T quantum critical fan above this QCP, scaling behavior is found that confirms the dynamic scaling exponent z = 1 to high accuracy. The anomalous dimension of the Ne´el field at this transition is determined to be ηN ≈ 0.35(3), almost an order of mag- nitude more than its value of 0.038 [13] for a conventional O(3) transition. For sufficiently large values of Q we find that the system enters a spin-gapped phase with VBS or- der. To the accuracy of our simulations, our results are fully consistent with a direct continuous QCP between the Ne´el and VBS phases, with a critical coupling be- tween J/Q ≈ 0.038 and J/Q ≈ 0.040. We have found no evidence for double-peaked distributions, indicating an absence of this sort of first-order behavior on the rela- tively large length scales studied here. It is interesting to compare our results to the only theory currently available for a continuous transition out of the Ne´el state into a quantum paramagnetic VBS state in an S = 1/2, SU(2) symmetric quantum magnet: the deconfined quantum criticality scenario [4], in which the Ne´el-VBS transition is described by the non-compact CP1 field theory. All of the qualitative observations above, including an unusu- ally large ηN [14] agree with the predictions of this theory. Indeed, our estimate for ηN is in remarkable numerical agreement with a recent field-theoretic computation [15] of this quantity which finds, ηN = 0.3381. With regard to a detailed quantitative comparison, we have provided the first step by computing many universal quantities, Xχ(x), XS(x), ηN ≈ 0.35 [Fig. 3], Y(x, 0), Z(x, 0) and Aρ √ Aχ ≈ 0.075 [Fig. 6] in the JQ model. Analogous computations in the CP1 model, although currently un- available [16] are highly desirable to further demonstrate that the JQ model realizes this new and exotic class of quantum criticality. We acknowledge scintillating discussions with S. Chan- drasekharan, A. del Maestro, T. Senthil, and especially S. Sachdev and A. Sandvik. This research (RGM) was sponsored by D.O.E. contract DE-AC05-00OR22725. RKK acknowledges financial support from NSF DMR- 0132874, DMR-0541988 and DMR-0537077. Computing resources were contributed by NERSC (D.O.E. contract DE-AC02-05CH11231), NCCS, the HYDRA cluster at Waterloo, and the DEAS and NNIN clusters at Harvard. [1] E. Manousakis, Rev. Mod. Phys. 63, 1 (1991). [2] F. D. M. Haldane, Phys. Rev. Lett. 61, 1029 (1988) [3] N. Read and S. Sachdev, Phys. Rev. B 42, 4568 (1990). [4] T. Senthil, A. Vishwanath, L. Balents, S. Sachdev, and M. P. A. Fisher, Science 303, 1490 (2004); Phys. Rev. B 70, 144407 (2004). [5] S. Sachdev, Quantum Phase Transitions (Cambridge University Press, New York, 1999). [6] A. W. Sandvik, Phys. Rev. Lett. 98, 227202 (2007). [7] O. F. Sylju˚asen and A. W. Sandvik, Phys. Rev. E 66, 046701 (2002); R. G. Melko and A. W. Sandvik, Phys. Rev. E 72, 026702 (2005). [8] The VBS order is also visible in measurements of corre- lation functions between off-diagonal terms [9]. [9] R. K. Kaul and R. G. Melko, unpublished. [10] A priori, it is unclear whether simple hyperscaling laws should hold for all quantities at a critical point with two diverging length scales. Empirically, we have deteced no violation of hyperscaling in our analysis. [11] A. V. Chubukov, S. Sachdev, and J. Ye, Phys. Rev. B 49, 11919 (1994). [12] A. W. Sandvik and R. G. Melko, cond-mat/0604451 (2006); Ann. Phys. (NY), 321, 1651 (2006). [13] M. Campostrini, M. Hasenbusch, A. Pelissetto, P. Rossi, and E. Vicari, Phys. Rev. B 65, 144520 (2002). [14] O. I. Motrunich and A. Vishwanath, Phys. Rev. B 70, 075104 (2004). spin stiffness, uniform susceptibility staggered spin structure factor and susceptibility What kind of VBS; columnar or plaquette? ➭ look at joint probability distribution P(Dx,Dy) Dx Dx Dy Dy |0〉 = ∑ k ck|Vk〉 The simulations sample the ground state; Graph joint probability distribution P (Dx, Dy) Dx = 〈Vk| 1N ∑N i=1(−1)xiSi · Si+xˆ|Vp〉 〈Vk|Vp〉 Dy = 〈Vk| 1N ∑N i=1(−1)yiSi · Si+yˆ|Vp〉 〈Vk|Vp〉 ➭ 4 peaks expected; Z4-symmetry unbroken in finite system critical VBS fluctuations in the theory of deconfined quantum-critical points ➣ plaquette and columnar VBS “degenerate” at criticality ➣ Z4 “lattice perturbation” irrelevant at critical point - and in the VBS phase for L<Λ∼ξa, a>1 ❨spinon confinement length❩ ➣ emergent U(1) symmetry ➣ ring-shaped distribution expected for L<Λ Dx Dx Dy Dy L=32 J=0 Loop updates in the valence-bond basis AWS and H. G. Evertz, ArXiv:0807.0682 (ai, bi) = (↑i↓j − ↓i↑j)/ √ 2 Put the spins back in a way compatible with the valence bonds and sample in a combined space of spins and bonds Loop updates similar to those in finite-T methods (world-line and stochastic series expansion methods) • valence-bond trial wave functions can be used • larger systems accessible • sample spins, but measure using the valence bonds |Ψ〉〈Ψ| A More efficient ground state QMC algorithm → larger lattices T=0 results with the improved valence-bond algorithm Universal exponents? Two different models: Studies of J-Q2 model and J-Q3 model on L×L lattices with L up to 64 D2 = 〈D2x +D2y〉, Dx = 1 N N∑ i=1 (−1)xiSi · Si+xˆ, Dy = 1 N N∑ i=1 (−1)yiSi · Si+yˆ !M = 1 N ∑ i (−1)xi+yi !SiM2 = 〈 !M · !M〉 Exponents ηs, ηd, and ν from the squared order parameters Cij = 14 − Si · Sj H2 = −Q2 ∑ 〈ijkl〉 CklCij H3 = −Q3 ∑ 〈ijklmn〉 CmnCklCij H1 = −J ∑ 〈ij〉 Cij J. Lou, A.W. Sandvik, N. Kawashima, arXiv:0908.0740 Now using coupling ratio J-Q2 model; qc=0.961(1) ηs = 0.35(2) ηd = 0.20(2) ν = 0.67(1) J-Q3 model; qc=0.600(3) ηs = 0.33(2) ηd = 0.20(2) ν = 0.69(2) q = Qp Qp + J , p = 2, 3 ηs, ν in perfect agreement with the finite-T results by Kaul and Melko •previous T=0 results may have been affected by scaling corrections in small latticed Experimental realizations of deconfined quantum-criticality? EtMe3Sb[Pd(dmit)2]2 shows no magnetic order • May be a realization of the deconfined quantum-critical point [Xu and Sachdev, PRB 79, 064405 (2009)] VBS state Layered triangular-lattice systems based on [Pd(dmit)2]2 dimers Y. Shimizu et al, J. Phys.: Condens. Matter 19, 145240 (2007) J = 200-250 K η ≈ 0.35 T. Itou et al, Phys. Rev. B 77, 104413 (2008) NMR spin-lattice relaxation rate is sensitive to η Quantum-critical scaling with exponent η in good agreement with the QMC calculations 1 T1 ∼ T η SU(N) generalization of the J-Q model Heisenberg model with SU(N) spins has VBS state for large N • Hamiltonian consisting of SU(N) singlet projectors • In large-N mean-field theory Nc≈5.5 (Read & Sachdev, PRL 1988) • QMC gives Nc≈4.5 (Tanabe & Kawashima, 2007; K. Beach et al. 2008) The valence-bond loop projector QMC has a simple generalization • N “colors” instead of 2 spin states • Each loop has N “orientations” • Stronger VBS order expected in SU(N) J-Q model J. Lou, A.W. Sandvik, N. Kawashima, arXiv:0908.0740 SU(2); qc=0.961(1) ηs = 0.35(2) ηd = 0.20(2) ν = 0.67(1) ηs = 0.42(5) ηd = 0.64(5) ν = 0.70(2) SU(4); qc=0.082(2) SU(4) SU(3) SU(N) J-Q2 criticality SU(3); qc=0.335(2) ηs = 0.38(3) ηd = 0.42(3) ν = 0.65(3) J-Q3 model SU(3) J-Q2 model q = 0.85 q = 0.65 Order parameter histograms P(Dx,Dy), L=32 q = 0.635 (qc ≈ 0.60) q = 0.45 (qc ≈ 0.33) D4 = ∫ rdr ∫ dφP (r,φ) cos(4φ) VBS symmetry cross-over Λ ∼ ξa ∼ q−aν Finite-size scaling gives U(1) (deconfinement) length-scale J-Q3 model SU(3) J-Q2 model α ≈ 1.3 Is it possible to dire