Quantum Phase Transitions in Low Dimensional Magnets

Transcription

1 Computational Approaches to Quantum Critical ISSP Quantum Phase Transitions in Low Dimensional Magnets Synge Todo ( ) Department of Applied Physics, University of Tokyo <wistaria@ap.t.u-tokyo.ac.jp>

2 Low-Dimensional Magnets Spin-S Heisenberg antiferromagnets H = J [S zi S zj 12 ] (S+i S j + S i S+j ) i,j (non-frustrated, no external field) 3D or higher: thermal phase transitions at finite T 2D: Néel long-range order at ground state 1D: Haldane conjecture S=1/2, 3/2,... : critical (i.e. quasi-lro) GS S=1,2,... : singlet GS with finite gap

3 2D Heisenberg Antiferromagnet H = i,j S 2i,j S 2i+1,j + α i,j S 2i+1,j S 2i+2,j + J i,j S i,j S i,j+1 spatial structure J 1! M.Matsumoto, et al (2002) S=1 S=1 ladder J VBS state decoupled dimers ground state phase diagram 0 H-II H-II 2D QCP AF AF antiferro LRO H-I bond alternating! 1D QCP chain isotropic square lattice Haldane chain

4 Agenda (Very brief) review on loop cluster quantum Monte Carlo Gap estimation by using loop cluster QMC extended moment method Topological order in 1D magnets twist order parameter and its improved estimator Quantum surface transition in 2D perturbation by using QMC

5 Path-Integral Representation H = H0 (diagonal part) + V (off-diagonal part) Z = tr e βh 0 U(β) with e β(h 0+V ) e βh 0 U(β) Time-dependent perturbation expansion du(τ)/dτ = V (τ)u(τ) U(β) = 1 = 1 β 0 β 0 [ Z = tr e βh V (τ)u(τ) dτ V (τ 1 ) dτ 1 + β ( 1) n n=1 β 0 0 τ1 0 0 τ1 V (τ) e τh 0 V e τh 0 V (τ 1 )V (τ 2 ) dτ 1 dτ 2 + τn 1 0 i=1 n V (τ i ) ] n dτ i i=1

6 World-Line Configuration Each term is specified by (α, n, {b i, τ i }) world-line configuration τ=β τ7 b=1 τ6 τ4 b=2 τ3 τ8 b=4 τ5 b=4 n=8 α = τ2 b=3 b=1 τ1 b=3 τ=0 Weight of each configuration can be calculated easily w(α, n, {b i, τ i }) ( 1) n α e βh 0 V b1 (τ 1 )V b2 (τ 2 ) V bn (τ n ) α

7 Path Integral Representation A d-dimensional quantum system is mapped to a (d+1)-dimensional classical system. additional dimension = imaginary time Up (down) spins form world lines due to Sz conservation. Weight of mapped configuration (world lines) is expressed as a product of local weights. Local updates or cluster updates (loop update) can be performed on world lines.

8 High Temperature Series Representation (SSE) Break up the Hamiltonian into diagonal and off-diagonal bond terms H = k=1,2 b H k,b diagonal H 1,b = Js z i s z j + h z (sz i + s z j ) off-diagonal H 2,b = J 2 (s+ i s j + s i s+ j ) Taylor expansion of density matrix (ip={k,b}) Z = α α e βh α = α n=0 S n βn n! α H i 1 H i2 H in α Sandvik and Kurkijarvi (1991)

9 World-Line Configuration in HTSR Each term is specified by Z = α α e βh α = α (α, {H ip }) n=0 S n operator string βn n! α H i 1 H i2 H in α

10 High Temperature Series Representation (SSE) High temperature series always converges due to the finiteness of the system (cf. HT series in thermodynamic limit) Similarity to path-integral representation Integer index instead of continuous variable in imaginary time direction

11 Non-Local Updates Non-local updates are necessary for simulating in grand canonical ensemble (different winding number of world lines) Strong restriction in world-line configuration: All possible changes of world lines form loops due to local Sz conservation. (Finding allowed local update is also very tough task!) Non-local updates may reduce auto-correlation time loop cluster update

12 One MCS of Loop Algorithm PI HTS breakup diagonal update loop flip loop flip

13 Loop Cluster Update Changes shape of world lines as well as their winding number Does not change expansion order n in HTSR Not ergodic by itself (used with diagonal update) Loop cluster update for spin-s XYZ models No loop solution in the presence of (longitudinal) external field (need to introduce global weight to loops) Both in PI and HTS representations

14 C++ implementation for S=1/2 Heisenberg antiferromagnet path integral for (unsigned int mcs = 0; mcs < therm + sweeps; ++mcs) { std::swap(operators, operators_p); operators.clear(); fragments.resize(nsites); std::fill(fragments.begin(), fragments.end(), fragment_t()); std::copy(boost::counting_iterator<int>(0), boost::counting_iterator<int>(nsites), current.begin()); double t = r_time(); for (std::vector<local_operator_t>::iterator opi = operators_p.begin(); t < 1 opi!= operators_p.end();) { if (opi == operators_p.end() t < opi->time) { unsigned int b = nbonds * random(); if (spins[left(nbonds, b)]!= spins[right(nbonds, b)]) { operators.push_back(local_operator_t(b, t)); t += r_time(); } else { t += r_time(); continue; } } else { if (opi->type == diagonal) { ++opi; continue; } else { operators.push_back(*opi); ++opi; } } std::vector<local_operator_t>::iterator oi = operators.end() - 1; unsigned int s0 = left(nbonds, oi->bond); unsigned int s1 = right(nbonds, oi->bond); oi->lower_loop = unify(fragments, current[s0], current[s1]); oi->upper_loop = current[s0] = current[s1] = add(fragments); if (oi->type == offdiagonal) { spins[s0] ^= 1; spins[s1] ^= 1; } } for (int s = 0; s < nsites; ++s) unify(fragments, s, current[s]); int nc = 0; for (std::vector<fragment_t>::iterator ci = fragments.begin(); ci!= fragments.end(); ++ci) if (ci->is_root()) ci->id = nc++; clusters.resize(nc); for (std::vector<fragment_t>::iterator ci = fragments.begin(); ci!= fragments.end(); ++ci) ci->id = cluster_id(fragments, *ci); for (std::vector<cluster_t>::iterator pi = clusters.begin(); pi!= clusters.end(); ++pi) *pi = cluster_t(random() < 0.5); for (std::vector<local_operator_t>::iterator oi = operators.begin(); oi!= operators.end(); ++oi) if (clusters[fragments[oi->lower_loop].id].to_flip ^ clusters[fragments[oi->upper_loop].id].to_flip) oi->flip(); for (unsigned int s = 0; s < nsites; ++s) if (clusters[fragments[s].id].to_flip) spins[s] ^= 1; } } high temperature series for (unsigned int mcs = 0; mcs < therm + sweeps; ++mcs) { int nop = operators.size(); std::swap(operators, operators_p); operators.clear(); fragments.resize(nsites); std::fill(fragments.begin(), fragments.end(), fragment_t()); std::copy(boost::counting_iterator<int>(0), boost::counting_iterator<int>(nsites), current.begin()); bool try_gap = true; for (std::vector<local_operator_t>::iterator opi = operators_p.begin(); try_gap opi!= operators_p.end();) { if (try_gap) { unsigned int b = nbonds * random(); if (spins[left(nbonds, b)]!= spins[right(nbonds, b)] && (nop + 1) * random() < lb2) { operators.push_back(local_operator_t(b)); ++nop; } else { try_gap = false; continue; } } else { if (opi->type == diagonal && lb2 * random() < nop) { --nop; ++opi; continue; } else { operators.push_back(*opi); ++opi; try_gap = true; } } std::vector<local_operator_t>::iterator oi = operators.end() - 1; unsigned int s0 = left(nbonds, oi->bond); unsigned int s1 = right(nbonds, oi->bond); oi->lower_loop = unify(fragments, current[s0], current[s1]); oi->upper_loop = current[s0] = current[s1] = add(fragments); if (oi->type == offdiagonal) { spins[s0] ^= 1; spins[s1] ^= 1; } } for (int s = 0; s < nsites; ++s) unify(fragments, s, current[s]); int nc = 0; for (std::vector<fragment_t>::iterator ci = fragments.begin(); ci!= fragments.end(); ++ci) if (ci->is_root()) ci->id = nc++; clusters.resize(nc); for (std::vector<fragment_t>::iterator ci = fragments.begin(); ci!= fragments.end(); ++ci) ci->id = cluster_id(fragments, *ci); for (std::vector<cluster_t>::iterator pi = clusters.begin(); pi!= clusters.end(); ++pi) *pi = cluster_t(random() < 0.5); for (std::vector<local_operator_t>::iterator oi = operators.begin(); oi!= operators.end(); ++oi) if (clusters[fragments[oi->lower_loop].id].to_flip ^ clusters[fragments[oi->upper_loop].id].to_flip) oi->flip(); for (unsigned int s = 0; s < nsites; ++s) if (clusters[fragments[s].id].to_flip) spins[s] ^= 1; } }

15 Improved Estimator Fortuin-Kasteleyn Representaion Z = c W (c) = c,g W (c, g) = c,g (c, g)f(g) In loop algorithm, graphs are generated according to W (g) W (c, g) = 2 N c f(g) c Expected value of a quantity: A(c) = Z 1 c A(c)W (c) A(c) = Z 1 c A(c) g (c, g)f(g) = Z 1 g A(g)W (g) A(g) c A(c) (c, g)

16 Improved Estimator A(c) N 1 MC A(g) e.g.) spin correlation function C i,j (c) = S z i S z j spin correlation represented in graph language { 1/4 if i and j are on the same loop C i,j (g) = 0 otherwise improved statistics (i.e. small statistical errors) loops (or graphs ) themselves have a physical meaning

17 Gap Estimation by using loop cluster QMC

18 How to Calculate Gap by QMC gap is easily calculated in exact diagonalization but not in world-line QMC fitting temperature dependence of susceptibility, etc χ exp( /T ) temperature range? error estimation? = E 1 E 0! ~ T ~ spin-1 ladder

19 A Tiny Gap in a Huge Gap frequency distribution at T 1 P 0 = d 0 d 0 + d 1 exp( 0 /T ) 1 ex) edge state in open S=1 chain effective interaction = singlet-triplet gap << bulk gap ~ 0.41J

20 Imaginary-Time Correlation Function We want a generic and systematic way of measuring gap Correlation function in imaginary-time direction C(τ) = i,j ( 1) i j S z i (0)S z j (τ) = a i cosh[(τ β/2)/ξ i ] ξ 0 = 1 > ξ 1 > However, numerical inverse Laplace transformation is ill-posed problem (c.f. MaxEnt) Systematic improvement of moment method

21 Moment Method Fourier transform of correlation function C(ω) = β 0 C(τ)e iωτ dτ Effectively measured by using improved estimator 2 τ = S i (τ)e iωτ j+1 dτ l = S i (τ j )e iω(τ j+1 τ j ) /ω 2 τ j j l Second-moment method (0-th order approximation) ˆξ (0) = β 2π C(0) [ C(2π/β) 1 ξ 0 1 i=1 a i 2a 0 ξ i ξ 0 ]

22 Higher-Order Approximations Second-order estimator ˆξ (2) = β 4π Fourth-order estimator ˆξ (4) = β 6π ξ 0 [ 1 i=1 convergence test 3 C(0) C(2π/β) C(2π/β) C(4π/β) 1 ξ 0 a i 2a 0 ( ) ] 5 xii ξ 0 S=1, L=128, T=1/64 [ 1 i= C(0) 4 C(2π/β) + 4 C(4π/β) 5 C(2π/β) 8 C(4π/β) + 3 C(6π/β) 1 a i 2a 0 ( ξi ξ 0 ) 3 ] 1/ˆξ (0) (2) 1/ˆξ (2) (6) 1/ˆξ (4) (1) ED (2)

23 Haldane Gap of S=3 Chain Estimation of extremely large correlation length and small gap 1000 S=3 Chain Results ξ = 637(1) = (3)J Simulation correlation length (! x,! " ) S=1 2 3! x! " L = TL=2S (fixed) T = T Todo and Kato (2001)

24 Haldane gap of S=1,2,3 chains ξ x method S= (7) MCPM (Nightingale-Blöte 1986) (2) 6.03(2) DMRG (White-Huse 1992) (2) 6.2 ED (Golinelli et al 1994) 0.408(12) QMC (Yamamoto 1995) (6) (2) QMC+loop (Todo-Kato 2001) S= (16) QMC (Yamamoto 1995) 0.055(15) DMRG (Nishiyama et al 1995) 0.085(5) 49(1) DMRG (Schollwöck-Jolicœur 1995) 0.090(5) 50(1) QMC+loop (Kim et al 1997) (13) DMRG (Wang et al 1999) (5) 49.49(1) QMC+loop (Todo-Kato 2001) S= (3) 637(1) QMC+loop (Todo-Kato 2001)

25 Full Dispersion Curve!(k) S = 1/2 L = 256 exact T=0.2J T=0.1J T=0.05J T=0.02J (In principle) moment method can be extended to any k-values But (in practice) suffers from large systematic and statistical errors 2 k / " S = 1/2 L = S = 1!(k) 1!(k) exact T=0.2J T=0.1J T=0.05J T=0.02J L=16 L=32 L= k / " k / "

26 Topological Order in 1D Magnets

27 Topological order in 1D magnets S=1 antiferromagnetic Heisenberg ladder 0.3 J J K no QPT between two limits Haldane state! ~ T ~ 1 dimer state 0.1! ~ " #1 R = K / J ST, M.Matsumoto, C.Yasuda, and H.Takayama (2001) R

28 Lieb-Schultz-Mattis Theorem Lieb-Schultz-Mattis overlap function z L Ψ 0 exp[i 2π L jsj z ] Ψ 0 L j=1 Ψ 0 : ground state exp[i 2π L jsj z ] Ψ 0 : twisted state with L j=1 E E 0 + O(L 1 ) z L 0 (L ) gapless or degenerating ground state E.Lieb, T. Schultz, D.Mattis 1961

29 Twist Order Parameter QPT in spin-s bond-alternating chain as a rearrangement of VBS pattern S=1 S=1/2 S=3/2 Expected value of LSM twistoperator = twist order parameter z L Ψ 0 exp[i 2π L L jsj z ] Ψ 0 j=1 M.Nakamura and ST (2002) (1) z L S=2 S=1-0.5 S= 3 2 S= δ

30 Sign of Twist Order Parameter contribution from a dimer i j S=1 dimer phase: (2,0) VBS state L-1 L Haldane phase: (1,1) VBS state..... cos[ π i j ] L [cos π L ]L L-1 L z L ( 1) b [cos π L ]L 1 [cos π (L 1)] 1 L

31 Effects of Randomness on zl weak bond randomness rearrangement of local VBS pattern local rearrangement of VBS pattern can not change the parity at boundary zl could be a good topological order parameter also for random systems T.Arakawa, ST, and H.Takayama (2005)

32 Twist Order Parameter in Loop Algorithm Split-spin representation of S=1 chain k=1 Twist order parameter z L = 2 j= L ( 2πi exp L Decomposing into contributions from each loop ( 2πi exp L L j=1 k=1,2 j ŝ z j,k ) L j=1 = exp k=1,2 ( πi L N c l=1 j ŝ z j,k ) X l ) = X l MCS N c l=1 exp ( πi ) L X l (j,k) C l 2 j ŝ z j,k

33 Averaging over 1 2 N c d 1 =±1 = N c l=1 Improved Estimator 2 N c N c d N =±1 l=1 configurations to obtain A(g) exp 1 [ ( πi ) exp 2 L X l ( πi L d lx l ) ( + exp Contribution from a loop ( π ) cos L X l πi )] L X l = N c l=1 ( π ) cos L X l for 2-site loop ( π ) cos L X l = cos ( π L i j ) = contribution from a dimer

34 Loops vs VBS picture loop defines a singlet pair (or group) of spins in the VBS picture loop across the periodic boundary 4n (4n+2) times contributes +1 (-1) to twist-order parameter local fluctuation of loop structure can not change the sign (as seen in the VBS picture for random system): well-defined (topologically protected) sign in each VBS phase. zl = 0 means the existence of global loop(s), i.e. correlation length ~ system size

35 Spin-1 ladder Plaquette singlet solid (PSS) state in S=1 ladder Haldane state J J K uniform susceptibility 0.3 PSS state dimer state ST, M.Matsumoto, C.Yasuda, and H.Takayama (2001)! ~ T ~

36 Quantum Surface Transition

37 Edge State of Open S=1 chain Open S=1 chain with odd L S z =0 and ± S z =±1 scaling plot frequency 0.6 S z =±1 0.4 S z =0 0.2 L = 7,9,..., T 0-dim surface of 1-dim system frequency S z =0 0.2! p = (Todo and Kato 2001) T exp[l/! p ] J eff exp( L/6.01) generalization to higher-dimensions?

38 Spin-1 Strips with Finite Width H = i,j S 2i,j S 2i+1,j + α i,j S 2i+1,j S 2i+2,j + J i,j S i,j S i,j+1 H-II J L 10 1 AF 1! Periodic boundary conditions in y-direction and L J What happens for open boundaries? 0 H-II ! AF H-I M.Matsumoto, et al (2002)

39 Open S=1 Strips H = i,j S 2i,j S 2i+1,j + α i,j S 2i+1,j S 2i+2,j + J i,j S i,j S i,j+1 Open boundaries in y-direction (multi-leg ladder) Consider odd L and J 1, α limit Edge spins form triplet, 0, on each rung if T 0.754J exp( L/6.01) Take first two terms (leg interaction) into account as weak perturbation

40 Effective Hamiltonian 9x9 Effective Hamiltonian between two rungs H eff = H H 0 H H 0 H 0 H H H 0 H H 12 non-zero elements among 81 H = K j S 1,j S 2,j K = 1 or α comparing with Hamiltonian of spin-1 dimer J = H J S 1 S 2 = K j S 1,j S 2,j = K j 1 S z 1,j 1 2 S z 2,j 2 1 S z 1,j 1 : z-component of j-th spin in total Sz=1 sector

41 QMC Results C = j S z j L = 7,9,..., T No temperature dependence at T < 0.1 effective interaction L = 15,17,..., / L T = 0.02 converges to a finite value for large L

42 Effective S=1 Chain Effective S=1 bond alternating chain for J 1, α H eff = Ji S i S i+1 Quantum phase transition at α c J = C or Cα Only spins near surfaces participate Quantum Surface Transition 10 1 H-II AF Phase transitions at finite J No quantum phase transition in semi-infinite system (i.e. system with only one surface) J AF H-II H-I

43 Summary (Very brief) review on loop cluster quantum Monte Carlo Gap estimation by using loop cluster QMC extended moment method Topological order in 1D magnets twist order parameter and its improved estimator Quantum surface transition in 2D perturbation by using QMC

44 Randomness effects to spin gapped state Poster presentations in symposium by C. Yasuda and M. Matsumoto Finite-temperature phase transition in quasi-1d magnets quantum critical point Oral presentation in symposium by ST

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