Coupled Cluster Method for Quantum Spin Systems

Transcription

1 Coupled Cluster Method for Quantum Spin Systems Sven E. Krüger Department of Electrical Engineering, IESK, Cognitive Systems Universität Magdeburg, PF 4120, Magdeburg, Germany MAMBT, Manchester, 31. August 2005 collaborations: J. Richter, R. Darradi (Magdeburg) D.J.J. Farnell (Liverpool) R.F. Bishop (Manchester) Sven Krüger, CCM for quantum spins p.1/23

2 Organization of the talk Heisenberg model and CCM Coupled cluster method application to quantum spin systems spin stiffness excited state Results J J model J 1 J 2 model square lattice, triangular lattice Conclusions Sven Krüger, CCM for quantum spins p.2/23

3 Heisenberg model, spin 1/2, two dimensional H = i,j J ij s i s j ground state HAFM on square and triangular lattice: Néel ordered important mechanisms to destroy magnetic long-range order (LRO): competition of bonds: higher quantum fluctuations suppression of long-range order, formation of local singlets; e.g., CaV 4 O 9, SrCu 2 (BO 3 ) 2 frustration my yield to PT to noncollinear (spiral) states in the classical model; quantum fluctuations favor collinear order my yield (together with quantum fluctuations) to quantum paramagnetic phase Sven Krüger, CCM for quantum spins p.3/23

4 Heisenberg model and CCM with CCM it can be calculated in higher orders: ground state energy, magnetization new: stiffness, gap CCM is able to describe (for example in the J J model) frustrated incommensurate spiral phase quantum phase transition without frustration here: quantum phase transition with frustration J 1 J 2 model Sven Krüger, CCM for quantum spins p.4/23

5 Coupled cluster method CCM I. choose a model state Φ and a set of creation operators (C + I ) C I Φ = 0 I 0, C + I Φ Φ C I = 1 I II. ansatz for the ground state Ψ with the correlation operator S Ψ = e S Φ, S = I 0 S I C + I ; Ψ = Φ Se S, S = 1 + I 0 S I C I III. ket-state equations (non linear), with H = Ψ H Ψ H S I = 0 Φ C I e S He S Φ = 0, ket coefficients S I, Ψ, E = Φ e S He S Φ IV. bra-state equations (linear): H S I = 0, bra coefficients S I, Ψ, Ā = Ψ A Ψ Sven Krüger, CCM for quantum spins p.5/23

6 CCM application on spin systems I. Selection of Φ : classical spin state S = i 1 S i1 s + i 1 + i 1 i 2 S i1 i 2 s + i 1 s + i 2 + II. approximation of S LSUBn i 1 i 2 i 3 S i1 i 2 i 3 s + i 1 s + i 2 s + i 3 + approximation of S is the only approximation in the CCM LSUBn: local approximation, including correlations with up to n spins hierarchical approximation, LSUB becomes exact Sven Krüger, CCM for quantum spins p.6/23

7 CCM Fundamental configurations example for square lattice: LSUB8 with 259 types of connected fundamental configurations Sven Krüger, CCM for quantum spins p.7/23

8 Spin stiffness spin stiffness ρ s measures the rigidity of the spins with respect to a small twist θ of the direction of spin between every pair of neighboring rows: ρ s = d2 E 0 (θ) dθ 2 N ρ s > 0 LRO, systems are stiff θ=0 ρ s = 0 no LRO, systems are not stiff Sven Krüger, CCM for quantum spins p.8/23

9 CCM calculation of the spin stiffness introducing the twist θ: appropriate changing of the classical spin state of Φ, doing the CCM, ground-state energy in dependence of θ twist θ for the square lattice: a) θ 0 + θ +2 θ θ 0 + θ +2 θ b) θ/2 cos(60 ) θ =+θ/2 +3θ/2 twist θ for the triangular lattice: 60 θ 0 +θ +2θ x twist is introduced along rows in x direction. Sven Krüger, CCM for quantum spins p.9/23

10 CCM Excited-State Formalism apply linearly an excitation operator X e to the ket-state wave function: Ψ e = X e Ψ = X e e S Φ, X e = I 0 X e I C+ I using Schrödinger equation E e Ψ e = H Ψ e gives for the excitation energy ɛ e E e E g apply Φ C I set of eigenvalue equations ɛ e X e Φ = e S [H, X e ]e S Φ ɛ e X e I = Φ C I e S [H, X e ]e S Φ, I 0 Sven Krüger, CCM for quantum spins p.10/23

11 CCM Excited-State Formalism: application on spin systems I. Selection of X e : classical spin state X e = i 1 II. approximation of X: X i1 s + i 1 + i 1 i 2 X i1 i 2 s + i 1 s + i 2 + similar to the ground state i 1 i 2 i 3 X i1 i 2 i 3 s + i 1 s + i 2 s + i 3 + but: choose configurations which change s T z by ±1 use the same approximation level (e.g., LSUBn) as for ground state Sven Krüger, CCM for quantum spins p.11/23

12 Applications and Results Sven Krüger, CCM for quantum spins p.12/23

13 J J model A B J = 1, J > J: quantum competition; at J = J s phase transition LRO dimerized paramagnetic phase with local singlets: J J J, J different signs: frustration spiral state known results with CCM: phase transition to the dimerized phase can be described by magnetization, gap, and spin stiffness frustrated region: quantum fluctuations favor collinear order PRB 61, (2000); PRB 64, (2001) Sven Krüger, CCM for quantum spins p.13/23

14 J J model: new results with CMM J = 1, J > J, transition to the dimerized paramagnetic phase influence of the Ising anisotropy s i s j s x i s x j + s y i sy j + sz i s z j on the position of the quantum critical point J s: linear relation J s( ) α with α R. Darradi, J. Richter and S.E. Krüger, J. Phys. Condens. Matter 16, (2004)) influence of the spin quantum number s on the position of the quantum critical point J s: J s s(s + 1) increase of J s with s diminishing of quantum effects R. Darradi, J. Richter and D.J.J. Farnell, J. Phys. Condens. Matter 17, (2005) Sven Krüger, CCM for quantum spins p.14/23

15 J 1 J 2 model A B J 2 1 J J = 1 antiferromagnetic J 2 > 0 parameter, frustration at J 2 = J2 c frustration (together with quantum fluctuations) destroys LRO quantum paramagnet (magnetically disordered phase) Sven Krüger, CCM for quantum spins p.15/23

16 J 1 J 2 model: magnetization sublattice magnetization M versus J 2 obtained by CCM-LSUBn Néel LRO disappears at J c with extr1, and at J c with extr2 new: up to LSUB10 with configuration (using the code of Damian M Farnell) LSUB4 LSUB6 LSUB8 LSUB10 ext1.lsub4-10 ext2.lsub J2 extr1 a 0 + a 1 (1/n) + a 2 (1/n) 2 extr2 a 0 + b 1 (1/n) b 2 at J 2 = 1 extr1 is better approximation at the critical point J c 2 extr2 seems to yield better results reason: scaling rules often change at a phase transition use an approximation (extr2) with variable exponents Sven Krüger, CCM for quantum spins p.16/23

17 J 1 J 2 model: spin stiffness spin stiffness ρ s versus J 2 obtained by CCM-LSUBn Néel LRO disappears at J c with extr1. and J c with extr2. ρ s LSUB2 LSUB4 LSUB6 LSUB8 extr1.lsub4-8 extr2.lsub4-8 extr1 a 0 + a 1 (1/n) + a 2 (1/n) extr2 a 0 + b 1 (1/n) b J2 again: extr1 better at J 2 = 1, extr2 better at J 2 = J2 c Sven Krüger, CCM for quantum spins p.17/23

18 J 1 J 2 model: gap Néel ordered state no gap quantum paramagnet gap J1-J2 Model gap LSUB4 LSUB6 LSUB8 extrapolated:4,6,8 number of configurations: LSUBn gs ex J2 Néel LRO disappears at J c Sven Krüger, CCM for quantum spins p.18/23

19 Conclusions: J 1 J 2 model already known: ground state (energy, magnetization) Bishop, Farnell, Parkinson PRB 58, 6394 new results: magnetization: CCM-LSUB10 extrapolation for calculating J c 2 spin stiffness gap with all three measures: transition from Néel LRO to quantum paramagnet (magnetically disordered phase) can be described approximation of J c 2 Sven Krüger, CCM for quantum spins p.19/23

20 Square lattice: magnetization CCM LSUBn approximation with n = {2, 4, 6, 8, 10} and extrapolated results N F number of fundamental configurations E g /N GS energy per spin M sublattice magnetisation N F E g /N M/M clas LSUB LSUB LSUB LSUB LSUB Extrapolated CCM rd order SWT QMC (5) (6) Hamer et al. PRB 46, 6276 (1992); Sandvik PRB 56, (1997) Sven Krüger, CCM for quantum spins p.20/23

21 Square lattice: spin stiffness CCM: comparison with other methods: LSUBn number eqs. stiffness ρ s extr method ρ s LSWT nd SWT rd SWT series exp exact diagon quantum Mone Carlo CCM in excellent agreement with the best results obtained by other means Sven Krüger, CCM for quantum spins p.21/23

22 Triangular lattice: spin stiffness parallel stiffness, i.e., the spins are rotated by the twist θ within the plane of the system CCM: comparison with other methods:: LSUBn number eqs. stiffness ρ s approx ρ s method exact diagonalization 0.05 LSWT Schwinger-boson approach CCM improved results by CCM (LSWT is to large) Sven Krüger, CCM for quantum spins p.22/23

23 Conclusions CCM leads to quite accurate results for quantum spin systems (ground state and first excitation) qualitativly correct description of GS order-disorder transitions no problems with frustration and incommensurate spiral phases higher spin s > 1/2 also possible some further things to do in high-order CCM: calculation of correlation function dimerized state as CCM ground state Sven Krüger, CCM for quantum spins p.23/23