# Quantum spin systems - models and computational methods

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1 Summer School on Computational Statistical Physics August 4-11, 2010, NCCU, Taipei, Taiwan Quantum spin systems - models and computational methods Anders W. Sandvik, Boston University Lecture outline Introduction to quantum spin systems Exact diagonalization methods - use of symmetries for block diagonalization - full diagonalization and the Lanczos method Quantum Monte Carlo - introduction to path integrals - the stochastic series expansion (SSE) method Studies of quantum phase transitions 1

2 Classical spin models Lattice models with spin degrees of freedom at the vertices Classified by type of spin: Ising model: discrete spins, normally two-state σi = -1, +1 XY model: planar vector spins (normally of length S=1) Heisenberg model: 3-dimensional vector spins (S=1) Statistical mechanics spin configurations C energy E(C) some quantity Q(C) temperature T (kb=1) Q = 1 Z Z = C Q(C)e E(C)/T E = ij J ij σ i σ j (Ising) C e E(C)/T E = ij J ij Si S j = ij J ij cos(θ i Θ j ) (XY) E = ij J ij Si S j (Heisenberg) 2

3 Phase transition: The 2D Ising model Ground state (T=0) depends on J, E(0)/N=-2J E = J i,j σ i σ j, σ i = ±1 +J and -J systems are trivially related by sublattice transformation σ i σ i for i on sublattice A Ising ferromagnet and antiferromagnet have identical properties T > 0, thermal expectation values, phase transition M (T c T ) β A = 1 Z M = 1 N A n e En/T, n T c /J 2.27 i ( σ i M = 1 N Z = e E n/t n ) σ i ( 1) x i+y i for AF i 3

4 Quantum spins Spin magnitude S; basis states S z 1,S z 2,...,S z N>, S z i = -S,..., S-1, S Commutation relations: [S x i,s y i ]=i Sz i (we set = 1) [S x i,s y j ] = [Sx i,s z j ]=... =[S z i,s z j ] = 0 (i j) Ladder (raising and lowering) operators: S + i = Si x + is y i, S i = Si x is y i S + i Sz i = S(S + 1) Si z(sz i + 1) Sz i +1, S i Sz i = S(S + 1) Si z(sz i 1) Sz i 1, Spin (individual) squared operator: S 2 i S z i = S(S + 1) S z i S=1/2 spins; very simple rules S z i =+ 1 2 = i, S z i = 1 2 = i S z i i =+ 1 2 i S i i = i S + i i =0 S z i i = 1 2 i S + i i = i S i i =0 4

5 Quantum spin models Ising, XY, Heisenberg hamiltonians the spins always have three (x,y,z) components interactions may contain 1 (Ising), 2 (XY), or 3 (Heisenberg) components H = J ij Si z Sj z = 1 4 J ij σ i σ j (Ising) ij ij H = ij J ij [S x i S x j + S y i Sy j ]= 1 2 J ij [S + i S j + S i S+ j ] ij (XY) H = ij J ij Si S j = ij J ij [S z i S z j (S+ i S j + S i S+ j )] (Heisenberg) Quantum statistical mechanics Q = 1 { } Z Tr Qe H/T Z = Tr { e H/T } = Large size M of the Hilbert space; M=2 N for S=1/2 - difficult problem to find the eigenstates and energies - we are also interested in the ground state (T 0) - for classical systems the ground state is often trivial M 1 n=0 e E n/t 5

6 Why study quantum spin systems? Solid-state physics localized electronic spins in Mott insulators (e.g., high-tc cuprates) large variety of lattices, interactions, physical properties search for exotic quantum states in such systems (e.g., spin liquid) Ultracold atoms (in optical lattices) spin hamiltonians can (?) be engineered some boson systems are very similar to spins (e.g., hard-core bosons) Quantum information theory / quantum computing possible physical realizations of quantum computers using interacting spins many concepts developed using spins (e.g., entanglement) Generic quantum many-body physics testing grounds for collective quantum behavior, quantum phase transitions identify Ising models of quantum many-body physics Particle physics / field theory / quantum gravity some quantum-spin phenomena have parallels in high-energy physics e.g., spinon (quark) confinement-deconfinement transition spin foams (?) 6

7 Prototypical Mott insulator; high-tc cuprates (antiferromagnets) CuO2 planes, localized spins on Cu sites - Lowest-order spin model: S=1/2 Heisenberg - Super-exchange coupling, J 1500K Many other quasi-1d and quasi-2d cuprates chains, ladders, impurities and dilution, frustrated interactions,... H = J i,j S i S j Cu (S = 1/2) Zn (S = 0) Ladder systems - even/odd effects non-magnetic impurities/dilution - dilution-driven phase transition 7

8 The antiferromagnetic (Néel) state and quantum fluctuations The ground state of the 2D Heisenberg model H = J ij S i S j = J ij [S z i S z j (S+ i S j + S i S+ j )] Does the staggered (Néel) order survive quantum fluctuations? order parameter: staggered (sublattice) magnetization m s = 1 N m s = 1 N N φ isi, φ i =( 1) x i+y i (2D square lattice) i=1 ( SA S B ) For S (classical limit) <ms> S what happens for small S (especially S=1/2)? <ms> 0.31 ( 60% of classical value 1/2) demonstrated using quantum Monte Carlo (Reger and Young, 1989) in good agreement with experiments on cuprates - neutron scattering, NMR,... 1D Heisenberg chain: no magnetic order (Mermin-Wagner theorem) 8

9 Non-magnetic states in 2D Consider two spins, i and j, in isolation: H ij = J ij Si S j = J ij [S z i S z j (S+ i S j + S i S+ j )] For Jij>0 the ground state is the singlet; φ s ij = i j i j 2, E ij = 3J ij /4 The Néel states have higher energy (expectation values; not eigenstates) φ N a ij = i j, φ N b ij = i j, H ij = J ij /4 The Néel states are product states: φ N a ij = i j = i j The singlet is a maximally entangled state (furthest from product state) 9

10 N>2: each spin tends to entangle with its neighbors entanglement is energetically favorable but cannot singlet-pair with more than 1 spin leads to fluctuating singlets (valence bonds) less entanglement, <Hij> > -3Jij/4 closer to a product state (e.g., Néel) non-magnetic states possible (N= ) resonating valence-bond (RVB) spin liquid valence-bond solid (VBS) Translationally invariant state no broken symmetries = ( )/ 2 Broken translational symmetry strong and weak correlations of neighbors S(r) S(r + ˆx ) S(r) S(r + ŷ ) 10

11 Quantum phase transitions (T=0; change in ground-state) Example: Dimerized S=1/2 Heisenberg models every spin belongs to a dimer (strongly-coupled pair) many possibilities, e.g., bilayer, dimerized single layer strong interactions weak interactions Singlet formation on strong bonds Neel - disordered transition Ground state (T=0) phases = spin gap s 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 11

12 S=1/2 Heisenberg chain with frustrated interactions = J 1 H = J 1 N S i S i+1 + J 2 N S i S i+2 = J 2 i=1 i=1 Different types of ground states, depending on the ratio g=j2/j1 (both >0) Antiferromagnetic quasi order (critical state) for g< 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> 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/ξ - singlet-product state is exact for g=0.5 (Majumdar-Gosh point) 12

13 Frustration in higher dimensions There are many (quasi-)2d and 3D materials with geometric spin frustration no classical spin configuration can minimize all bond energies triangular (hexagonal) Kagome A single triangular cell: 6-fold degenerate Ising model 120 º Néel order for vectors SrCu2(BO3)2 (Shastry-Sutherland) Pyroclore Infinite triangular lattice highly degenerate Ising model (no order) 120 º Néel (3-sublattice) order for vectors S=1/2 quantum triangular Heisenberg model the classical 3-sublattice order most likely survives [White and Chernyshev, PRL 2007] S=1/2 Kagome system very challenging, active research field; VBS or spin liquid? 13

14 Frustration due to longer-range antiferromagnetic interactions in 2D Quantum phase transitions as some coupling (ratio) is varied J1-J2 Heisenberg model is the prototypical example H = i,j J ij Si S j = J 1 = J 2 g = J 2 /J 1 Ground states for small and large g are well understood Standard Néel order up to g 0.45 collinear magnetic order for g>0.6 0 g< g<0.6 g>0.6 A non-magnetic state exists between the magnetic phases Most likely a VBS (what kind? Columnar or plaquette?) Some calculations (interpretations) suggest spin liquid 2D frustrated models are challenging no generally applicable unbiased methods (numerical or analytical) 14

15 Finite-lattice calculations Numerically exact calculations (no approximations) for finite lattices extrapolate to infinite size, to learn about the ground state and excitations nature of quantum phase transitions associated T>0 physics Example: Dimerized Heisenberg model QMC results for L L lattices It is often known how various quantities should depend on L in a Néel state, spin-wave theory m 2 s(l) = m 2 s( ) + a/l +... use finite-size scaling theory to study the quantum-critical point 15

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