Billiard ball model for structure factor in 1-D Heisenberg anti-ferromagnets

Billiard ball model for structure factor in 1-D Heisenberg anti-ferromagnets Shreyas Patankar 1 Chennai Mathematical Institute August 5, 2010 1 Project with Prof. Kedar Damle, TIFR and Girish Sharma, Satyasai U Shreyas Patankar (CMI) Spin-1 Heisenberg antiferromagnets August 5, 2010 1 / 21

Motivation Inelastic neutron scattering Inelastic neutron scattering used to study magnetic solids Magnetic moment of neutrons acts on spin states Apparatus to measure q = p 1 p 2 (change in momentum) (1) = E 1 E 2 (change in energy) (2) Experiment used to calculate the structure factor S(q, ω) (ω gives measure of energy) Shreyas Patankar (CMI) Spin-1 Heisenberg antiferromagnets August 5, 2010 2 / 21

Motivation Plan We have 1-D Heisenberg spin S = 1 chain with antiferromagnetic interaction (Nearest neighbour interaction) Use fancy math to simplify Hamiltonian to quantum rotor model Count to get two point correlation function analytically Fourier transform to give dynamical structure factor S(k, ω) Throughout we use units with = k B = 1 Shreyas Patankar (CMI) Spin-1 Heisenberg antiferromagnets August 5, 2010 3 / 21

The model Hamltonian Spin system - path integral Consider 1-D system of spins (S = 1) with Hamiltonian H = J ij S i S j (3) with J > 0 We can write partition function for system as a path integral Z = Tr e βh = Dα(τ) e β 0 S[α(τ)]dτ (4) α(0)=α(β) where α(τ) are states in the Hilbert space, τ is parameter called imaginary time and S[α(τ)] = lim ɛ 0 α(τ + ɛ) βh α(τ) ɛ Shreyas Patankar (CMI) Spin-1 Heisenberg antiferromagnets August 5, 2010 4 / 21

The model Hamltonian Coherent state representation Let us write α states in more commonly used notation: N = e iφŝ3 e iθŝ2 e iψŝ3 (5) notice the Euler rotation angles Satisfies the property that N S N = SN (6) where N is the unit vector corresponding to the coherent state Thus partition function can be written as [ Z = DN(τ) e β 0 N(τ) d dτ N(τ) dτ+ β 0 H(S ] N(τ)) N(0)= N(β) (7) Shreyas Patankar (CMI) Spin-1 Heisenberg antiferromagnets August 5, 2010 5 / 21

Berry Phase The model Hamltonian Observe that the first term in exponent is purely imaginary [ β dτ N(τ) d ] [ β dτ N(τ) = dτ N(τ) d ] dτ N(τ) 0 This term is known as the Berry phase It gives surface area of part of sphere bound by the path N(τ) β 0 0 This can be re-written as S B = is β 0 dτ 1 with u as additional parameter 0 du N(u, τ) ( dn du d N ) dτ Observe: Berry phase has nothing to do with Hamiltonian Shreyas Patankar (CMI) Spin-1 Heisenberg antiferromagnets August 5, 2010 6 / 21 (8)

The model Hamltonian Long distance effective model Approximation: antiferromagnetic orientation vector varies gradually. Split Hamiltonian as N(x j, τ) = ( 1) j n(x j, τ) 1 a2d S 2 L 2 + ad S L(x j, τ) (9) where n is slowly varying, n L and L is small We can think of L as the total angular momentum density Substituting, we can write Z as Z = DN(τ) e S B β 0 dτ ( d d x ρs2 ( τ ˆn) 2 +S 2 L 2 ) 2χ N(0)= N(β) (10) Shreyas Patankar (CMI) Spin-1 Heisenberg antiferromagnets August 5, 2010 7 / 21

Quantum rotor model The model Hamltonian Quantum Rotor Model Re-discretizing we can write H = [ gl 2 i 2 + J ] 2 ( n i+1 n i ) 2 i (11) Hamiltonian known as the Rotor model Think of it as chain of rigid rotors, interaction favoring nearest-neighbour alignment Can be shown that even for small values of g, first term dominates the Hamiltonian Eigenstates of the unperturbed Hamiltonian are i where l i, m i are eigenvalues of L 2 i, L i;z respectively l i, m i Shreyas Patankar (CMI) Spin-1 Heisenberg antiferromagnets August 5, 2010 8 / 21

The model Hamltonian Quantum Rotor Model Perturbation Rewrite Hamiltonian as H = i [ gl 2 i 2 + J ] 2 n i n i+1 Treat n i n i+1 term as a perturbation Ground state is all sites with l = 0 (Degenerate) first excited state has l = 1, m = 0, ±1 for one site Need to use degenerate perturbation Shreyas Patankar (CMI) Spin-1 Heisenberg antiferromagnets August 5, 2010 9 / 21

The model Hamltonian Quantum Rotor Model Dispersion relation To diagonalize, choose basis k = 1 e ikaq q N q is label for first excited states Expectation value gives energy as function of k We have dispersion relation q ε(k) = E 1 2J 3 cos(ka) (12) we can call k as momentum For small k, ε(k) = + c2 k 2 2 +..., identify as mass c2 We have excitations as quasiparticles Shreyas Patankar (CMI) Spin-1 Heisenberg antiferromagnets August 5, 2010 10 / 21

The model Hamltonian Quantum Rotor Model Quasi-particles Each quasi-particle has l = 1 (by definition) and comes in three colors : m = 0, ±1 Let n be the rotor position (vector) operator Define ˆn 0 = ˆn z ˆn + = ˆn x + iˆn y ˆn = ˆn x iˆn y We can show[1] that ˆn 0 is sum of creation and annihilation operators for m = 0 quasiparticles Thus, ˆn 0;i 0 will produce a particle of m = 0 at site i Shreyas Patankar (CMI) Spin-1 Heisenberg antiferromagnets August 5, 2010 11 / 21

Structure factor Two-point correlation function We wish to calculate structure factor It can be shown[5] that structure factor is Fourier transform of two point correlation function S(k, ω) = dxdt C(x, t)e i(kx+ωt) where C(x, t) = ˆn(x, t)ˆn(0, 0) ˆn(x, t) can be thought of as operator acting at (x, t) Write (Heisenberg picture) n 0 (x, t) = e iht n 0 (x, 0)e iht Shreyas Patankar (CMI) Spin-1 Heisenberg antiferromagnets August 5, 2010 12 / 21

Semi-classical approximation Quasi-particle collision Assume small temperature: T we have small number of excitations It can be shown[1] that S matrix for quasi-particle collision is just 1 We can visualize C(x, t) = e iht n 0 (x, 0)e ihtˆn(0, 0) as 1 Creation of quasi particle at (0,0) 2 Time evolution of system by t 3 Annihilation of particle at (x,t) 4 Reverse time evolution of background by t For non-zero contribution, initial background state must match with final background state Further, created and annihilated particle must have same m = 0 Shreyas Patankar (CMI) Spin-1 Heisenberg antiferromagnets August 5, 2010 13 / 21

Collision contd. Semi-classical approximation We can depict pictorially as x t plot Shreyas Patankar (CMI) Spin-1 Heisenberg antiferromagnets August 5, 2010 14 / 21

Semi-classical approximation Trajectories Semiclassically, think of quasi-particles having definite trajectories In ensemble average for C(x, t), each state is a configuration of trajectories Dotted trajectory is one created by action of ˆn 0 Suppose dotted trajectory has r right and l left collisions Then annihilated particle is r l positions to right of created particle For solid lines to be reversal invariant, all intermediate particles must have same m = 0 Shreyas Patankar (CMI) Spin-1 Heisenberg antiferromagnets August 5, 2010 15 / 21

Response function Semi-classical approximation Suppose we take temperature T = 0 Then C(x, t) is just the free (quasi-)particle propogator K(x, t)[2] At finite temperature, there is term due to collisions (S-matrix and probability) ( ) 1 r l From above observations, these contributions are ( 1) r+l, 3 respectively Thus, we have C(x, t) = R(x, t)k(x, t) (13) ( 1 ) r l where R(x, t) = 3 C Shreyas Patankar (CMI) Spin-1 Heisenberg antiferromagnets August 5, 2010 16 / 21

Counting collisions Semi-classical approximation Suppose configuration of trajectories is given by x ν = x ν + v ν t and dotted trajectory is x = x 0 + v 0 t Probability that there are n collisions on this trajectory is: [ ] P (n) = δ θ(x x ν (t)) θ( x ν ), n This gives R(x, t) = n= ν (14) [ ] ( 1) n 3 n δ (θ(x x ν (t)) θ( x ν )), n ν {x ν,v ν} Shreyas Patankar (CMI) Spin-1 Heisenberg antiferromagnets August 5, 2010 17 / 21

Expression for R(x, t) Semi-classical approximation Expression can be simplified(!) to give[2]: R( x, t) = π π ( ) dφ φ) π 1 e u2 +u erf(u) e t (1 cos 2π 4 cos( x sin φ) 5 + 3 cos φ where x, t are in suitably normalized units, u = x t Shreyas Patankar (CMI) Spin-1 Heisenberg antiferromagnets August 5, 2010 18 / 21

Conclusion Features Pretty picture: Figure: Real part of C(x, t) vs. x, t Observe interesting behaviour over t Analytic expression itself gives assymetry in S(q, ω) about q Shreyas Patankar (CMI) Spin-1 Heisenberg antiferromagnets August 5, 2010 19 / 21

Comparison Conclusion Essler, Konik calculated S(q, ω) using NLσM field theory methods Figure: Susceptibility vs. frequency at different temperatures It is claimed that the assymetry is a purely quantum effect and cannot be reproduced semi-classically Task: To verify this claim! Shreyas Patankar (CMI) Spin-1 Heisenberg antiferromagnets August 5, 2010 20 / 21

References References Damle, Kedar; Sachdev, Subir (1998), Phys. Rev. B, 57, 14 Rapp, Ákos; Zaránd, Gergely (2006), Phys. Rev. B, 74, 1 Damle, Kedar; Sachdev, Subir (2005), Phys. Rev. Lett., 18, 95 Essler, Fabian H. L.; Konik, Robert M. (2008), Phys. Rev. B, 78, 10 van Hove, Léon (1954) Phys. Rev., 95, 1 Sen, Arnab (2007), Deconfined Quantum Critical Points (MSc. thesis) Shreyas Patankar (CMI) Spin-1 Heisenberg antiferromagnets August 5, 2010 21 / 21