The Bethe Ansatz Heisenberg Model and Generalizations F. Alexander Wolf University of Augsburg June 22 2011 1 / 33 Contents 1 Introduction 2 Ferromagnetic 1D Heisenberg model 3 Antiferromagnetic 1D Heisenberg model 4 Generalizations 5 Summary and References 2 / 33 Although eigenvalues and eigenstates of a finite system may be obtained from brute force numerical diagonalization Two important advantages of the Bethe ansatz all eigenstates characterized by set of quantum numbers ! distinction according to specific physical properties in many cases: possibility to take thermodynamic limit, no system size restrictions One shortcoming structure of obtained eigenstates in practice often to complicated to obtain correlation functions Introduction Bethe ansatz Hans Bethe (1931): particular parametrization of eigenstates of the 1D Heisenberg model Bethe, ZS. f. Phys. (1931) Today: generalized to whole class of 1D quantum many-body systems 3 / 33 Introduction Bethe ansatz Hans Bethe (1931): particular parametrization of eigenstates of the 1D Heisenberg model Bethe, ZS. f. Phys. (1931) Today: generalized to whole class of 1D quantum many-body systems Although eigenvalues and eigenstates of a finite system may be obtained from brute force numerical diagonalization Two important advantages of the Bethe ansatz all eigenstates characterized by set of quantum numbers ! distinction according to specific physical properties in many cases: possibility to take thermodynamic limit, no system size restrictions One shortcoming structure of obtained eigenstates in practice often to complicated to obtain correlation functions 3 / 33 Contents 1 Introduction 2 Ferromagnetic 1D Heisenberg model 3 Antiferromagnetic 1D Heisenberg model 4 Generalizations 5 Summary and References 4 / 33 Basic remarks: eigenstates reference basis: {jσ1 : : : σN i} Bethe ansatz is basis tansformation z PN z z rotational symmetry z-axis ST ≡ n=1 Sn conserved: [H; ST ] = 0 z ) block diagonalization by sorting basis according to hST i = N=2 − r where r = number of down spins Ferromagnetic 1D Heisenberg model Goal obtain exact eigenvalues and eigenstates with their physical properties N X H = −J Sn · Sn+1 n=1 N X 1 + − − + = −J S S + S S + Sz Sz 2 n n+1 n n+1 n n+1 n=1 5 / 33 Ferromagnetic 1D Heisenberg model Goal obtain exact eigenvalues and eigenstates with their physical properties N X H = −J Sn · Sn+1 n=1 N X 1 + − − + = −J S S + S S + Sz Sz 2 n n+1 n n+1 n n+1 n=1 Basic remarks: eigenstates reference basis: {jσ1 : : : σN i} Bethe ansatz is basis tansformation z PN z z rotational symmetry z-axis ST ≡ n=1 Sn conserved: [H; ST ] = 0 z ) block diagonalization by sorting basis according to hST i = N=2 − r where r = number of down spins 5 / 33 block r = 1: one-particle excitations N X 1 ikn − j i = jki ≡ a(n)jni where a(n) ≡ p e and jni ≡ Sn jFi n=1 N with energy E = J(1 − cos k) + E0 magnons jki N one-particle excitations correspond to elementary particles “magnons” with one particle states jki Note: not the lowest excitations! Intuitive states Lowest energy states intuitively obtained block r = 0: groundstate jFi ≡ j " ::: "i with energy E0 = −JN=4 6 / 33 Intuitive states Lowest energy states intuitively obtained block r = 0: groundstate jFi ≡ j " ::: "i with energy E0 = −JN=4 block r = 1: one-particle excitations N X 1 ikn − j i = jki ≡ a(n)jni where a(n) ≡ p e and jni ≡ Sn jFi n=1 N with energy E = J(1 − cos k) + E0 magnons jki N one-particle excitations correspond to elementary particles “magnons” with one particle states jki Note: not the lowest excitations! 6 / 33 ! N block r = 2: dim= = N(N − 1)=2 2 X − − j i = a(n1; n2)jn1; n2i where jn1; n2i ≡ Sn1 Sn2 jFi 1≤n1<n2≤N Hj i = Ej i , 2[E − E0]a(n1; n2) = J[4a(n1; n2)−a(n1 −1; n2) − a(n1 +1; n2) − a(n1; n2 −1) − a(n1; n2 +1)] for n2 > n1 +1 2[E − E0]a(n1; n2) = J[2a(n1; n2) − a(n1 −1; n2) − a(n1; n2 +1)] for n2 = n1 +1 Systematic proceeding to obtain eigenstates block r = 1: dim= N N X j i = a(n)jni n=1 Hj i = Ej i , 2[E − E0]a(n) = J[2a(n) − a(n − 1) − a(n + 1)] 7 / 33 2[E − E0]a(n1; n2) = J[4a(n1; n2)−a(n1 −1; n2) − a(n1 +1; n2) − a(n1; n2 −1) − a(n1; n2 +1)] for n2 > n1 +1 2[E − E0]a(n1; n2) = J[2a(n1; n2) − a(n1 −1; n2) − a(n1; n2 +1)] for n2 = n1 +1 Systematic proceeding to obtain eigenstates block r = 1: dim= N N X j i = a(n)jni n=1 Hj i = Ej i , 2[E − E0]a(n) = J[2a(n) − a(n − 1) − a(n + 1)] ! N block r = 2: dim= = N(N − 1)=2 2 X − − j i = a(n1; n2)jn1; n2i where jn1; n2i ≡ Sn1 Sn2 jFi 1≤n1<n2≤N Hj i = Ej i , 7 / 33 Systematic proceeding to obtain eigenstates block r = 1: dim= N N X j i = a(n)jni n=1 Hj i = Ej i , 2[E − E0]a(n) = J[2a(n) − a(n − 1) − a(n + 1)] ! N block r = 2: dim= = N(N − 1)=2 2 X − − j i = a(n1; n2)jn1; n2i where jn1; n2i ≡ Sn1 Sn2 jFi 1≤n1<n2≤N Hj i = Ej i , 2[E − E0]a(n1; n2) = J[4a(n1; n2)−a(n1 −1; n2) − a(n1 +1; n2) − a(n1; n2 −1) − a(n1; n2 +1)] for n2 > n1 +1 2[E − E0]a(n1; n2) = J[2a(n1; n2) − a(n1 −1; n2) − a(n1; n2 +1)] for n2 = n1 +1 7 / 33 Note: only for A = A0 interpretation as direct product of two one-particle states, i.e. of two non-interacting magnons To summarize rewrite: i(k n +k n + 1 θ) i(k n +k n − 1 θ) θ k1 k2 a(n ; n ) = e 1 1 2 2 2 + e 1 2 2 1 2 where 2 cot = cot − cot 1 2 2 2 2 Translational invariance: Nk1 = 2πλ1 + θ; Nk2 = 2πλ2 − θ where λi 2 f0; 1;:::; N − 1g with λi the integer (Bethe) quantum numbers Two magnon excitations – eigenstates Solution by parametrization i(k1n1+k2n2) 0 i(k1n2+k2n1) a(n1; n2) = Ae + A e where A ei(k1+k2) + 1 − 2eik1 ≡ eiθ = − A0 ei(k1+k2) + 1 − 2eik2 with energy E = J(1 − cos k1) + J(1 − cos k2) + E0 8 / 33 To summarize rewrite: i(k n +k n + 1 θ) i(k n +k n − 1 θ) θ k1 k2 a(n ; n ) = e 1 1 2 2 2 + e 1 2 2 1 2 where 2 cot = cot − cot 1 2 2 2 2 Translational invariance: Nk1 = 2πλ1 + θ; Nk2 = 2πλ2 − θ where λi 2 f0; 1;:::; N − 1g with λi the integer (Bethe) quantum numbers Two magnon excitations – eigenstates Solution by parametrization i(k1n1+k2n2) 0 i(k1n2+k2n1) a(n1; n2) = Ae + A e where A ei(k1+k2) + 1 − 2eik1 ≡ eiθ = − A0 ei(k1+k2) + 1 − 2eik2 with energy E = J(1 − cos k1) + J(1 − cos k2) + E0 Note: only for A = A0 interpretation as direct product of two one-particle states, i.e. of two non-interacting magnons 8 / 33 Translational invariance: Nk1 = 2πλ1 + θ; Nk2 = 2πλ2 − θ where λi 2 f0; 1;:::; N − 1g with λi the integer (Bethe) quantum numbers Two magnon excitations – eigenstates Solution by parametrization i(k1n1+k2n2) 0 i(k1n2+k2n1) a(n1; n2) = Ae + A e where A ei(k1+k2) + 1 − 2eik1 ≡ eiθ = − A0 ei(k1+k2) + 1 − 2eik2 with energy E = J(1 − cos k1) + J(1 − cos k2) + E0 Note: only for A = A0 interpretation as direct product of two one-particle states, i.e. of two non-interacting magnons To summarize rewrite: i(k n +k n + 1 θ) i(k n +k n − 1 θ) θ k1 k2 a(n ; n ) = e 1 1 2 2 2 + e 1 2 2 1 2 where 2 cot = cot − cot 1 2 2 2 2 8 / 33 Two magnon excitations – eigenstates Solution by parametrization i(k1n1+k2n2) 0 i(k1n2+k2n1) a(n1; n2) = Ae + A e where A ei(k1+k2) + 1 − 2eik1 ≡ eiθ = − A0 ei(k1+k2) + 1 − 2eik2 with energy E = J(1 − cos k1) + J(1 − cos k2) + E0 Note: only for A = A0 interpretation as direct product of two one-particle states, i.e. of two non-interacting magnons To summarize rewrite: i(k n +k n + 1 θ) i(k n +k n − 1 θ) θ k1 k2 a(n ; n ) = e 1 1 2 2 2 + e 1 2 2 1 2 where 2 cot = cot − cot 1 2 2 2 2 Translational invariance: Nk1 = 2πλ1 + θ; Nk2 = 2πλ2 − θ where λi 2 f0; 1;:::; N − 1g with λi the integer (Bethe) quantum numbers 8 / 33 Figure for N = 32 Karbach and Müller, Computers in Physics (1997) + = 3N 2 1 2 I + = N 1 2 I N(N − 1)=2 solutions: class 1 (red): λ1 = 0 + = N 2 1 2 I ) k1 = 0; k2 = 2πλ2=N; θ = 0 1 class 2 (white): λ − λ ≥ 2 2 1 ) real solutions k ; k 1 2 class 3 (blue): λ2 − λ1 < 2 ) complex solutions 0 0 N 1 k k 2 k1 ≡ 2 + iv; k2 ≡ 2 − iv Two magnon excitations – eigenstates Rewrite constraints θ k k 2 cot = cot 1 − cot 2 2 2 2 Nk1 = 2πλ1 + θ Nk2 = 2πλ2 − θ 9 / 33 + = 3N 2 1 2 I + = N 1 2 I + = N 2 1 2 I 1 0 0 N 1 2 Two magnon excitations – eigenstates Rewrite constraints Figure for N = 32 Karbach and Müller, Computers in Physics (1997) θ k k 2 cot = cot 1 − cot 2 2 2 2 Nk1 = 2πλ1 + θ Nk2 = 2πλ2 − θ N(N − 1)=2 solutions: class 1 (red): λ1 = 0 ) k1 = 0; k2 = 2πλ2=N; θ = 0 class 2 (white): λ2 − λ1 ≥ 2 ) real solutions k1; k2 class 3 (blue): λ2 − λ1 < 2 ) complex solutions k k k1 ≡ 2 + iv; k2 ≡ 2 − iv 9 / 33 Two magnon excitations – dispersion Nk1 = 2πλ1 + θ Nk2 = 2πλ2 − θ ) k = k1 + k2 = 2π(λ1 + λ2)=N Figure for N = 32 Karbach and Müller, Computers in Physics (1997) 4 C1 C2 λ λ C3 , 1= 2 3 λ λ C3 , 1= 2-1 )/J 0 2 (E - E 1 0 0 1 2 3 k 10 / 33 Figure for N = 32 Karbach and Müller, Computers in Physics (1997) 4 C1 C2 free magnons 3 )/J 0 2 (E - E 1 0 0 1 2 3 k Two magnon excitations – physical properties classification class 1 + 2: almost free scattering states, i.e.