Student Seminar: Classical and Quantum Integrable Systems

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i.e. the L! (L−1)!1! = L allowed values of the pseudo-momenta are p = 2πk with k = 0, · · · , L − 1 . L Further, we have the eigenvalue equation H|ψ〉 = − JA L∑ ] e [S ipm n + Sn+1 − + Sn − S n+1 + + 2S 3 2 nSn+1 3 |m〉 = E(p)|ψ〉 . m,n=1 To work out the l.h.s. we have to use the formulae as well as S + n S − n+1|m〉 = δ nm |m + 1〉 , S − n S + n+1|m〉 = δ n+1,m |m − 1〉 2SnS 3 n+1|m〉 3 = 1 |m〉 , for m ≠ n, n + 1 , 2 2SnS 3 n+1|m〉 3 = − 1 |m〉 , for m = n, or m = n + 1 . 2 Taking this into account we obtain H|ψ〉 = − JA [ ∑ L ( ) e ipn |n + 1〉 + e ip(n+1) |n〉 + 1 2 2 − 1 2 n=1 L∑ e ipn |n〉 − 1 2 n=1 L∑ n=1 ] e ip(n+1) |n + 1〉 . Using periodicity conditions we finally get H|ψ〉 = − JA L∑ ( e ip(n−1) + e ip(n+1) + L − 4 2 2 n=1 From here we read off the eigenvalue L∑ ( ∑ L m=1 ) e ipn |n〉 = − J ( 2 E − E 0 = J(1 − cos p) = 2J sin 2 p 2 , n=1 n≠m,m−1 ) e ipm |m〉 e −ip + e ip + L − 4 2 ) |ψ〉 . where E 0 = − JL . Excitation of the spin chain around the pseudo-vacuum |F 〉 4 carrying the pseudo-momentum p is called a magnon 12 . Thus, magnon can be viewed 12 The concept of a magnon was introduced in 1930 by Felix Bloch in order to explain the reduction of the spontaneous magnetization in a ferromagnet. At absolute zero temperature, a ferromagnet reaches the state of lowest energy, in which all of the atomic spins (and hence magnetic moments) point in the same direction. As the temperature increases, more and more spins deviate randomly from the common direction, thus increasing the internal energy and reducing the net magnetization. If one views the perfectly magnetized state at zero temperature as the vacuum state of the ferromagnet, the low-temperature state with a few spins out of alignment can be viewed as a gas of quasiparticles, in this case magnons. Each magnon reduces the total spin along the direction of magnetization by one unit of and the magnetization itself by , where g is the gyromagnetic ratio. The quantitative theory of quantized spin waves, or magnons, was developed further by Ted Holstein and Henry Primakoff (1940) and Freeman Dyson (1956). By using the formalism of second quantization they showed that the magnons behave as weakly interacting quasiparticles obeying the Bose-Einstein statistics (the bosons). – 61 –
as the pseudo-particle with the momentum p = 2πk , k = 0, . . . , L − 1 and the energy L E = 2J sin 2 p 2 . The last expression is the dispersion relation for one-magnon states. Let us comment on the sign of the coupling constant. If J < 0 then E k < 0 and |F 〉 is not the ground state, i.e. a state with the lowest energy. In other words, in this case, |F 〉 is not a vacuum, but rather a pseudo-vacuum, or “false” vacuum. The true ground state in non-trivial and needs some work to be identified. The case J < 0 is called the anti-ferromagnetic one. Oppositely, if J > 0 then |F 〉 is a state with the lowest energy and, therefore, is the true vacuum. Later on we will see that the anti-ferromagnetic ground state corresponds M = 1 L and, therefore, it 2 is spinless. The ferromagnetic ground state corresponds to M = 0 and, therefore, carries maximal spin S 3 = 1 2 L.13 where Let us now turn to the more complicated case M = 2. Here we have ∑ |ψ〉 = a(n 1 , n 2 )|n 1 , n 2 〉 , 1≤n 1
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Preprint typeset in JHEP style - HY
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7. Homework exercises 95 7.1 Semina
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which can be written as follows {q
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To get more insight on the Liouvill
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4. The equations of motion with Ham
Page 11 and 12: Let us show that the variables (I i
Page 13 and 14: We have H = p2 2 p2 + V (x) = 2 + V
Page 15 and 16: We can now see how the solution can
Page 17 and 18: Thus, and, therefore, the half-peri
Page 19 and 20: This is the so-called focal equatio
Page 21 and 22: to show that ˙⃗ R = 0. The last
Page 23 and 24: is the bounded kinetic energy). Acc
Page 25 and 26: One can study the structure of the
Page 27 and 28: Substituting these formula into the
Page 29 and 30: Upper−half plane 01 01 0 0 1 1 01
Page 31 and 32: Thus, the combination ˙θ 2 − 2m
Page 33 and 34: Then using the eoms ˙q = p and ṗ
Page 35 and 36: where we have assumed that the eige
Page 37 and 38: are called the Euler-Arnold equatio
Page 39 and 40: Thus, g(λ)Q(λ)g(λ) −1 = ( = g
Page 41 and 42: We see that the there is one more v
Page 43 and 44: 4.1 General remarks Remarkably, the
Page 45 and 46: We thus obtain u 2 x = k − 2eu +
Page 47 and 48: Here ɛ 0 = ±1. This solution can
Page 49 and 50: where σ i are the Pauli matrices 9
Page 51 and 52: Let g ≡ g(x, t, λ) be a regular
Page 53 and 54: Indeed, ∂ t L x − ∂ x L t + [
Page 55 and 56: ecomes a differential equation for
Page 57 and 58: • Coordinate Bethe ansatz. This t
Page 59 and 60: The first interesting observation i
Page 61: This state is an eigenstate of the
Page 65 and 66: This allows one to determine the ra
Page 67 and 68: The first problem is to find all po
Page 69 and 70: 5.2 Algebraic Bethe Ansatz Here we
Page 71 and 72: Semi-classical limit and quantizati
Page 73 and 74: Thus, the transfer matrix is also p
Page 75 and 76: To write down the fundamental commu
Page 77 and 78: provided W A n + W D n = 0 for all
Page 79 and 80: and if k < j the contribution will
Page 81 and 82: Now we have to specify which of M p
Page 83 and 84: • Pseudo-orthogonal groups SO(p,
Page 85 and 86: 3. The Jacobi identity [[ξ, η],
Page 87 and 88: • Special linear group SL(n, R) o
Page 89 and 90: where A is an element of the corres
Page 91 and 92: Here the Jacobi identity has been u
Page 93 and 94: Note that both ad tz and ad y are d
Page 95 and 96: if α + β is a root, [e α , e −
Page 97 and 98: • case 3: • case 4: g2 U(q) =
Page 99 and 100: 7.4 Seminar 4 Exercise 1 (K. Bohlin
Page 101 and 102: 7.5 Seminar 5 Exercise 1 (Open Toda
Page 103 and 104: 7.6 Seminar 6 Exercise 1 Prove that
Page 105 and 106: is a representation of the group SU
Page 107 and 108: Exercise 4 Consider the non-linear
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Student Seminar: Classical and Quantum Integrable Systems

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