Student Seminar: Classical and Quantum Integrable Systems

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corresponding to equal eigenvalues form a representation of su(2) with spin s = 1 and the state ⎛ ⎞ 0 vs=0 hw = ⎜ −1 ⎟ ⎝ 1 ⎠ } 0 {{ } h.w. which corresponds to 3 J is a singlet of su(2). Indeed, the generators of the global 2 su(2) are realized as ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 0 1 1 0 0 0 0 0 1 0 0 0 S + = ⎜ 0 0 0 1 ⎟ ⎝ 0 0 0 1 ⎠ , S− = ⎜ 1 0 0 0 ⎟ ⎝ 1 0 0 0 ⎠ , S3 = ⎜ 0 0 0 0 ⎟ ⎝ 0 0 0 0 ⎠ . 0 0 0 0 0 1 1 0 0 0 0 −1 The vectors vs=1 hw and vs=0 hw are the highest-weight vectors of the s = 1 and s = 0 representations respectively, because they are annihilated by S + and are eigenstates of S 3 . In fact, vs=0 hw is also annihilated by S − which shows that this state has zero spin. Thus, we completely understood the structure of the Hilbert space for L = 2. In general, the Hamiltonian can be realized as 2 L × 2 L symmetric matrix which means that it has a complete orthogonal system of eigenvectors. The Hilbert space split into sum of irreducible representations of su(2). Thus, for L being finite the problem of finding the eigenvalues of H reduces to the problem of diagonalizing a symmetric 2 L ×2 L matrix. This can be easily achieved by computer provided L is sufficiently small. However, for the physically interesting regime L → ∞ corresponding to the thermodynamic limit new analytic methods are required. In what follows it is useful to introduce the following operator: P = 1 2 ( I ⊗ I + ∑ α ) ( 1 σ α ⊗ σ α = 2 4 I ⊗ I + ∑ α S α ⊗ S α ) which acts on C 2 ⊗ C 2 as the permutation: P (a ⊗ b) = b ⊗ a. Indeed, we have It is appropriate to call S 3 the operator of the total spin. On a state |ψ〉 with M spins down we have S 3 |ψ〉 = ( 1 2 (L − M) − 1 ) ( 1 ) 2 M |ψ〉 = 2 L − M |ψ〉 . Since [H, S 3 ] = 0 the Hamiltonian can be diagonalized within each subspace of the full Hilbert space with a given total spin (which is uniquely characterized by the number of spins down). Let M < L be a number of overturned spins. If M = 0 we have a unique state |F 〉 = | ↑ · · · ↑〉. – 59 –
This state is an eigenstate of the Hamiltonian with the eigenvalue E 0 = − JL 4 : H|F 〉 = −J L∑ SnS 3 n+1| 3 ↑ · · · ↑〉 = − JL 4 | ↑ · · · ↑〉 . n=1 L! Let M be arbitrary. Since the M-th space has the dimension one should (L−M)!M! find the same number of eigenvectors of H in this subspace. So let us write the eigenvectors of H in the form ∑ |ψ〉 = a(n 1 , . . . , n M )|n 1 , . . . , n M 〉 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
Page 9 and 10: 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: The first interesting observation i
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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
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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
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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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