Student Seminar: Classical and Quantum Integrable Systems

More documents

Recommendations

Info

The last equations are called “Bethe equations”. quantization conditions for momenta p k . They are nothing else but the Let us note the following useful representation for the S-matrix. We have ( e ip 2 − 1 ) + 1 − e ip 1 = − eip2 e i 2 p1 ( e i 2 p1 − e − i 2 p1 ) + e i S(p 2 , p 1 ) = − eip2 ( e ip 1 − 1 ) + 1 − e ip2 e ip 1 = − e i 2 p2 sin p1 2 − e− i 2 p1 sin p2 2 e i 2 p 1 sin p 2 2 − e − i 2 p 2 sin p 1 2 = e ip i 1 e 2 p 2 ( cos p 2 = − cos p 2 2 sin p 1 2 − cos p 1 2 sin p 2 2 + 2i sin p 1 2 sin p 2 2 cos p 1 2 sin p 2 2 − cos p 2 2 sin p 1 2 + 2i sin p 1 2 sin p 2 Thus, we obtained ( i e 2 p 2 − e − i 2 p 2 ) sin p 1 ) + e i 2 p 1 2 + i sin p2 2 2 − ( cos p1 ( 2 cos p 1 ) 2 + i sin p1 2 sin p 2 2 − ( cos p2 2 2 = 1 2 S(p 1 , p 2 ) = cot p 1 − 1 cot p 2 + i 2 2 2 1 cot p 1 − 1 cot p 2 − i . 2 2 2 2 1 ( ) 2 p2 e − i 2 p2 − e i 2 p2 ( e − i 2 p i ) 1 − e 2 p 1 ) p1 − i sin 2 sin p 2 ) 2 sin p 1 − i sin p2 2 2 cot p 2 2 − 1 2 cot p 1 2 + i 1 2 cot p 2 2 − 1 2 cot p 1 2 − i . It is therefore convenient to introduce the variable λ = 1 cot p which is called 2 2 rapidity and get S(λ 1 , λ 2 ) = λ 1 − λ 2 + i λ 1 − λ 2 − i . Hence, on the rapidity plane the S-matrix depends only on the difference of rapidities of scattering particles. 2 Taking the logarithm of the Bethe equations we obtain Lp 1 = 2πm 1 + θ(p 1 , p 2 ) , Lp 2 = 2πm 2 + θ(p 2 , p 1 ) , where the integers m i ∈ {0, 1, . . . , L − 1} are called Bethe quantum numbers. The Bethe quantum numbers are useful to distinguish eigenstates with different physical properties. Furthermore, these equations imply that the total momentum is Writing the equations in the form p 1 = 2πm 1 L } {{ } P = p 1 + p 2 = 2π L (m 1 + m 2 ) . + 1 L θ(p 1, p 2 ) , p 2 = 2πm 2 + 1 } {{ L } L θ(p 2, p 1 ) , we see that the magnon interaction is reflected in the phase shift θ and in the deviation of the momenta p 1 , p 2 from the values of the underbraced one-magnon wave numbers. What is very interesting, as we will see, that magnons either scatter off each other or form the bound states. – 65 –
The first problem is to find all possible Bethe quantum numbers (m 1 , m 2 ) for which Bethe equations have solutions. The allowed pairs (m 1 , m 2 ) are restricted to 0 ≤ m 1 ≤ m 2 ≤ L − 1 . This is because switching m 1 and m 2 simply interchanges p 1 and p 2 and produces the same solution. There are 1 L(L + 1) pairs which meet this restriction but only 2 1L(L − 1) of them yield a solution of the Bethe equations. Some of these solutions 2 have real p 1 and p 2 , the others yield the complex conjugate momenta p 2 = p ∗ 1. The simplest solutions are the pairs for which one of the Bethe numbers is zero, e.g. m 1 = 0, m = m 2 = 0, 1, . . . , L − 1. For such a pair we have Lp 1 = θ(p 1 , p 2 ) , Lp 2 = 2πm + θ(p 2 , p 1 ) , which is solved by p 1 = 0 and p 2 = 2πm. Indeed, for p L 1 = 0 the phase shift vanishes: θ(0, p 2 ) = 0. These solutions have the dispersion relation E − E 0 = 2J sin 2 p 2 , p = p 2 which is the same as the dispersion for the one-magnon states. These solutions are nothing else but su(2)-descendants of the solutions with M = 1. One can show that for M = 2 all solutions are divided into three distinct classes Descendents } {{ } , Scattering States , Bound } {{ } } {{ States } L L(L−5) L−3 +3 2 so that L(L − 5) L + + 3 + L − 3 = 1 L(L − 1) 2 2 gives a complete solution space of the two-magnon problem. Pseudo−vacuum F L one−magnon states 01 01 01 01 01 01 01 01 01 01 L(L−1) 2 two−magnon states 0 0 1 01 01 0 0 1 0 01 1 101 0 1 1 01 0 01 01 01 01 0 1 1 01 1 01 0 1 – 66 –
Page 1 and 2:
Preprint typeset in JHEP style - HY
Page 3 and 4:
7. Homework exercises 95 7.1 Semina
Page 5 and 6:
which can be written as follows {q
Page 7 and 8:
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 and 60: The first interesting observation i
Page 61 and 62: This state is an eigenstate of the
Page 63 and 64: as the pseudo-particle with the mom
Page 65: This allows one to determine the ra
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
show all

Student Seminar: Classical and Quantum Integrable Systems

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?