The Bethe/Gauge Correspondence

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4 Chapter 1. OverviewPseudovacuum and magnons. The total spin operator in the α-direction, S α := ∑ Ll=1 Sα l ,acts on the global spin space H. Using the su(2) algebra (1.1) it’s easy to check that the totalspin operator commutes with the Hamiltonian:[H , S α ] = 0 . (1.4)This means that H and one of the S α can be simultaneously diagonalized; we choose S z andquantize along the z-axis as usual. In particular, (1.4) tells us that the spectrum of H will bedegenerate: all states in a su(2)-multiplet will have the same energy. Therefore, the Hamiltoniancan be written in block-diagonal form. Let’s take a closer look at these blocks.The smallest block is easy: it’s spanned by a single vector|Ω〉 = | ↑ · · · ↑ 〉 :=1 LL⊗|↑ 〉 ll=1which is the highest weight vector for S z :S z |Ω〉 = 1 2 L |Ω〉 and H |Ω〉 = 1 4 J L |Ω〉 .The second equation shows that |Ω〉 has energy E 0 = J L/4. In the ferromagnetic case, J < 0and |Ω〉 is a ground state of the system; on the other hand, in the antiferromagnetic case J > 0so |Ω〉 is a state with highest energy. For this reason, |Ω〉 is generically called a pseudovacuum.In either case, we can use |Ω〉 to construct a convenient basis for H.The next block corresponds to the L vectors with one spin down:|l〉 := | ↑1· · · ↑↓l↑ · · · ↑ 〉 .LTo diagonalize this L × L-block we can use our boundary conditions, which imply that H isinvariant under (discrete) translations. Thus, the solution |Ψ 1 〉 can be expanded in terms ofplane waves as|Ψ 1 〉 = √ 1 ∑L e i p l |l〉 (1.5)Ll=1with wave number p = 2πk/L, for 0 ≤ k ≤ L − 1 (recall that we use units in which the latticespacing is equal to one). The eigenvalue E 1 of these solutions is given byE 1 − E 0 = −J(1 − cos p) = −2 J sin 2 p 2 . (1.6)They describe excitations around the pseudovacuum called magnons:wavelength 2π/p travelling along the chain.spin waves withN-particle sector. To see what happens in general, we define the usual su(2) ladder operatorsS ± l:= S x l ± iS y lsatisfying[S z k, S ± l ] = ± S± k δ kland [S + k , S− l ] = 2 Sz k δ kl.The Hamiltonian can then be written asH = JL∑l=112 (S+ l S− l+1 + S− l S+ l+1 ) + Sz l S z l+1 , (1.7)which is convenient for computing the energy of states via the usual rules:S + l |↑ 〉 l = 0 , S − l |↑ 〉 l = |↓ 〉 l , S z l |↑ 〉 l = + 1 2 |↑ 〉 l ,S + l |↓ 〉 l = |↑ 〉 l , S − l |↓ 〉 l = 0 , S z l |↓ 〉 l = − 1 2 |↑ 〉 l .
1.1. Bethe: quantum integrable models 5By repeatedly acting on |Ω〉 with the lowering operators S − lwe obtain a basis for the globalspin space H, given by the configurations with spin down at lattice sites 1 ≤ l 1 < · · · < l N ≤ L:|l 1 , · · ·, l N 〉 := S − l 1· · · S − l N|Ω〉 = | ↑ · · · ↑ ↓ ↑ · · · ↑ ↓ ↑ · · · ↑1 l 1 l N L〉 , (1.8)where N runs through {0, · · ·, L}. A one-line computation shows that these spin configurationsare eigenvectors of the total spin operator:S z |l 1 , · · ·, l N 〉 = ( 12 L − N) |l 1 , · · ·, l N 〉 . (1.9)The linear span of the |l 1 , · · ·, l N 〉, for fixed 0 ≤ N ≤ L, is called the N-particle sector of thesystem. The decompositionH =L⊕ (H (N) in terms of dimensions: 2 L =N=0L∑ ( LN) )corresponds to the block-diagonal form of the Hamiltonian matrix. As we have seen in the caseN = 1, the basis elements (1.8) themselves are not eigenvectors of H. We have to look for amore general solution of (1.3) of the form|Ψ N 〉 = ∑l 1
Page 1 and 2: Institute for Theoretical PhysicsMa
Page 4 and 5: 4 Bethe/gauge 614.1 The corresponde
Page 6 and 7: viPrefaceA note on notationMany asp
Page 9 and 10: Chapter 1OverviewIn quantum field t
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54 Chapter 3. GaugeLooking at the t
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56 Chapter 3. Gaugethe effective tw
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58 Chapter 3. GaugeMore matter repr
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Chapter 4Bethe/gaugeNow that we hav
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4.1. The correspondence 63the spin
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4.2. Further topics 654.1.3 Diction
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4.2. Further topics 67N = 2 symin f
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70 Chapter 4. Bethe/gauge• descri
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[17] N. Dorey, T. Hollowood, and D.
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[56] U. Lindström, “Supersymmetr
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The Bethe/Gauge Correspondence

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