S Leurent, V. Kazakov. 6 July 2010

Transcription

1 NLIE for SU(N) SU(N) Principal Chiral Field via Hirota dynamics S Leurent, V. Kazakov 6 July 2010

2 1 Thermodynaamic Bethe Ansatz (TBA) and Y -system Ground state energy Excited states 2 3 Principal Chiral Field Other models AdS/CFT Y -system

3 Outline Ground state energy Excited states 1 Thermodynaamic Bethe Ansatz (TBA) and Y -system Ground state energy Excited states 2 3 Principal Chiral Field Other models AdS/CFT Y -system

4 Model definition Ground state energy Excited states The principal chiral field (PCF) is a 1+1 D field theory on the cylinder 0 x < L, t R S PCF = 1 2e0 2 dt dx tr(h 1 α h) 2. (1) Where h SU(N) SU(N) L SU(N) R symmetry Integrable theory with rational S matrix, identified by Zamolodchikov : χ CDD (θ) S 0 (θ) ˆR(θ) θ i S 0 (θ) ˆR(θ) θ i = =

5 Ground state energy Excited states Ground state energy : double Wick rotation Spatial periodicity L time-periodicity R : Path integral dominated by Ground state Z e RE 0(L) Spatial periodicity R time-periodicity L (finite temperature) free Energy : f (L) = E 0 (L)

6 Solution for large spatial period L Ground state energy Excited states solutions described by particules having rapidities θ j : p j = m j sinh( 2π N θ j), E = N j=1 E j = N j=1 m jcosh( 2π N θ j) bound states with mass m a = m sin aπ N sin π N periodicity condition e imr sinh(πθj) = S(θ j ) QR N 1 (θ j+i/2) Q L N 1 (θ j+i/2) QN 1 R (θ j i/2) QN 1 L (θ j i/2) θ j θ j+1 θ j+2 S(θ) = j S2 0 (θ θ j)χ CDD (θ θ j ) magnons 1 = QR k 1 (u(k) j (1 k N 1) Q R k 1 (u(k) j i/2) +i/2) Q R k (u(k) j Q R k (u(k) j +i)qk+1 R (u(k) j i/2) i)qk+1 R (u(k) j +i/2)

7 String hypothesis Ground state energy Excited states due to Q R k 1 (u(k) j Q R k 1 (u(k) j i/2) +i/2) Q R k (u(k) j Q R k (u(k) j +i)qk+1 R (u(k) j i/2) i)qk+1 R (u(k) j +i/2) large number of magnons roots are organized as strings. = 1, the u (n) j,a = u(n) j + i 1 2 (n + 1) ia, a = 1,...,n. Such strings scatter with a shifted product of the original matrix the right configuration (described by one density for each type of string) is identified by minimization of the free entropy. u (n) j,1.. u (n) j,n 1 u (n) j u (n) j,2 u (n) j,n. u (m) k.

8 Y-system Ground state energy Excited states a N, 0 N 1, 0 2, 0 1, 0 0, 0 s Y + a,s Y a,s = (a, s) Z {1, 2,, N 1}, θ C, 1 + Y a,s (Y a+1,s ) Y a,s (Y a 1,s ) 1, f ± = f (θ ± i/2) p a = cosh( 2θπ N Y a,s e L pa(θ)δ s,0 const a,s, θ 1 Y 0,s = Y N,s = E = 1 N 1 N a=1 p a(θ)log (1 + Y a,0 (θ)) dθ )sin( aπ N ) sin( π N )

9 Continuation to excited states Ground state energy Excited states The Y -system equation is the same as for Vacuum, only the analyticity is different Vacuum : most analytic state, no pole on the physical strip Excited states characterized by the pole structure Once the Y -system is solved, the energy is given by E = 1 N 1 N a=1 C p a(θ)log (1 + Y a,0 (θ)) dθ where the right contour of integration has to be identified. use the asymptotic limit (L ) as a guide-line. Claim : the sigma model is completely recast into Y a,s + Ya,s = 1+Y a,s+1 1+Y a,s 1 1+(Y a+1,s ) 1 1+(Y a 1,s on a given ) 1 domain in (a, s) Large θ asymptotic Contour of integration

10 Outline 1 Thermodynaamic Bethe Ansatz (TBA) and Y -system Ground state energy Excited states 2 3 Principal Chiral Field Other models AdS/CFT Y -system

11 Hirota equation If we define Y a,s = T a,s+1t a,s 1 T a+1,s T a 1,s, then Y + a,s Y a,s = 1 + Y a,s (Y a+1,s ) Y a,s (Y a 1,s ) 1 T + a,st a,s = T a+1,s T a 1,s + T a,s+1 T a,s 1 a N, 0 N 1, 0 2, 0 T a,s (θ) is a function of a {0, 1, N}, s Z, θ C 1, 0 0, 0 Y a,s is invariant w.r.t. the gauge transformation T a,s χ [a+s] 1 χ [a s] 2 χ [ a+s] 3 χ [ a s] 4 T a,s f [±k] f (θ ± ki/2) s

12 Hirota equation for integrable spin chains For XXX Heisenberg spin chain, the transfer matrix T λ (θ) is a family of commuting operators, defined as ) T {λ} (θ) = tr λ (R N λ (θ θ L) R1 λ (θ θ 1 )π λ (g) (2) Notation : T a,s (u) when λ is a rectular young tableau a π λ (g) N, 0 N 1, 0 2, 0 In this context, T a,s satisfies the hirota equation, which describes the fusion relations between representations. 1, 0 0, 0 s

13 Hirota equation for integrable spin chains The general solution of Hirota is given by the Bazhanov-Reshetikin determinant s=0 T a,s = det 1 j,k a T 1,s+k j (θ + (a + 1 k j)i/2) Π a 1 ( ϕ [ s N/2] ) (3) where Π k [f ](θ) = (k 1)/2 j= (k 1)/2 f (θ + i j) For spins chains, there is a generating series for symmetric representations : T 1,s (θ+(s 1)i/2) ( 1 X (N) (θ)e i θ ϕ(θ Ni/4) e is θ = ) 1 ( 1 X (N 1) (θ)e i θ ) 1 ( ) X (1) (θ)e i θ

14 Hirota equation for integrable spin chains s=0 For spins chains, there is a generating series for symmetric representations : T 1,s (θ+(s 1)i/2) ( 1 X (N) (θ)e i θ ϕ(θ Ni/4) e is θ = ) 1 ( 1 X (N 1) (θ)e i θ X (k) = Q k 1 [N/2 k 1] [N/2 k+2] Q k [N/2 k+1] [N/2 k] Q k 1 Q k T and Q functions are polynomials. In particular T a,0 ϕ [a N/2], where ϕ = j (θ θ j) ) 1 ( ) X (1) (θ)e i θ We will assume that for PCF in the L limit, where Y a,0 T a,1 T a, 1 1, there is a gauge where T a, 1 1 and where T a,s 0 is described by such a spin chain solution of hirota.

15 Middle node equations 1 + Y a,s = T + a,st a,s T a+1,s T a 1,s Y a,s + Y a,s = 1+Y a,s+1 1+Y a,s 1 (Y a+1,s ) 1 δa,n 1 (Y a 1,s ) 1 δ a,1 (1+Y a+1,s 1 δ a,n 1 (1+Y a 1,s ) 1 δ a,1 ( ) Ya,0 = T a,1 (T (L) ) a, 1 T Ta+1,0 T + N,0 T δa,n 1 ( ) N,0 T 0,0 + T δa,1 0,0 a 1,0 T N,1 T (L) N, 1 T 0,1 T (L) 0, 1 where F a := F + a F a (F a+1 ) 1 δ a,n 1 (F a 1 ) 1 δ a,1 Inversion of gives ( [ Y a,0 = e L pa(θ) T a,1 T (L) a, 1 T a+1,0 T a 1,0 Π N a T 0,0 + T 0,0 T 0,1 T (L) N, 1 ]Π a [ K N is inverse to Π N : f regular, (Π N [f ]) K N = f K N = tan( 1 π( 1 2 N 2itθ N ))+tan( 1 π( 1 2 N + 2iθ N )) 2N T + N,0 T N,0 T N,1 T (L) 0, 1 ]) KN

16 Asymptotic Bethe Ansatz crossing relation for S(θ) = j S2 0 (θ θ j)χ CDD (θ θ j ) : Π N S(θ) = ( Q j u θ j i N 1 2 Qj u θ j+i N 1 2 ) 2 at L, T a,0 ϕ [a N/2], and S(θ) appears ( ) T 0,0 eg in + KN, giving T 0,0 Y a,0 (θ) e L pa T a,1 T (v) a, 1 ϕ [ N/2 a+1] ϕ [ N/2 a+1] 1 T a+1,0 T a 1,0 ϕ [ N/2+a 1] ϕ [ N/2+a+1] Π a [S [ N/2] ] 2 χ [ N/2] CDD 1 + Y 1,0 (θ j + in/4) = 0 gives the asymptotic Bethe equation : 1 = e ilsinh 2π N θ j 1 Q (R) N 1 (θ j i/2) χ CDD (θ j )S(θ j ) 2 Q (R) N 1 (θ j+i/2) Q (L) N 1 (θ j i/2) Q (L) N 1 (θ j+i/2)

17 General solution of Hirota equation Any solution of hirota on the lattice a {0, 1,, N}, s Z is gauge equivalent to a wronskian determinant ( ( q [s+a+1+ N 2k]) j 2 ) 1 j N,1 k a T a,s = q [ s+a+1+ N 2 2k] j 1 j N,a<k N It involves 2N functions (q j ) j=1,...,n and (q j ) j=1,...,n, that we will now express through NLIE. On this wronskian form, the solution has two gauge freedoms left, writen as q j (θ) q j (θ) g(θ) q j (θ) g(θ) q j (θ)

18 Analyticity strips large L cosh(θ) : Y a,s e Lpa(θ)δ s,0 p a = cosh( 2θπ N )sin( aπ N ) sin( π N ) therefore, T a,0 ϕ [a N/2] should hold on the following strips T 0,0 L cosh( 2πθ N ) ϕ [ N/2] when Im(θ) < N 4 T a,0 0<a<N L cosh( 2πθ N ) ϕ [+a N/2] when Im(θ) < N T N,0 L cosh( 2πθ N ) ϕ [+N/2] when Im(θ) > N 4 that is compatible with requiring that q j (θ) is analytic for Im(θ) < 0, and q j (θ) is analytic for Im(θ) > 0

19 Introduction of the jump densities Reality condition, and a choice of gauge (q 1 = 1) leaves N 1 unknown functions, described as a known polynomial asymptotic at large θ plus finite size corrections described by densities f j 1 L+ θ q j q j f j q j (θ) = P j (θ) + F j (θ) { where 1 f j (ψ) 2iπ θ ψ dψ = Fj (θ) if Im(θ) < 0 F j (θ) if Im(θ) > 0 Hence q j [+0] q [ 0] j lim ǫ 0 q j [+ǫ] q [ ǫ] j = f j

20 Closed equations on the densities T a, 1 is small because in the determinant, there is a full column of q [+0] j AND a full column of q [ 0] j. Column substraction and expension w.r.t. these columns give ( T a, 1 θ i N 2a ) = d a,j (θ)f j (θ) 4 j on the other hand Y a,0 = T a,1t a, 1 T a+1,0 T a 1,0 was expressed by inversion of the middle node equations. Hence this system of N 1 equations j d N 2a Lpa(θ i a,j(θ)f j (θ) = e 4 ) T [a N/2] T [ N/2+1] 0,0 T [ N/2 1] 0,0 T [N/2 1] N,0 T [N/2+1] N,0 iterative resolution (T [ 3N/2 a+1] 0,0 T [ 3N/2+a 1] 0,0 a,1 ) [a 1] K N ( ) T [3N/2+N a+1] [ N+a+1] K N N,0 T [3N/2 N+a 1] N,0

21 Regularity conditions One of the steps of this iterative solution is to invert the matrix (d a,j ). When its determinant is zero, that could induce a pole ine the f j s. The requirement that the denominator cancels at the same position (i.e. that f j s don t have such poles) gives finite size Bethe equations this procedure gives simple poles to the Y -functions, and it can be explicitely checked that it reproduces χ CDD at L

22 Numerical results θ 0 : One particule at rest θ 1 : One particule, momentum=1 (e 2iπ in the Bethe Equation) θ 2 : One particule, momentum=2 (e 4iπ in the Bethe Equation) θ 2 5 E L 2π θ 1 1 θ 0,0 : Two particules at rest θ 00 0 θ 0 vacuum Energies of vacuum, of mass gap and of some excited states as functions of L, at N = 3 L

23 Checks for mass gap The µ term (Lüscher correction) is reproduced from the imaginary part of the Bethe root (green dotted line). 3 E L 2π L The first logarithmic corrections, in the conformal limit (red dashed line), are also reproduced.

24 Principal Chiral Field Other models AdS/CFT Y-system Outline 1 Thermodynaamic Bethe Ansatz (TBA) and Y -system Ground state energy Excited states 2 3 Principal Chiral Field Other models AdS/CFT Y -system

25 Principal Chiral Field Other models AdS/CFT Y-system PCF is not over The numerics still deserve to be improved better precision, study of N 4, states outside the U(1) sector the choice of the contour isn t completely understood yet It could be interesting to understand the UV limit in terms of these densities The N limit also requires some investigations...

26 Principal Chiral Field Other models AdS/CFT Y-system Other models having a known Y -system cosh( pq) O(4) s-model SU(2) Chiral Gross-Neveu O(3) s-model 1... n 2 Sine-Gordon with b =8p(1-1/n) n-1... Super-sine-Gordon -k+1-1 Current-current perturbation of SU k(2) WZW Sausage models 1 2 SU(N+1) Principal Chiral Field N N

27 Principal Chiral Field Other models AdS/CFT Y-system AdS/CFT Y -system 4,-2 4,-1 4,0 4,1 4,2 3,-2 3,-1 3,0 3,1 3,2 2,-4 2,-3 2,-2 2,-1 2,0 2,1 2,2 2,3 2,4 1,-4 1,-3 1,-2 1,-1 1,0 1,1 1,2 1,3 1,4 0,-4 0,-3 0,-2 0,-1 0,0 0,1 0,2 0,3 0,4

28 Principal Chiral Field Other models AdS/CFT Y-system AdS/CFT Y -system non relativistic dispersion relation : ǫ a (u) = a + 2ig where u g = x + 1 x the mapping u x has zhukowski cuts 2ig x [+a] x [+a] the hirota equation has a wronskian solution, which explains quite well the analyticity strips infinite number of middle nodes different reality conditions

29 Principal Chiral Field Other models AdS/CFT Y-system Conclusion 1 Thermodynaamic Bethe Ansatz (TBA) and Y -system Ground state energy Excited states 2 3 Principal Chiral Field Other models AdS/CFT Y -system

30 Principal Chiral Field Other models AdS/CFT Y-system Thanks! Thank you