Bogolyubov Limit of Bethe Ansatz. and Dark Solitons

Transcription

1 Bogolyubov Limit of Bethe Ansatz and Dark Solitons Konstantin Zarembo (LPT) arxiv:0702:???? Journées de la FRIF,

2 Large-N expansion of YM theory String theory AdS/CFT correspondence Maldacena 97 Great progress in understanding 4d QFT using methods of (1+1)d integrable systems: spin chains exact S-matrices Bethe ansatz

3 Main outstanding problems: to identify elementary degrees of freedom continuum vs. lattice description (worldsheet sigma model vs. spin chain) finite-size effects

4 Motivation I: Giant Magnons world-sheet solitons of string theory in AdS 5 xs 5 believed to be elementary building blocks of the spectrum in AdS/CFT correspondence Hofman,Maldacena 06

5 Puzzles Continuum / Lattice Classical / Quantum Janik,Łukowski 07 Elementary / Composite Minahan,Tirziu,Tseytlin 06 Minahan,OhlssonSax 08

6 Dark Solitons Charged field: background charge density: twisted boundary conditions:

7 Thermodynamic T Bethe Ansatz In an integrable system S-matrix, wave functions, are 2-body factorizable. Dispersion relation: S-matrix: Problem: true vacuum can be a many-body state

8 Bethe equations Periodicity condition for multiparticle wavefunction: Physics is that of interacting fermions

9 particles holes - pseudo-energy (the energy of an elementary excitation on top of the vacuum) Holes are solitons

10 Thermodynamical limit Yang,Yang 66 chemical potential Ground state (free) energy density:

11 Excitations dispersion law of elementary excitations

12 TBA equations are seldom solvable (except for B= or very large Hulthén 38 ) BAE reduce in some cases to singular intergal equations: Large-N limit Semiclassical approximation Conformal limit Polchinski,Mann 05 Another solvable case: Fateev,Kazakov,Wiegmann 94 Gromov,Kazakov,Vieira 06; Gromov,Kazakov 06 Sutherland 95; Dhar,Shastry 00 Bogolyubov limit: weakly interacting bosons Beisert,Minahan,Staudacher,Z. 03 Kazakov,Marshakov,Minahan,Z. 04

13 Plan Weakly repulsive Bose gas O(N) sigma-model at large N Nearly isotropic XXZ spin chain in external magnetic field

14 Weakly repulsive Bose gas: non-linear Schrödinger model Ground state: Excitations: interpolate between sound waves (with ) and free particles with Bogolyubov 47

15 Bethe-ansatz solution Lieb,Liniger 63

16 Weak-coupling limit

17 How to determine B? => Minimize free energy.

18 Matrix model analogy: (critical point!)

19 Excitation spectrum Particles and holes are very different!

20 Particles => Bogolyubov branch of the spectrum Holes are solitons (Faddeev,Kulish,Manakov 76):

21 Dark soliton Tsuzuki 71 Zakharov,Shabat 73

22 vacuum energy/momentum Ground state with twisted boundary conditions: agrees with Bethe ansatz Faddeev,Kulish,Manakov 76

23 O(N) sigma-model O(N) charges: Chemical potentials:

24 Take, all other

25 Large N Integrate out z I I = 2 N/2 Exclude σ via its equations of motion (fluctuations are suppressed at N ) At zero μ this gives vev to σ => all z I acquire equal mass

26 Effective action for φ: Non-zero μ Large-N = classical limit: Excitations are phonons with

27 Thermodynamic Bethe Ansatz S-matrix: Zamolodchikov,Zamolodchikov 79 TBA: Hasenfratz,Niedermayer 90; Polyakov,Wiegmann 83; Hasenfratz,Maggiore,Niedermayer 90

28 Large-N limit of TBA

29 Large-N solitons?

30 Giant magnons (classical) Hofman,Maldacena 06 Periodicity in momentum

31 Target space

32 Giant magnons as dark solitons charge <=> angular momentum Then Dark soliton!

33 Giant magnons: large-n non-local! Ansatz:

34 Non-locality: Equations of motion

35 Exact solution Solitons of non-local large-n equations are in some cases known Dashen,Hasslacher,Neveu 74; Makeenko 94; Feinberg 95

36 Dispersion relation: again agrees with Bethe ansatz

37 XXZ spin chain in magnetic field TBA: Near-isotropic limit:

38 Effective matrix-model potential: Critical magnetic field:

39 Solution: Ground-state energy density:

40 Conclusions Thermodynamic Bethe Ansatz equations are solvable in the Bogolyubov limit. Solution exhibits some universal features: TBA => singular integral equation ( matrix model ) spectrum has two branches: sound waves (Bogolyubov) dark solitons E soliton (p) < E sound (p) In quantum theory: dark solitons <=> holes in the Fermi sea sound waves <=> particles For solitons p <p F : Typically E soliton (p+2p F )=E(p) Giant magnons in O(N) sigma-model are dark solitons. Implications for AdS/CFT?