Non-linear integral equation approach to sl ( 2 | 1 ) integrable - PowerPoint PPT Presentation

Non-linear integral equation approach to sl ( 2 | 1 ) integrable network models Andreas Kl¨ umper University of Wuppertal Non-linear integral equation approach to sl ( 2 | 1 ) integrable network models – p.1/23

Contents Outline • Quantum Hall systems, electrons in random potentials; black hole CFTs • R -matrices for fundamental representations of sl ( 2 | 1 ) • transfer matrices and Hamiltonians • Bethe ansatz, short review of work by Gade and Essler, Frahm, Saleur • derivation of non-linear integral equations tJ -model thermodynamics network model Work in collaboration with M. Brockmann Non-linear integral equation approach to sl ( 2 | 1 ) integrable network models – p.2/23

Integrable network models: R -matrices, Yang-Baxter equation Consider R -matrix acting on tensor products of “standard” fundamental representation of sl ( 2 | 1 ) R ( u , v ) = P − 1 2 ( u − v ) I P : graded permutation operator, u and v are complex variables, and indices α , β , µ , ν take three values. R -matrix satisfies Yang-Baxter equation u u = v v w w Generalization to mixed representations (standard fundamental and its conjugate visualized by left and right or up and down pointing arrows) possible! In fact, the three new R -matrices are essentially obtained from rotations of above R -matrix by 90 , 180 , and 270 degrees. Yang-Baxter equation still holds where only arrow directions differ from above pictorial visualization (Gade 1998; Links, Foerster 1999; Abad, Rios 1999; Derkachov, Karakhanyan, Kirschner 2000). Hamiltonian – p.3/23

Transfer matrices, Hamiltonians 1) Product of R -matrices with same representations v 0 0 0 0 0 0 L defines transfer matrix whose logarithmic derivative yields Hamiltonian of supersymmetric tJ -model ( 2 t = J ) P ( c † j , σ c j + 1 , σ + c † ( � S j � H = − t ∑ j + 1 , σ c j , σ ) P + J ∑ S j + 1 − n j n j + 1 / 4 ) , j , σ j 2) Product of R -matrices with alternating representations yields “quantum transfer matrix” whose largest eigenvalue yields free energy of supersymmetric tJ -model Hamiltonian – p.4/23

Transfer matrices, Hamiltonians 3) Transfer matrix with two rows and alternation of representations from column to column (and row to row) v −v 0 +v v 0 +v −v +v −v +v −v 0 0 0 0 0 0 2L defines transfer matrix whose logarithmic derivative yields a local Hamiltonian. Alternatively: lattice constructed from repeated application of double row yields realization of an integrable Chalker-Coddington network with or without relevance for spin-quantum Hall effect; black hole CFTs, emerging non-compact degrees of freedom, continuous spectrum (Saleur, Jacobsen, Ikhlef; Frahm, Seel). Derivation and proof of integrability by R. Gade (1998); extensive investigations of spectrum by Essler, Frahm, Saleur (2005) Our goal: Analytical calculation of largest eigenvalues of T 1 ( v + v 0 ) T 2 ( v − v 0 ) where T 1 and T 2 are transfer matrices with “standard” and conjugated fundamental representations of sl ( 2 | 1 ) in auxiliary space. Hamiltonian – p.5/23

Bethe Ansatz Eigenvalues of transfer matrices T 1 ( v ) and T 2 ( v ) (...Links, Foerster 1999; Göhmann, Seel 2004) Λ 1 ( v ) = λ ( − ) ( v )+ λ ( 0 ) 1 ( v )+ λ (+) Λ 2 ( v ) = λ ( − ) ( v )+ λ ( 0 ) 2 ( v )+ λ (+) ( v ) , ( v ) , 1 1 2 2 where ( v ) = e − i ϕ Φ + ( v + i / 2 ) Φ − ( v + 3i / 2 ) q u ( v − 3i 2 ) λ ( − ) 1 q u ( v + i 2 ) 1 ( v ) = 1 · Φ + ( v + i / 2 ) Φ − ( v − i / 2 ) q u ( v − 3i q γ ( v + 3i 2 ) 2 ) λ ( 0 ) 2 ) , ( ϕ → π ) q u ( v + i q γ ( v − i 2 ) ( v ) = e + i ϕ Φ + ( v − 3i / 2 ) Φ − ( v − i / 2 ) q γ ( v + 3i 2 ) λ (+) 1 q γ ( v − i 2 ) and formulas for λ ( ± , 0 ) are obtained from those above by simultaneous exchange Φ + ↔ Φ − and q u ↔ q γ 2 “Vacuum functions” Φ ± and q -functions in terms of Bethe ansatz rapidities u j and γ α N M Φ ± ( v ) : = ( v ± v 0 ) L , ∏ ∏ q u ( v ) : = ( v − u k ) , q γ ( v ) : = ( v − γ β ) , k = 1 β = 1 Bethe Ansatz – p.6/23

Bethe Ansatz equations Eigenvalue functions have to be analytic → cancellation of poles by zeros yielding Bethe ansatz equations Φ − ( u j − i ) = − e i ϕ q γ ( u j + i ) Φ − ( u j + i ) q γ ( u j − i ) , j = 1 ,..., N Φ + ( γ α + i ) Φ + ( γ α − i ) = − e i ϕ q u ( γ α + i ) q u ( γ α − i ) , α = 1 ,..., M These equations are the same for the QTM of the tJ model and for the supersymmetric network model. Characterization of largest eigenvalue differs: tJ : maximum value of Λ 1 network model: maximum value(s) of Λ 1 · Λ 2 “strange strings” (Essler, Frahm, Saleur 2005) Bethe Ansatz – p.7/23

Bethe Ansatz: root distributions Some results from Essler, Frahm, Saleur (2005) (numerical work for L up to approx. 5000): • groundstate for ϕ = π given by “degenerate solution” u j = − v 0 , γ α = + v 0 for all j , α = 1 ,..., L . groundstate energy is E 0 = − 4 L and hence central charge c = 0 . • excited states are given by seas of “strange strings”, i.e. one u and one γ rapidity with condition Im u = + 1 2 + ε , Im γ = − 1 Re u = Re γ 2 − ε ; and or Im u = − 1 2 + ε , Im γ = + 1 Re u = Re γ 2 − ε and • infinite number of excited states with same scaling dimension, differing by logarithmic corrections 2.0 ∆ N =0 (TB) ∆ N =1 ∆ N =2 ∆ N =3 1.0 ∆ N =5 ∆ N =7 2 indec. (TB) L ( E 8 - E 0 )/2 π 0.5 1/4 0.1 0 0.2 0.4 0.6 1/log( L ) • For special case v 0 = 0 : simplification for states with identical sets of u rapidities and γ rapidities, u j = γ j ( j = 1 ,..., N ) two sets of BA equations coincide as Φ + = Φ − and q u = q γ remaining set of BA equations equivalent to Takhtajan-Babujian solution of spin-1 su ( 2 ) chain Bethe Ansatz – p.8/23

Functional equations: Definition of auxiliary functions tJ model motivated ansatz of suitable auxiliary functions b : = λ ( 0 ) 1 + λ (+) B : = 1 + b = λ ( − ) + λ ( 0 ) 1 + λ (+) 1 1 1 , , λ ( − ) λ ( − ) 1 1 b : = λ ( − ) + λ ( 0 ) b = λ ( − ) + λ ( 0 ) 1 + λ (+) ¯ 1 1 B : = 1 + ¯ 1 1 ¯ , , λ (+) λ (+) 1 1 � � � �� � λ ( 0 ) λ ( − ) + λ ( 0 ) 1 + λ (+) λ ( − ) + λ ( 0 ) λ ( 0 ) 1 + λ (+) 1 1 1 1 1 1 c : = , C : = 1 + c = , λ ( − ) λ (+) λ ( − ) λ (+) 1 1 1 1 Factorization into “elementary factors” ... log ( 1 + b ) = log ( 1 + e − L ε ) etc. ... yields integral equations for logs: log b = : − L ε , Bethe Ansatz – p.9/23

Functional equations: factorization Factorization into “elementary factors” q u , q γ , D u , D γ , Λ 1 b ( v ) = e i ϕ Φ − ( v − i / 2 ) q γ ( v + 3i / 2 ) D γ ( v − i / 2 ) q u ( v + i / 2 ) Λ 1 ( v ) B ( v ) = e i ϕ Φ + ( v + i / 2 ) Φ − ( v + 3i / 2 ) q u ( v − 3i / 2 ) , Φ + ( v + i / 2 ) Φ − ( v + 3i / 2 ) q u ( v − 3i / 2 ) q γ ( v − i / 2 ) Λ 1 ( v ) b ( v ) = e − i ϕ Φ + ( v + i / 2 ) q u ( v − 3i / 2 ) D u ( v + i / 2 ) B ( v ) = e − i ϕ ¯ ¯ Φ − ( v − i / 2 ) Φ + ( v − 3i / 2 ) q γ ( v + 3i / 2 ) , Φ − ( v − i / 2 ) Φ + ( v − 3i / 2 ) q γ ( v + 3i / 2 ) D u ( v + i / 2 ) D γ ( v − i / 2 ) Λ 1 ( v ) c ( v ) = Φ + ( v − 3i / 2 ) Φ − ( v + 3i / 2 ) , C ( v ) = Φ + ( v − 3i / 2 ) Φ − ( v + 3i / 2 ) , where 1 � � Φ − ( v − i ) q γ ( v + i )+ e − i ϕ Φ − ( v + i ) q γ ( v − i ) D u ( v ) : = q u ( v ) 1 � � Φ + ( v + i ) q u ( v − i )+ e i ϕ Φ + ( v − i ) q u ( v + i ) D γ ( v ) : = q γ ( v ) are polynomials due to the Bethe ansatz equations. Usual treatment: taking logarithm and then Fourier transform. However, from the three expressions for B , ¯ B , and C the functions q u , q γ , D u , D γ and Λ 1 can not be resolved! Apparent reason: too many unknowns (5) in comparison to number of equations (3) Bethe Ansatz – p.10/23

Solution of functional equations: tJ -Model Interesting case: thermodynamics of tJ -model (Jüttner, AK, J. Suzuki 1997) • q u and D u are free of zeros above the real axis, q γ and D γ are free of zeros below the real axis, • “effective number” of unknowns: 3 Concrete calculations are done for Fourier transforms of logarithms of all involved functions. Final equations are integral equations of convolution type with kernels κ ( x ) = 1 1 x 2 + 1 / 4 , κ ± ( x ) = κ ( x ± i / 2 ) , 2 π β log b ( x ) = − x 2 + 1 / 4 + β ( µ + h / 2 ) − κ + ∗ log B − κ ∗ log C , β log b ( x ) = − x 2 + 1 / 4 + β ( µ − h / 2 ) − κ − ∗ log B − κ ∗ log C , 2 β log c ( x ) = − x 2 + 1 + 2 β µ − κ ∗ log B − κ ∗ log B − ( κ + + κ − ) ∗ log C 0.35 0.35 1 n=0.604 0.3 0.3 n=0.697 0.8 n=0.079 n=0.776 0.8 Specific heat Compressibility n=0.162 n=0.839 0.25 0.25 n=0.306 n=0.921 n=0.502 0.6 n=0.604 0.6 0.2 0.2 S(T) k(T) c(T) c(T) n=0.226 n=0.502 0.4 n=0.226 0.15 0.15 n=0.604 0.4 n=0.502 n=0.776 n=0.604 n=0.921 0.1 0.1 n=0.776 0.2 n=0.921 0.2 0.05 0.05 0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5 3 T T T T Bethe Ansatz – p.11/23