Two-Cut Solutions of the Heisenberg Ferromagnet and Stability Till Bargheer Feb 13, 2008 Work with Niklas Beisert (to appear) Feb. ’08: Two-Cut Solutions of the Heisenberg Ferromagnet and Stability 1 / 27
Contents Motivation The Landau-Lifshitz Model The Heisenberg Spin Chain and the Thermodynamic Limit A Single Cut Cuts Interact: The Two-Cut Solution Feb. ’08: Two-Cut Solutions of the Heisenberg Ferromagnet and Stability 2 / 27
Heisenberg Spin Chain, Landau-Lifshitz model, Integrability LL model: Limit of ultrarelativistic rotating string on R × S 3 . [ hep-th/0311203 ] Kruczenski ◮ Subspace of AdS 5 × S 5 → AdS/CFT correspondence. ◮ Quantization around certain solutions → Unstable excitation modes. Heisenberg Spin Chain: Quantum-mechanical model for 1D magnet ◮ Equivalent to SU(2) sector of planar N = 4 SYM (one-loop). [ Minahan Zarembo ] ◮ Solved (in principle) by the Bethe equations. ◮ Matches R × S 3 sector of AdS 5 × S 5 string theory. ◮ Thermodynamic limit ( ∞ long chain): Equivalent to the LL model → Heisenberg Chain is quantized version of LL model. [ hep-th/0311203 ] Kruczenski Integrability: ◮ LL model: Classical integrability → Spectral curves . ◮ Spin Chain: Quantum integrability → Bethe equations . Feb. ’08: Two-Cut Solutions of the Heisenberg Ferromagnet and Stability 3 / 27
Aim of the Project Goal: Find the spectrum of the Heisenberg ferromagnet in the thermodynamic limit. ◮ Understand the phase space of solutions to the Bethe equations. ◮ In particular: Investigate unstable modes of the LL model. ◮ Measurements on 1D-magnets?! Feb. ’08: Two-Cut Solutions of the Heisenberg Ferromagnet and Stability 4 / 27
Contents Motivation The Landau-Lifshitz Model The Heisenberg Spin Chain and the Thermodynamic Limit A Single Cut Cuts Interact: The Two-Cut Solution Feb. ’08: Two-Cut Solutions of the Heisenberg Ferromagnet and Stability 5 / 27
The Landau-Lifshitz Model ◮ Classical non-relativistic sigma model on the sphere S 2 with fields φ , θ and Lagrange function φ, θ ] = − L � φ d σ − π � ( θ ′ 2 + sin 2 θφ ′ 2 ) d σ L [ φ, ˙ cos θ ˙ 4 π 2 L ◮ Effective model for the exact description of strings on R × S 3 that rotate at highly relativistic speed. [ hep-th/0311203 ] Kruczenski ◮ Equations of motion: cos( θ ) φ ′ 2 − csc( θ ) θ ′′ � ˙ φ = (2 π/L ) 2 � , 2 cos( θ ) θ ′ φ ′ + sin( θ ) φ ′′ � ˙ θ = (2 π/L ) 2 � . ◮ Charges: Momentum P , spin α , energy ˜ E : P = 1 � � α = 1 � � φ ′ d σ , 1 − cos( θ ) � 1 − cos( θ ) � d σ , 2 4 π E = π � � θ ′ 2 + sin 2 ( θ ) φ ′ 2 � ˜ d σ . 2 Feb. ’08: Two-Cut Solutions of the Heisenberg Ferromagnet and Stability 6 / 27
A Simple Solution ◮ Vacuum: Energy minimized by string localized at a point: P = α = ˜ θ ( σ, τ ) ≡ 0 , E = 0 . ◮ A simple solution: Constant latitude θ 0 , n windings: φ ( σ, τ ) = nσ + (2 π/L ) 2 n 2 cos( θ 0 ) τ . θ ( σ, τ ) ≡ θ 0 , Momentum P and energy ˜ E can be expressed in terms of spin α and mode number n : ˜ E = 4 π 2 n 2 α (1 − α ) . α = 1 2 (1 − cos( θ 0 )) , P = 2 πnα , ◮ Semiclassical quantization → Contact with Heisenberg model. Feb. ’08: Two-Cut Solutions of the Heisenberg Ferromagnet and Stability 7 / 27
Fluctuation Modes ◮ Add excitations with mode number k to the simple solution: θ ( σ, τ ) = θ 0 + εθ + ( τ ) e ikσ + εθ − ( τ ) e − ikσ + ε 2 θ c ( τ ) , φ ( σ, τ ) = φ 0 + εφ + ( τ ) e ikσ + εφ − ( τ ) e − ikσ + ε 2 φ c ( τ ) . φ, θ ] to order ε 2 yields two coupled HO’s with ◮ Expanding L [ φ, ˙ charges δα = 1 /L , δP = 2 π ( n + k ) /L , � � δ ˜ E = (2 π ) 2 /L n ( n + 2 k )(1 − 2 α ) + k 2 � 1 − 4 n 2 α (1 − α ) /k 2 . ⇒ The classical solution becomes unstable when � 2 n α (1 − α ) > k = 1 , that is for large n and α . Feb. ’08: Two-Cut Solutions of the Heisenberg Ferromagnet and Stability 8 / 27
Contents Motivation The Landau-Lifshitz Model The Heisenberg Spin Chain and the Thermodynamic Limit A Single Cut Cuts Interact: The Two-Cut Solution Feb. ’08: Two-Cut Solutions of the Heisenberg Ferromagnet and Stability 9 / 27
The Heisenberg Spin Chain ◮ One of the oldest quantum mechanical models, set up by Heisenberg in 1928, describes a 1D-magnet with nearest-neighbor interaction of L spin-1/2 particles. [ Z. Phys. A49, 619 ] Heisenberg ◮ Describes SU(2) -sector of planar N = 4 SYM at one loop. [ Minahan Zarembo ] ◮ Hilbert space H is the tensor product of L single-spin spaces C 2 : C 2 � ⊗ L , H = C 2 ⊗ C 2 ⊗ · · · ⊗ C 2 = � e.g. | ↓↓↑↓↓↓↑↑↓↑↓ � ∈ H . ◮ The Hamiltonian H : H → H is periodic L L H = 1 � � (1 − σ k · σ k +1 ) = (1 − P k,k +1 ) . 2 k =1 k =1 ◮ The energy spectrum is bounded between [ Hulthén 1938 ] ◮ The ferromagnetic ground state | ↓↓↓↓↓↓↓↓ � , energy E = 0 . ◮ The antiferromangetic ground state “ | ↓↑↓↑↓↑↓↑ � ”, E ≈ L log 4 . Feb. ’08: Two-Cut Solutions of the Heisenberg Ferromagnet and Stability 10 / 27
Bethe Equations The Heisenberg spin chain is exactly solvable: [ Z. Phys. 71, 205 ] Bethe ◮ Fundamental excitations: Magnons with definite momentum p and rapidity u = 1 2 cot( p/ 2) . ◮ All eigenstates can be explicitly constructed as combinations of multiple magnons (Bethe ansatz). ◮ Only requirement: Constituent magnons u 1 , . . . , u M must satisfy Bethe equations (ensure periodicity of wave function): � L M u k − u j + i � u k + i/ 2 � = u k − u j − i , k = 1 , . . . , M . u k − i/ 2 j =1 j � = k ◮ Momentum and Energy: M M u k + i/ 2 1 e iP = � � u k − i/ 2 , E = k − 1 / 4 . u 2 k =1 k =1 Feb. ’08: Two-Cut Solutions of the Heisenberg Ferromagnet and Stability 11 / 27
The Thermodynamic Limit ◮ Bethe equations hard to solve for more than a few excitations u k . Sutherland ◮ Problem simplifies in thermodynamic limit : [ 74, 816 (1995) ][ Staudacher, Zarembo ] Beisert, Minahan Phys. Rev. Lett. ◮ Length of the chain (number of sites) L → ∞ . ◮ Number of excitations (flipped spins) M → ∞ . ◮ Filling fraction α = M/L fixed. ◮ Keep only low-energy excitations (IR modes, coherent states), energies E = ˜ E/L ∼ 1 /L . ◮ In this limit, contact with the LL model is established: In coherent states, the expectation values of single spins form paths on S 2 : − → ⇒ Classical LL model on S 2 is an effective model for the Heisenberg chain in the ferromagnetic thermodynamic limit. [ hep-th/0311203 ] Kruczenski Feb. ’08: Two-Cut Solutions of the Heisenberg Ferromagnet and Stability 12 / 27
Root Condensation ◮ In the thermodynamic limit, rescaled roots x k = u k /L of coherent states condense on contours in the complex plane: [ Staudacher, Zarembo ] Beisert, Minahan → → ◮ In the strict limit L ⇒ ∞ , the Bethe equations turn into integral eqations which describe the positions of the contours and the root density along them. [ Kazakov, Marshakov Minahan, Zarembo ] The contours constitute the branch cuts of the spectral curve, which is the solution of the LL model. Feb. ’08: Two-Cut Solutions of the Heisenberg Ferromagnet and Stability 13 / 27
Small Filling ◮ General picture for small fillings α i : Positions of cuts: Integer mode numbers n i , x k ∼ 1 /n i . Small densities, weak interaction between individual cuts. ◮ When the filling of a cut grows, the cut gets longer and its density increases. ◮ Questions : ◮ What happens when root density approaches | ρ | = 1 ? ◮ Singularity in the Bethe equations; construction of the spectral curve no longer valid!? ◮ Relation to unstable modes in LL model? ◮ Do corresponding solutions to the Bethe equations exist for all spectral curves? Feb. ’08: Two-Cut Solutions of the Heisenberg Ferromagnet and Stability 14 / 27
Contents Motivation The Landau-Lifshitz Model The Heisenberg Spin Chain and the Thermodynamic Limit A Single Cut Cuts Interact: The Two-Cut Solution Feb. ’08: Two-Cut Solutions of the Heisenberg Ferromagnet and Stability 15 / 27
From Small to Large Filling First: Study the configuration with a single cut in detail. [ TB, Beisert to appear ] ◮ As the cut grows, it attracts the neighboring fluctuation points: 3 2 3 2 1 1 5 4 1’ 3 2’ 3 ◮ Expect something interesting when [ Beisert, Tseytlin ][ Hernández, López Periáñez, Sierra ][ Beisert Freyhult ] Zarembo ◮ Fluctuation point collides with cut: Density reaches | ρ | = 1 , Bethe equations singular. ◮ Two successive fluctuation points collide and diverge into the complex plane. Instability? Coincides with filling where instability appears in the classical LL model. Feb. ’08: Two-Cut Solutions of the Heisenberg Ferromagnet and Stability 16 / 27
Bethe Strings 4 ◮ For the a priori infinite chain, individ- ual magnon rapidities are arbitrary (no periodicity → no Bethe equations), but 2 generically must be real. ◮ Only exception: “Bethe strings”. Bound states on the infinitely long chain, com- 0 plex rapidities with regular pattern. ◮ Guess/Expect: Bethe strings appear when density on contour reaches | ρ | = 1 . � 2 � 4 0 1 2 3 4 Feb. ’08: Two-Cut Solutions of the Heisenberg Ferromagnet and Stability 17 / 27
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