Two-Cut Solutions of the HeisenbergFerromagnet 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× S3. [ Kruczenskihep-th/0311203]

I Subspace of AdS5 × S5 → AdS/CFT correspondence.I Quantization around certain solutions → Unstable excitation modes.

Heisenberg Spin Chain: Quantum-mechanical model for 1D magnetI Equivalent to SU(2) sector of planar N = 4 SYM (one-loop). [Minahan

Zarembo]I Solved (in principle) by the Bethe equations.I Matches R× S3 sector of AdS5 × S5 string theory.I Thermodynamic limit (∞ long chain): Equivalent to the LL model→ Heisenberg Chain is quantized version of LL model. [ Kruczenski

hep-th/0311203]Integrability:

I LL model: Classical integrability → Spectral curves.I 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 thethermodynamic limit.

I Understand the phase space of solutions to the Bethe equations.I In particular: Investigate unstable modes of the LL model.I 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 ModelI Classical non-relativistic sigma model on the sphere S2 with fields φ,

θ and Lagrange function

L[φ, φ, θ] = − L

4π

∮cos θφdσ − π

2L

∮(θ′2 + sin2 θφ′2) dσ

I Effective model for the exact description of strings on R× S3 thatrotate at highly relativistic speed. [ Kruczenski

hep-th/0311203]I Equations of motion:

φ = (2π/L)2(cos(θ)φ′2 − csc(θ)θ′′

),

θ = (2π/L)2(2 cos(θ)θ′φ′ + sin(θ)φ′′

).

I Charges: Momentum P , spin α, energy E:

P =12

∫ (1− cos(θ)

)φ′ dσ , α =

14π

∫ (1− cos(θ)

)dσ ,

E =π

2

∫ (θ′2 + sin2(θ)φ′2

)dσ .

Feb. ’08: Two-Cut Solutions of the Heisenberg Ferromagnet and Stability 6 / 27

A Simple Solution

I Vacuum: Energy minimized by string localized at a point:

θ(σ, τ) ≡ 0 , P = α = E = 0 .

I A simple solution: Constant latitude θ0, n windings:

θ(σ, τ) ≡ θ0 , φ(σ, τ) = nσ + (2π/L)2n2 cos(θ0)τ .

Momentum P and energy E can be expressed in terms of spin α andmode number n:

α = 12 (1− cos(θ0)) , P = 2πnα , E = 4π2n2α(1− α) .

I Semiclassical quantization → Contact with Heisenberg model.

Feb. ’08: Two-Cut Solutions of the Heisenberg Ferromagnet and Stability 7 / 27

Fluctuation Modes

I Add excitations with mode number k to the simple solution:

θ(σ, τ) = θ0 + εθ+(τ)eikσ + εθ−(τ)e−ikσ + ε2θc(τ) ,

φ(σ, τ) = φ0 + εφ+(τ)eikσ + εφ−(τ)e−ikσ + ε2φc(τ) .

I Expanding L[φ, φ, θ] to order ε2 yields two coupled HO’s withcharges

δα = 1/L ,

δP = 2π(n + k)/L ,

δE = (2π)2/L(n(n + 2k)(1− 2α) + k2

√1− 4n2α(1− α)/k2

).

⇒ The classical solution becomes unstable when2n

√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

I One of the oldest quantum mechanical models, set up by Heisenbergin 1928, describes a 1D-magnet with nearest-neighbor interaction ofL spin-1/2 particles. [ Heisenberg

Z. Phys. A49, 619]I Describes SU(2)-sector of planar N = 4 SYM at one loop. [Minahan

Zarembo]I Hilbert space H is the tensor product of L single-spin spaces C2:

H = C2 ⊗ C2 ⊗ · · · ⊗ C2 =(C2

)⊗L, e.g. |↓↓↑↓↓↓↑↑↓↑↓〉 ∈ H .

I The Hamiltonian H : H → H is periodic

H =12

L∑k=1

(1− σk · σk+1) =L∑

k=1

(1− Pk,k+1) .

I The energy spectrum is bounded between [Hulthén1938 ]

I The ferromagnetic ground state |↓↓↓↓↓↓↓↓〉, energy E = 0.I 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: [ BetheZ. Phys. 71, 205]

I Fundamental excitations: Magnons with definite momentum p andrapidity u = 1

2 cot(p/2).I All eigenstates can be explicitly constructed as combinations of

multiple magnons (Bethe ansatz).I Only requirement: Constituent magnons u1, . . . , uM must satisfy

Bethe equations (ensure periodicity of wave function):(uk + i/2uk − i/2

)L

=M∏

j=1j 6=k

uk − uj + i

uk − uj − i, k = 1, . . . ,M .

I Momentum and Energy:

eiP =M∏

k=1

uk + i/2uk − i/2

, E =M∑

k=1

1u2

k − 1/4.

Feb. ’08: Two-Cut Solutions of the Heisenberg Ferromagnet and Stability 11 / 27

The Thermodynamic Limit

I Bethe equations hard to solve for more than a few excitations uk.I Problem simplifies in thermodynamic limit: [ Sutherland

Phys. Rev. Lett.74, 816 (1995)

][ Beisert, MinahanStaudacher, Zarembo]

I Length of the chain (number of sites) L →∞.I Number of excitations (flipped spins) M →∞.I Filling fraction α = M/L fixed.I Keep only low-energy excitations (IR modes, coherent states),

energies E = E/L ∼ 1/L.I In this limit, contact with the LL model is established: In coherent

states, the expectation values of single spins form paths on S2:

−→

⇒ Classical LL model on S2 is an effective model for the Heisenbergchain in the ferromagnetic thermodynamic limit. [ Kruczenski

hep-th/0311203]

Feb. ’08: Two-Cut Solutions of the Heisenberg Ferromagnet and Stability 12 / 27

Root Condensation

I In the thermodynamic limit, rescaled roots xk = uk/L of coherentstates condense on contours in the complex plane: [ Beisert, Minahan

Staudacher, Zarembo]

→ →

I In the strict limit L ⇒∞, the Bethe equations turn into integraleqations which describe the positions of the contours and the rootdensity along them. [Kazakov, Marshakov

Minahan, Zarembo ]The contours constitute the branch cuts of the spectral curve, whichis the solution of the LL model.

Feb. ’08: Two-Cut Solutions of the Heisenberg Ferromagnet and Stability 13 / 27

Small Filling

I General picture for small fillings αi:

Positions of cuts: Integer mode numbers ni, xk ∼ 1/ni.Small densities, weak interaction between individual cuts.

I When the filling of a cut grows, the cut gets longer and its densityincreases.

I Questions:I What happens when root density approaches |ρ| = 1?I Singularity in the Bethe equations; construction of the spectral curve

no longer valid!?I Relation to unstable modes in LL model?I 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, Beisertto appear ]

I As the cut grows, it attracts the neighboring fluctuation points:

3

3

1245

3

3

1

2

2’

1’

I Expect something interesting when [Beisert, TseytlinZarembo ][Hernández, López

Periáñez, Sierra ][ BeisertFreyhult]

I Fluctuation point collides with cut: Density reaches |ρ| = 1, Betheequations singular.

I Two successive fluctuation points collide and diverge into thecomplex plane. Instability? Coincides with filling where instabilityappears in the classical LL model.

Feb. ’08: Two-Cut Solutions of the Heisenberg Ferromagnet and Stability 16 / 27

Bethe Strings

I For the a priori infinite chain, individ-ual magnon rapidities are arbitrary (noperiodicity → no Bethe equations), butgenerically must be real.

I Only exception: “Bethe strings”. Boundstates on the infinitely long chain, com-plex rapidities with regular pattern.

I Guess/Expect: Bethe strings appearwhen density on contour reaches |ρ| = 1.

0 1 2 3 4

-4

-2

0

2

4

Feb. ’08: Two-Cut Solutions of the Heisenberg Ferromagnet and Stability 17 / 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 18 / 27

Construction of the Two-Cut Spectral Curve

I Spectral curve that encodes the contours is a function with genusg = 1 (for two cuts). Find analytic expression.

I Generalize the symmetric solution [Beisert,Dippel,Staudacher ]

Map two symmetric cuts to two general cuts via Möbiustransformation µ:

The parameters of µ and the position of the symmetric cuts leaveenough freedom for placing the cuts at will.

I Need to solve for specific solutions that obey the integral equations.Possible numerically.

Feb. ’08: Two-Cut Solutions of the Heisenberg Ferromagnet and Stability 19 / 27

Collision of Two CutsWhen the filling grows, cuts can collide and form condensates withdensity |ρ| = ∆u = 1:

−→ =

Cuts can even pass through each other:

The passing cut changes its mode number: n2 → 2n1 − n2.Feb. ’08: Two-Cut Solutions of the Heisenberg Ferromagnet and Stability 20 / 27

Closed Loop Cuts

Consider a very small second cut:

−→ −→ −→ −→

Compare this to a bare excitation point that passes through:

−→ −→ −→ −→

A closed loop with a condensate appears naturally.

Feb. ’08: Two-Cut Solutions of the Heisenberg Ferromagnet and Stability 21 / 27

Collision of Fluctuation Points: InstabilityLook at the point where unstable modes in the LL model appear:

−→ −→

Consider again small cuts instead of bare excitation points:

−→ −→

Implication for closed cut (one-cut solution):

−→ −→

Feb. ’08: Two-Cut Solutions of the Heisenberg Ferromagnet and Stability 22 / 27

Phase transition

I Excitation of a mode means: Regular point → Two branch points.I Fluctuation point real: Excitation = Addition of roots.I Fluctuation points complex with loop cut: Excitation means taking

roots away.⇒ Behind instability point: Single cut + loop → two cuts:

Feb. ’08: Two-Cut Solutions of the Heisenberg Ferromagnet and Stability 23 / 27

Further Features of SolutionsWhat if the fluctuation point is already excited at the instability point?

BPs can cycle around each other until they reach the imagin. line, P = π:

Feb. ’08: Two-Cut Solutions of the Heisenberg Ferromagnet and Stability 24 / 27

Phase Space of Consecutive Mode Numbers

Feb. ’08: Two-Cut Solutions of the Heisenberg Ferromagnet and Stability 25 / 27

Summary and Conclusions

I TD limit of Heisenberg Ferromagnet is equivalent to LL model.I Macroscopic excitations are contours in complex plane.I Apparent singularity of the Bethe equations in the TD limit is always

hidden in a condensate.I Moduli space is connected: Mode numbers can change.I Phase space is “filled”: For all values of the moduli there are

corresponding solutions to the Bethe equations.I Unstable classical solutions are degenerate three-cut solutions that

“decay” into two-cut solutions.

Feb. ’08: Two-Cut Solutions of the Heisenberg Ferromagnet and Stability 26 / 27

Numerical Results

Numerical verification: [TB, Beisert, Gromovto appear ]

Feb. ’08: Two-Cut Solutions of the Heisenberg Ferromagnet and Stability 27 / 27