The XXX chain: from spin flips to Bethe equations

How does an interacting quantum system turn into a problem about a few wave numbers? The spin-½ XXX chain offers a concrete entrance. A single flipped spin propagates as a wave. With two flipped spins, the interaction changes how the waves combine; taking either excitation around a ring then constrains its wave number. Those constraints are the Bethe equations.

This sequence develops that argument and checks actual eigenvectors of a six-site Hamiltonian. You will construct selected two-magnon states, rather than assume that a list of roots automatically gives every state. You do not need to complete the Toda sequence first.

The spin-chain problem

Place $N\geq3$ spin-½ degrees of freedom on a ring. Set $\hbar=1$ and the lattice spacing to one. The model record fixes the Hamiltonian

$$H=J\sum_{n=1}^{N}\left(\frac14-\mathbf S_n\cdot\mathbf S_{n+1}\right), \qquad \mathbf S_{N+1}=\mathbf S_1,\qquad J\gt0.$$

Here $\mathbf S_n=\boldsymbol{\sigma}_n/2$: spin operators are half the Pauli matrices. $J$ is an energy scale. The fully aligned state $|F\rangle={|\uparrow\cdots\uparrow\rangle}$ has energy zero. A down spin relative to this reference is a spin flip; a delocalized spin-flip eigenstate is a magnon. The total number $M$ of down spins is conserved, so the Hamiltonian can be studied in smaller fixed-$M$ spaces.

The first result is the one-magnon dispersion $E(k)=J(1-\cos k)$. The second is more subtle: for two magnons the energy is still a sum of these expressions, but the allowed wave numbers depend on a scattering amplitude. An additive energy formula does not make the interacting eigenstate a product of independent waves.

Check your starting point

You need complex numbers, vectors and matrix multiplication. Familiarity with spin helps, but the first lesson builds the required operators. The numerical project uses Python and NumPy; all preceding calculations can be done by hand.

If complex inner products, tensor factors or commutators are unfamiliar, begin with Eigenvalues, commutators & tensor products. Its small worked matrices prepare the normalization and symmetry-sector calculations used here.

Try these preparation checks.

1. If $v=(1,i)/\sqrt2$, compute $v^\dagger v$. Why is the complex conjugate necessary?
2. How many basis states of six spins contain exactly two down spins?
3. What does $e^{ikN}=1$ imply for real $k$, counted modulo $2\pi$?

Solutions and preparation

The squared norm is $(1+(-i)i)/2=1$. An inner product conjugates the first vector; using a transpose alone would incorrectly give zero.

There are $\binom62=15$ choices of two distinct down-spin sites, compared with $2^6=64$ unrestricted basis states. A symmetry reduces the matrix dimension before any diagonalization.

The allowed values are $k=2\pi m/N$, with $m=0,\ldots,N-1$. This condition will apply to one magnon. The two-magnon problem includes an additional scattering phase, so do not reuse this grid without checking the boundary equation.

For an unfamiliar step, use the entry check and repair in the corresponding lesson. The convention reference also explains spin normalization, energy shifts and amplitude orientation.

Follow the six-step sequence

Step	Work	Capability
1	This orientation	Specify the finite quantum problem and the scope of the result.
2	Construct a small spin-chain Hamiltonian	Build the bond operator and identify fixed-magnetization sectors.
3	Solve the one-magnon sector	Derive the hopping equation, dispersion and ring quantization.
4	Solve two-magnon scattering	Match adjacent spin flips and calculate the amplitude ratio.
5	Quantize magnon momenta on a ring	Derive the boundary equations and construct a regular solution.
6	Test finite-chain Bethe solutions	Compare independent Hamiltonian constructions and verify a normalized eigenvector.

Each of the four lessons has a worked calculation, guided practice, an independent task and a changed-setting task, with hints and solutions. The project combines them. Readers comfortable with spin chains can begin with the Library derivation, then use the project to test the formulas.

The central example uses $N=6$ and the pair $k_1=-k_2=2\pi/5$. Its energy is

$$\frac{E}{J}=2\left(1-\cos\frac{2\pi}{5}\right) =\frac{5-\sqrt5}{2}.$$

These wave numbers lie off the six-site one-magnon grid. Explaining why they nevertheless produce a periodic two-magnon state is one of the sequence’s key outcomes.

Demonstrate what you have learned

Keep a calculation sheet and the output of the numerical project. By the end, you should be able to:

• Recover the bond matrix from either spin matrices or the spin-swap operator.
• Derive the one-magnon energy and explain its zero-momentum state.
• Obtain the two-magnon contact condition and state exactly which amplitude ratio you call the scattering factor.
• Derive both ring equations, including the reciprocal in the first equation.
• Reconstruct a nonzero vector, normalize it and test $\|H\psi-E\psi\|_2$.
• Explain why finding one correct energy, solving a root equation or plotting a small residual does not establish spectral completeness.
• Predict how a uniform longitudinal field shifts a fixed-$M$ energy and why removing a bond changes the boundary problem.

Use a failed check to locate the missing reasoning, then repeat the calculation with the second momentum pair in the project. Solutions support practice; they do not replace attempting the calculation.

Classical and quantum structures

The Toda and XXX sequences share a habit of separating claims, even though their mathematical tests differ.

Question	Open Toda	Periodic XXX chain
What evolves or is solved?	Canonical positions and momenta	Quantum states and energy eigenvalues
What does the first spectral construction provide?	Conserved eigenvalues of a Lax matrix	Candidate eigenstates from wave amplitudes
What must still be checked?	Involution and independence of enough integrals	Contact and boundary equations, nonzero states, and the scope of completeness
What does the computation establish?	Trajectory accuracy in tested cases	Finite-matrix agreement in tested sectors

The algebraic follow-up now develops a different part of the structure: verify a rational R-matrix, construct monodromy, prove that transfer matrices commute, recover this same Hamiltonian by differentiation, and construct regular Bethe vectors by cancelling unwanted transfer terms. The Library proof gives the complete finite-chain argument, and the R-matrix reference translates its conventions. You can take that four-lesson route after this sequence or enter it directly with the necessary matrix background.

This coordinate sequence treats one- and two-magnon states. Complex roots, singular limits, general particle number, thermodynamic limits and correlation functions need additional arguments.

The Quantum Integrability overview maps those later topics. Begin now with the spin-chain Hamiltonian.