Two-magnon Bethe ansatz for the periodic XXX chain

How can two interacting spin reversals be described by plane waves? In the periodic spin-half XXX chain, the bulk equation fixes an additive energy, the contact equation fixes the ratio of two amplitudes, and carrying one spin reversal around the ring quantizes the two wave numbers together. We derive a sufficient construction of regular two-magnon eigenvectors and an exact six-site example. The result is a sector-level eigenvector construction; it is not a proof that all spin-chain states have been found.

Required background. The XXX model defines the Hamiltonian and fixed-magnetization basis; solve the one-magnon sector explains its plane-wave eigenvectors. Helpful background. Solve two-magnon scattering and quantize magnon momenta on a ring provide guided versions of the two main derivations.

Two reversed spins on a ring

Use the periodic Hamiltonian with $N\geq3$,

$$H=\frac J2\sum_{n=1}^N(1-P_{n,n+1}), \qquad J\gt0,$$

with spin operators $\mathbf S=\boldsymbol\sigma/2$, $\hbar=1$, unit lattice spacing, and all-up energy zero. Work in the $M=2$ sector:

$$|\Psi\rangle=\sum_{1\leq x\lt y\leq N} \psi(x,y)|x,y\rangle.$$

There is one coefficient per unordered pair of distinct sites. To express boundary equations without repeatedly reordering the pair, extend the coefficients to ordered coordinates on the covering lattice subject to

$$\psi(x,y)=\psi(y,x+N),\qquad x\lt y\lt x+N.$$

This is the same physical pair described after taking the left coordinate once around the ring. Applying the identification twice gives simultaneous periodicity. Values at coincident coordinates will be used only as algebraic extensions of an ansatz, never as physical double occupancy.

The separated and contact equations

When the down spins are not neighboring sites on the circle, four bonds exchange an up spin and a down spin. The eigenvalue equation is

$$\begin{aligned} E\psi(x,y)=2J\psi(x,y)-\frac J2\bigl[&\psi(x-1,y)+\psi(x+1,y)\\ &+\psi(x,y-1)+\psi(x,y+1)\bigr]. \end{aligned}$$

The separation satisfies $1\lt y-x\lt N-1$ in the chosen covering interval. In particular $(1,N)$ is an adjacent pair, not a bulk pair. For $N=3$ there are no separated physical pairs; the contact and cyclic equations still apply.

For adjacent sites $y=x+1$, the intervening down–down bond acts as zero. Only the two outer bonds contribute:

$$E\psi(x,x+1)=J\psi(x,x+1) -\frac J2\bigl[\psi(x-1,x+1)+\psi(x,x+2)\bigr].$$

This missing pair of hopping terms is the interaction that a naive product of free waves fails to account for. It also changes the diagonal coefficient from $2J$ to $J$.

The corresponding equations in Karbach and Müller 1997, arXiv v1 PDF, p. 2, equations (9)–(12) use $E-E_0$ because their Hamiltonian omits our constant $JN/4$. The derivation here follows directly by acting with each permutation bond.

Two plane waves

Take finite, nonzero complex numbers $z_1,z_2$ and write

$$\psi(x,y)=A_{12}z_1^xz_2^y+A_{21}z_2^xz_1^y.$$

Substitution into the separated equation gives

$$E=\frac J2\left(4-z_1-z_1^{-1}-z_2-z_2^{-1}\right).$$

For $z_j=e^{ik_j}$, this is $E=J(2-\cos k_1-\cos k_2)$. It is an algebraic identity for each plane wave on the covering lattice, including formal coincident coordinates. We now impose the physical contact equation as well.

Subtract the contact equation from the formal separated equation at $(x,x+1)$. The result is

$$\psi(x,x)+\psi(x+1,x+1)=2\psi(x,x+1).$$

These diagonal values do not add basis states. They are a concise way of expressing the constraint on the two amplitudes. Substitution and division by $(z_1z_2)^x$ give

$$\begin{aligned} 0={}&A_{12}(1+z_1z_2-2z_2)\\ &+A_{21}(1+z_1z_2-2z_1). \end{aligned}$$

If $A_{12}\ne0$ and the denominator below is nonzero, define the scattering amplitude

$$S_{12}\equiv\frac{A_{21}}{A_{12}} =-\frac{1+z_1z_2-2z_2}{1+z_1z_2-2z_1}.$$

An overall amplitude remains arbitrary until normalization. Exchanging the labels gives $S_{21}=S_{12}^{-1}$ whenever both expressions are defined. Our $S_{12}$ is the reciprocal of the ratio $A/A'$ used in Karbach and Müller 1997, arXiv v1 PDF, p. 3, equations (13)–(16).

Periodicity with the coordinates reordered

The boundary identification is not $\psi(x+N,y)=\psi(x,y)$ with the same coordinate ordering: moving $x$ around the ring changes its position in the ordered pair. Instead,

$$\begin{aligned} \psi(y,x+N) ={}&A_{12}z_1^yz_2^{x+N}\\ &+A_{21}z_2^yz_1^{x+N}. \end{aligned}$$

For distinct $z_1,z_2$, matching the coefficients of the two waves to $\psi(x,y)$ yields

$$A_{12}=A_{21}z_1^N,\qquad A_{21}=A_{12}z_2^N.$$

Thus the regular two-magnon Bethe equations in this amplitude convention are

$$z_1^N=\frac1{S_{12}},\qquad z_2^N=S_{12}.$$

Their product gives $(z_1z_2)^N=1$. Total wave number $K=k_1+k_2$ is therefore quantized modulo $2\pi$, while the individual wave numbers are coupled through the contact amplitude. With $T|x,y\rangle=|x+1,y+1\rangle$ and cyclic reordering understood, the translation eigenvalue is $e^{-iK}$ for the positive-exponent coefficient convention.

The cyclic relation also supplies the contact equation across the closing bond $(N,1)$ and the missing hopping terms at the coordinate seam. It is essential to the finite ring. The boundary equations correspond to Karbach and Müller 1997, arXiv v1 PDF, p. 3, equations (17)–(18); their phase satisfies $e^{i\theta}=1/S_{12}$.

A sufficient eigenvector criterion

Suppose $z_1,z_2$ are finite, nonzero, distinct; the denominator defining $S_{12}$ is nonzero; and both Bethe equations hold. Form the coefficients with $A_{12}=1$ and $A_{21}=S_{12}$. If

$$\mathcal N=\sum_{1\leq x\lt y\leq N}|\psi(x,y)|^2\gt0,$$

then $\mathcal N^{-1/2}|\Psi\rangle$ is an eigenvector with the energy above. To prove this, apply the separated equation at each separated pair and the contact equation at each adjacent pair, using cyclic identification whenever a bond crosses the seam. These cases exhaust the basis. For $N=3$ only the second case occurs. This proves the entire matrix equation, not merely the equations away from the boundary.

The norm condition is necessary: a zero vector satisfies a homogeneous equation but is not a state. Because the physical Hamiltonian is Hermitian, any nonzero vector passing this criterion has real $E$, even when intermediate parameters are complex. The criterion is sufficient; exceptional states excluded by its divisions require separate treatment.

Rapidity variables and real scattering phases

For $z_j\ne1$, introduce

$$\lambda_j=\frac i2\frac{z_j+1}{z_j-1}, \qquad z_j=\frac{\lambda_j+i/2}{\lambda_j-i/2}.$$

If $k_j$ is real and nonzero modulo $2\pi$, this is $\lambda_j=\tfrac12\cot(k_j/2)$. Direct substitution gives

$$S_{12}=\frac{\lambda_1-\lambda_2-i} {\lambda_1-\lambda_2+i}.$$

For real rapidities, numerator and denominator are complex conjugates, so $|S_{12}|=1$. Write $S_{12}=e^{i\delta}$; then

$$Nk_1+\delta=2\pi m_1,\qquad Nk_2-\delta=2\pi m_2,$$

with integers $m_j$ and a consistent branch of $\delta$. The multiplicative equations avoid a premature phase-branch choice.

Equivalently, for $j=1,2$ and the other index $\ell\ne j$,

$$\left(\frac{\lambda_j+i/2}{\lambda_j-i/2}\right)^N =\frac{\lambda_j-\lambda_\ell+i}{\lambda_j-\lambda_\ell-i}, \qquad E=\frac J2\sum_{j=1}^2\frac1{\lambda_j^2+1/4}.$$

These rational equations inherit the nonzero-denominator restrictions. The zero-wave-number limit corresponds to infinite rapidity. Roots at $\lambda=\pm i/2$ are outside this finite-$z$ parametrization; inserting them into a rational formula as ordinary numbers is invalid. The conventions reference keeps this rapidity normalization separate from integer Bethe labels.

An exact six-site eigenvector

Choose $N=6$, $k_1=-k_2=\theta=2\pi/5$, and $z=e^{i\theta}$. Then $z_1=z$, $z_2=z^{-1}$, and

$$S_{12}=-\frac{2-2z^{-1}}{2-2z}=z^{-1}.$$

The first periodic equation becomes $z^6=z$, which holds because $z^5=1$; the second is its inverse. The amplitudes depend only on the separation $d=y-x$:

$$\begin{aligned} \psi(x,y)&=z^{-d}+z^{d-1}\\ &=2e^{-i\theta/2}\cos\!\left[\theta\left(d-\frac12\right)\right]. \end{aligned}$$

For each $d=1,\ldots,5$ there are $6-d$ ordered pairs. Hence

$$\mathcal N=4\sum_{d=1}^{5}(6-d) \cos^2\!\left[\theta\left(d-\frac12\right)\right]=30.$$

Removing the common phase and dividing by $\sqrt{30}$ gives the real normalized coefficients

$$\psi_{\mathrm{norm}}(x,y)=\sqrt{\frac{2}{15}} \cos\!\left[\theta\left(y-x-\frac12\right)\right].$$

They are nonzero; for example the unnormalized coefficient at $d=1$ is $1+z^{-1}$. The energy is exactly

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

Replacing $\theta$ by $4\pi/5$ gives another state with the same norm formula and $E=(5+\sqrt5)J/2$. The finite-chain project checks both constructions using the complete $15\times15$ matrix in the two-magnon sector, including the closing bond.

Complex roots and omitted states

Complex wave numbers are allowed in the algebraic ansatz. They do not represent individually measurable complex momenta: only the reconstructed finite-chain state and its physical eigenvalues are observables. For a conjugate pair $k_1=K/2+i\kappa$, $k_2=K/2-i\kappa$ with real $K,\kappa$, the two terms vary exponentially with separation, and

$$E=2J\bigl[1-\cos(K/2)\cosh\kappa\bigr].$$

Arbitrary $K,\kappa$ do not satisfy the boundary equations. Only admissible pairs can describe states, and localization in the reconstructed separation distribution is needed for a bound-pair interpretation. The finite-ring analysis of complex solutions appears in Karbach and Müller 1997, arXiv v1 PDF, pp. 4–5, equations (20)–(25) and the bound-state discussion. We have not classified those solutions here.

Three limitations matter even before seeking a complete spectrum:

• Coincident parameters can give zero vectors. If $z_1=z_2=z\ne1$, the contact formula gives $S_{12}=-1$, so the two terms cancel identically. For example, $z^N=-1$ solves the displayed multiplicative equations but produces no eigenstate.
• A singular ratio is not a missing physical state by itself. At $z_1=z_2=1$, both numerator and denominator vanish. Nevertheless $(S_{\mathrm{tot}}^-)^2|\Omega\rangle$ is a nonzero zero-energy state for $N\geq2$, with constant pair coefficients. It follows from $[H,S_{\mathrm{tot}}^-]=0$, without dividing by the scattering denominator.
• Counting roots requires counting states. Exchanging $z_1,z_2$ changes amplitudes only by an overall factor when $S_{12}\ne0$. Other degeneracies, singular limits, complex solutions, and descendants generated by $S_{\mathrm{tot}}^-$ must be treated before claiming a complete basis. Checking two examples is not such a count.

The scalar two-body amplitude is also not a proof of the commuting transfer-matrix construction or of consistency for arbitrary numbers of magnons. The model record states that wider structure with its source and normalization.

Check your understanding

Change the size of the ring. Take $z_1=z$, $z_2=z^{-1}$ with $z\ne1$. Show that the Bethe equations reduce to $z^{N-1}=1$. Explain why $z=-1$, when it satisfies that equation, is not a regular two-wave state.

Solution

The contact ratio is $S_{12}=z^{-1}$, so $z^N=z$. The second equation gives the same condition. But $z=-1$ makes $z_1=z_2$, and $S_{12}=-1$ then cancels the entire vector. A root equation can be correct while its state is zero. The regular construction therefore excludes this coincident pair.

Check a boundary bond. Why would validating only the separated equation fail to distinguish periodic and open chains?

Solution

Their interior hopping equations agree. The open chain lacks $P_{N,1}$, so its endpoint diagonal and hopping terms differ. The cyclic identification supplies those periodic terms; omitting the seam from a residual check leaves precisely the boundary condition untested.

References

• Karbach, Michael, and Gerhard Müller. “Introduction to the Bethe ansatz I.” Computers in Physics 11, 36–43 (1997). DOI. Author version arXiv:cond-mat/9809162v1, submitted 1998, 8 pp.; page locators refer to this version. Version record. Open PDF.