XXX spin-½ chain

The XXX chain is a quantum lattice model with the same exchange coupling in all three spin directions. The periodic spin-half chain provides a concrete entrance to quantum integrability: one reversed spin propagates as a plane wave, while two reversed spins require a scattering amplitude and interacting momentum quantization. This record defines the ferromagnetic normalization used in the XXX learning sequence, identifies its commuting-transfer-matrix structure, and separates those statements from spectral completeness and other boundary regimes.

Required background. Form tensor products and act with a matrix on a vector; construct a small spin-chain Hamiltonian develops those operations. Helpful background. The spin-chain conventions collect the sign, normalization, and scattering conversions used here.

The periodic spin-half Hamiltonian

Fix $N\geq3$ sites on a ring, a real coupling $J\gt0$, and Hilbert space

$$\mathcal H=(\mathbb C^2)^{\otimes N}.$$

Set $\hbar=1$ and the lattice spacing to one. Each site carries $S_n^\alpha=\sigma_n^\alpha/2$, where $\sigma^\alpha$ are the Pauli matrices, $\alpha=x,y,z$, and operators on different sites commute. The site index is periodic, $\mathbf S_{N+1}=\mathbf S_1$. Our zero-field Hamiltonian is

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

$J$ sets the energy scale and $J^{-1}$ the time scale. The additive constant makes the all-up state

$$|\Omega\rangle=|\uparrow\cdots\uparrow\rangle, \qquad H|\Omega\rangle=0.$$

This is the Hamiltonian of Karbach and Müller 1997, arXiv v1 PDF, pp. 1–2, equations (1)–(5) shifted by $JN/4$: their vacuum energy is $-JN/4$, and their $E-E_0$ is our $E$. The restriction $N\geq3$ avoids the two-site convention in which the periodic sum counts the same unordered bond twice.

Exchange as a permutation

Let $P_{n,n+1}$ exchange the two local spin states. On the triplet subspace, $\mathbf S_n\cdot\mathbf S_{n+1}=1/4$ and $P=1$; on the singlet, these eigenvalues are $-3/4$ and $-1$. Therefore

$$P_{n,n+1}=2\mathbf S_n\cdot\mathbf S_{n+1}+\frac12, \qquad H=\frac J2\sum_n(1-P_{n,n+1}).$$

In the ordered local basis $\uparrow\uparrow,\uparrow\downarrow,\downarrow\uparrow,\downarrow\downarrow$, one bond is

$$h_{n,n+1}=\frac J2 \begin{pmatrix} 0&0&0&0\\ 0&1&-1&0\\ 0&-1&1&0\\ 0&0&0&0 \end{pmatrix}.$$

Its eigenvalues are $0,0,0,J$. Each bond is positive semidefinite, so $H\geq0$ and the all-up state is a ground state. It is not the unique ground state: the fully symmetric spin multiplet also has zero energy. In particular, the all-down state has zero energy.

Symmetry and magnetization sectors

The total spin is $\mathbf S_{\mathrm{tot}}=\sum_n\mathbf S_n$. Isotropy gives $[H,S_{\mathrm{tot}}^\alpha]=0$. This does not make the three spin components mutually commuting: they still satisfy the spin commutation relations.

The number of down spins is the operator

$$\widehat M=\frac N2-S_{\mathrm{tot}}^z.$$

Since $[H,\widehat M]=0$, the matrix splits into sectors with integer $M=0,\ldots,N$ and dimensions $\binom NM$. For $M=2$, an orthonormal basis is

$$|x,y\rangle=S_x^-S_y^-|\Omega\rangle, \qquad 1\leq x\lt y\leq N, \qquad S_n^-=S_n^x-iS_n^y.$$

No site can be lowered twice: $(S_n^-)^2=0$. Coordinates $x,y$ label overturned spins on distinct lattice sites, not continuous particle positions. Translation symmetry is also present; periodicity quantizes total lattice wave number modulo $2\pi$.

One magnon and an interacting pair

For $|x\rangle=S_x^-|\Omega\rangle$, the two incident bonds give

$$H|x\rangle=J|x\rangle-\frac J2 \bigl(|x-1\rangle+|x+1\rangle\bigr).$$

The normalized plane wave and its energy are

$$|k\rangle=\frac1{\sqrt N}\sum_{x=1}^N e^{ikx}|x\rangle, \qquad k=\frac{2\pi m}{N}, \qquad E(k)=J(1-\cos k).$$

Here $m=0,\ldots,N-1$. The $k=0$ state is a symmetry partner of the vacuum, not a positive-energy excitation. If translation is defined by $T|x\rangle=|x+1\rangle$, then $T|k\rangle=e^{-ik}|k\rangle$; the sign belongs to the chosen coefficient convention.

For two magnons, use $z_j=e^{ik_j}$ and

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

For regular solutions, the contact equation and periodic boundary condition require

$$S_{12}\equiv\frac{A_{21}}{A_{12}} =-\frac{1+z_1z_2-2z_2}{1+z_1z_2-2z_1}, \qquad z_1^N=S_{12}^{-1},\quad z_2^N=S_{12}.$$

The corresponding energy is

$$E=J\bigl(2-\cos k_1-\cos k_2\bigr).$$

The sum resembles two one-magnon energies, but the allowed $k_j$ are coupled. Their separate values generally are not integer multiples of $2\pi/N$. The Library derivation supplies the contact calculation, the cyclic ordering argument, regularity conditions, and the check that the vector is nonzero.

A six-site check

For $N=6$, choose $k_1=-k_2=\theta=2\pi/5$ and $z=e^{i\theta}$. Then $S_{12}=z^{-1}$ and $z^5=1$ establish both periodic equations. An explicitly normalized, real representative is

$$\psi(x,y)=\sqrt{\frac{2}{15}} \cos\!\left[\theta\left(y-x-\frac12\right)\right], \qquad E=\frac{5-\sqrt5}{2}J.$$

Its total wave number is zero, although neither individual wave number is on the free six-site grid. The finite-chain project reconstructs this state and tests the eigenvector equation against an independently assembled Hamiltonian. That check verifies this state; it does not count every state in the $15$-dimensional sector.

The commuting transfer structure

The periodic homogeneous XXX chain has a stronger structure than the two-magnon calculation alone. Introduce a two-dimensional auxiliary space $a$, its swap $P_{an}$ with site $n$, and

$$\begin{aligned} R_{ab}(u)&=uI+iP_{ab},\\ L_{an}(u)&=(u-i/2)I+iP_{an},\\ \tau(u)&=\operatorname{tr}_a \bigl[L_{aN}(u)\cdots L_{a1}(u)\bigr]. \end{aligned}$$

The rational exchange relation gives $[\tau(u),\tau(v)]=0$. At $u=i/2$, $\tau$ is an invertible scalar multiple of the cyclic shift, and

$$H=\frac{JN}{2}I- \left.\frac{iJ}{2}\tau(u)^{-1}\tau'(u)\right|_{u=i/2}.$$

These are the conventions of Faddeev 1996, § 3, eqs. (31)–(65), PDF, with $H=-JH_{\mathrm{Faddeev}}$. The Library derivation proves the exchange relation, commuting traces and Hamiltonian extraction; the algebraic learning route develops the steps with exercises and a short-chain computation. The R-matrix convention reference explains the spectral shift, product order and scalar normalization. Neither commutativity alone nor successful finite examples establish completeness of a particular Bethe parametrization.

Exchange sign, field, and boundaries

These variations must be distinguished before importing a formula.

Variation	Precise change	Consequence for this treatment
Antiferromagnetic convention	$H_{\mathrm{AF}}=K\sum_n\mathbf S_n\cdot\mathbf S_{n+1}$, $K\gt0$	$H_{\mathrm{AF}}=-(K/J)H+KN/4$. Eigenvectors agree, but energy ordering reverses; the all-up state is no longer a lowest-energy reference.
Uniform longitudinal field	$H_h=H-hS_{\mathrm{tot}}^z$, real $h$	In a fixed-$M$ sector, the added term is a scalar. Energy relative to the all-up state becomes $E+hM$.
Open chain	Sum only $n=1,\ldots,N-1$	The closing bond disappears. Endpoint equations and quantization change; the periodic equations above cannot be reused.
Infinite chain or thermodynamic limit	Specify an infinite state or a sequence of finite systems and observables	A finite-sector eigenvector calculation supplies neither an infinite-system state nor a thermodynamic limit.

The field statement follows directly from $S_{\mathrm{tot}}^z=N/2-M$. A spatially varying field is not a scalar in each magnetization sector and does not inherit this argument. Likewise, changing spin $1/2$ to a higher local spin while retaining only bilinear exchange does not follow from the permutation construction used here.

What this entrance establishes

The lessons lead from an explicit matrix to one-magnon eigenstates and regular two-magnon eigenstates. The Library article proves why the latter construction works and explains its exclusions. For a specific singular case, the four-site benchmark and five-site rejection show how a limiting Bethe vector needs an additional physicality check; the reproduction project carries out that calculation.

The observables sequence uses a regular five-site state to calculate spin matrix elements and an exact finite correlation with a complete one-magnon final sector. A separate four-site completeness proof combines regular and singular states with symmetry descendants to span the two-down-spin sector. General chain-length completeness, thermodynamics, and correlations in larger systems require further arguments. Keeping these questions separate makes finite-chain calculations useful preparation for the broader theory.

References

• Faddeev, L. D. How Algebraic Bethe Ansatz works for integrable model. Les Houches lectures, arXiv:hep-th/9605187v1, 1996, 59 pp. Version record. Open PDF.
• 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 above refer to this version. Version record. Open PDF.