XXZ Spin Chain

The spin-$1/2$ XXZ chain is the nearest-neighbor Heisenberg chain with different exchange strengths along the distinguished $z$ axis and in the transverse $xy$ plane. In a standard convention,

$$H = J \sum_j \left( s_j^x s_{j+1}^x + s_j^y s_{j+1}^y + \Delta s_j^z s_{j+1}^z \right) - h \sum_j s_j^z.$$

Here $J$ and $h$ have units of energy, the $s_j^\alpha$ are dimensionless spin-$1/2$ operators, and $\Delta$ is the dimensionless exchange anisotropy. This one parameter connects a free-fermion chain, a gapless interacting quantum liquid, the isotropic antiferromagnet, a gapped Néel phase, and a ferromagnet.

The model is unusually valuable because several descriptions meet without becoming interchangeable. It is a local spin Hamiltonian, an interacting spinless-fermion model after a Jordan–Wigner transformation, an integrable many-body system solved by factorized scattering, and a compact-boson field theory at low energy in its critical regime. Each description answers a different class of questions.

Helpful background. Heisenberg exchange supplies the isotropic limit; thermodynamic limits explains the difference between finite spectra and phases.

Axial exchange and spin normalization

There is one spin one half at each of $L$ sites, so $\mathcal H=(\mathbb C^2)^{\otimes L}$. Throughout the phase discussion $J>0$, the field is longitudinal, and the lattice spacing is $a$. The lattice momenta $k,q$ below are dimensionless; their physical wave vectors are $k/a,q/a$.

The operators satisfy $[s_i^\alpha,s_j^\beta]=i\delta_{ij}\epsilon_{\alpha\beta\gamma}s_j^\gamma$ and $\mathbf s_j^2=3/4$. Physical angular momentum is $\hbar\mathbf s_j$. With $s_j^\alpha=\sigma_j^\alpha/2$, the same Hamiltonian reads

$$H=\frac J4\sum_j(\sigma_j^x\sigma_{j+1}^x+\sigma_j^y\sigma_{j+1}^y +\Delta\sigma_j^z\sigma_{j+1}^z)-\frac h2\sum_j\sigma_j^z.$$

The factors of four and two must be kept when comparing exchange and field scales. The XXZ chain reference collects the model’s exact benchmarks; the Heisenberg model develops isotropic exchange on more general lattices.

Open and Periodic Chains

For an open chain,

$$H_{\mathrm{OBC}} = J \sum_{j=1}^{L-1} \left( s_j^x s_{j+1}^x + s_j^y s_{j+1}^y + \Delta s_j^z s_{j+1}^z \right) - h\sum_{j=1}^{L}s_j^z.$$

For a periodic chain,

$$s_{L+1}^\alpha = s_1^\alpha,$$

and, for $L\ge3$, the exchange sum has $L$ distinct bonds. Open chains have $L-1$ bonds. This difference is order one in the total energy but vanishes in the bulk energy density as $L\to\infty$.

Periodic spin boundary conditions do not become one universal periodic boundary condition for Jordan–Wigner fermions. The fermion boundary term depends on total fermion parity. Ignoring that sector dependence can shift finite-size momenta and apparent gaps.

Exchange Anisotropy

Write the exchange tensor as

$$J_x=J_y=J, \qquad J_z=J\Delta.$$

The parameter $\Delta$ compares longitudinal and transverse exchange:

Regime	Conventional name	Dominant tendency for $J>0$
$\Delta=0$	XX limit	transverse exchange and spin transport
$\lvert\Delta\rvert<1$	easy plane	fluctuating $xy$ correlations
$\Delta=1$	isotropic Heisenberg point	full rotationally invariant exchange
$\Delta>1$	antiferromagnetic easy axis	alternating $z$ alignment
$\Delta<-1$	ferromagnetic easy axis	uniform $z$ polarization

The phrases easy plane and easy axis describe the exchange anisotropy, not a proof that a finite one-dimensional ground state has a classical orientation. Quantum fluctuations, symmetry, and the thermodynamic limit still determine the phase.

Raising and Lowering Form

Define

$$s_j^\pm = s_j^x \pm i s_j^y.$$

Then

$$s_j^x s_{j+1}^x + s_j^y s_{j+1}^y = \frac12 \left( s_j^+s_{j+1}^- + s_j^-s_{j+1}^+ \right).$$

The Hamiltonian becomes

$$H = {} \frac{J}{2} \sum_j \left( s_j^+s_{j+1}^- + s_j^-s_{j+1}^+ \right) + J\Delta \sum_j s_j^z s_{j+1}^z - h\sum_j s_j^z.$$

The transverse term moves a down spin relative to an up-spin background. The longitudinal term assigns an interaction energy according to neighboring $z$ projections. This separation anticipates the hopping-plus-density-interaction fermion form.

Continuous Spin Symmetry

For generic $\Delta$, simultaneous rotations of every spin about the $z$ axis leave the Hamiltonian invariant. The generator is

$$S_{\mathrm{tot}}^z = \sum_{j=1}^{L}s_j^z.$$

Because every transverse exchange term raises one site and lowers its neighbor,

$$[H,S_{\mathrm{tot}}^z] = 0.$$

The Hilbert space decomposes into fixed-magnetization sectors. If $M$ sites are down relative to the all-up state, then

$$S_{\mathrm{tot}}^z = \frac{L}{2}-M,$$

and the sector dimension is

$$\dim\mathcal H_M = \binom{L}{M}.$$

This $U(1)$ symmetry is both physical and computational: exact diagonalization can work one magnetization block at a time.

Symmetry Enhancement at the Isotropic Point

At $\Delta=1$ and $h=0$,

$$H = J\sum_j \mathbf s_j\cdot\mathbf s_{j+1},$$

so all components of total spin commute with $H$:

$$[H,S_{\mathrm{tot}}^\alpha] = 0, \qquad \alpha=x,y,z.$$

The internal symmetry is enhanced from axial $U(1)$ to $SU(2)$. A longitudinal field retains rotations about $z$ but lifts degeneracies between different values of $S_{\mathrm{tot}}^z$.

The SU(2) reference develops the group structure. The XXZ page uses only the consequence that $\Delta=1$ is a symmetry-enhanced point, not a generic representative of the entire anisotropic family.

Discrete and Lattice Symmetries

At $h=0$, a global $\pi$ rotation about the $x$ axis sends

$$s_j^z \longmapsto -s_j^z, \qquad s_j^y \longmapsto -s_j^y,$$

while leaving $s_j^x$ unchanged. It reverses the total magnetization and preserves the Hamiltonian.

For uniform periodic couplings, the model also has lattice translation and spatial inversion symmetry. At $h=0$ it is time-reversal invariant. Boundaries, bond alternation, disorder, Dzyaloshinskii–Moriya exchange, or site-dependent fields can remove some of these symmetries even if $S_{\mathrm{tot}}^z$ remains conserved.

Zero-field phases and their bulk gaps

Assume $J>0$, $h=0$, short-range nearest-neighbor exchange, and the thermodynamic limit.

Anisotropy	Ground-state regime	Gap	Long-distance structure
$\Delta<-1$	$z$-ferromagnet	gapped away from $\Delta=-1$	saturated magnetization
$-1<\Delta<1$	critical easy-plane phase	gapless	algebraic correlations, $c=1$
$\Delta=1$	isotropic critical point	gapless	algebraic correlations with logarithmic corrections
$\Delta>1$	easy-axis antiferromagnet	gapped	thermodynamic Néel order

Here “gapped” means a gap above the thermodynamic ground-state manifold; degenerate partners are excluded. The critical interval is a phase, not one isolated critical point. Its exponents vary continuously with $\Delta$ through the Luttinger parameter $K$.

Finite chains require more careful language. A finite even periodic chain in the critical or antiferromagnetic regimes may have a unique symmetry eigenstate. Thermodynamic order is diagnosed through size scaling, long-distance correlations, symmetry-partner structure, or a symmetry-breaking source with an explicit order of limits.

For $J>0$ and $h=0$, the gapless interval lies between the ferromagnetic endpoint at $\Delta=-1$ and the BKT boundary at $\Delta=1$. Within it, $K$ decreases through the free-fermion value $K=1$ at $\Delta=0$ and approaches $1/2$ at the isotropic antiferromagnet. The diverging $K$ near $-1$ accompanies a vanishing velocity; the endpoint itself is not an ordinary conformal liquid.

Ferromagnetic Regime

For $\Delta<-1$, the all-up and all-down product states minimize every zero-field bond and are exact finite-chain ground states on a connected chain. For a periodic chain,

$$\lvert F_\uparrow\rangle = \lvert\uparrow\uparrow\cdots\uparrow\rangle, \qquad \lvert F_\downarrow\rangle = \lvert\downarrow\downarrow\cdots\downarrow\rangle.$$

Their exchange energy is

$$E_F = \frac{J\Delta L}{4}.$$

The transverse exchange annihilates each fully polarized state. A single overturned spin can propagate, but at $h=0$ its thermodynamic minimum excitation energy is

$$\Delta_{\mathrm{mag}} = J(-\Delta-1), \qquad \Delta<-1.$$

This gap closes as $\Delta\to-1^-$.

The Endpoint at Minus One

On an open chain of any length, or an even periodic ring, rotate every other spin by $\pi$ about the $z$ axis. The transformation changes the signs of $s^x$ and $s^y$ on one endpoint of every bond while leaving $s^z$ unchanged. At $\Delta=-1$,

$$H(-1) \longmapsto -J \sum_j \mathbf s_j\cdot\mathbf s_{j+1}.$$

Thus the endpoint is unitarily equivalent to an isotropic ferromagnetic Heisenberg chain. Its finite-size ground space is much more degenerate than the generic critical ground state.

For spin $1/2$, the ferromagnetic total-spin multiplet has

$$S_{\mathrm{tot}} = \frac{L}{2}, \qquad \dim\mathcal G = 2S_{\mathrm{tot}}+1 = L+1.$$

The staggered rotation preserves this dimension at $\Delta=-1$ for these bipartite boundary conditions. An odd ring retains a frustrated closing bond.

The low-energy one-magnon dispersion near its minimum is quadratic, not relativistic. If $k=\pi+q$ at $h=0$,

$$\varepsilon(\pi+q) = J\left[1-\cos q\right] \sim \frac{Jq^2}{2}.$$

Consequently $\Delta=-1$ is not included as an ordinary $c=1$ conformal point of the interval $-1<\Delta<1$. Both the Luttinger velocity and the assumptions behind the linear low-energy theory become singular there.

Critical Easy-Plane Regime

For

$$-1 < \Delta < 1,$$

the chain is gapless. It has no conventional long-range transverse order, but spin correlations decay algebraically. Low-energy excitations live near two Fermi points in the fermion description and become collective density and phase modes once interactions are included.

This phase is a Luttinger liquid. Luttinger-liquid theory develops bosonization and perturbation criteria; the formulas below give the XXZ parameters in that convention. The phase has:

• linear low-energy dispersion;
• central charge $c=1$;
• continuously varying correlation exponents;
• power-law finite-size gaps of order $1/L$;
• logarithmic interval entanglement;
• interacting fermion correlations without an ordinary Fermi-liquid quasiparticle pole, except at the free XX point. At that point the Jordan–Wigner fermion has a sharp pole even though a local spin-flip response includes its nonlocal string.

The word liquid here names a universality class of one-dimensional gapless systems. It does not imply a continuum fluid at the microscopic scale.

The Isotropic Antiferromagnet

At $\Delta=1$, the spin-$1/2$ antiferromagnetic Heisenberg chain remains gapless. Its low-energy velocity is

$$v = \frac{\pi J a}{2\hbar},$$

where $a$ is the lattice spacing.

The continuum fixed point has $c=1$ and the limiting Luttinger value $K=1/2$ in the convention used below. A marginally irrelevant operator produces multiplicative logarithmic corrections. Finite-size fits that ignore those corrections can converge slowly and imitate shifted exponents.

The exact ground-state energy density is

$$\frac{e_0}{J} = \frac14 - \ln2.$$

This number is a useful normalization and numerical benchmark.

Antiferromagnetic Easy-Axis Regime

For $\Delta>1$, longitudinal exchange favors the two alternating configurations

$$\lvert N_1\rangle = \lvert\uparrow\downarrow\uparrow\downarrow\cdots\rangle,$$

and

$$\lvert N_2\rangle = \lvert\downarrow\uparrow\downarrow\uparrow\cdots\rangle.$$

At large $\Delta$, the transverse exchange creates quantum fluctuations around these Ising-like configurations. The thermodynamic phase has staggered $z$ order and a nonzero bulk gap.

A convenient order parameter is the staggered magnetization

$$m_{\mathrm{st}} = \lim_{L\to\infty} \frac1L \sum_{j=1}^{L} (-1)^j \langle s_j^z\rangle,$$

with a symmetry-breaking prescription understood. In a finite translation-invariant state, $\langle s_j^z\rangle$ may vanish even when

$$\lim_{r\to\infty} (-1)^r \langle s_j^z s_{j+r}^z\rangle \ne 0$$

after the thermodynamic limit.

The Transition at One

The boundary at $\Delta=1$ is of Berezinskii–Kosterlitz–Thouless type. The critical side approaches $K=1/2$, while an umklapp perturbation becomes relevant on the easy-axis side and pins the low-energy field.

The gap does not open as a simple power

$$\Delta_{\mathrm{gap}} \not\propto (\Delta-1)^\nu$$

with an ordinary finite exponent $\nu$. It has an essential singularity. This makes numerical location of the transition delicate: modest sizes can remain much shorter than the rapidly growing correlation length.

Quantum phase transitions explains scaling; renormalization group explains fixed lines and BKT flow. Here the important model-specific fact is that the first-order boundary at $\Delta=-1$ and the BKT boundary at $\Delta=1$ require different diagnostics.

A One-Magnon Check

Let

$$\lvert F_\uparrow\rangle = \lvert\uparrow\uparrow\cdots\uparrow\rangle$$

and define a one-down-spin basis

$$\lvert j\rangle = s_j^- \lvert F_\uparrow\rangle.$$

For a periodic chain, the polarized-state energy is

$$E_F = \frac{J\Delta L}{4} - \frac{hL}{2}.$$

Acting on $\lvert j\rangle$ gives

$$(H-E_F)\lvert j\rangle = {} (h-J\Delta)\lvert j\rangle + \frac{J}{2} \left( \lvert j-1\rangle + \lvert j+1\rangle \right).$$

The momentum state

$$\lvert k\rangle = \frac1{\sqrt L} \sum_{j=1}^{L} e^{ikj}\lvert j\rangle$$

has excitation energy

$$\varepsilon(k) = h - J\Delta + J\cos k.$$

In the bulk, or on an even ring where $k=\pi$ is allowed, the one-magnon minimum is at $k=\pi$. For $\Delta\ge-1$ the exact positive saturation threshold is

$$h \ge h_{\mathrm{sat}} = J(1+\Delta).$$

The one-magnon calculation proves instability below this field. The positive-bond decomposition in the exercise proves global stability at and above it; a one-particle stability test alone would not exclude a many-particle instability in a generic model. An odd finite ring misses $k=\pi$ and needs its own spectrum. For $\Delta<-1$, the zero-field chain is already ferromagnetic: $h=0$ is the boundary between its up and down branches, not the negative value $J(1+\Delta)$.

Exact Two-Site Spectrum

Consider one open bond at $h=0$,

$$H_2 = J \left( s_1^x s_2^x + s_1^y s_2^y + \Delta s_1^z s_2^z \right).$$

The parallel states have energies

$$E_{\uparrow\uparrow} = E_{\downarrow\downarrow} = \frac{J\Delta}{4}.$$

In the zero-magnetization sector, define

$$\lvert t_0\rangle = \frac{ \lvert\uparrow\downarrow\rangle + \lvert\downarrow\uparrow\rangle }{\sqrt2}$$

and

$$\lvert s\rangle = \frac{ \lvert\uparrow\downarrow\rangle - \lvert\downarrow\uparrow\rangle }{\sqrt2}.$$

Their energies are

$$E_{t_0} = \frac{J(2-\Delta)}{4}, \qquad E_s = -\frac{J(2+\Delta)}{4}.$$

At $\Delta=1$, the three triplet states become degenerate at $J/4$, while the singlet lies at $-3J/4$. At $\Delta=-1$, the parallel states cross the singlet. This dimer detects the ferromagnetic boundary, but it cannot reproduce the thermodynamic distinction between a Luttinger liquid and a Néel phase.

Jordan–Wigner Dictionary

Introduce spinless fermions satisfying

$$\{c_i,c_j^\dagger\} = \delta_{ij}, \qquad \{c_i,c_j\} = 0.$$

One useful convention is

$$s_j^z = n_j - \frac12, \qquad n_j = c_j^\dagger c_j,$$

and

$$s_j^+ = c_j^\dagger \exp\!\left( i\pi \sum_{\ell<j}n_\ell \right).$$

The nonlocal string makes spin operators on different sites commute while the $c_j$ anticommute. For adjacent sites, the strings cancel locally in the exchange term.

For an open chain,

$$H = {} \frac{J}{2} \sum_{j=1}^{L-1} \left( c_j^\dagger c_{j+1} + c_{j+1}^\dagger c_j \right) + J\Delta \sum_{j=1}^{L-1} \left(n_j-\frac12\right) \left(n_{j+1}-\frac12\right) - h \sum_{j=1}^{L} \left(n_j-\frac12\right).$$

Thus

$$t = \frac{J}{2}, \qquad V = J\Delta,$$

where $t$ denotes the hopping magnitude. A staggered phase reverses the hopping sign on an open chain or even ring; on an odd ring it changes the boundary twist. In this convention fermions count up spins: $N_f=L-M$ when $M$ counts the down-spin Bethe particles used below. The density interaction remains for $\Delta\ne0$; changing variables does not diagonalize it. Spinless fermion chains explains its filling-dependent physics, and Jordan–Wigner derives the string and parity boundary term.

Free-Fermion XX Limit

At $\Delta=0$, the interaction vanishes. After a staggered gauge transformation if desired, the dispersion can be written

$$\epsilon(k) = -J\cos k - h.$$

At $h=0$, the ground state is half filled, with Fermi points

$$k_F = \pm\frac{\pi}{2}.$$

The energy density is

$$e_0 = -\frac{J}{\pi},$$

and the Fermi velocity is

$$v_F = \frac{Ja}{\hbar}.$$

For example, in the infinite half-filled ground state Wick’s rule gives, for integer $r\ne0$,

$$\langle s_0^z s_r^z\rangle=-|\langle c_0^\dagger c_r\rangle|^2 =-\frac{\sin^2(\pi r/2)}{\pi^2r^2}.$$

This vanishes at nonzero even separations and is negative at odd separations. It verifies the uniform and staggered $K=1$ amplitudes below. Transverse spin correlations have a different power because a spin-flip operator contains the Jordan–Wigner string.

Why the Interacting Chain Is Integrable

The uniform nearest-neighbor XXZ chain belongs to a family generated by the six-vertex transfer matrix. The Yang–Baxter relation implies commuting transfer matrices,

$$[T(u),T(v)] = 0,$$

for spectral parameters $u$ and $v$. Expanding a logarithm of $T(u)$ generates mutually commuting conserved quantities. The Hamiltonian is one member of that hierarchy.

This algebraic statement is stronger than energy conservation and total-magnetization conservation. It constrains many-body scattering so that it factorizes into compatible two-body processes.

Integrability is fragile under generic perturbations. Next-nearest-neighbor exchange, generic disorder, transverse fields, or arbitrary staggered couplings usually destroy the commuting hierarchy even when some ordinary symmetries survive.

The distinction between localization and slow finite-time dynamics in the interacting random-field problem is treated in Many-Body Localization Preview.

Coordinate Bethe Ansatz

Choose the all-up state as a reference and work in a sector with $M$ down spins at ordered positions

$$1 \le x_1 < x_2 < \cdots < x_M \le L.$$

Away from collisions, the wavefunction is a superposition of plane waves:

$$\psi(x_1,\ldots,x_M) = \sum_{P\in S_M} A(P) \exp\!\left( i\sum_{a=1}^{M} k_{P_a}x_a \right).$$

The amplitudes for permutations are related by two-body scattering phases. Periodicity requires each quasiparticle to accumulate both its free phase around the ring and its scattering phases through every other quasiparticle. Schematically,

$$e^{ik_jL} = \prod_{\ell\ne j} S(k_j,k_\ell).$$

These coupled quantization conditions are the Bethe equations. The energy remains additive in the Bethe momenta,

$$E = E_F + \sum_{j=1}^{M} \left( h - J\Delta + J\cos k_j \right),$$

but the allowed $k_j$ are strongly correlated by the scattering equations.

A Rapidity Form of the Bethe Equations

In the critical regime, parameterize

$$\Delta = \cos\gamma, \qquad 0<\gamma<\pi.$$

A representative rapidity convention uses

$$e^{ik(\lambda)} = \frac{ \sinh(\lambda+i\gamma/2) }{ \sinh(\lambda-i\gamma/2) }.$$

The periodic Bethe equations then take the form

$$\left[ \frac{ \sinh(\lambda_j+i\gamma/2) }{ \sinh(\lambda_j-i\gamma/2) } \right]^L = \prod_{\ell\ne j} \frac{ \sinh(\lambda_j-\lambda_\ell+i\gamma) }{ \sinh(\lambda_j-\lambda_\ell-i\gamma) }.$$

Different sources shift, rescale, or negate rapidities and may include an overall phase in the equations. Those forms can be equivalent. A convention must be checked by reconstructing momentum, energy, and the one-magnon limit.

What Exact Solvability Does and Does Not Give

The Bethe construction determines finite-volume spectral data, with singular roots, symmetry descendants and special anisotropies requiring appropriate limiting or completeness prescriptions. In practice:

• the number of roots grows with system size;
• roots can form complex patterns associated with bound states;
• selecting the ground-state root distribution requires care;
• finite-temperature thermodynamics leads to coupled nonlinear integral equations;
• norms and matrix elements require determinant formulas;
• dynamical correlation functions need sums over many intermediate states;
• special anisotropies can carry additional degeneracies and root subtleties.

“Exactly solvable” therefore does not mean “every observable has an elementary closed form.” It means that a nonperturbative spectral structure is available and can anchor controlled analytic and numerical work.

Thermodynamic Bethe Ansatz Preview

As $L,M\to\infty$ at fixed density, discrete Bethe roots condense into distributions. Sums become integrals,

$$\frac1L \sum_{j=1}^{M} f(\lambda_j) \longrightarrow \int d\lambda\, \rho(\lambda)f(\lambda).$$

The logarithmic Bethe equations become integral equations for root and hole densities. At finite temperature, minimizing the free energy subject to those constraints produces thermodynamic Bethe-ansatz equations for dressed energies.

This procedure is model-specific. Ordinary canonical and grand-canonical ensembles remain the statistical framework; integrability supplies an exact way to evaluate their thermodynamics for this special Hamiltonian. Post-quench stationary root densities and the charge-completeness problem are developed in Integrability and Generalized Gibbs Ensembles Preview.

Luttinger-Liquid Hamiltonian

Throughout $-1<\Delta<1$, the long-wavelength sector is described by conjugate compact fields $\phi(x)$ and $\theta(x)$. One common normalization is

$$[\phi(x),\partial_y\theta(y)] = i\pi\delta(x-y).$$

The universal quadratic Hamiltonian is

$$H_{\mathrm{LL}} = \frac{\hbar v}{2\pi} \int dx \left[ K(\partial_x\theta)^2 + \frac1K(\partial_x\phi)^2 \right].$$

The velocity $v$ sets the low-energy light cone. The dimensionless Luttinger parameter $K$ sets scaling dimensions and the relative stiffness of density and phase fluctuations.

The numerical value called $K$ depends on field normalization. Formulas must be compared as a complete package: commutator, Hamiltonian, operator dictionary, and correlation exponents.

Exact Luttinger Parameters at Zero Magnetization

For

$$\Delta = \cos\gamma, \qquad 0<\gamma<\pi,$$

the zero-field XXZ chain has

$$K = \frac{\pi}{2(\pi-\gamma)}$$

and

$$v = \frac{Ja}{\hbar} \frac{\pi\sin\gamma}{2\gamma}.$$

Useful checks are

$$\begin{array}{c|c|c} \Delta & K & v \\ \hline 0 & 1 & Ja/\hbar \\ 1^- & 1/2 & \pi Ja/(2\hbar) \\ -1^+ & \infty & 0 \end{array}$$

The divergent $K$ and vanishing $v$ at $\Delta\to-1^+$ signal the failure of an ordinary finite-velocity conformal description at the endpoint.

Long-Distance Correlations

At zero magnetization and away from endpoint subtleties, the leading forms are

$$\begin{aligned} \langle s_0^z s_r^z\rangle \sim {}& -\frac{K}{2\pi^2r^2} + A_z \frac{(-1)^r}{r^{2K}}, \\ \langle s_0^+s_r^-\rangle \sim {}& A_\perp \frac{(-1)^r}{r^{1/(2K)}}. \end{aligned}$$

The amplitudes $A_z$ and $A_\perp$ are nonuniversal. The exponents are universal once the $K$ convention is fixed.

At the XX point, $K=1$, so the leading transverse exponent is $1/2$. At the isotropic limit, $K=1/2$, and the leading staggered longitudinal and transverse powers both approach $1/r$, as required by restored spin rotation symmetry. Multiplicative logarithms modify the pure powers at $\Delta=1$.

Correlation Does Not Mean Order

An algebraic correlator tends to zero as $r\to\infty$. The gapless phase therefore has quasi-long-range order rather than a nonzero transverse order parameter. By contrast, in the easy-axis Néel phase,

$$(-1)^r \langle s_0^z s_r^z\rangle \longrightarrow m_{\mathrm{st}}^2$$

in an appropriate thermodynamic state.

Distinguishing these behaviors requires distance and size scaling. A large short-distance staggered correlator on one small chain is not enough.

Entanglement and Central Charge

For a periodic critical chain of physical circumference $\mathcal L=La$, the ground-state entropy of an interval of length $\ell$ is

$$S(\ell) = \frac{c}{3} \ln\!\left[ \frac{\mathcal L}{\pi a_0} \sin\!\left( \frac{\pi\ell}{\mathcal L} \right) \right] + s_1 + \cdots.$$

Here $a_0$ is a short-distance cutoff, $s_1$ is nonuniversal, and

$$c = 1$$

throughout the Luttinger-liquid regime. Open chains have the leading coefficient $c/6$ for an interval adjacent to a boundary.

In either gapped phase, interval entanglement saturates once $\ell$ greatly exceeds the correlation length, apart from finite cat-state or boundary contributions. Entanglement entropy explains the scaling limits and numerical caveats.

Finite-Size Conformal Spectrum

For the zero-magnetization ground state of an even periodic critical chain,

$$E_0(L) = Le_\infty - \frac{\pi\hbar v c}{6La} + o(L^{-1}).$$

Low-energy gaps behave as

$$E_n(L)-E_0(L) = \frac{2\pi\hbar v}{La} x_n + o(L^{-1}),$$

where $x_n$ are scaling dimensions, with momentum and winding quantum numbers determined by the compact boson.

These formulas provide a numerical route to $v$, $c$, and $K$, but subleading corrections matter. Near $\Delta=1$, marginal logarithms make naive straight-line extrapolations unreliable.

Spin Current

Conservation of $S_{\mathrm{tot}}^z$ implies a lattice continuity equation. With

$$\dot s_j^z = \mathcal J_{j-1}^z - \mathcal J_j^z,$$

the current of dimensionless $z$ spin is

$$\mathcal J_j^z = \frac{J}{\hbar} \left( s_j^x s_{j+1}^y - s_j^y s_{j+1}^x \right).$$

This current has units of inverse time; multiply by $\hbar$ for angular-momentum flow. The anisotropy term does not move $z$ magnetization; the transverse exchange does. Transport in integrable chains is subtle because conserved quantities can protect ballistic components in some regimes and ensembles. A transport claim must specify temperature, field, order of limits, and the measured response. Transport Coefficients Preview supplies the generic Drude-weight, diffusion, Green–Kubo, and finite-size dictionary; this page retains the model-specific current.

Longitudinal Field

The field $h$ is conjugate to total magnetization. In the fermion language it is a chemical potential. It changes the filling and therefore changes the Luttinger parameter away from the zero-magnetization formula quoted above.

For $-1<\Delta\le1$, a finite interval below saturation remains a gapless Luttinger liquid. At

$$|h| = h_{\mathrm{sat}} = J(1+\Delta),$$

the ground state becomes fully polarized in the stated convention.

For $\Delta>1$, the zero-field Néel phase survives up to a lower critical field set by its gap. Between that field and saturation lies a field-induced gapless regime. The exact lower critical field and dressed parameters require Bethe-ansatz thermodynamics; they are not given by the one-magnon saturation calculation.

Finite Temperature

The zero-field phases above describe $T=0$. A one-dimensional short-range chain has no nonzero-temperature transition into true ferromagnetic or Néel long-range order. Thermal domain walls disrupt the discrete ordered phases; in the critical regime thermal low-energy modes also make spin correlations decay on a finite thermal length.

Low-temperature observables can nevertheless retain quantum-critical scaling over large windows when

$$k_BT \ll J$$

and other gaps or crossover scales are controlled. Calling such a window a phase transition would conflate a crossover with the zero-temperature phase boundary.

Boundary Conditions and Parity

Three boundary issues should be kept separate:

1. Open and periodic spin chains have different bond counts and finite-size spectra.
2. Jordan–Wigner turns a periodic spin boundary into a fermion boundary term containing total parity.
3. Bethe equations depend on periodicity, twists, magnetization sector, and rapidity convention.

For bulk thermodynamics these differences often become subextensive. For exact spectra, momentum quantum numbers, entanglement cuts, and edge physics they remain essential.

A twist can be introduced by

$$s_{L+1}^\pm = e^{\pm i\Phi} s_1^\pm.$$

The dependence of the ground-state energy on $\Phi$ probes spin stiffness. The normalization of stiffness or Drude weight must be stated because conventions differ by factors of $L$, $\pi$, and $\hbar$.

Model Variants

Common extensions include:

• bond-alternating or dimerized XXZ chains;
• next-nearest-neighbor and frustrated exchange;
• random fields or random bonds;
• long-range exchange;
• Dzyaloshinskii–Moriya interactions;
• open chains with boundary fields or impurities;
• higher-spin anisotropic chains;
• ladders and coupled chains;
• quenches and periodic driving.

Some variants retain integrability for special boundary or coupling choices, but generic extensions do not. Phase diagrams and low-energy theories must be re-established rather than inherited automatically from the uniform nearest-neighbor chain.

Exercises

Fermion mapping of one bond

Using $s_j^z=n_j-1/2$, show that the longitudinal exchange becomes a density interaction and identify its one-body and constant pieces when expanded.

Solution

One bond gives

$$J\Delta s_j^z s_{j+1}^z = J\Delta \left(n_j-\frac12\right) \left(n_{j+1}-\frac12\right).$$

Expanding,

$$J\Delta s_j^z s_{j+1}^z = {} J\Delta n_jn_{j+1} - \frac{J\Delta}{2} (n_j+n_{j+1}) + \frac{J\Delta}{4}.$$

The first term is the nearest-neighbor interaction. In a uniform periodic chain, the one-body pieces combine into a chemical-potential shift because each site belongs to two bonds. The final term is a constant. Keeping the centered form avoids losing these convention-dependent shifts.

XX ground-state energy

At $\Delta=h=0$, use the free-fermion dispersion $\epsilon(k)=-J\cos k$ and half filling to compute the bulk ground-state energy density.

Solution

At half filling, the occupied momenta satisfy

$$-\frac{\pi}{2} < k < \frac{\pi}{2}.$$

Therefore

$$\begin{aligned} e_0 &= \int_{-\pi/2}^{\pi/2} \frac{dk}{2\pi} (-J\cos k) \\ &= -\frac{J}{2\pi} \left[ \sin k \right]_{-\pi/2}^{\pi/2} \\ &= -\frac{J}{\pi}. \end{aligned}$$

The slope at either Fermi point has magnitude $J$ in lattice units, giving $v_F=Ja/\hbar$.

Saturation requires global stability

For an even periodic ring with $J>0$ and $\Delta\ge-1$, show that $h\ge J(1+\Delta)$ makes the all-up state a global ground state, not merely stable against one overturned spin. Distribute half the field on each end of every bond.

Solution

For one bond, subtract the energy $J\Delta/4-h/2$ of its up-up state. The other eigenvalues relative to it are

$$E_{\downarrow\downarrow}-E_{\uparrow\uparrow}=h,\qquad E_s-E_{\uparrow\uparrow}=\frac{h-J(1+\Delta)}2,\qquad E_{t_0}-E_{\uparrow\uparrow}=\frac{h-J(\Delta-1)}2.$$

All are nonnegative at the stated field. Every shifted bond operator is therefore positive semidefinite and annihilates the all-up product state, proving global minimality of their sum. Below the threshold the allowed $k=\pi$ one-magnon state has lower energy. At the threshold degeneracy is possible, so saturation does not mean that every ground state must already be fully polarized.

Interpreting a finite-chain spectrum

A numerical study of an even periodic chain finds a unique ground state, a vanishing one-point staggered magnetization, a low gap, and a large staggered structure-factor peak. Does this prove the chain is in the Luttinger-liquid phase?

Solution

No. A finite symmetric ground state can have zero one-point staggered magnetization in both the critical and thermodynamic Néel regimes. A low gap on one size can be a critical gap, a symmetry-partner splitting, or a crossover effect near the BKT boundary. A large structure-factor peak also occurs in systems with large but finite correlation length.

One should compare several sizes and test:

• whether the relevant bulk gaps scale as $1/L$ or approach a nonzero value;
• whether staggered correlations decay algebraically or approach a constant;
• whether entanglement follows a $c=1$ logarithm or saturates;
• whether near-degenerate symmetry partners separate from bulk excitations;
• whether logarithmic and BKT crossover corrections have been controlled.

These complementary limits distinguish the phases.

References

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• F. Franchini, An Introduction to Integrable Techniques for One-Dimensional Quantum Systems, Springer, 2017.
• M. Gaudin, The Bethe Wavefunction, translated by J.-S. Caux, Cambridge University Press, 2014.
• T. Giamarchi, Quantum Physics in One Dimension, Oxford University Press, 2004.
• F. D. M. Haldane, “‘Luttinger Liquid Theory’ of One-Dimensional Quantum Fluids. I. Properties of the Luttinger Model and Their Extension to the General 1D Interacting Spinless Fermi Gas”, Journal of Physics C 14, 2585–2609, 1981.
• A. Luther and I. Peschel, “Calculation of Critical Exponents in Two Dimensions from Quantum Field Theory in One Dimension”, Physical Review B 12, 3908–3917, 1975.
• R. Orbach, “Linear Antiferromagnetic Chain with Anisotropic Coupling”, Physical Review 112, 309–316, 1958.
• M. Takahashi, Thermodynamics of One-Dimensional Solvable Models, Cambridge University Press, 1999.
• C. N. Yang and C. P. Yang, “One-Dimensional Chain of Anisotropic Spin-Spin Interactions. I. Proof of Bethe’s Hypothesis for Ground State in a Finite System”, Physical Review 150, 321–327, 1966.
• C. N. Yang and C. P. Yang, “One-Dimensional Chain of Anisotropic Spin-Spin Interactions. II. Properties of the Ground-State Energy per Lattice Site for an Infinite System”, Physical Review 150, 327–339, 1966.
• C. N. Yang and C. P. Yang, “One-Dimensional Chain of Anisotropic Spin-Spin Interactions. III. Applications”, Physical Review 151, 258–264, 1966.