Jordan–Wigner Maps and the Lattice–Continuum Dictionary

The Jordan–Wigner transformation turns a one-dimensional spin chain into a fermion problem exactly; linearization then turns the low-energy lattice fermions into right- and left-moving continuum fields. The exact map, the boundary-parity sector, and the range of the continuum expansion must all be retained: dropping any one of them can change the spectrum or the allowed perturbations.

Required background. Second-Quantized Bosons and Fermions supplies canonical anticommutation relations and the occupation-number representation. Helpful background. Bethe Quantization and Finite-Volume Spectra provides the finite-size momentum quantization used to check boundary sectors.

The Jordan–Wigner string

For spin $1/2$ on an ordered chain, choose

$$S_j^z=n_j-\frac12, \qquad S_j^+=c_j^\dagger \exp\!\left(i\pi\sum_{\ell<j}n_\ell\right), \qquad n_j=c_j^\dagger c_j .$$

The exponential is the fermion-parity operator to the left of site $j$. It is not decorative: for $j<k$, moving $c_j$ through the string in $S_k^+$ supplies the minus sign needed for spin operators on different sites to commute. At the same site, $n_j^2=n_j$ gives the spin-$1/2$ algebra. This is the convention introduced in the original transformation Jordan and Wigner 1928, pp. 631–651.

Applied to the XXZ chain

$$H=J\sum_j\left(S_j^xS_{j+1}^x+S_j^yS_{j+1}^y +\Delta S_j^zS_{j+1}^z\right)-h\sum_jS_j^z,$$

the bulk Hamiltonian becomes

$$H=\frac J2\sum_j(c_j^\dagger c_{j+1}+c_{j+1}^\dagger c_j) +J\Delta\sum_j\left(n_j-\frac12\right) \left(n_{j+1}-\frac12\right)-h\sum_j\left(n_j-\frac12\right).$$

Thus the XY exchange is hopping and the Ising exchange is a nearest-neighbor density interaction. A staggered gauge transformation $c_j\mapsto(-1)^jc_j$ reverses the hopping sign, so that sign alone is not physical on an open bipartite chain.

The closing bond on a ring

On a periodic spin chain the bond $(L,1)$ crosses the entire string. With the definitions above its fermionic hopping carries the total parity $P=(-1)^{N_f}$; equivalently the fermions obey a parity-dependent twist. The precise periodic/antiperiodic assignment can change under a gauge convention, but the invariant statement is that one must diagonalize each fixed-$P$ sector with its corresponding boundary condition. The classic exact solution keeps this sector dependence explicitly Lieb, Schultz, and Mattis 1961, §§ II–III.

From the lattice to chiral fields

Let the lattice spacing be $a$ and $x=ja$. Near two Fermi points,

$$\frac{c_j}{\sqrt a} =e^{ik_Fx}\psi_R(x)+e^{-ik_Fx}\psi_L(x)+O(a\,\partial_x\psi).$$

The factor $a^{-1/2}$ gives $\psi_r$ dimension $L^{-1/2}$. Substituting into a smooth bilinear separates slowly varying terms from harmonics near $2k_F$:

$$\frac{n_j}{a} =\rho_R+\rho_L +e^{-2ik_Fx}\psi_R^\dagger\psi_L +e^{2ik_Fx}\psi_L^\dagger\psi_R+\cdots .$$

For a dispersion $\varepsilon(k)$, writing $k=rk_F+q$ with $r=+1$ for $R$ and $r=-1$ for $L$ gives

$$\varepsilon(rk_F+q)-\mu=r v_Fq+\frac{q^2}{2m_*}+\cdots, \qquad H_0=-iv_F\int dx\, (\psi_R^\dagger\partial_x\psi_R-\psi_L^\dagger\partial_x\psi_L).$$

The linear theory is valid for $|q|\ll a^{-1}$ and energies below both the bandwidth and any scale at which curvature matters. Curvature is often irrelevant for equilibrium scaling dimensions but is essential for threshold line shapes and late-time dynamics. Likewise, an oscillatory operator may be discarded only after comparing its wave vector with reciprocal-lattice vectors: at commensurate filling, an apparently rapid harmonic can become an allowed umklapp term.

Translation rules and checks

The microscopic translation $T_a$ sends $\psi_R\mapsto e^{ik_Fa}\psi_R$ and $\psi_L\mapsto e^{-ik_Fa}\psi_L$, apart from the shift of their smooth arguments. This phase is the quickest way to decide which continuum monomials respect the lattice. Particle number is

$$N_f=\sum_jn_j\longrightarrow\int dx\,(\rho_R+\rho_L),$$

while lattice momentum contains the large pieces $\pm k_F$ as well as the small chiral momenta. These translation and charge assignments must agree before a continuum perturbation is accepted.

The map is special to a one-dimensional ordering. In more than one spatial dimension a simple string becomes nonlocal in a direction-dependent way. Even in one dimension, a local spin operator transverse to $z$ maps to a nonlocal fermion operator; locality is representation dependent, although spectra and properly transformed correlators agree.

Exercises

1. Verify explicitly that $[S_j^+,S_k^+]=0$ for $j<k$.

Solution

Write $K_j=\exp(i\pi\sum_{\ell<j}n_\ell)$. For $j<k$, $K_k$ contains $(-1)^{n_j}=1-2n_j$, which anticommutes with $c_j^\dagger$, whereas $c_j^\dagger$ anticommutes with $c_k^\dagger$. The two minus signs cancel, giving $c_j^\dagger K_jc_k^\dagger K_k=c_k^\dagger K_kc_j^\dagger K_j$. Hence the spin-raising operators commute at distinct sites.

2. For nearest-neighbor hopping $\varepsilon(k)=J\cos(ka)$, find the half-filled Fermi points and velocity magnitude.

Solution

At half filling and zero chemical potential, $\cos(k_Fa)=0$, so $k_F=\pi/(2a)$ modulo reciprocal lattice vectors. Since $\partial_k\varepsilon=-Ja\sin(ka)$, the two slopes are $\mp Ja$ and $v_F=|J|a$. A staggered gauge change interchanges the slope assignment but not $v_F$.

References

• Jordan, P., and E. Wigner. “Über das Paulische Äquivalenzverbot.” Zeitschrift für Physik 47 (1928): 631–651. DOI.
• Lieb, E., T. Schultz, and D. Mattis. “Two Soluble Models of an Antiferromagnetic Chain.” Annals of Physics 16 (1961): 407–466. DOI.