Spinful and Multicomponent Luttinger Liquids

Several conserved species produce several compact bosonic modes. For spin-$1/2$ fermions with spin-rotation symmetry, the universal basis separates total charge from spin; more generally, the normal modes are linear combinations determined by the interaction and compactification lattice. Diagonalizing the quadratic form is only part of the problem: allowed nonlinear operators decide which modes remain gapless.

Required background. Luttinger Liquids supplies the single-mode Hamiltonian and dimensions. Helpful background. Zero Modes, Klein Factors, and Compactification supplies the global charge lattice and interspecies fermion signs.

Charge and spin fields

Bosonize each spin projection $\sigma=\uparrow,\downarrow$ with fields $\phi_\sigma,\theta_\sigma$, then define

$$\phi_c=\frac{\phi_\uparrow+\phi_\downarrow}{\sqrt2}, \quad \phi_s=\frac{\phi_\uparrow-\phi_\downarrow}{\sqrt2}, \qquad \theta_c=\frac{\theta_\uparrow+\theta_\downarrow}{\sqrt2}, \quad \theta_s=\frac{\theta_\uparrow-\theta_\downarrow}{\sqrt2}.$$

The orthogonal transformation preserves $[\phi_\nu(x),\partial_y\theta_{\nu'}(y)]=i\pi\delta_{\nu\nu'}\delta(x-y)$. Away from commensurability, the fixed-point Hamiltonian is

$$H=\sum_{\nu=c,s}\frac{u_\nu}{2\pi}\int dx \left[K_\nu(\partial_x\theta_\nu)^2 +K_\nu^{-1}(\partial_x\phi_\nu)^2\right].$$

Distinct velocities $u_c$ and $u_s$ imply spin–charge separation. Exact SU(2) spin symmetry pins the infrared value $K_s=1$ in this normalization, although a marginally irrelevant current interaction produces logarithmic corrections. The charge parameter $K_c$ remains interaction dependent. The separation and the marginal spin correction follow from the weak-coupling $g$-ology flow Solyom 1979, §§ 3–5.

Operators and exponents

With independent Klein factors $\eta_{r\sigma}$,

$$\psi_{r\sigma}\propto\eta_{r\sigma} \exp\left[-\frac{i}{\sqrt2} \left(r\phi_c-\theta_c+\sigma(r\phi_s-\theta_s)\right)\right],$$

where $\sigma=+1$ for $\uparrow$ and $-1$ for $\downarrow$. Its bulk dimension is

$$\Delta_\psi=\frac18 \left(K_c+K_c^{-1}+K_s+K_s^{-1}\right).$$

For an SU(2)-invariant gapless spin sector, the bulk tunneling density-of-states exponent is

$$\alpha_{\mathrm{bulk}}=2\Delta_\psi-1 =\frac{K_c+K_c^{-1}-2}{4}.$$

The leading order parameters factor into charge and spin vertices. Schematically,

$$O_{\mathrm{CDW}}\sim e^{-i\sqrt2\phi_c}\cos(\sqrt2\phi_s), \qquad O_{\mathrm{SS}}\sim e^{i\sqrt2\theta_c}\cos(\sqrt2\phi_s).$$

When the spin sector is gapless, their equal-time exponents are $K_c+1$ and $K_c^{-1}+1$, respectively, up to logarithms. Repulsion therefore favors density correlations, attraction singlet pairing. If spin backscattering becomes relevant and pins $\phi_s$, the spin factor acquires a nonzero expectation value: the Luther–Emery liquid has one gapless charge mode, exponentially decaying single-particle correlations, and competing CDW and pairing powers $K_c$ and $K_c^{-1}$ Luther and Emery 1974, pp. 589–592.

Beyond two components

For $M$ species, collect fields in vectors $\boldsymbol\phi,\boldsymbol\theta$. If parity or time reversal forbids density–current mixing—or after an allowed canonical transformation removes it—the stable quadratic Hamiltonian can be written with positive matrices,

$$H=\frac1{2\pi}\int dx\left[ (\partial_x\boldsymbol\theta)^{\mathsf T}V_\theta (\partial_x\boldsymbol\theta) +(\partial_x\boldsymbol\phi)^{\mathsf T}V_\phi (\partial_x\boldsymbol\phi)\right].$$

A canonical linear transformation simultaneously reduces the dynamics to normal modes. It need not be an ordinary orthogonal rotation when $V_\theta$ and $V_\phi$ do not commute. Without the stated symmetry or transformation, a general quadratic theory may also contain $(\partial_x\boldsymbol\phi)^{\mathsf T}W(\partial_x\boldsymbol\theta)$ and its transpose. Correlator exponents are quadratic forms in the integer vertex-charge vectors. The canonical map must carry the integral compactification lattice to its image in the normal-mode basis; diagonalizing the real matrices while treating that lattice as continuous can admit nonphysical vertices.

Mode separation is exact only at the fixed point. Velocity degeneracies, symmetry-breaking interactions, intermode drag, and nonlinear dispersion can mix observables even when the quadratic Hamiltonian is diagonal. Relevant cosine operators may gap some combinations and leave others gapless. A central-charge measurement counts surviving modes but does not identify their microscopic quantum numbers.

Checks and limits

At the noninteracting point $K_c=K_s=1$ and $u_c=u_s=v_F$, giving $2\Delta_\psi=1$ and a constant bulk density of states. Spin–charge separation is nevertheless not merely the inequality of velocities: it is factorization into sectors carrying different conserved quantum numbers. In a finite system, selection rules correlate charge and spin zero modes, so their oscillator spectra factorize more readily than the entire Hilbert space.

Exercises

1. Derive the spinful bulk tunneling exponent for SU(2) symmetry.

Solution

Insert $K_s=1$ into $2\Delta_\psi=(K_c+K_c^{-1}+K_s+K_s^{-1})/4$. The local spectral exponent is $2\Delta_\psi-1$, hence $\alpha=(K_c+K_c^{-1}+2)/4-1=(K_c+K_c^{-1}-2)/4$.

2. Compare CDW and singlet-pair exponents in a spin-gapped liquid at $K_c=2$.

Solution

Pinning $\phi_s$ removes its algebraic factor. CDW correlations decay as $|x|^{-K_c}=|x|^{-2}$; singlet pairing decays as $|x|^{-1/K_c}=|x|^{-1/2}$. Pairing is dominant, though still only quasi-long-range.

References

• Luther, A., and V. J. Emery. “Backward Scattering in the One-Dimensional Electron Gas.” Physical Review Letters 33 (1974): 589–592. DOI.
• Sólyom, J. “The Fermi Gas Model of One-Dimensional Conductors.” Advances in Physics 28 (1979): 201–303. DOI.