One-Dimensional Quantum Matter

Strong infrared fluctuations reshape quantum matter in one dimension. Bosonization connects microscopic particles to collective charge and spin fields, making power laws, interaction-induced gaps, and boundary transport accessible.

Enter this chapter

Begin with Jordan–Wigner Maps and the Lattice–Continuum Dictionary to translate a spin chain into fermions before taking its continuum limit.

Helpful background. Jordan–Wigner Maps and the Lattice–Continuum Dictionary and Free Bosons and Vertex Operators develop the definitions used by the chapter’s opening pages.

Topics

1. Jordan–Wigner Maps and the Lattice–Continuum Dictionary — How do Jordan–Wigner strings and linearization map spin chains or lattice fermions to continuum right- and left-moving fields?
2. The Abelian Bosonization Dictionary — How are one-dimensional chiral fermions, currents, densities, and vertex operators translated into compact bosonic fields in a convention-complete way?
3. Zero Modes, Klein Factors, and Compactification — How do compact zero modes, winding sectors, boundary conditions, and Klein factors complete the bosonization dictionary beyond oscillator modes?
4. Luttinger Liquids — Why do generic gapless one-dimensional fluids form a Luttinger-liquid universality class rather than a Fermi liquid?
5. Spinful and Multicomponent Luttinger Liquids — How do charge, spin, flavor, and coupled modes diagonalize in spinful or multicomponent Luttinger liquids?
6. Sine–Gordon Perturbations, Commensurability, and Duality — How do cosine perturbations encode gaps, locking, solitons, and commensurate transitions in one-dimensional matter?
7. Non-Abelian Bosonization and Spin Sectors — How do affine currents and WZW fields encode spin and flavor sectors of multicomponent one-dimensional fermions?
8. Spin–Charge Separation and Spectral Observables — How does spin–charge separation appear in spectral, tunneling, and dynamical-correlation observables rather than only in diagonalized fields?
9. Impurities, Junctions, and Transport in One Dimension — How do local scatterers and junctions drive boundary RG flows and modify conductance in a Luttinger liquid?

How the ideas fit together

The Jordan–Wigner transformation is exact on the lattice, but linearizing the spectrum selects a low-energy regime. Bosonization then trades fermionic operators for compact collective fields. Their stiffness fixes correlation exponents, while allowed nonlinear operators determine whether the gapless liquid remains stable Haldane 1981, pp. 2585–2609.

Zero modes and Klein factors carry global charges and anticommutation information that local oscillator correlators alone miss. Likewise, a clean bulk power law does not determine transport through an impurity: boundary perturbations have their own scaling and can qualitatively change the measured conductance.

The comparison below separates the assumptions and tests needed for these different conclusions.

Convention and validity table

The Gaussian entries assume positive velocities and stiffnesses, $u>0$ and $K>0$ for each gapless mode, in the low-energy window where linearization applies. Bulk powers are zero-temperature, long-distance fixed-point results; marginal interactions, finite temperature, and finite size require their own corrections or crossovers. Allowed vertices must lie on the physical compactification lattice. “Relevant” refers to a weak perturbation at the stated bulk or boundary fixed point; a strong-coupling phase still requires symmetry and compactification checks.

Low-energy operator dimensions and their validity conditions. The comparison keeps local scaling data together with the global or boundary information needed to use them.

Object or claim	This chapter’s convention	Scaling statement	Global or boundary datum	Direct check	Failure if omitted
Smooth density	$\delta\rho=-\partial_x\phi/\pi$	derivative operator	$\phi(x+L)=\phi(x)-\pi N$	integrated charge is $N$	fractional or sign-reversed charge
Spinless fermion	$\psi_r\propto\eta_r e^{-i(r\phi-\theta)}$	$2\Delta_\psi=(K+K^{-1})/2$	Klein factors and parity twist	$K=1$ gives $1/x$	wrong interspecies signs or ring spectrum
General vertex	$V_{m,n}=e^{i(m\phi+n\theta)}$	$\Delta=(m^2K+n^2/K)/4$	$(m,n)$ must lie on the physical vertex lattice	charge, momentum, exchange phase	spurious local operators
Density locking	$\cos(2p\phi)$	bulk relevant for $p^2K<2$	commensurability and translation	kink charge magnitude $1/p$	a rapidly oscillating term is treated as a gap
Phase locking	$\cos(2q\theta)$	bulk relevant for $q^2/K<2$	charge symmetry must permit it	duality under $K\leftrightarrow K^{-1}$	simultaneous sharp locking of conjugate fields
Single spinless-channel backscattering	$\cos[2\phi(0)]$	boundary dimension $K$ at the transmitting fixed point	single channel; specified reservoirs and boundary conditions	two identical open ends: weak-link dimension $1/K$	bulk and boundary relevance are confused
Spinful electron	charge and spin vertices divided by $\sqrt2$	$\alpha_{\mathrm{bulk}}=(K_c+K_c^{-1}-2)/4$ for a gapless SU(2) spin fixed point	correlated charge–spin zero modes	free limit $K_c=1$	incorrect tunneling exponent
Spectral function	$A=-2\operatorname{Im}G^R$	generic interacting gapless thresholds; the free coincident-velocity limit has a pole	temperature, curvature, transverse hopping	$\int d\omega\,A/(2\pi)=1$ for the complete canonical fermion over all frequencies	intensity is mistaken for normalized spectral weight

The gapless spin qualification is essential: SU(2) symmetry alone does not exclude a spin gap Kane and Fisher 1992, § II, pp. 15236–15237. The weak-barrier and weak-link dimensions refer to the two stated spinless boundary fixed points, not a generic multichannel junction Kane and Fisher 1992, §§ III A–B, pp. 15238–15240. For the spectral row, $K_c=K_s=1$ and $u_c=u_s=v_F$ recover the free pole; a truncated low-energy spectral window need not carry unit weight.

Download the comparison data (JSON).

Where to go next

Continue to Quantum Magnetism and Frustration to derive effective spin interactions from the microscopic hierarchy.

References

• Haldane, F. D. M. “‘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: Solid State Physics 14 (1981): 2585–2609. DOI.

• Kane, C. L., and M. P. A. Fisher. “Transmission through Barriers and Resonant Tunneling in an Interacting One-Dimensional Electron Gas.” Physical Review B 46 (1992): 15233–15262. DOI.