Luttinger Liquids

A Luttinger liquid is the generic stable phase of a compressible, translation-invariant one-dimensional quantum fluid with one gapless mode. Its low-energy excitations are collective density waves, not long-lived fermionic quasiparticles. Two parameters—the sound velocity $u$ and dimensionless Luttinger parameter $K$ —control thermodynamics and an infinite family of correlation exponents.

Required background. The Abelian Bosonization Dictionary fixes the fields and vertex dimensions; Fermi-Gas Surface Kinematics supplies the two-Fermi-point linearization; Complex Coordinates and Local Conformal Transformations supplies the cylinder map used for finite-temperature correlations.

The Gaussian fixed line

For one conserved density mode, locality, U(1) symmetry, and parity allow the leading Hamiltonian

H=\frac{u}{2\pi}\int dx\left[K(\partial_x\theta)^2 +K^{-1}(\partial_x\phi)^2\right], \qquad [\phi(x),\partial_y\theta(y)]=i\pi\delta(x-y).

The density and current are

\delta\rho=-\frac1\pi\partial_x\phi, \qquad j=\frac1\pi\partial_t\phi=\frac{uK}{\pi}\partial_x\theta.

Hamilton’s equations give $\partial_t^2\phi=u^2\partial_x^2\phi$ : the infrared spectrum is a linearly dispersing sound mode. The parameter $u$ changes units between space and time, whereas $K$ fixes fluctuations. Repulsive spinless fermions normally have $K<1$ , attraction gives $K>1$ , and the free value is $K=1$ in this convention. These inequalities are useful tendencies, not definitions; long-range interactions and unusual microscopic constraints require direct matching.

The theory is a fixed line: forward-scattering interactions change $u$ and $K$ without opening a gap. Haldane identified this universality for both fermionic and bosonic fluids Haldane 1981, pp. 2585–2609.

Measuring $u$ and $K$

A uniform density change $\delta n=N/L$ costs

\frac{E_N-E_0}{L}=\frac{\pi u}{2K}(\delta n)^2.

Therefore the zero-temperature compressibility is $\kappa=K/(\pi u n^2)$ if $\kappa=n^{-2}\partial n/\partial\mu$ , or $\partial n/\partial\mu=K/(\pi u)$ without the conventional $n^{-2}$ . A phase twist costs an energy proportional to $uK$ , defining the charge or superfluid stiffness. Their product and ratio determine $u$ and $K$ independently.

Galilean invariance imposes the additional identity $uK=\pi n/m$ for particles of mass $m$ with $\hbar=1$ . A lattice breaks Galilean invariance, so $uK$ must then be measured rather than inferred. Thermodynamics gives the universal low-temperature energy density

\frac{E(T)-E(0)}{L}=\frac{\pi T^2}{6u}+o(T^2),

for a single mode of central charge $c=1$ .

Power laws instead of quasiparticles

For $V_{m,n}=e^{i(m\phi+n\theta)}$ ,

\Delta_{m,n}=\frac14\left(m^2K+\frac{n^2}{K}\right).

The leading $2k_F$ density and pairing correlations are consequently

\begin{aligned} \langle\rho_{2k_F}(x)\rho_{2k_F}(0)\rangle&\sim \cos(2k_Fx)|x|^{-2K},\\ \langle\Delta^\dagger(x)\Delta(0)\rangle&\sim |x|^{-2/K}. \end{aligned}

Density correlations dominate for $K<1$ , while pairing correlations dominate for $K>1$ ; one-dimensional continuous symmetries still have no true long-range order in the ground state of this short-range system. The fermion correlator decays as $|x|^{-(K+K^{-1})/2}$ , so its momentum distribution has a power-law cusp rather than a Fermi-liquid jump. The bulk tunneling density of states behaves as

\nu(\omega)\propto |\omega|^{\alpha_{\mathrm{bulk}}}, \qquad \alpha_{\mathrm{bulk}}=\frac{K+K^{-1}-2}{2}

for a spinless liquid. Boundary exponents differ because left and right movers are related at an open end.

At temperature $T$ , the conformal substitution $x\mapsto (u/\pi T)\sinh(\pi Tx/u)$ turns each zero-temperature power law into exponential decay beyond the thermal length $u/T$ . Exact amplitudes remain microscopic; only exponents and scaling functions within the linear regime are universal Cazalilla 2004, §§ 3–5.

Stability and its boundaries

The Gaussian theory is stable only against perturbations allowed by microscopic symmetries. At incommensurate filling, umklapp operators oscillate and average away. At commensurability a cosine can become relevant and gap the density mode. Pairing, spin backscattering, disorder, and a lattice impurity have their own dimensions. Long-range Coulomb interactions can make $K$ and the velocity scale dependent, while band curvature controls nonlinear spectral edges. Thus “Luttinger liquid” specifies an infrared regime and its operator content, not all energies of a one-dimensional material.

Exercises

Derive the static compressibility from the finite-size charge energy.

Solution

With $N=L\delta n$ , $E/L=(\pi u/2K)(\delta n)^2$ . Differentiating gives $\delta\mu=(\pi u/K)\delta n$ , hence $\partial n/\partial\mu=K/(\pi u)$ . Multiplying by $n^{-2}$ gives the thermodynamic convention quoted above.

Which correlation decays more slowly at $K=1/2$ ?