Sine–Gordon Perturbations, Commensurability, and Duality

A compact Luttinger field becomes a sine–Gordon theory when a symmetry-allowed vertex is added. The cosine can lock a density phase, open a gap, create solitons, or drive a commensurate–incommensurate transition. Which of these occurs is decided by the vertex charge, compactification, and Luttinger parameter—not by the word “cosine” alone.

Required background. Luttinger Liquids supplies vertex dimensions and Gaussian correlations; Relevant, Marginal, and Irrelevant Directions supplies the linearized RG criterion. Helpful background. Free Bosons and Vertex Operators supplies the compact vertex lattice.

Density locking and its scaling dimension

Use

$$H_0=\frac{u}{2\pi}\int dx \left[K(\partial_x\theta)^2+K^{-1}(\partial_x\phi)^2\right].$$

At a commensurability of order $p$, a density-locking perturbation has the form

$$H_g=\frac{g}{(2\pi a_0)^2}\int dx\, \cos(2p\phi-\delta x).$$

$\delta$ measures the mismatch from commensurability. Since $e^{2ip\phi}$ has dimension $p^2K$, the leading flow at $\delta=0$ is

$$\frac{dg}{d\ell}=(2-p^2K)g+O(g^3).$$

Thus weak $g$ is relevant for $p^2K<2$. For integer-filled bosons the leading lattice term has $p=1$, giving the Mott threshold $K<2$. For half-filled spinless fermions the leading umklapp is $\cos4\phi$, so $p=2$ and the threshold is $K<1/2$. These different numbers are the same formula applied to different microscopic operator lattices.

The coupled Berezinskii–Kosterlitz–Thouless flow also renormalizes $K$ at order $g^2$. Close to the separatrix the gap is therefore exponentially small rather than a simple power of the bare distance. The Mott transition and its universal $K$ criterion were developed by Haldane 1981, pp. 2585–2609 and reviewed for both fixed filling and doping-driven transitions by Giamarchi 1997, pp. 975–980.

Locked vacua and solitons

When $g$ flows strong, $\phi$ is pinned at minima of the cosine. Small oscillations are massive. A kink between adjacent minima changes $\phi$ by $\Delta\phi=\pi/p$ and therefore carries charge

$$Q=-\frac1\pi\int dx\,\partial_x\phi=-\frac1p$$

up to orientation. Fractional kink charge does not mean an isolated fractional particle always exists: boundary conditions and confinement can require kinks in combinations with integer total microscopic charge.

The mismatch $\delta$ can be removed from the cosine by shifting $\phi$, at the cost of a term linear in $\partial_x\phi$. It acts as a chemical potential for solitons. Below a critical mismatch the phase remains locked; above it a finite soliton density forms an incommensurate Luttinger liquid. Near the dilute threshold the solitons behave as impenetrable particles and the density grows with a square-root law in the ideal one-dimensional continuum case Pokrovsky and Talapov 1979, pp. 65–67.

Dual locking and mutual exclusion

A phase-locking perturbation

$$\widetilde H=\frac{\widetilde g}{(2\pi a_0)^2}\int dx\,\cos(2q\theta)$$

has dimension $q^2/K$ and is relevant for $q^2/K<2$. Under

$$\phi\longleftrightarrow\theta, \qquad K\longleftrightarrow K^{-1},$$

the Gaussian theory exchanges density and phase locking. Because $\phi$ and $\theta$ are conjugate, they cannot both be sharply pinned. Competing relevant cosines can produce an Ising critical point, a first-order transition, or an intermediate phase depending on their allowed charges and symmetries; duality alone does not choose among these outcomes.

At special couplings the sine–Gordon model refermionizes. In the present normalization, a bulk vertex’s dimension must first be matched to the chosen fermion mass convention; quoting a “free-fermion point” without that translation is ambiguous. The exact sine–Gordon spectrum contains solitons, antisolitons, and, in the attractive regime, bound-state breathers Coleman 1975, pp. 2088–2097.

Validity boundaries

Perturbative dimensions decide the initial RG direction near the Gaussian fixed line. They do not determine a strong-coupling gap amplitude, the order of a transition between competing locked phases, or whether extra microscopic modes intervene. Oscillatory phases, open boundaries, disorder, and retarded interactions must be included before applying the relevance test. Compactification then determines the number of distinct vacua and the quantum numbers of kinks.

Exercises

1. Determine when $\cos(6\phi)$ is relevant and find its elementary kink charge.

Solution

$\cos(6\phi)$ has $2p=6$, so $p=3$. Its dimension is $9K$ and it is relevant for $K<2/9$. Adjacent minima differ by $\Delta\phi=\pi/3$, giving $|Q|=|\Delta\phi|/\pi=1/3$.

2. Find the relevance condition for $\cos(4\theta)$.

Solution

Here $2q=4$, so $q=2$ and the dimension is $4/K$. Relevance requires $4/K<2$, or $K>2$. This is the dual of the $p=2$ density cosine under $K\leftrightarrow K^{-1}$.

References

• Coleman, S. “Quantum Sine-Gordon Equation as the Massive Thirring Model.” Physical Review D 11 (1975): 2088–2097. DOI.
• Giamarchi, T. “Mott Transition in One Dimension.” Physica B: Condensed Matter 230–232 (1997): 975–980. DOI.
• 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.
• Pokrovsky, V. L., and A. L. Talapov. “Ground State, Spectrum, and Phase Diagram of Two-Dimensional Incommensurate Crystals.” Physical Review Letters 42 (1979): 65–67. DOI.