Model-dependent content and range of validity of the harmonic-fluid (Luttinger-liquid) description of one-dimensional quantum fluids

Haldane's 1981 harmonic-fluid approach established that the low-energy physics of any one-dimensional quantum fluid, Bose or Fermi, is governed by a universal quadratic theory of long-wavelength density fluctuations whose two velocities $v_N$ and $v_J$ fix all correlation exponents. The same paper acknowledges what this universal theory cannot deliver: the model-dependent amplitude coefficients $A_m$ , $B_m$ , $C_m$ and multiplicative prefactors of the asymptotic correlation expansions; explicit real-space correlation functions of Bethe-ansatz-integrable fluids beyond special cases; the parameters of non-Galilean-invariant lattice systems and axially symmetric spin chains; and quantitative control of crossover regimes and of the range of validity of the harmonic-fluid (Luttinger-liquid) description at intermediate coupling.

Background

In a one-dimensional (1D) quantum fluid — a gas of bosons or spinless fermions with short-ranged interactions, mean density $\rho_0 = N/L$ — no Landau quasiparticle survives interactions: the low-energy excitations are collective density waves, and equilibrium correlation functions decay as power laws rather than exponentials. Haldane's "harmonic-fluid approach" (the short companion of the longer paper that named this framework "Luttinger liquid theory") is the standard effective description. One writes the Hamiltonian as a quadratic ("harmonic") theory of the long-wavelength density-fluctuation field $\Pi(x)$ and its conjugate phase $\varphi(x)$ ; it is parametrized by two velocities, $v_J = \pi\hbar\rho_0/m$ (fixed by Galilean invariance; $m$ is the particle mass) and $v_N = 1/(\pi\hbar\kappa)$ (with $\kappa = \partial\rho/\partial\mu$ the compressibility per unit length), with sound velocity $v_s = (v_N v_J)^{1/2}$ . Zero-temperature correlation functions then take universal power-law forms; e.g., the density correlator contains oscillatory harmonics $A_m (\rho_0 x)^{-m^2\eta}\cos(2\pi m\rho_0 x)$ , the single-particle correlator of the boson field decays as $(\rho_0 x)^{-1/\eta}$ (with further harmonics of amplitudes $B_m$ ), and the fermion-field correlator involves amplitudes $C_m$ — all controlled by a single exponent $\eta = 2(v_J/v_N)^{1/2}$ . The exponents are universal (they depend only on the density $\rho_0$ , the compressibility $\kappa$ , and the harmonic index $m$ ), but the amplitude coefficients $A_m$ , $B_m$ , $C_m$ and the overall multiplicative factors of the bosonized operator expansions are not: they depend on short-wavelength physics and the high-energy cutoff, and the 1981 paper states explicitly that the harmonic theory leaves them undetermined.

Two anchors make the theory testable. First, integrable models solved by the Bethe ansatz (an exact method, going back to Bethe's 1931 solution of the Heisenberg chain, that yields exact spectra of certain 1D interacting models). The archetype is the Lieb-Liniger model, a 1D Bose gas with repulsive contact interaction of strength $c$ and dimensionless coupling $\gamma = mc/(\hbar^2\rho_0)$ ; $\gamma \to \infty$ is the impenetrable (Tonks-Girardeau) limit, equivalent to free spinless fermions via a Jordan-Wigner-type mapping (an equivalence of spectra and density correlations, not of field-operator correlators). Haldane showed that Bethe-ansatz solutions reproduce the harmonic spectrum and determine $\eta(\gamma)$ through a Fredholm integral equation (a linear integral equation of the second kind; derived in his companion Physics Letters A paper on the "Luttinger liquid" character of Bethe-ansatz-soluble models). Second, the weak-coupling Bogoliubov approximation — diagonalizing a quadratic boson Hamiltonian around a putative condensate — is valid only at small $\gamma$ ; Haldane noted that the natural boundary of this weak-coupling regime is $\gamma \sim \pi^2$ , since the Bogoliubov sound velocity $v_s = v_J(\gamma/\pi^2)^{1/2}$ reaches the strong-coupling scale when $\gamma^{1/2}/\pi \sim 1$ (Lieb had earlier found the Bogoliubov ground-state-energy estimate numerically indistinguishable from the exact one up to $\gamma \sim 10 \approx \pi^2$ ).

Against this backdrop, the 1981 paper acknowledges four things the universal harmonic description does not deliver: (i) the model-dependent amplitudes and multiplicative prefactors; (ii) explicit real-space correlation functions of Bethe-ansatz-integrable fluids apart from the special cases then known (the Calogero-Sutherland model with $1/r^2$ interactions at $\eta = 1$ and $4$ , the hard-core limit, and free fermions) — called "an outstanding, up-to-now intractable problem"; (iii) the extension to systems without Galilean invariance, where $v_J$ is no longer fixed by a symmetry but must be computed — the paper asserts that its Bethe-ansatz treatment applies equally well to lattice fluids such as the spinless-fermion chain (equivalently the anisotropic XXZ spin-1/2 Heisenberg chain), whose $v_J$ phase diagram had already been computed, and defers to a companion "to be published" only the treatment of axially symmetric spin chains with arbitrary spin quantum number $S$ ; and (iv) quantitative control of crossover regimes and of the range of validity of the harmonic-fluid description at intermediate couplings, beyond the weak-coupling Bogoliubov estimates. These four items share one scientific objective: supplying the microscopic content the universal low-energy theory cannot fix, so that the harmonic-fluid description becomes quantitatively predictive rather than only asymptotically constraining.

Problem Statement

Determine the model-dependent content that the harmonic-fluid (Luttinger-liquid) description of one-dimensional quantum fluids leaves undetermined, and the quantitative range of validity of that description, with the Lieb-Liniger Bose gas (dimensionless coupling $\gamma$ ) as the paradigmatic test case. The question has four linked parts:

(1) Compute the short-distance amplitude coefficients $A_m$ , $B_m$ , $C_m$ and the overall multiplicative prefactors (e.g., the undetermined overall factor of the field-operator expansion) in closed form at generic coupling for the paradigmatic microscopic model, the Lieb-Liniger gas; these are set by high-energy cutoff and short-wavelength physics that the harmonic low-energy theory does not fix.

(2) Obtain explicit real-space correlation functions of Bethe-ansatz-integrable one-dimensional fluids beyond the special cases known in 1981 (Calogero-Sutherland at $\eta = 1$ and $4$ , the hard-core/Tonks-Girardeau limit, and free fermions) — the problem Haldane called "outstanding" and "up-to-now intractable".

(3) Supply the treatment the paper defers to a companion paper ("will be reported elsewhere"): axially symmetric spin chains with arbitrary spin $S$ along harmonic-fluid lines, in systems without Galilean invariance where $v_J$ is no longer coupling independent and must be computed microscopically, determining how $v_J$ and the other low-energy parameters deviate from their Galilean-invariant values.

(4) Establish quantitative control of the crossover regimes and of the range of validity of the harmonic-fluid (Luttinger-liquid) description at intermediate coupling strengths for the Lieb-Liniger gas: the coupling and energy (momentum) scale at which corrections to the leading harmonic prediction for the excitation spectrum and the correlation functions become comparable with the leading term, and the corrections beyond that scale. The weak-coupling Bogoliubov expansion, whose domain ends near $\gamma \sim \pi^2$ , is a separate approximation.

The question asks for parts (1), (2) and (4), all for the Lieb-Liniger gas at generic coupling: part (1) for the amplitude and prefactor coefficients $A_m$ , $B_m$ , $C_m$ ; part (2) for the full real-space correlation functions, whose long-distance power-law asymptotics alone are not the answer; and part (4) for the coupling and energy scale at which the corrections to the leading harmonic prediction become comparable with it, and the corrections beyond that scale. Part (3), the spin-chain extension deferred to a companion paper, is included as background.

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