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arXiv:cond-mat/0105463

Scaling theory in Tomonaga-Luttinger liquid with $1/r^β$ type long-range interactions

Hitoshi Inoue

Published May 23, 2001• 0 citations
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Abstract

We discuss effects of $1/r^β$ type long-range (LR) interactions in a tight-binding model by utilizing the bosonization technique, renormalization group and conformal field theory (CFT). We obtain the low energy action known for Kibble's model which generates the mass gap in 3 dimension when $β=1$, the Coulomb force case. In one dimension, the dispersion relations predict that the system remains gapless even for $β=1$ and the existences of Tomonaga-Luttinger liquid (TLL) when $β> 1$. When $β=1$, the LR interactions break TLL in the long wavelength limits, even if the strength is very small. We make the more precise arguments from the stand point of the renormalization group and CFT. Finally we derive the accurate finite size scaling of energies and thermodynamics. Moreover we proceed to numerical calculations, considering the LR umklapp process terms. We conclude that the TLL phase become wider in the strength space of interactions as the power $β$ approaches to 1.

DOI
10.48550/arxiv.cond-mat/0105463

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Scaling theory in Tomonaga-Luttinger liquid with 1/rβ1/r^β1/rβ type long-range interactions
We discuss effects of 1/rβ1/r^β1/rβ type long-range (LR) interactions in a tight-binding model by utilizing the bosonization technique, renormalization group and conformal field theory (CFT). We obtain the low energy action known for Kibble's model which generates the mass gap in 3 dimension when β=1β=1β=1, the Coulomb force case. In one dimension, the dispersion relations predict that the system remains gapless even for β=1β=1β=1 and the existences of Tomonaga-Luttinger liquid (TLL) when β>1β> 1β>1. When β=1β=1β=1, the LR interactions break TLL in the long wavelength limits, even if the strength is very small. We make the more precise arguments from the stand point of the renormalization group and CFT. Finally we derive the accurate finite size scaling of energies and thermodynamics. Moreover we proceed to numerical calculations, considering the LR umklapp process terms. We conclude that the TLL phase become wider in the strength space of interactions as the power βββ approaches to 1.
We attempt to construct U (1) conformal field theory (CFT) in the Tomonaga–Luttinger (TL) liquid with the 1/ r β long-range interaction (LRI). Treating the long-range forward scattering as a perturbation and applying CFT to it, we derive finite size scalings which depend on the power of the LRI. The obtained finite size scalings give nontrivial behaviours when β is odd and is close to 2. We find consistency between the analytical arguments and numerical results in the finite size scaling of energy.
The concept of Tomonaga--Luttinger liquids (TLL) on the basis of the free-boson models is ubiquitous in theoretical descriptions of low-energy properties in one-dimensional quantum systems. In this work, we develop a squeezed-field path-integral description for gapless one-dimensional systems beyond the free-boson picture of the TLL paradigm. In the squeezed-field description, the parameter of the Bogoliubov transformation for the TL Hamiltonian becomes a dynamical squeezing field, and its fluctuations give rise to corrections to the free-boson results. We derive an effective nonlinear Lagrangian describing the dispersion relation of the squeezing field, and interactions between the excitations of the TLL and the squeezing modes. Using the effective Lagrangian, we analyze the imaginary-time correlation function of a vertex operator in the non-interacting limit. We show that a side-band branch emerges due to the fluctuation of the squeezing field, in addition to the standard branch of the free-boson model of the TLL paradigm. Furthermore, we perturbatively analyze the spectral function of the density fluctuations for an ultracold Bose gas in one dimension. We evaluate the renormalized values of the phase velocities and spectral weights of the TLL and side-band branches due to the interaction between the TLL and the squeezing modes. At zero temperature, the renormalized dispersion relations are linear in the momentum, but at nonzero temperatures, these acquire a nonlinear dependence on the momentum due to the thermal population of the excitation branches.
Quantum fluctuations can disrupt long-range order in one-dimensional systems and replace it with the universal paradigm of the Tomonaga-Luttinger liquid (TLL), a critical phase of matter characterized by power-law decaying correlations and linearly dispersing excitations. Using a Rydberg quantum simulator, we study how TLL physics manifests in the low-energy properties of a spin chain, interacting under either the ferromagnetic or the antiferromagnetic dipolar XY Hamiltonian. Following quasiadiabatic preparation, we directly observe the power-law decay of spin-spin correlations in real space, allowing us to extract the Luttinger parameter. In the presence of an impurity, the chain exhibits tunable Friedel oscillations of the local magnetization. Moreover, by utilizing a quantum quench, we directly probe the propagation of correlations, which exhibit a light-cone structure related to the linear sound mode of the underlying TLL. Our measurements demonstrate the influence of the long-range dipolar interactions, renormalizing the parameters of TLL with respect to the case of nearest-neighbor interactions. Finally, comparison to numerical simulations exposes the high sensitivity of TLLs to doping and finite-size effects.
Interacting one-dimensional bosons with long range hopping decaying as a power law r−αr^{-α}r−α with distance rrr are considered with the renormalization group and the self-consistent harmonic approximation. For α≥3α\ge 3α≥3, the ground state is always a Tomonaga-Luttinger liquid, whereas for α<3α<3α<3, a ground state with long range order breaking the continuous global gauge symmetry becomes possible for sufficiently weak repulsion. At positive temperature, continuous symmetry breaking becomes restricted to α<2α<2α<2, and for 2<α<32<α<32<α<3, a Tomonaga-Luttinger liquid with the Tomonaga-Luttinger exponent diverging at low temperature is found.
Resta proposed a definition of the electric polarization in one-dimensional systems in terms of the ground-state expectation value of the large gauge transformation operator. Vanishing of the expectation value in the thermodynamic limit implies that the system is a conductor. We study Resta's polarization amplitude (expectation value) in the S=1/2S=1/2S=1/2 XXZ chain and its several generalizations, in the gapless conducting Tomonaga-Luttinger liquid phase. We obtain an analytical expression in the lowest-order perturbation theory about the free fermion point (XY chain) and an exact result for the Haldane-Shastry model with long-range interactions. We also obtain numerical results, mostly using the exact diagonalization method. We find that the amplitude exhibits a power-law scaling in the system size (chain length) and vanishes in the thermodynamic limit. On the other hand, the exponent depends on the model even when the low-energy limit is described by the Tomonaga-Luttinger liquid with the same Luttinger parameter. We find that a change in the exponent occurs when the Umklapp term(s) are eliminated, suggesting the importance of the Umklapp terms.
Abstract Long-range interactions in one-dimensional quantum systems can qualitatively modify low-energy critical behavior and provide a natural setting for exploring unconventional phase transitions beyond short-range Tomonaga—Luttinger liquid physics. In this work, we study a one-dimensional XXZ spin chain with transverse long-range interactions and an explicit dimerization perturbation. Using bosonization and weak-coupling renormalization group (RG) analysis, we derive the RG flow equations. The resulting weak-coupling phase diagram contains a gapless Tomonaga—Luttinger liquid, an explicitly dimerized phase, and a continuous symmetry breaking (CSB) phase induced by the transverse long-range interaction. We further analyze the collapse point at which the two perturbations become simultaneously marginal. A stability analysis of the corresponding doubly marginal Gaussian fixed point shows that it is weakly unstable rather than a stable multicritical infrared fixed point, thereby giving rise to a pseudo-multicritical crossover over large length scales. Similar physics may be accessible in Rydberg-atom systems, offering a controllable theoretical and experimental platform for exploring long-range interaction effects and crossover behavior in one-dimensional quantum systems.