Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Modern Physics Letters B

EXPRESSION OF THE PARTITION FUNCTION AND RENORMALIZATION EQUATION OF SPIN-1/2 TOMONAGA–LUTTINGER CHAINS

Y. M. Li, Y. YU, N. d’Ambrumenil, Lu Yu, Z. B. SU

Published May 20, 1994• 2 citations
Publisher

Abstract

We derive an exact expression for the partition function of two spin-1/2 Tomonaga–Luttinger chains, and obtain the renormalization group equation for the normal state. Anderson's confinement is then discussed. We find that spin–charge separation is irrelevant to the confinement.
DOI
10.1142/s0217984994000753

Related papers

Renormalization group treatment of spin-charge separation effects

Álvaro Antônio Bandeira Ferraz

We examine the spin-charge separation effects which are manifest in the Tomonaga-Luttinger model (TLM) in the framework of the field theoretical renormalization group (RG) theory. Initially, it is shown how far one can go with the perturbative RG method. Following that, we take into full account the Ward identities [Phys. Rev. 78, 172 (1950)]. This allows us to calculate the spin and charge velocities exactly. Next, we construct the Callan-Symanzik equation (CSE) [Phys. Rev. D 5, 3202 (1972); Commun. Math. Phys. 23, 49 (1971)] for the renormalized fermionic one-particle Green's function. Finally, making use of those perturbative results for the TLM and taking the two characteristic velocities as basic ingredients, we solve this CSE in qualitative good agreement with the exact result.

Physical Review B•Jun 18, 2007•6 citations

Field theory renormalization group: the Tomonaga–Luttinger model revisited

A Ferraz

We apply field theoretical renormalization group (RG) methods to describe the Tomonaga–Luttinger model as an important test ground to deal with spin–charge separation effects in higher spatial dimensions. We calculate the RG equations for the renormalized forward couplings g 2 R and g 4 R up to two-loop order and demonstrate that they do not flow in the vicinities of the Fermi points (FPs). We solve the Callan–Symanzik equation for G ( a ) R in the vicinities of the FPs. We calculate the related spectral function and the momentum distribution function at p = k F , p 0 = ω. We compute the renormalized one-particle irreducible function Γ (2) (+) R ( p , p 0 = 0; Λ) and show that it carries important spin–charge separation effects in agreement with well-known results. Finally, we discuss the implementation of the RG scheme taking into account the important simplifications produced by the Ward identities.

arXiv:cond-mat/0512098•Journal of Physics A Mathematical and General•Jun 7, 2006•5 citations

Determination of Tomonaga-Luttinger parameters for a two-component liquid

Olabode Mayodele Sule, Hitesh J. Changlani, Isao Maruyama +1 more

We provide evidence for the mapping of critical spin-1 chains, in particular the $\mathrm{SU}(3)$ symmetric bilinear-biquadratic model with additional interactions, to free boson theories using exact diagonalization and the density-matrix renormalization-group algorithm. Using the correspondence with a conformal field theory with central charge $c=2$, we determine the analytic formulas for the scaling dimensions in terms of four Tomonaga-Luttinger liquid parameters. By matching the lowest scaling dimensions, we numerically calculate these field-theoretic parameters and track their evolution as a function of the parameters of the lattice model.

arXiv:1502.02228•Physical Review B•Aug 18, 2015•11 citations

One-dimensional Hubbard model in a magnetic field and the multicomponent Tomonaga-Luttinger model

Karlo Penc, Jenő Sólyom

The one-dimensional Hubbard model in a magnetic field is equivalent under renormalization-group transformation to a multicomponent Tomonaga-Luttinger model. The mapping between the Coulomb repulsion U of the Hubbard model and the couplings of the Tomonaga-Luttinger model is obtained for aribitrary magnetic field h and band filling n from a comparison of the correlation function exponents. These quantities are calculated for the Hubbard model from the finite-size corrections to the excitation energies, since the assumption of conformal invariance relates them to the critical exponents. On the other hand, the correlation functions of the multicomponent Tomonaga-Luttinger model are determined by solving the exact equation of motion derived by the use of generalized Ward identities.

Physical review. B, Condensed matter•Mar 15, 1993•85 citations

Renormalization group flow of the Luttinger-Ward functional: Conserving approximations and application to the Anderson impurity model

J. F. Rentrop, V. Meden, Severin G. Jakobs

We study the renormalization group flow of the Luttinger-Ward functional and of its two-particle-irreducible vertex functions, given a cutoff in the two-particle interaction. We derive a conserving approximation to the flow and relate it to the fluctuation exchange approximation as well as to nonconserving approximations introduced in an earlier publication [J. F. Rentrop, S. G. Jakobs, and V. Meden, J. Phys. A: Math. Theor. 48, 145002 (2015)]. We apply the different approximate flow equations to the single-impurity Anderson model in thermal equilibrium at vanishing temperature. Numerical results for the effective mass, the spin susceptibility, the charge susceptibility, and the linear conductance reflect the similarity of the methods to the fluctuation exchange approximation. We find the majority of the approximations to deviate stronger from the exact results than one-particle-irreducible functional renormalization group schemes. However, we identify a simple static two-particle-irreducible flow scheme which performs remarkably well and produces an exponential Kondo-like scale in the renormalized level position.

arXiv:1602.06120•Physical review. B./Physical review. B•May 31, 2016•21 citations

Properties of One-Particle Excitation for Two-Chain Tomonaga–Luttinger Model

Ken Yokoyama

The one-particle Green's function for the two-chain Tomonaga–Luttinger (TL) model coupled by interchain hopping including spin degrees of freedom is calculated as a function of distance and time by the higher-dimensional bosonization method. In the short distance and time region, the one-dimensional TL liquid behavior is reproduced, while the long distance and time behavior is different from that for either TL or Fermi liquid. As the interaction becomes stronger, the behavior of Green's function becomes closer to that for the one-dimensional TL liquid, showing that the one-dimensionality is enhanced for stronger electron interaction, and is interpreted as the precursor of the “confinement”.

Journal of the Physical Society of Japan•Oct 1, 2001•2 citations
We examine the spin-charge separation effects which are manifest in the Tomonaga-Luttinger model (TLM) in the framework of the field theoretical renormalization group (RG) theory. Initially, it is shown how far one can go with the perturbative RG method. Following that, we take into full account the Ward identities [Phys. Rev. 78, 172 (1950)]. This allows us to calculate the spin and charge velocities exactly. Next, we construct the Callan-Symanzik equation (CSE) [Phys. Rev. D 5, 3202 (1972); Commun. Math. Phys. 23, 49 (1971)] for the renormalized fermionic one-particle Green's function. Finally, making use of those perturbative results for the TLM and taking the two characteristic velocities as basic ingredients, we solve this CSE in qualitative good agreement with the exact result.
We apply field theoretical renormalization group (RG) methods to describe the Tomonaga–Luttinger model as an important test ground to deal with spin–charge separation effects in higher spatial dimensions. We calculate the RG equations for the renormalized forward couplings g 2 R and g 4 R up to two-loop order and demonstrate that they do not flow in the vicinities of the Fermi points (FPs). We solve the Callan–Symanzik equation for G ( a ) R in the vicinities of the FPs. We calculate the related spectral function and the momentum distribution function at p = k F , p 0 = ω. We compute the renormalized one-particle irreducible function Γ (2) (+) R ( p , p 0 = 0; Λ) and show that it carries important spin–charge separation effects in agreement with well-known results. Finally, we discuss the implementation of the RG scheme taking into account the important simplifications produced by the Ward identities.
We provide evidence for the mapping of critical spin-1 chains, in particular the SU(3)\mathrm{SU}(3)SU(3) symmetric bilinear-biquadratic model with additional interactions, to free boson theories using exact diagonalization and the density-matrix renormalization-group algorithm. Using the correspondence with a conformal field theory with central charge c=2c=2c=2, we determine the analytic formulas for the scaling dimensions in terms of four Tomonaga-Luttinger liquid parameters. By matching the lowest scaling dimensions, we numerically calculate these field-theoretic parameters and track their evolution as a function of the parameters of the lattice model.
The one-dimensional Hubbard model in a magnetic field is equivalent under renormalization-group transformation to a multicomponent Tomonaga-Luttinger model. The mapping between the Coulomb repulsion U of the Hubbard model and the couplings of the Tomonaga-Luttinger model is obtained for aribitrary magnetic field h and band filling n from a comparison of the correlation function exponents. These quantities are calculated for the Hubbard model from the finite-size corrections to the excitation energies, since the assumption of conformal invariance relates them to the critical exponents. On the other hand, the correlation functions of the multicomponent Tomonaga-Luttinger model are determined by solving the exact equation of motion derived by the use of generalized Ward identities.
We study the renormalization group flow of the Luttinger-Ward functional and of its two-particle-irreducible vertex functions, given a cutoff in the two-particle interaction. We derive a conserving approximation to the flow and relate it to the fluctuation exchange approximation as well as to nonconserving approximations introduced in an earlier publication [J. F. Rentrop, S. G. Jakobs, and V. Meden, J. Phys. A: Math. Theor. 48, 145002 (2015)]. We apply the different approximate flow equations to the single-impurity Anderson model in thermal equilibrium at vanishing temperature. Numerical results for the effective mass, the spin susceptibility, the charge susceptibility, and the linear conductance reflect the similarity of the methods to the fluctuation exchange approximation. We find the majority of the approximations to deviate stronger from the exact results than one-particle-irreducible functional renormalization group schemes. However, we identify a simple static two-particle-irreducible flow scheme which performs remarkably well and produces an exponential Kondo-like scale in the renormalized level position.
The one-particle Green's function for the two-chain Tomonaga–Luttinger (TL) model coupled by interchain hopping including spin degrees of freedom is calculated as a function of distance and time by the higher-dimensional bosonization method. In the short distance and time region, the one-dimensional TL liquid behavior is reproduced, while the long distance and time behavior is different from that for either TL or Fermi liquid. As the interaction becomes stronger, the behavior of Green's function becomes closer to that for the one-dimensional TL liquid, showing that the one-dimensionality is enhanced for stronger electron interaction, and is interpreted as the precursor of the “confinement”.