We studied the correlated quasi-one-dimensional systems by one-loop renormalization group techniques in weak coupling. In contrast to conventional g-ology approach, we formulate the theory in terms of bilinear currents and obtain all possible interaction vertices. Furthermore, the one-loop renormalization group equations are derived by operator product expansions of these currents at short length scale. It is rather remarkable that these coupled non-linear equations, after appropriate rescaling, can be casted into potential flows. The existence of what we nicknamed “RG potential ” provides a natural explanation of the emergent symmetry enhancement in ladder systems. Further implications arisen from the RG potential are also discussed at the end. Quasi-one-dimensional (Q1D) systems have attracted extensive attentions from both experimental and theoretical aspects. 1),2) Due to the low dimensionality, strong quantum fluctuations often give rise to surprising behaviors, which are rather different from our intuitions built in higher dimensions. It is then exciting to exploreWe studied the correlated quasi-one-dimensional systems by one-loop renormalization group techniques in weak coupling. In contrast to conventional g-ology approach, we formulate the theory in terms of bilinear currents and obtain all possible interaction vertices. Furthermore, the one-loop renormalization group equations are derived by operator product expansions of these currents at short length scale. It is rather remarkable that these coupled non-linear equations, after appropriate rescaling, can be casted into potential flows. The existence of what we nicknamed “RG potential ” provides a natural explanation of the emergent symmetry enhancement in ladder systems. Further implications arisen from the RG potential are also discussed at the end. Quasi-one-dimensional (Q1D) systems have attracted extensive attentions from both experimental and theoretical aspects. 1),2) Due to the low dimensionality, strong quantum fluctuations often give rise to surprising behaviors, which are rather different from our intuitions built in higher dimensions. It is then exciting to explore

Jan 1, 20056 citations

Potential flow of renormalization group in quasi-one-dimensional systems

Wei Chen, Ming-Shyang Chang, Hsiu‐Hau Lin +2 more

We address the issue why the phase diagrams for quasi-one-dimensional systems are rather simple, while the renormalization-group equations behind the scene are nonlinear and messy looking. The puzzle is answered in two steps---we first demonstrate that the complicated coupled flow equations are simply described by a potential $V({h}_{i})$ , in an appropriate basis for the interaction couplings ${h}_{i}$ . The renormalization-group potential is explicitly constructed by introducing the Majorana fermion representation. The existence of the potential prevents chaotic behaviors and other exotic possibilities such as limit cycles. Once the potential is obtained, the ultimate fate of the flows is described by a special set of fixed-ray solutions and the phase diagram is determined by Abelian bosonization. Extension to the strong-coupling regime and comparison with the Zamolodchikov $c$ theorem are discussed at the end.We address the issue why the phase diagrams for quasi-one-dimensional systems are rather simple, while the renormalization-group equations behind the scene are nonlinear and messy looking. The puzzle is answered in two steps---we first demonstrate that the complicated coupled flow equations are simply described by a potential $V({h}_{i})$, in an appropriate basis for the interaction couplings ${h}_{i}$. The renormalization-group potential is explicitly constructed by introducing the Majorana fermion representation. The existence of the potential prevents chaotic behaviors and other exotic possibilities such as limit cycles. Once the potential is obtained, the ultimate fate of the flows is described by a special set of fixed-ray solutions and the phase diagram is determined by Abelian bosonization. Extension to the strong-coupling regime and comparison with the Zamolodchikov $c$ theorem are discussed at the end.

arXiv:cond-mat/0306159Physical Review BNov 12, 200411 citations

Two-loop renormalization group theory for the quasi-one-dimensional Hubbard model at half filling

Masahisa Tsuchiizu

We derive two-loop renormalization group equations for the half filled one-dimensional Hubbard chains coupled by the interchain hopping. Our renormalization group scheme for the quasi-one-dimensional electron system is a natural extension of that for the purely one-dimensional systems in the sense that transverse-momentum dependences are introduced in the $g$ -ological coupling constants and we regard the transverse momentum as a patch index. We develop symmetry arguments for the particle-hole symmetric half filled Hubbard model and obtain constraints on the $g$ -ological coupling constants by which resultant renormalization equations are given in a compact form. By solving the renormalization group equations numerically, we estimate the magnitude of excitation gaps and clarify that the charge gap is suppressed due to the interchain hopping but is always finite even for the relevant interchain hopping. To show the validity of the present analysis, we also apply this to the two-leg ladder system. By utilizing the field-theoretical Bosonization and Fermionization method, we derive low-energy effective theory and analyze the magnitude of all the excitation gaps in detail. It is shown that the low-energy excitations in the two-leg Hubbard ladder have $\mathrm{SO}(3)\ifmmode\times\else\texttimes\fi{}\mathrm{SO}(3)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1)$ symmetry when the interchain hopping exceeds the magnitude of the charge gap.We derive two-loop renormalization group equations for the half filled one-dimensional Hubbard chains coupled by the interchain hopping. Our renormalization group scheme for the quasi-one-dimensional electron system is a natural extension of that for the purely one-dimensional systems in the sense that transverse-momentum dependences are introduced in the $g$-ological coupling constants and we regard the transverse momentum as a patch index. We develop symmetry arguments for the particle-hole symmetric half filled Hubbard model and obtain constraints on the $g$-ological coupling constants by which resultant renormalization equations are given in a compact form. By solving the renormalization group equations numerically, we estimate the magnitude of excitation gaps and clarify that the charge gap is suppressed due to the interchain hopping but is always finite even for the relevant interchain hopping. To show the validity of the present analysis, we also apply this to the two-leg ladder system. By utilizing the field-theoretical Bosonization and Fermionization method, we derive low-energy effective theory and analyze the magnitude of all the excitation gaps in detail. It is shown that the low-energy excitations in the two-leg Hubbard ladder have $\mathrm{SO}(3)\ifmmode\times\else\texttimes\fi{}\mathrm{SO}(3)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1)$ symmetry when the interchain hopping exceeds the magnitude of the charge gap.

arXiv:cond-mat/0603644Physical Review BOct 13, 20069 citations

N -chain Hubbard model in weak coupling

Hsiu‐Hau Lin, Leon Balents, Matthew P. A. Fisher

We present a systematic weak-coupling renormalization group (RG) technique for studying a collection of $N$ coupled one-dimensional interacting electron systems, focusing on the example of $N$ -leg Hubbard ladders. For $N=2,3,$ we recover previous results, and find that also more generally broad regions of the phase space of these models are unstable to pairing, usually with approximate $d$ -wave symmetry. We show how these instabilities can be understood in terms of a fairly conventional ``gap'' function \ensuremath{\Delta} at the discretized Fermi surface, and describe how this function is calculated. The dimensional crossovers as $N\ensuremath{\rightarrow}\ensuremath{\infty}$ and as many such ladders are weakly coupled together are also discussed.We present a systematic weak-coupling renormalization group (RG) technique for studying a collection of $N$ coupled one-dimensional interacting electron systems, focusing on the example of $N$-leg Hubbard ladders. For $N=2,3,$ we recover previous results, and find that also more generally broad regions of the phase space of these models are unstable to pairing, usually with approximate $d$-wave symmetry. We show how these instabilities can be understood in terms of a fairly conventional ``gap'' function \ensuremath{\Delta} at the discretized Fermi surface, and describe how this function is calculated. The dimensional crossovers as $N\ensuremath{\rightarrow}\ensuremath{\infty}$ and as many such ladders are weakly coupled together are also discussed.

arXiv:cond-mat/9703055Physical review. B, Condensed matterSep 15, 1997160 citations

The renormalization group (RG) approach: The critical theory near four dimensions

Jean Zinn‐Justin

Abstract In Chapter 14, the singular behavior of ferromagnetic systems with O(N) symmetry and short-range interactions, near a second order phase transition has been determined in the mean-field approximation, which is also a quasi-Gaussian approximation. The mean-field approximation predicts a set of universal properties, properties independent of the detailed structure of the microscopic Hamiltonian, the dimension of space, and, to a large extent, of the symmetry of systems. However, the leading corrections to the mean-field approximation, in dimensions smaller than or equal to four, diverge at the critical temperature, and the universal predictions of the mean-field approximation cannot be correct. Such a problem originates from the non-decoupling of scales and leads to the question of possible universality. In Chapter 9, the question has been answered in four dimensions using renormalization theory, and related renormalization group (RG) equations. Moreover, below four dimensions, in an expansion around the mean-field, the most singular terms near criticality can be also formally recovered from a continuum, low-mass φ4 field theory. More generally, following Wilson, to understand universality beyond the mean-field approximation, it is necessary to build a general renormalization group in the form of flow equations for effective Hamiltonians and to find fixed points of the flow equations. Near four dimensions, the flow equations can be approximated by the renormalization group of quantum field theory (QFT), and the fixed points and critical behaviours derived within the framework of the Wilson-Fisher ϵ expansion.Abstract In Chapter 14, the singular behavior of ferromagnetic systems with O(N) symmetry and short-range interactions, near a second order phase transition has been determined in the mean-field approximation, which is also a quasi-Gaussian approximation. The mean-field approximation predicts a set of universal properties, properties independent of the detailed structure of the microscopic Hamiltonian, the dimension of space, and, to a large extent, of the symmetry of systems. However, the leading corrections to the mean-field approximation, in dimensions smaller than or equal to four, diverge at the critical temperature, and the universal predictions of the mean-field approximation cannot be correct. Such a problem originates from the non-decoupling of scales and leads to the question of possible universality. In Chapter 9, the question has been answered in four dimensions using renormalization theory, and related renormalization group (RG) equations. Moreover, below four dimensions, in an expansion around the mean-field, the most singular terms near criticality can be also formally recovered from a continuum, low-mass φ4 field theory. More generally, following Wilson, to understand universality beyond the mean-field approximation, it is necessary to build a general renormalization group in the form of flow equations for effective Hamiltonians and to find fixed points of the flow equations. Near four dimensions, the flow equations can be approximated by the renormalization group of quantum field theory (QFT), and the fixed points and critical behaviours derived within the framework of the Wilson-Fisher ϵ expansion.

Apr 15, 20210 citations

Two-loop functional renormalization group approach to the one- and two-dimensional Hubbard model

A. A. Katanin

We consider the application of the two-loop functional renormalization group (fRG) approach to study the low-dimensional Hubbard model. This approach accounts for both the universal and nonuniversal contributions to the RG flow. While the universal contributions were studied previously within the field-theoretical RG for the one-dimensional Hubbard model with linearized electronic dispersion and the two-dimensional Hubbard model with flat Fermi surface, the nonuniversal contributions to the flow of the vertices and susceptibilities appear to be important at large momenta scales. The two-loop fRG approach is also applied to the two-dimensional Hubbard model with a curved Fermi surface and the van Hove singularities near the Fermi level. The vertices and susceptibilities in the end of the flow of the two-loop approach are suppressed in comparison with both the one-loop approach with vertex projection and the modified one-loop approach with corrected vertex projection errors. The results of the two-loop approach are closer to the results of one-loop approach with the projected vertices, the difference of the results of one- and two-loop fRG approaches decreases when going away from the van Hove band filling. The quasiparticle weight remains finite in two dimensions at not too low temperatures above the paramagnetic ground state.We consider the application of the two-loop functional renormalization group (fRG) approach to study the low-dimensional Hubbard model. This approach accounts for both the universal and nonuniversal contributions to the RG flow. While the universal contributions were studied previously within the field-theoretical RG for the one-dimensional Hubbard model with linearized electronic dispersion and the two-dimensional Hubbard model with flat Fermi surface, the nonuniversal contributions to the flow of the vertices and susceptibilities appear to be important at large momenta scales. The two-loop fRG approach is also applied to the two-dimensional Hubbard model with a curved Fermi surface and the van Hove singularities near the Fermi level. The vertices and susceptibilities in the end of the flow of the two-loop approach are suppressed in comparison with both the one-loop approach with vertex projection and the modified one-loop approach with corrected vertex projection errors. The results of the two-loop approach are closer to the results of one-loop approach with the projected vertices, the difference of the results of one- and two-loop fRG approaches decreases when going away from the van Hove band filling. The quasiparticle weight remains finite in two dimensions at not too low temperatures above the paramagnetic ground state.

arXiv:0806.0300Physical Review BJun 11, 200936 citations

V({h}_{i})

, in an appropriate basis for the interaction couplings

{h}_{i}

. The renormalization-group potential is explicitly constructed by introducing the Majorana fermion representation. The existence of the potential prevents chaotic behaviors and other exotic possibilities such as limit cycles. Once the potential is obtained, the ultimate fate of the flows is described by a special set of fixed-ray solutions and the phase diagram is determined by Abelian bosonization. Extension to the strong-coupling regime and comparison with the Zamolodchikov

c

theorem are discussed at the end.

g

-ological coupling constants and we regard the transverse momentum as a patch index. We develop symmetry arguments for the particle-hole symmetric half filled Hubbard model and obtain constraints on the

g

-ological coupling constants by which resultant renormalization equations are given in a compact form. By solving the renormalization group equations numerically, we estimate the magnitude of excitation gaps and clarify that the charge gap is suppressed due to the interchain hopping but is always finite even for the relevant interchain hopping. To show the validity of the present analysis, we also apply this to the two-leg ladder system. By utilizing the field-theoretical Bosonization and Fermionization method, we derive low-energy effective theory and analyze the magnitude of all the excitation gaps in detail. It is shown that the low-energy excitations in the two-leg Hubbard ladder have $\mathrm{SO}(3)\ifmmode\times\else\texttimes\fi{}\mathrm{SO}(3)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1)$ symmetry when the interchain hopping exceeds the magnitude of the charge gap.

We present a systematic weak-coupling renormalization group (RG) technique for studying a collection of

N

coupled one-dimensional interacting electron systems, focusing on the example of

N

-leg Hubbard ladders. For

N=2,3,

we recover previous results, and find that also more generally broad regions of the phase space of these models are unstable to pairing, usually with approximate

d

-wave symmetry. We show how these instabilities can be understood in terms of a fairly conventional ``gap'' function \ensuremath{\Delta} at the discretized Fermi surface, and describe how this function is calculated. The dimensional crossovers as $N\ensuremath{\rightarrow}\ensuremath{\infty}$ and as many such ladders are weakly coupled together are also discussed.