arXiv:0801.1837 · Physical Review B

Test of the low-energy model for one-dimensional interacting Fermi systems

Andrey V. Chubukov, Dmitrii L. Maslov, Fabian H. L. Eßler

Published Apr 14, 2008• 4 citations

Abstract

Bosonization predicts that the specific heat $C(T)$ of a one-dimensional interacting Fermi system is a sum of the specific heats of free collective charge and spin excitations, plus the term with the running backscattering amplitude which flows to zero logarithmically with decreasing $T$. We verify whether this result is reproduced in the $g$-ology model. Of specific interest are the anomalous terms in $C(T)$ that depend on the bare backscattering amplitude. We show that these terms can be incorporated into a renormalized spin velocity. We do this by proving the equivalence of the results for $C(T)$ obtained within the $g$-ology model and by bosonization with velocities obtained by the numerical solution of the Bethe-ansatz equations for the Hubbard model.

DOI: 10.1103/physrevb.77.161102

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arXiv:0801.1837 · Physical Review B

Test of the low-energy model for one-dimensional interacting Fermi systems

Andrey V. Chubukov, Dmitrii L. Maslov, Fabian H. L. Eßler

Published Apr 14, 2008• 4 citations

arXiv HTML PDF

Abstract

Bosonization predicts that the specific heat

C(T)

of a one-dimensional interacting Fermi system is a sum of the specific heats of free collective charge and spin excitations, plus the term with the running backscattering amplitude which flows to zero logarithmically with decreasing

T

. We verify whether this result is reproduced in the

g

-ology model. Of specific interest are the anomalous terms in

C(T)

that depend on the bare backscattering amplitude. We show that these terms can be incorporated into a renormalized spin velocity. We do this by proving the equivalence of the results for

C(T)

obtained within the

g

-ology model and by bosonization with velocities obtained by the numerical solution of the Bethe-ansatz equations for the Hubbard model.

DOI: 10.1103/physrevb.77.161102

Related papers

Specific heat of a one-dimensional interacting Fermi system: Role of anomalies

Andrey V. Chubukov, Dmitrii L. Maslov, Ronojoy Saha

We revisit the issue of the temperature dependence of the specific heat $C(T)$ for interacting fermions in one dimension. The charge component ${C}_{c}(T)$ scales linearly with $T$ , but the spin component ${C}_{s}(T)$ displays a more complex behavior with $T$ as it depends on the backscattering amplitude, ${g}_{1}$ , which scales down under renormalization group transformation and eventually behaves as ${g}_{1}(T)\ensuremath{\sim}1∕\mathrm{log}\phantom{\rule{0.2em}{0ex}}T$. We show, however, by direct perturbative calculations that ${C}_{s}(T)$ is strictly linear in $T$ to order ${g}_{1}^{2}$ as it contains the renormalized backscattering amplitude not on the scale of $T$ , but at the cutoff scale set by the momentum dependence of the interaction around $2{k}_{F}$ . The running amplitude ${g}_{1}(T)$ appears only at third order and gives rise to an extra $T∕{\mathrm{log}}^{3}\phantom{\rule{0.2em}{0ex}}T$ term in ${C}_{s}(T)$ . This agrees with the results obtained by a variety of bosonization techniques. We also show how to obtain the same expansion in ${g}_{1}$ within the sine-Gordon model.We revisit the issue of the temperature dependence of the specific heat $C(T)$ for interacting fermions in one dimension. The charge component ${C}_{c}(T)$ scales linearly with $T$, but the spin component ${C}_{s}(T)$ displays a more complex behavior with $T$ as it depends on the backscattering amplitude, ${g}_{1}$, which scales down under renormalization group transformation and eventually behaves as ${g}_{1}(T)\ensuremath{\sim}1∕\mathrm{log}\phantom{\rule{0.2em}{0ex}}T$. We show, however, by direct perturbative calculations that ${C}_{s}(T)$ is strictly linear in $T$ to order ${g}_{1}^{2}$ as it contains the renormalized backscattering amplitude not on the scale of $T$, but at the cutoff scale set by the momentum dependence of the interaction around $2{k}_{F}$. The running amplitude ${g}_{1}(T)$ appears only at third order and gives rise to an extra $T∕{\mathrm{log}}^{3}\phantom{\rule{0.2em}{0ex}}T$ term in ${C}_{s}(T)$. This agrees with the results obtained by a variety of bosonization techniques. We also show how to obtain the same expansion in ${g}_{1}$ within the sine-Gordon model.

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K. B. Efetov, I. L. Aleǐner

We discuss a recent bosonization method developed to study clean Fermi gases with a repulsion in any dimensions. The method enables one to consider both density and spin excitations. It is demonstrated that due to a non-abelian structure of the effective theory, the spin excitations interact with each other, which leads to new logarithmic in temperature corrections to physical quantities. Using a renormalization group scheme constructed for the effective low energy field theory these logarithms are summed up in all orders. Temperature dependent corrections to the specific heat and spin susceptibility are obtained for all dimensions d=1,2,3.We discuss a recent bosonization method developed to study clean Fermi gases with a repulsion in any dimensions. The method enables one to consider both density and spin excitations. It is demonstrated that due to a non-abelian structure of the effective theory, the spin excitations interact with each other, which leads to new logarithmic in temperature corrections to physical quantities. Using a renormalization group scheme constructed for the effective low energy field theory these logarithms are summed up in all orders. Temperature dependent corrections to the specific heat and spin susceptibility are obtained for all dimensions d=1,2,3.

arXiv:cond-mat/0610345Journal of Physics Condensed MatterMay 30, 20072 citations

Bosonization predicts that the specific heat

C(T)

T

. We verify whether this result is reproduced in the

g

-ology model. Of specific interest are the anomalous terms in

C(T)

that depend on the bare backscattering amplitude. We show that these terms can be incorporated into a renormalized spin velocity. We do this by proving the equivalence of the results for

C(T)

obtained within the

g

-ology model and by bosonization with velocities obtained by the numerical solution of the Bethe-ansatz equations for the Hubbard model.

We revisit the issue of the temperature dependence of the specific heat

C(T)

for interacting fermions in one dimension. The charge component

{C}_{c}(T)

scales linearly with

T

, but the spin component

{C}_{s}(T)

displays a more complex behavior with

T

as it depends on the backscattering amplitude,

{g}_{1}

, which scales down under renormalization group transformation and eventually behaves as ${g}_{1}(T)\ensuremath{\sim}1∕\mathrm{log}\phantom{\rule{0.2em}{0ex}}T$. We show, however, by direct perturbative calculations that

{C}_{s}(T)

is strictly linear in

T

to order

{g}_{1}^{2}

as it contains the renormalized backscattering amplitude not on the scale of

T

, but at the cutoff scale set by the momentum dependence of the interaction around

2{k}_{F}

. The running amplitude

{g}_{1}(T)

appears only at third order and gives rise to an extra

T∕{\mathrm{log}}^{3}\phantom{\rule{0.2em}{0ex}}T

term in

{C}_{s}(T)

. This agrees with the results obtained by a variety of bosonization techniques. We also show how to obtain the same expansion in

{g}_{1}

within the sine-Gordon model.

A(T)

is expressed exactly in terms of the fully renormalized backscattering amplitude, which includes the renormalization in the Cooper channel. In contrast to the one-dimensional case, both charge and spin components of the backscattering amplitudes are subject to this renormalization. We show that the logarithmic renormalization of the charge amplitude vanishes for a flat Fermi surface when the system becomes effectively one dimensional.