We calculate the dynamic structure factor $S(\ensuremath{\omega},q)$ of spinless fermions in one dimension with quadratic energy dispersion ${k}^{2}/2m$ and long-range density-density interaction whose Fourier transform ${f}_{q}$ is dominated by small momentum transfers $q\ensuremath{\lesssim}{q}_{0}⪡{k}_{F}$. Here ${q}_{0}$ is a momentum-transfer cutoff and ${k}_{F}$ is the Fermi momentum. Using functional bosonization and the known properties of symmetrized closed fermion loops, we obtain an expansion of the inverse irreducible polarization to second order in the small parameter ${q}_{0}/{k}_{F}$ . In contrast to perturbation theory based on conventional bosonization, our functional bosonization approach is not plagued by mass-shell singularities. For interactions which can be expanded as ${f}_{q}={f}_{0}+{f}_{0}^{\ensuremath{''}}{q}^{2}/2+O({q}^{4})$ with ${f}_{0}^{\ensuremath{''}}\ensuremath{\ne}0$, we show that the momentum scale ${q}_{c}=1/|m{f}_{0}^{\ensuremath{''}}|$ separates two regimes characterized by a different $q$ dependence of the width ${\ensuremath{\gamma}}_{q}$ of the collective zero sound mode and other features of $S(\ensuremath{\omega},q)$. For ${q}_{c}⪡q⪡{k}_{F}$ we find that the line shape is non-Lorentzian with an overall width ${\ensuremath{\gamma}}_{q}\ensuremath{\propto}{q}^{3}/(m{q}_{c})$ and a threshold singularity ${[(\ensuremath{\omega}\ensuremath{-}{\ensuremath{\omega}}_{q}^{\ensuremath{-}}){\text{ln}}^{2}(\ensuremath{\omega}\ensuremath{-}{\ensuremath{\omega}}_{q}^{\ensuremath{-}})]}^{\ensuremath{-}1}$ at the lower edge $\ensuremath{\omega}\ensuremath{\rightarrow}{\ensuremath{\omega}}_{q}^{\ensuremath{-}}=vq\ensuremath{-}{\ensuremath{\gamma}}_{q}$, where $v$ is the velocity of the zero sound mode. Assuming that higher orders in perturbation theory transform the logarithmic singularity into an algebraic one, we find for the corresponding threshold exponent ${\ensuremath{\mu}}_{q}=1\ensuremath{-}2{\ensuremath{\eta}}_{q}$ with ${\ensuremath{\eta}}_{q}\ensuremath{\propto}{q}_{c}^{2}/{q}^{2}$. Although for $q\ensuremath{\lesssim}{q}_{c}$ we have not succeeded to explicitly evaluate our functional bosonization result for $S(\ensuremath{\omega},q)$, we argue that for any one-dimensional model belonging to the Luttinger liquid universality class, the width of the zero sound mode scales as ${q}^{2}/m$ for $q\ensuremath{\rightarrow}0$.We calculate the dynamic structure factor $S(\ensuremath{\omega},q)$ of spinless fermions in one dimension with quadratic energy dispersion ${k}^{2}/2m$ and long-range density-density interaction whose Fourier transform ${f}_{q}$ is dominated by small momentum transfers $q\ensuremath{\lesssim}{q}_{0}⪡{k}_{F}$. Here ${q}_{0}$ is a momentum-transfer cutoff and ${k}_{F}$ is the Fermi momentum. Using functional bosonization and the known properties of symmetrized closed fermion loops, we obtain an expansion of the inverse irreducible polarization to second order in the small parameter ${q}_{0}/{k}_{F}$. In contrast to perturbation theory based on conventional bosonization, our functional bosonization approach is not plagued by mass-shell singularities. For interactions which can be expanded as ${f}_{q}={f}_{0}+{f}_{0}^{\ensuremath{''}}{q}^{2}/2+O({q}^{4})$ with ${f}_{0}^{\ensuremath{''}}\ensuremath{\ne}0$, we show that the momentum scale ${q}_{c}=1/|m{f}_{0}^{\ensuremath{''}}|$ separates two regimes characterized by a different $q$ dependence of the width ${\ensuremath{\gamma}}_{q}$ of the collective zero sound mode and other features of $S(\ensuremath{\omega},q)$. For ${q}_{c}⪡q⪡{k}_{F}$ we find that the line shape is non-Lorentzian with an overall width ${\ensuremath{\gamma}}_{q}\ensuremath{\propto}{q}^{3}/(m{q}_{c})$ and a threshold singularity ${[(\ensuremath{\omega}\ensuremath{-}{\ensuremath{\omega}}_{q}^{\ensuremath{-}}){\text{ln}}^{2}(\ensuremath{\omega}\ensuremath{-}{\ensuremath{\omega}}_{q}^{\ensuremath{-}})]}^{\ensuremath{-}1}$ at the lower edge $\ensuremath{\omega}\ensuremath{\rightarrow}{\ensuremath{\omega}}_{q}^{\ensuremath{-}}=vq\ensuremath{-}{\ensuremath{\gamma}}_{q}$, where $v$ is the velocity of the zero sound mode. Assuming that higher orders in perturbation theory transform the logarithmic singularity into an algebraic one, we find for the corresponding threshold exponent ${\ensuremath{\mu}}_{q}=1\ensuremath{-}2{\ensuremath{\eta}}_{q}$ with ${\ensuremath{\eta}}_{q}\ensuremath{\propto}{q}_{c}^{2}/{q}^{2}$. Although for $q\ensuremath{\lesssim}{q}_{c}$ we have not succeeded to explicitly evaluate our functional bosonization result for $S(\ensuremath{\omega},q)$, we argue that for any one-dimensional model belonging to the Luttinger liquid universality class, the width of the zero sound mode scales as ${q}^{2}/m$ for $q\ensuremath{\rightarrow}0$.

arXiv:0802.0970Physical Review BAug 14, 200816 citations

Transport in Luttinger Liquids

Thierry Giamarchi, Thomas Nattermann, Pierre Le Doussal

Springer eBooksJan 1, 20040 citations

Exact Green's Functions and Bosonization of a Luttinger Liquid Coupled to Impedances

K.-V. Pham

The exact Green's functions of a Luttinger Liquid (LL) connected to impedances are computed at zero and finite temperature. It is also shown that if the resistances are equal to the characteristic impedance of the Luttinger liquid, then the finite Luttinger liquid connected to resistors is equivalent to an infinite Luttinger liquid. Impedance boundary conditions (IBC) include also as a special limit the case of open boundary conditions, which are explicitly recovered. Finally bosonization for a LL with IBC is proven to hold.The exact Green's functions of a Luttinger Liquid (LL) connected to impedances are computed at zero and finite temperature. It is also shown that if the resistances are equal to the characteristic impedance of the Luttinger liquid, then the finite Luttinger liquid connected to resistors is equivalent to an infinite Luttinger liquid. Impedance boundary conditions (IBC) include also as a special limit the case of open boundary conditions, which are explicitly recovered. Finally bosonization for a LL with IBC is proven to hold.

arXiv:0709.0475Progress of Theoretical PhysicsSep 1, 20070 citations

Transport in Luttinger Liquids

Thierry Giamarchi, T. Nattermann, Pierre Le Doussal

arXiv:cond-mat/0403487NATO science series. Series II, Mathematics, physics and chemistryJan 1, 20040 citations

We calculate the dynamic structure factor $S(\ensuremath{\omega},q)$ of spinless fermions in one dimension with quadratic energy dispersion

{k}^{2}/2m

and long-range density-density interaction whose Fourier transform

{f}_{q}

is dominated by small momentum transfers $q\ensuremath{\lesssim}{q}_{0}⪡{k}_{F}$. Here

{q}_{0}

is a momentum-transfer cutoff and

{k}_{F}

is the Fermi momentum. Using functional bosonization and the known properties of symmetrized closed fermion loops, we obtain an expansion of the inverse irreducible polarization to second order in the small parameter

{q}_{0}/{k}_{F}

. In contrast to perturbation theory based on conventional bosonization, our functional bosonization approach is not plagued by mass-shell singularities. For interactions which can be expanded as ${f}_{q}={f}_{0}+{f}_{0}^{\ensuremath{''}}{q}^{2}/2+O({q}^{4})$ with ${f}_{0}^{\ensuremath{''}}\ensuremath{\ne}0$, we show that the momentum scale ${q}_{c}=1/|m{f}_{0}^{\ensuremath{''}}|$ separates two regimes characterized by a different

q

dependence of the width ${\ensuremath{\gamma}}_{q}$ of the collective zero sound mode and other features of $S(\ensuremath{\omega},q)$. For

{q}_{c}⪡q⪡{k}_{F}

we find that the line shape is non-Lorentzian with an overall width ${\ensuremath{\gamma}}_{q}\ensuremath{\propto}{q}^{3}/(m{q}_{c})$ and a threshold singularity ${[(\ensuremath{\omega}\ensuremath{-}{\ensuremath{\omega}}_{q}^{\ensuremath{-}}){\text{ln}}^{2}(\ensuremath{\omega}\ensuremath{-}{\ensuremath{\omega}}_{q}^{\ensuremath{-}})]}^{\ensuremath{-}1}$ at the lower edge $\ensuremath{\omega}\ensuremath{\rightarrow}{\ensuremath{\omega}}_{q}^{\ensuremath{-}}=vq\ensuremath{-}{\ensuremath{\gamma}}_{q}$, where

v

is the velocity of the zero sound mode. Assuming that higher orders in perturbation theory transform the logarithmic singularity into an algebraic one, we find for the corresponding threshold exponent ${\ensuremath{\mu}}_{q}=1\ensuremath{-}2{\ensuremath{\eta}}_{q}$ with ${\ensuremath{\eta}}_{q}\ensuremath{\propto}{q}_{c}^{2}/{q}^{2}$. Although for $q\ensuremath{\lesssim}{q}_{c}$ we have not succeeded to explicitly evaluate our functional bosonization result for $S(\ensuremath{\omega},q)$, we argue that for any one-dimensional model belonging to the Luttinger liquid universality class, the width of the zero sound mode scales as

{q}^{2}/m

for $q\ensuremath{\rightarrow}0$.