Non-Fermi liquid behavior of quasi one-dimensional pseudogap materials
Lorenz Bartosch
We study the spectral function of the pseudogap phase of quasi one-dimensional charge density wave materials. Using a stochastic approach and emphasizing an exact treatment of non-Gaussian order parameter fluctuations we will go beyond a usual perturbative calculation. %Generalizing Halperin's stochastic method to calculate the spectral function of a particle in a white noise potential towards a backscattering problem with complex order parameter fluctuations determined by a generalized Ginzburg-Landau theory, we find exact solutions for the spectral function by solving two-dimensional Fokker-Planck equations. Our results give a good fit to ARPES data and explain the absence of the Fermi edge in charge density wave materials even above the Peierls transition, indicating non-Fermi liquid behavior.
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\title{Non-Fermi liquid behavior of quasi one-dimensional pseudogap materials}
%\title{Spectral properties of the one-dimensional pseudogap phase}
\author{Lorenz Bartosch}

\affiliation{Institut f\"{u}r Theoretische Physik, Universit\"{a}t Frankfurt,
Robert-Mayer-Strasse 8, 60054 Frankfurt, Germany}
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\date{August 26, 2002}
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\begin{abstract}
We study the spectral function of the pseudogap phase of quasi one-dimensional charge density wave materials. 
Using a stochastic approach and emphasizing an exact treatment of non-Gaussian order parameter fluctuations we will go beyond a usual perturbative calculation.
%Generalizing Halperin's stochastic method to calculate the spectral function of a particle in a white noise potential towards a backscattering problem with complex order parameter fluctuations determined by a generalized Ginzburg-Landau theory, we find exact solutions for the spectral function by solving two-dimensional Fokker-Planck equations.
Our results give a good fit to ARPES data and explain the absence of the Fermi edge in charge density wave materials even above the Peierls transition, indicating non-Fermi liquid behavior.
\end{abstract}
%
\pacs{71.15.-m, 71.23.-k, 71.45.Lr, 79.60.-i}
%
%
%   71.15.-m : Methods of electronic structure calculations
%   71.23.-k : Electronic structure of disordered solids
%   71.45.Lr : Charge-density-wave systems
%   79.60.-i : Photoemission and photoelectron spectra
%
%
\maketitle


%\section{Introduction}

%High resolution {\em angle-resolved photoemission spectroscopy} (ARPES) is an excellent tool to probe the spectral properties of quasi low-dimensional strongly correlated electron systems. The experimentally measured energy- and momentum-resolved ARPES line shape is closely related to the spectral function which 
%%itself is given by the imaginary part of the single particle Green function and 
%contains information about all single-particle properties. 
%ARPES can be used to map out the Fermi surface, it reveals the opening of gaps or pseudogaps, and it also provides information about the energy and momentum dependence of quasiparticles or the absence of these. 


Non-Fermi liquid behavior is a central topic of modern condensed matter physics.
A paradigmatic example represents the universality class of Luttinger liquids \cite{Haldane81,Voit94} whose spectral properties have been calculated exactly \cite{Meden92} and which applies to the low-energy physics of a variety of one-dimensional ($1d$) gapless systems.
However, recent {\em angle-resolved photoemission spectroscopy} (ARPES) measurements of charge density wave materials like the {\em blue bronze} $\text{K}_{0.3} \text{Mo} \text{O}_3$ or $(\text{Ta}\, \text{Se}_4)_2\,\text{I}$ \cite{Dardel91,Claessen95,Gweon96,Gweon01,Perfetti02} show that these materials cannot be understood in terms of Luttinger liquid theory, even at temperatures far above the Peierls transition. 
It turns out that the 
pseudogap observed in the charge density wave materials is similar to that found in the underdoped phase of high temperature superconductors \cite{Loeser96,Damascelli02}.
It is this analogy which makes a better understanding of the $1d$ pseudogap phase even more interesting.



%One of the most studied quasi one-dimensional ($1d$) solids is the {\em blue bronze} ($\text{K}_{0.3} \text{Mo} \text{O}_3$), which at the critical temperature $T_c = 183\,$K undergoes a Peierls transition and establishes an incommensurate charge density wave \cite{Gruener94}. The experimentally measured 
%%energy- and momentum-resolved 
%{\em angle-resolved photoemission spectroscopy} (ARPES) line shape \cite{Dardel91,Claessen95,Gweon96,Gweon01} is closely related to the spectral function which 
%%itself is given by the imaginary part of the single particle Green function and 
%contains information about all single-particle properties. Especially interesting is the pseudogap phase above $T_c$. As pointed out in Refs.\ \onlinecite{Dardel91,Claessen95,Gweon96,Gweon01}, standard charge density wave theory is not capable of explaining the non-Fermi liquid line shape of $\text{K}_{0.3} \text{Mo} \text{O}_3$. 
%A comparison of experiment with Luttinger liquid theory 
%based on the Tomonaga-Luttinger model (with electron-electron interactions but no electron-phonon interactions) 
%turns out to be unsatisfactory either.
%%This does not seem to be such a big surprise since the Tomonaga-Luttinger model is based on electron-electron interactionsbut not electron-phonon interactions which are generally accepted to be responsible for the Peierls transition and, hence, should also be important in the pseudogap phase.
%Here we shall show that the spectral properties of the pseudogap phase of $\text{K}_{0.3} \text{Mo} \text{O}_3$ or similar compounds with strong electron-phonon interactions can be understood within a strictly $1d$ model based on static order parameter fluctuations if the order parameter fluctuations are handled exactly.




%The success of a mean field theory depends critically on the effective dimension of the system considered. 
%%While the predictions of a mean field theory are usually qualitatively correct for isotropic three dimensional materials like BCS superconductors, 
%In two dimensions and below, order parameter fluctuations prohibit a phase transition at the critical mean field temperature $T_c^{\text MF}$ and therefore cannot be neglected.
%%In fact, as shown by Mermin and Wagner \cite{Mermin66}, a continuous symmetry cannot be broken in dimensions less or equal to $2$.
%In quasi two-dimensional high temperature superconductors the growing order parameter fluctuations can lead to a suppression of the single-particle spectral weight at the Fermi level which is called a pseudogap.
%Such a pseudogap is also known from quasi one-dimensional Peierls systems whose pseudogap phase comprises a large temperature range \cite{Gruener94}. 

%Even though there is no long-range order in the pseudogap phase, the correlation length of the order parameter fluctuations can become very large. This large correlation length enhances small but finite three dimensional couplings and thus triggers a crossover to three dimensional ordering, resulting in a phase transition at temperature $T_c^{3d} \ll T_c^{\text{MF}}$.

%%If the temperature is further lowered, the large correlation length enhances small but finite three dimensional couplings and thus triggers a crossover to three dimensional ordering, resulting in a phase transition at temperature $T_c^{3d} \ll T_c^{\text{MF}}$. 











The relative proximity of the pseudogap phase of charge density wave materials to the Peierls transition indicates that strong electron-phonon interactions should play a dominant role.
In the early seventies, Lee, Rice, and Anderson \cite{Lee73} studied the single-particle Green function of a $1d$ model based on order parameter fluctuations %describing incommensurate Peierls systems 
within the second-order Born approximation.
Although this calculation could qualitatively explain the observed pseudogap phenomenon \cite{Shannon00}, its validity hinges on its perturbative nature.
As is well known for correlated $1d$ electronic systems, interference effects due to perfect nesting between the two Fermi points invalidate a perturbative calculation near the Fermi level. Therefore, one should rather try to find a more accurate or even exact solution. A widely used partial fraction expansion of the single-particle Green function presented by Sadovskii \cite{Sadovskii79} was intended to be exact for Gaussian order parameter fluctuations but turned out to be only an approximate solution to the problem raised by Lee, Rice, and Anderson \cite{Tchernyshyov99}.

Numerical simulations based on exact diagonalization methods \cite{Millis00b} or the phase formalism \cite{Bartosch99d,Bartosch00d} gave exact results for the density of states (DOS) for Gaussian order parameter fluctuations but showed also that Gaussian order parameter fluctuations are inapplicable at low temperatures where amplitude fluctuations should get frozen out and a hard gap should appear in the limit $T \to 0$ if quantum fluctuations are neglected. 
As pointed out in Ref.\ \onlinecite{Bartosch00d}, the right low-temperature physics can be described within a model based on classical phase fluctuations.
Using the phase formalism outlined in Ref.\ \onlinecite{Bartosch00d}, the DOS was also calculated numerically for the crossover regime to non-Gaussian order parameter fluctuations using a standard Ginzburg-Landau theory \cite{Monien01}.
How to find non-perturbative results for the complete spectral function which could be compared with ARPES data,
%or other quantities being more complicated than the DOS, 
however, remained an open question.

In this work we shall turn to this problem and calculate the spectral function of the $1d$ pseudogap phase of charge density wave materials for order parameter fluctuations given by a generalized Ginzburg-Landau functional.
%which involves all terms up to second order in the gradient of the order parameter field. 
%To obtain the spectral function we have to solve a two-dimensional Fokker-Planck equation and an inhomogeneous two-dimensional partial differential equation numerically.
We generalize a stochastic method originated by Halperin \cite{Halperin65,Lifshits88} for a particle in a white noise potential towards pseudogap systems involving disorder of the colored noise type.








As a starting point, we use the fluctuating gap model \cite{Lee73,Sadovskii79,Lifshits88} 
%Ovchinnikov77,Golub79,Bartosch99d,Bartosch00d,Bartosch01b,Monien01
which couples %spinless fermions close to the Fermi points 
the electronic degrees of freedom 
to a complex order parameter field $\tilde\Delta(x) \equiv \Delta(x) e^{i\vartheta(x)}$ with amplitude $\Delta(x)$ and phase $\vartheta(x)$. The two-component wave functions $\bm{\psi}_n(x)$ satisfy a Dirac-like equation \footnote{In this work we use units such that $\hbar = v_F = k_B = 1$. Later we will reintroduce the Fermi velocity $v_F$.},
\begin{equation}
  \label{eq:Dirac}
  [-i\bm{\sigma}_3 \partial_x + \tilde\Delta(x)  \bm{\sigma}_{+} + \tilde\Delta^{\ast}(x) \bm{\sigma}_{-} ] \bm{\psi}_n(x) = \omega_n \bm{\psi}_n(x) \;. 
\end{equation}
Here $\bm{\sigma}_{\pm} \equiv (\bm{\sigma}_{1} \pm i \bm{\sigma}_{2})/2$ and $\bm{\sigma}_3$ are the usual Pauli matrices.
Since the Hamiltonian corresponding to Eq.\ (\ref{eq:Dirac}) is form invariant under the charge conjugation operation $\bm{H} \to \bm{\sigma}_{1} \bm{H}^{\ast} \bm{\sigma}_{1}$ it is possible to use boundary conditions which imply that all eigenstates $\bm{\psi}_n(x)$ are invariant under the charge conjugation operation $\bm{\psi}_n(x) \to \bm{\sigma}_{1} \bm{\psi}_n^{\ast}(x)$. 
All these eigenstates may be found by making the Eikonal ansatz
%Using boundary conditions which imply that both components of the wave function $\bm{\psi}(x) = (\psi_1(x),\psi_2(x))^T$ are real and equal at $x = \mp L/2$ all eigenstates are even under charge conjugation and may be found by making the Eikonal ansatz
%Using boundary conditions which respect the symmetry of the Hamiltonian , we may find all eigenstates to this equation by making the Eikonal ansatz
\begin{equation}
  \label{eq:wavefunction}
  \bm{\psi}(x) = \left(
    \begin{array}{c}
      \psi(x) \\
      \psi^{\ast}(x) \\
    \end{array} \right) =
A e^{\zeta(x)}\left(
    \begin{array}{c}
      i e^{i\varphi(x)/2} \\
      -i e^{-i\varphi(x)/2} \\
    \end{array} \right)\;.
\end{equation}
In the thermodynamic limit $L \to \infty$, where $L$ is the length of the chain,
%which we will finally consider here, 
all physical quantities we will be interested in are independent of these boundary conditions \cite{Lifshits88}.
The logarithmic amplitude $\zeta(x)$ and the phase $\varphi(x)$ satisfy the following equations of motion \cite{Bartosch00d}, %01b
\begin{subequations}
  \label{eq:motion}
\begin{align}
  \label{eq:phi}
  \partial_x \varphi(x) & = 2 \omega + 2 \Delta(x) \cos\left[\varphi(x)-\vartheta(x)\right] \;, \\
  \label{eq:zeta}
  \partial_x \zeta(x) & = \Delta(x) \sin\left[\varphi(x)-\vartheta(x)\right] \;.
\end{align}
\end{subequations}
$\varphi(x)$ and $\zeta(x)$ are simply related to the DOS $\rho(\omega)$ and the inverse localization length (or Lyapunov exponent) $\ell^{-1}(\omega)$ by \cite{Bartosch00d}
\begin{equation}
  \rho(\omega) = \displaystyle{\frac{1}{2\pi}} \partial_\omega \langle \partial_x \varphi(x) \rangle \;,\;
  \ell^{-1}(\omega) = \langle \partial_x \zeta(x) \rangle \;.
\end{equation}
Here and in the following, $\langle \dots \rangle$ indicates averaging with respect to the probability distribution of $\tilde \Delta(x)$ which we will specify below.

The spectral function for right- or left-moving electrons ($\alpha = \pm 1$)
% of a system of length $L$ 
is defined in terms of the appropriately normalized wave functions $\bm{\psi}_{n}(x)$ as
\begin{align}
  & A^{\alpha}(k;\omega)\equiv A (\alpha (k_F+k);\omega) \nonumber \\
  & \quad \equiv \frac{1}{L} \left\langle \sum_{n}  \left|\int_{-L/2}^{L/2} dx\, e^{-ikx} \psi_{n}(x)\right|^2 \delta(\omega-\omega_n) \right\rangle\;.
  \label{eq:spectralfunction}
\end{align}
Note that we measure the deviation $k$ of a wave vector $p = \alpha(k_F + k)$ from one of the two Fermi wave vectors $\pm k_F$ locally outwards such that due to inversion symmetry $A^{\alpha}(k;\omega)$ is independent of $\alpha$.

Instead of using $\int |\bm{\psi}_n|^2\, dx = 1$ to normalize the wave functions let us now use the normalization condition $|\psi_n(0)|=1$.
Since the process of averaging restores translational invariance, it follows
\begin{equation}
  A^{\alpha}(k;\omega) 
  = \left\langle \sum_{n} \frac{\int_{-L/2}^{L/2} dx\, e^{-ikx} \psi_{n}(x)\psi_{n}^{\ast}(0)}{2 \int_{-L/2}^{L/2} dx\, \left|\psi_{n}(x)\right|^2}\, \delta(\omega-\omega_n) \right\rangle\;.
  \label{eq:spectralfunctionII}
\end{equation}
%The factor of $2$ in the denominator is due to the fact that $\int |\bm{\psi}_n(x)|^2\, dx = 2 \int |\psi_n(x)|^2\, dx$.  

Of course, not every solution to Eqs.\ (\ref{eq:wavefunction}) and (\ref{eq:motion}) is an eigenstate of the system. To construct the eigenstates, we follow Ref.\ \onlinecite{Lifshits88} and integrate the equations of motion (\ref{eq:motion}) from the boundaries at $-L/2$ and $L/2$ to $x=0$ where we try to match these solutions. Let us call the solutions to Eqs.\ (\ref{eq:phi}) and (\ref{eq:zeta}) when advancing the solution from $\mp L/2$ to the right (left) $\varphi_{\pm}(x)$ and $\zeta_{\pm}(x)$.
%Analogously we define $\varphi_{-}(x)$ and $\zeta_{-}(x)$ to be the solutions when integrating Eqs.\ (\ref{eq:phi}) and (\ref{eq:zeta}) from $L/2$ to the left. 
$\psi(x)$ is then given by
\begin{equation}
  \label{eq:psiplus}
  \psi (x) = \left\{
    \begin{array}{ll}
     i e^{-\zeta_{+}(0) + \zeta_{+}(x) + i\varphi_{+}(x)/2} \;, \;x < 0 \\
     i e^{-\zeta_{-}(0) + \zeta_{-}(x) + i\varphi_{-}(x)/2} \;,\;x > 0
    \end{array} \right. \;.
\end{equation}
The eigenstates $\bm{\psi}_n(x)$ are determined by the energies $\omega_n$ for which the matching condition $\varphi_{+}(0)=\varphi_{-}(0)$ (up to multiples of $2\pi$) is satisfied. In this case $\varphi_{+}(x)=\varphi_{-}(x)$ for all $x$, and the eigenstates have their maxima somewhere in the bulk from where the wave functions fall off exponentially.

Let us also introduce
\begin{equation}
  \label{eq:U}
  U_{\pm}(x) \equiv \frac{e^{ikx}}{\psi(x)}\int_{\mp L/2}^{x} dx^{\prime} \, e^{-ikx^{\prime}} \psi(x^{\prime}) \;,
\end{equation}
%\begin{align}
%  U_{\pm}(x) & = e^{ikx-\zeta_{+}(0)/2 - i\varphi_{+}(0)/2} \nonumber \\
%  & \times \int_{\pm L/2}^{x} dx^{\prime} e^{-ikx^{\prime} +\zeta_{+}(x)/2 + i\varphi_{+}(x)/2} \;,
%\end{align}
which satisfy the additional equation of motion
\begin{equation}
  \label{eq:U}
  \partial_x U (x) = i[k-\omega-\Delta(x) e^{-i[\varphi(x)-\vartheta(x)]} ] U(x) + 1 \;.
\end{equation}
%With $\int_{-L/2}^{L/2} dx\, e^{-ikx} \psi_{n}(x) = U_{+}(0)-U_{-}(0)$ and 
%$\delta(\varphi_{+}(0) - \varphi_{-}(0) = (2 \int_{-L/2}^{L/2} dx\, \left|\psi_{n}(x)\right|^2)^{-1} \delta(\omega-\omega_n)$ we find
It is now possible to rewrite the spectral function as
\begin{align}
  A^{\alpha}(k;\omega) 
  & = \left\langle \left[U_{+}(0) - U_{-}(0) \right] \, \delta(\varphi_{+}(0)-\varphi_{-}(0)) \right\rangle \nonumber \\
  & = 2 \, \text{Re} \left\langle U_{+}(0) \, \delta(\varphi_{+}(0)-\varphi_{-}(0)) \right\rangle\;.
  \label{eq:spectralfunctionIII}
\end{align}
%This expression is valid for an arbitrary probability distribution of the complex order parameter field $\tilde \Delta(x)$. 
%A similar expression has been found for the spectral function of the much simpler problem of a particle in a one-dimensional random potential \cite{Halperin65,Lifshits88}. 
One may explicitly verify that the spectral function $A(k;\omega) = A^{\alpha}(|k|-k_F;\omega)$ satisfies the usual sum rules,
\begin{align}
  \int_{-\infty}^{\infty} d\omega \, A(k;\omega) & = 1 \;, \\
  \int_{-\infty}^{\infty} \frac{dk}{2\pi} \, A(k;\omega) & = 2 \langle \delta(\varphi_{+}(0) - \varphi_{-}(0) ) \rangle =  \rho(\omega) \;.
\end{align}


%\begin{align}
%  \int_{-\infty}^{\infty} d\omega \, A^{\alpha}(k;\omega) & = 1 \;, \\
%  \sum_{\alpha} \int_{-\infty}^{\infty} \frac{dk}{2\pi} \, A^{\alpha}(k;\omega) & = 2 \langle (\delta(\varphi_{+}(0) - \varphi_{-}(0))) \rangle \nonumber \\
% & =  \rho(\omega) \;.
%\end{align}





%\section{Generalized Ginzburg-Landau functional}

The order parameter fluctuations of $\tilde \Delta (x) = \Delta(x) e^{i\vartheta(x)}$ are determined by a free energy functional which has been derived in the context of superconductivity \cite{Kosztin98}. Expanding the free energy functional in powers of the order parameter field $\tilde \Delta(x)$ and derivatives thereof one obtains the usual Ginzburg-Landau functional. A better approximation which is not restricted to small $\Delta$ and hence is valid at arbitrary temperatures can be obtained by expanding only in gradients of the order parameter field. The {\em generalized} Ginzburg-Landau free energy functional including all terms up to second order in the gradients of the phase and the amplitude of the order parameter has the form \cite{Kosztin98}
%\end{multicols}
\begin{align}
  \label{eq:Ginzburg_Landau}
  \beta F \{\tilde \Delta\} & =  \int_{-L/2}^{L/2} dx \, \left[ \frac{M(\Delta(x))}{2} \left(\partial_x \Delta(x)\right)^2 \right. \nonumber \\
  & \left. {} + \frac{n(\Delta(x))}{2} \left(\partial_x \vartheta (x)\right)^2 + V(\Delta(x)) \right] \;,
\end{align}
where
\begin{subequations}
\label{eq:MnV}
\begin{align}
  \label{eq:M}
  M(\Delta) & = \sum_{\tilde \omega_n > 0} \frac{\tilde \omega_n^2}{(\tilde \omega_n^2 + \Delta^2)^{5/2}} \;,\\
  n(\Delta) & = \sum_{\tilde \omega_n > 0} \frac{\Delta^2}{(\tilde \omega_n^2 + \Delta^2)^{3/2}} \;,\\
  V(\Delta) & = - 4 \!\!\!\! \sum_{0<\tilde \omega_n \lesssim \epsilon_0} \!\! \left( \sqrt{\tilde \omega_n^2 + \Delta^2} - \tilde \omega_n \right) + \frac{\Delta^2}{\pi \lambda T} \;.
  \label{eq:V}
\end{align}
\end{subequations}
Here, $T=1/\beta$ is the temperature of the system, $\lambda \ll 1$ is the dimensionless electron-phonon coupling constant, $\epsilon_0$ is an ultraviolet (bandwidth) cutoff, and $\tilde \omega_n = (2n+1)\pi T$ are fermionic Matsubara frequencies. In the weak coupling limit $\lambda \ll 1$, $V(\Delta)$ and hence $F \{\tilde \Delta\}$ depend on $\lambda$ and $\epsilon_0$ only via the critical mean-field temperature $T_c^{\text{MF}} = 1.134\, \epsilon_0 \exp(-1/\lambda)$.
% which we will set equal to one. This amounts to measuring temperatures and energies in units of the critical mean-field temperature.
%Note that $M(\Delta)$, $m(Delta)$ as well as $V(\Delta)$ are independent of the phase $\vartheta$ of the order parameter field.

The probability for a given realization of the order parameter field $\tilde \Delta(x)$ is proportional to $\exp(-\beta F \{\tilde \Delta\})$. This probability can be described as a stochastic process which due to the locality of $F \{\tilde \Delta\}$ has the Markov property. The probability $P(\Delta)$ that $\Delta(x)$ assumes the value $\Delta$ and the transition probabilities $T(\tilde \Delta(x) \! \gets \! \tilde \Delta(x^{\prime}))$ can be calculated using the transfer matrix method \cite{Scalapino72}. %Negele88,Shankar94} 
%as pointed out in Refs.\ \onlinecite{Bartosch00d} and \onlinecite{Monien01}.
%and evaluating the ground state wave function $\phi_0(\Delta)$ of the following imaginary time Schr{\"o}dinger equation for an isotropic two-dimensional anharmonic oscillator,
%\begin{equation}
%  \label{eq:Schroedinger}
%  -\frac{1}{2M} \left(\partial_{\Delta}^2+\frac{\partial_{\Delta}}{\Delta}\right) \phi_0 + V(\Delta) \phi_0 = E_0 \phi_0\;.
%\end{equation}
%The wave function $\phi_0(\Delta)$ is subject to the boundary conditions $\phi_0^{\prime}(0)=\phi_0(\infty)=0$ and can 
%%for an arbitrary potential $V(\Delta)$ 
%be evaluated numerically. We normalize $\phi_0$ such that
%\begin{equation}
%  \label{eq:P}
%  P(\Delta) = \Delta \phi_0^2(\Delta)
%\end{equation}
%is the desired probability distribution satisfying $\int_0^{\infty} d\Delta \, P(\Delta) =1$.
At this point we would like to note that the integral and derivatives in Eq.\ (\ref{eq:Ginzburg_Landau}) should be considered as a sum and finite differences. In the case of a generalized Ginzburg-Landau theory as considered here, $P(\Delta)$ and $T(\tilde \Delta(x) \gets \tilde \Delta(x^{\prime}))$ actually do depend on the lattice constant $d$.
In Fig.\ \ref{fig:correlationlength} we show $\sqrt{\langle \Delta^2 \rangle}$ and $\xi^{-1}$ as functions of temperature. For small temperatures we have $\xi \sim v_F/\pi T$ and $\sqrt{\langle \Delta^2 \rangle} \approx 1.76 \, T_c^{\text{MF}}$, consistent with the predictions of a strictly $1d$ theory based on classical phase fluctuations \cite{Gruener94,Bartosch00d}.
Since $\sqrt{\langle \Delta^2 \rangle} \gtrsim 1.76 \, T_c^{\text{MF}}$, our {\em generalized} Ginzburg-Landau functional given in Eqs. (\ref{eq:Ginzburg_Landau}) and (\ref{eq:MnV}) may not simply be replaced by a usual Ginzburg-Landau functional by expanding in $\Delta$.
In contrast to predictions based on standard Ginzburg-Landau theory as used in Refs.\ \onlinecite{Monien01} and \onlinecite{Scalapino72}, our calculations show that $\sqrt{\langle \Delta^2 \rangle}$ grows with increasing temperature.

\begin{figure}[tb]
\begin{center}
%\psfrag{omega}{\hspace{1.0mm}$T/T_c^{\text{MF}}$}
%\psfrag{Delta}{\hspace{-5.6mm}$\sqrt{\langle \Delta^2 \rangle}/T_c^{\text{MF}}$}
%\psfrag{xim1}{\hspace{-4.3mm}$v_F/T_c^{\text{MF}} \xi$}
%\psfrag{0}{\hspace{-0.3mm}$0$}
%\psfrag{0.2}[c][c]{$0.2$}
%\psfrag{0.4}[c][c]{$0.4$}
%\psfrag{0.6}[c][c]{$0.6$}
%\psfrag{0.8}[c][c]{$0.8$}
%\psfrag{1}[c][c]{$1$}
%\psfrag{2}[c][c]{$2$}
%\psfrag{3}[c][c]{$3$}
%\psfrag{4}[c][c]{$4$}
%\psfrag{5}[c][c]{$5$}
\epsfig{file=Fig1.eps} %,width=6cm
\end{center}
\vspace{-1mm}
\caption{Temperature dependence of $\sqrt{\langle \Delta^2(x) \rangle}$ and $\xi^{-1}$ for a lattice spacing $d = 0.2\, v_F/T_c^{\text{MF}}$ such that $\xi \gtrsim d$ for $T \lesssim T_c^{\text{MF}}$.}
\label{fig:correlationlength}
\end{figure}
The stochastic process corresponding to $T(\tilde \Delta(x) \gets\tilde \Delta(x^{\prime}))$ can be described by the Langevin equations
\begin{subequations}
\begin{align}
  \label{eq:Delta}
  \partial_x \Delta (x) & = a(\Delta)  + b(\Delta) \, \eta_{\Delta}(x) \;, \\
  \label{eq:theta}
  \partial_x \vartheta (x) & = \left(\frac{1}{n(\Delta(x))}\right)^{1/2} \, \eta_{\vartheta}(x) \;.
\end{align}
\end{subequations}
Here $\eta_{i}(x)$ with $i=\Delta,\vartheta$ represent Gaussian white noise with $\langle \eta_{i}(x)\eta_{j}(x^{\prime})\rangle=\delta_{i,j} \delta(x-x^{\prime})$.
$a(\Delta)$, $b(\Delta)$ and $P(\Delta)$ are simply related to 
$T(\Delta \! \gets \! \Delta^{\prime})$
and satisfy $a(\Delta) = b(\Delta){\partial_{\Delta} (b(\Delta) P(\Delta))}/{2 P(\Delta)}$.



To make further progress we absorb the phase of the order parameter field $\vartheta(x)$ by the gauge transformation \cite{Brazovskii76,Bartosch00d}
%\begin{equation}
%  \label{eq:gauge_transformation}
%  \varphi(x) \to \varphi(x) + \vartheta(x) \;.
%\end{equation}
$\varphi(x) \to \varphi(x) + \vartheta(x)$.
We now obtain the equations of motion
\begin{subequations}
\begin{align}
  \label{eq:phiII}
  \partial_x \varphi(x) & = 2 \omega -  \partial_x \vartheta(x) + 2 \Delta(x) \cos\left[\varphi(x)\right] \;, \\
  \label{eq:zetaII}
  \partial_x \zeta(x) & = \Delta(x) \sin\left[\varphi(x)\right] \;, \\
  \label{eq:UII}
  \partial_x U (x) & = i [k-\omega-\Delta(x) e^{-i\varphi(x)} ] U(x) + 1 \;.
\end{align}
\end{subequations}
Since the stochastic process for a given realization of the order parameter field has the Markov property we may write Eq.\ (\ref{eq:spectralfunctionIII}) as
\begin{align}
  A^{\alpha}(k;\omega) 
   = \, & 2 \, \text{Re} \int_0^{\infty} d\Delta \int_{0}^{2\pi} d\varphi \, \frac{1}{P(\Delta)} \nonumber \\
   \times  &  P_1(\Delta,\varphi;0)   P_0(\Delta,-\varphi;0)\;.
  \label{eq:spectralfunctionIV}
\end{align}
Here
\begin{subequations}
\begin{align}
  P_0(\Delta,\varphi;x) = & \,
  \langle \delta(\varphi-\varphi_{+}(x)) \delta(\Delta-\Delta(x)) \rangle \;, \\
  P_1(\Delta,\varphi;x) = & \,
  \langle U_{+}(x) \, \delta(\varphi - \varphi_{+}(x) \delta(\Delta-\Delta(x)) \rangle \;.
\end{align}
\end{subequations}
%and we have used the fact that $\langle \delta(\varphi-\varphi_{-}(x)) \delta(\Delta-\Delta(x)) \rangle = \langle \delta(\varphi+\varphi_{+}(x)) \delta(\Delta-\Delta(x)) \rangle$.
An expression similar to Eq.\ (\ref{eq:spectralfunctionIV}) (with $P_1$ replaced by $P_0$) does hold for the DOS.
Making use of the Gaussian white noise property of the variables $\eta_i(x)$, one may show that $P_0$ and $P_1$ obey (with $\partial_{\Delta}$ acting on everything to its right)
\begin{subequations}
\begin{align}
  \partial_x P_0 = \big[ & \frac{\partial_{\varphi}^2}{2n} - 2 \partial_{\varphi} (\omega + \Delta\cos \varphi) + \frac{(\partial_{\Delta} b)^2}{2}  - \partial_{\Delta} a \big] P_0 \;,
 \label{eq:FP:P0}\\
  \partial_x P_1 = \big[ & \frac{\partial_{\varphi}^2}{2n} - 2 (\omega + \Delta\cos \varphi) \partial_{\varphi} + i (k-\omega -\Delta e^{i\varphi}) \big. \nonumber \\
 \big. {} + & \frac{(\partial_{\Delta} b)^2}{2}  - \partial_{\Delta} a \big] P_1 + P_0 \;.  \label{eq:PDE:P1}
\end{align}
\end{subequations}
The stationary solutions to the $2d$ Fokker-Planck equation (\ref{eq:FP:P0}) and the $2d$ partial differential equation (\ref{eq:PDE:P1}) may be found numerically. We then obtain the spectral function $A^{\alpha}(k;\omega)$ (as well as the DOS and the inverse localization length) by simple quadrature.
The spectral function, the DOS and the inverse localization length are therefore simultaneously directly calculable without any approximations. There is no need to simulate a long chain as was done for the DOS (and the inverse localization length) in Refs.\ \onlinecite{Millis00b,Bartosch99d,Bartosch00d,Monien01}.
%01b
%and \onlinecite{Monien01}.

As can be seen in Fig.\ \ref{fig:SF}, at low temperatures $A^{\alpha}(k;\omega)$ exhibits two peaks instead of a quasi particle peak.
\begin{figure}[b]
\begin{center}
%\psfrag{omega}{\hspace{1.5mm}$\omega/T_c^{\text{MF}}$}
%\psfrag{SF1}{\hspace{-10.5mm}$A^{\alpha}(k=0;\omega) \cdot T_c^{\text{MF}}$}
%\psfrag{SF2}{\hspace{-11.5mm}$A^{\alpha}(k=0.5;\omega) \cdot T_c^{\text{MF}}$}
%\psfrag{0}{$0$}
%\psfrag{0.1}[c][c]{$0.1$}
%\psfrag{0.2}[c][c]{$0.2$}
%\psfrag{0.3}[c][c]{$0.3$}
%\psfrag{0.4}[c][c]{$0.4$}
%\psfrag{0.5}[c][c]{$0.5$}
%\psfrag{2}[c][c]{$2$}
%\psfrag{4}[c][c]{$4$}
%\psfrag{-2}[c][c]{$-2$}
%\psfrag{-4}[c][c]{$-4$}
\epsfig{file=Fig2.eps} %,width=8.4cm
\end{center}
\vspace{-1mm}
\caption{Spectral function $A^{\alpha}(k;\omega)$ for $k=0$ and $k=0.5\,T_c^{\textrm{MF}}/v_F$. The temperature varies form $T/T_c^{\text{MF}}=0.1$ to $1.0$ in steps of $0.1$.}
\label{fig:SF}
\end{figure}
Although the qualitative features of the spectral function agree with perturbation theory or the Sadovskii approximation \cite{McKenzie96b}, on a quantitative level there are large deviations, being due to the non-Gaussian nature of the order parameter fluctuations.
To compare our results with experiments, we note that for large photon energies the conventional interpretation of the ARPES line shape predicts that it is essentially proportional to \cite{Gweon01,Joynt99}
\begin{equation}
  \label{eq:ARPESintensity}
  I(k;\omega) = \sum_{k^{\prime}\approx k} \sum_{\omega^{\prime}\approx \omega} A^{<}(k^{\prime};\omega^{\prime})\;,
\end{equation}
where $A^{<}(k;\omega) \equiv A^{\alpha}(k;\omega)/(e^{\beta \omega} +1)$ is the spectral function multiplied with the Fermi function. 
In Fig.\ \ref{fig:ARPES}(a) we show a plot of $A^{<}(k;\omega)$ for $T=0.5\, T_c^{\text{MF}}$ and different values of $k$. $A^{<}(k;\omega)$ agrees reasonably well with the ARPES spectra presented in Fig.\ 1(a) of Ref.\ \onlinecite{Perfetti02} which were taken at $T=210\,$K.
%$T=220\,$K and $T=250\,$K for the A band of $\text{K}_{0.3} \text{Mo} \text{O}_3$ presented in Figs.\ 15 and 16 of Ref.\ \cite{Gweon01}.
\begin{figure}[tb]
\begin{center}
%\psfrag{omega}{\hspace{1.5mm}$\omega/T_c^{\text{MF}}$}
%\psfrag{ARPES}{\hspace{-9mm}$A^{<}(k;\omega)$ (arb. units)}
%\psfrag{rho}{\hspace{-3.5mm}$\rho^{<}(\omega)/\rho_0$}
%\psfrag{a}{(a)}
%\psfrag{b}{(b)}
%\psfrag{0.1}{\small$0.1$}
%\psfrag{0.50}{\small$0.5$}
%\psfrag{0.3}{\small$0.3$}
%\psfrag{0.8}{\hspace{-1.2mm}\small$0.8$}
%\psfrag{-2.0}{\small$-2$}
%\psfrag{-2}[r][r]{$-2$}
%\psfrag{-4}[r][r]{$-4$}
%\psfrag{-6}[r][r]{$-6$}
%\psfrag{-8}[r][r]{$-8$}
%\psfrag{0.5}[r][r]{$0.5$}
%\psfrag{0}[r][r]{$0$}
%\psfrag{1}[r][r]{$1$}
%\psfrag{1.5}[r][r]{$1.5$}
\epsfig{file=Fig3.eps} %,width=84mm
\end{center}
\vspace{-1mm}
\caption{(a) $A^{<}(k;\omega)$ for $T=0.5\, T_c^{\text{MF}}$ and $v_F k/T_c^{\textrm{MF}} =-2$ to $2$ in steps of $0.25$. Thick line: $k=0$, i.e.\ $p=k_F$. The spectra are normalized to the same maximum hight. (b) Solid lines: $\rho^{<}(\omega) = \int \frac{dk}{2\pi} A^{<}(k;\omega)$ for $T/T_c^{\textrm{MF}} = 0.3$ to $0.8$ in steps of $0.1$. Comparison with experimental data: Empty circles (Ref.\ \onlinecite{Gweon01}): $T=250$\,K. Empty triangles (Ref.\ \onlinecite{Gweon96}): $T=300$\,K. Filled triangles (Ref.\ \onlinecite{Dardel91}): $T=313$\,K. As a fit parameter we have used $T_c^{\textrm{MF}} = 60$\,meV$=696$\,K.}
\label{fig:ARPES}
\end{figure}
In Fig.\ \ref{fig:ARPES}(b) we compare angle integrated photoemission data with our theory. The only fit parameter used is $T_c^{\textrm{MF}} = 60$\,meV$=696$\,K. This value is less than a factor of two larger than $T_c^{\textrm{MF}} \approx 400$\,K, as obtained from optical conductivity measurements \cite{Gruener94}. The deviation might be due to a renormalization of $\lambda$ by electron-electron interactions and the simplicity of the model. 
Nonetheless, $\rho^{<}(\omega) = \int \frac{dk}{2\pi} A^{<}(k;\omega)$ does give a good fit to experimental data. We conclude that order parameter fluctuations seem to dominate the pseudogap phase.




In summary, we have calculated the spectral function of a $1d$ electronic system coupled to order parameter fluctuations determined by a generalized Ginzburg-Landau theory. Our results show that the essential physics of the pseudogap phase of quasi $1d$ charge density wave systems can be captured within a strictly $1d$ model based on order parameter fluctuations, treating their non-Gaussian statistics exactly. These order parameter fluctuations lead to non-Fermi liquid behavior and the experimentally observed absence of a Fermi edge.
\begin{acknowledgments}
The author would like to thank P.\ Kopietz, H.\ Monien, and R.\ H.\ McKenzie for discussions.
This work was partially supported by the DFG via Forschergruppe FOR 412, Project No. KO 1442/5-1.

\end{acknowledgments}




%\vspace{-1.5mm}


\begin{thebibliography}{18}


%\vspace{-1.5mm}











\expandafter\ifx\csname natexlab\endcsname\relax\def\natexlab#1{#1}\fi
\expandafter\ifx\csname bibnamefont\endcsname\relax
  \def\bibnamefont#1{#1}\fi
\expandafter\ifx\csname bibfnamefont\endcsname\relax
  \def\bibfnamefont#1{#1}\fi
\expandafter\ifx\csname citenamefont\endcsname\relax
  \def\citenamefont#1{#1}\fi
\expandafter\ifx\csname url\endcsname\relax
  \def\url#1{\texttt{#1}}\fi
\expandafter\ifx\csname urlprefix\endcsname\relax\def\urlprefix{URL }\fi
\providecommand{\bibinfo}[2]{#2}
\providecommand{\eprint}[2][]{\url{#2}}

\bibitem{Haldane81} F.\ D.\ M.\ Haldane, 
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\bibitem{Voit94} J.\ Voit, Rep.\ Prog.\ Phys.\ {\bf 57}, 977 (1994).  

\bibitem{Meden92} V.\ Meden and K.\ Sch{\"o}nhammer, Phys.\ Rev.\ B
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\bibitem{Dardel91} B.\ Dardel {\em et al.}, Phys.\ Rev.\ Lett.\ {\bf 67}, 3144 (1991);
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\bibitem{Claessen95}
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\bibitem{Gweon96}
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\bibitem{Gweon01}
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L.\ Perfetti {\em et al.}, Phys.\ Rev.\ B {\bf 66}, 075107 (2002).

\bibitem{Loeser96}
A.\ G.\ Loeser {\em et al.}, Science \textbf{273}, 325 (1996);
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\bibitem{Damascelli02}
A.\ Damascelli, Z.-X.\ Shen, and Z.\ Hussain, Rev.\ Mod.\ Phys.\ (in press), cond-mat/0208504.

\bibitem[{\citenamefont{Lee et~al.}(1973)\citenamefont{Lee, Rice, and
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\bibinfo{author}{\bibfnamefont{P.~A.} \bibnamefont{Lee}},
  \bibinfo{author}{\bibfnamefont{T.~M.} \bibnamefont{Rice}}, \bibnamefont{and}
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\bibitem{Shannon00} N.\ Shannon and R.\ Joynt, Solid State Commun. {\bf 115}, 411 (2000).


\bibitem[{\citenamefont{Sadovskii}(1979)}]{Sadovskii79}
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  \bibinfo{journal}{Zh.~Eksp.~Teor.~Fiz.} \textbf{\bibinfo{volume}{77}},
  \bibinfo{pages}{2070} (\bibinfo{year}{1979}), \bibinfo{note}{[Sov.~Phys.~JETP
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\bibitem[{\citenamefont{Tchernyshyov}(1999)}]{Tchernyshyov99}
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\bibinfo{author}{\bibfnamefont{L.}~\bibnamefont{Bartosch}} \bibnamefont{and}
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\bibitem{Bartosch00d}
\bibinfo{author}{\bibfnamefont{L.}~\bibnamefont{Bartosch}} \bibnamefont{and}
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  \bibinfo{pages}{799} (\bibinfo{year}{2001}).

%\bibitem[{\citenamefont{Bartosch}(2001)}]{Bartosch01b}
%\bibinfo{author}{\bibfnamefont{L.}~\bibnamefont{Bartosch}},
%  \bibinfo{journal}{Ann.\ Phys.\ (Leipzig)} \textbf{\bibinfo{volume}{10}},
%  \bibinfo{pages}{799} (\bibinfo{year}{2001}).

%\bibitem[{\citenamefont{Bartosch and Kopietz}(2000)}]{Bartosch00d}
%\bibinfo{author}{\bibfnamefont{L.}~\bibnamefont{Bartosch}} \bibnamefont{and}
%  \bibinfo{author}{\bibfnamefont{P.}~\bibnamefont{Kopietz}},
%  \bibinfo{journal}{Phys.\ Rev.\ B} \textbf{\bibinfo{volume}{62}},
%  \bibinfo{pages}{R16223} (\bibinfo{year}{2000}).

\bibitem[{\citenamefont{Monien}(2001)}]{Monien01}
\bibinfo{author}{\bibfnamefont{H.}~\bibnamefont{Monien}},
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  \bibinfo{pages}{126402} (\bibinfo{year}{2001}).

\bibitem[{\citenamefont{Halperin}(1965)}]{Halperin65}
\bibinfo{author}{\bibfnamefont{B.~I.} \bibnamefont{Halperin}},
  \bibinfo{journal}{Phys.\ Rev.} \textbf{\bibinfo{volume}{139}},
  \bibinfo{pages}{A104} (\bibinfo{year}{1965}).

\bibitem[{\citenamefont{Lifshits et~al.}(1988)\citenamefont{Lifshits,
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\bibinfo{author}{\bibfnamefont{I.~M.} \bibnamefont{Lifshits}},
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\bibitem[{\citenamefont{Kosztin et~al.}(1998)\citenamefont{Kosztin, Kos, Stone,
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For a recent account of the subject, see
\bibinfo{author}{\bibfnamefont{I.} \bibnamefont{Kosztin}},
  \bibinfo{author}{\bibfnamefont{S.}~\bibnamefont{Kos}},
  \bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Stone}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{A.~J.} \bibnamefont{Leggett}},
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\bibitem[{\citenamefont{Scalapino et~al.}(1972)\citenamefont{Scalapino, Sears,
  and Ferrell}}]{Scalapino72}
\bibinfo{author}{\bibfnamefont{D.~J.} \bibnamefont{Scalapino}},
  \bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Sears}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{R.~A.} \bibnamefont{Ferrell}},
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%\bibitem[{\citenamefont{Negele and Orland}(1988)}]{Negele88}
%\bibinfo{author}{\bibfnamefont{J.~W.} \bibnamefont{Negele}} \bibnamefont{and}
%  \bibinfo{author}{\bibfnamefont{H.}~\bibnamefont{Orland}},
%  \emph{\bibinfo{title}{Quantum Many-Particle Physics}}
%  (\bibinfo{publisher}{Addison-Wesley, Redwood City}, \bibinfo{year}{1988}).

%\bibitem[{\citenamefont{Shankar}(2nd ed., 1994)}]{Shankar94}
%\bibinfo{author}{\bibfnamefont{R.}~\bibnamefont{Shankar}},
%  \emph{\bibinfo{title}{Principles of quantum mechanics}}
%  (\bibinfo{publisher}{Plenum Press, New York}, \bibinfo{year}{2nd ed., 1994}).

\bibitem[{\citenamefont{Gr{\"{u}}ner}(1994)}]{Gruener94}
\bibinfo{author}{\bibfnamefont{G.}~\bibnamefont{Gr{\"{u}}ner}},
  \emph{\bibinfo{title}{Density Waves in Solids}}
  (\bibinfo{publisher}{Addison-Wesley, Reading}, \bibinfo{year}{1994}).

\bibitem[{\citenamefont{Brazovskii and Dzyaloshinskii}(1976)}]{Brazovskii76}
\bibinfo{author}{\bibfnamefont{S.~A.} \bibnamefont{Brazovskii}}
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{I.~E.}
  \bibnamefont{Dzyaloshinskii}}, \bibinfo{journal}{Zh-Eksp.\ Teor.\ Fiz.}
  \textbf{\bibinfo{volume}{71}}, \bibinfo{pages}{2338} (\bibinfo{year}{1976}),
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%
%\bibitem{McKenzie95a}
%R.\ H.\ McKenzie, Phys.\ Rev.\ B {\bf{52}}, 16428 (1995).
%
\bibitem{McKenzie96b}
R.\ H.\ McKenzie and D.\ Scarratt,
Phys.\ Rev.\ B {\bf{54}}, R12709 (1996).

\bibitem{Joynt99} For a different interpretation of the ARPES signal, see R.\ Joynt, Science \textbf{284}, 777 (1999).


\end{thebibliography}



%\bibliography{peierls}





\end{document}


k
cond-mat0208499.extracted_bib
\begin{thebibliography}{18}


%\vspace{-1.5mm}











\expandafter\ifx\csname natexlab\endcsname\relax\def\natexlab#1{#1}\fi
\expandafter\ifx\csname bibnamefont\endcsname\relax
  \def\bibnamefont#1{#1}\fi
\expandafter\ifx\csname bibfnamefont\endcsname\relax
  \def\bibfnamefont#1{#1}\fi
\expandafter\ifx\csname citenamefont\endcsname\relax
  \def\citenamefont#1{#1}\fi
\expandafter\ifx\csname url\endcsname\relax
  \def\url#1{\texttt{#1}}\fi
\expandafter\ifx\csname urlprefix\endcsname\relax\def\urlprefix{URL }\fi
\providecommand{\bibinfo}[2]{#2}
\providecommand{\eprint}[2][]{\url{#2}}

\bibitem{Haldane81} F.\ D.\ M.\ Haldane, 
Phys.\ Rev.\ Lett.\ {\bf 45}, 1358 (1980); {\it ibid.} {\bf 47}, 1840 (1981);
J.\ Phys.\ C {\bf 14}, 2585 (1981).

\bibitem{Voit94} J.\ Voit, Rep.\ Prog.\ Phys.\ {\bf 57}, 977 (1994).  

\bibitem{Meden92} V.\ Meden and K.\ Sch{\"o}nhammer, Phys.\ Rev.\ B
  {\bf 46}, 15753 (1992); 
J.\ Voit, Phys.\ Rev.\ B {\bf 47}, 6740 (1993);
D.\ Orgad, Philos.\ Mag.\ B {\bf 81}, 375 (2001).

\bibitem{Dardel91} B.\ Dardel {\em et al.}, Phys.\ Rev.\ Lett.\ {\bf 67}, 3144 (1991);
  \bibinfo{journal}{Europhys.~Lett.} \textbf{\bibinfo{volume}{19}},
  \bibinfo{pages}{525} (\bibinfo{year}{1992}).

\bibitem{Claessen95}
R.\ Claessen {\em et al.},
  \bibinfo{journal}{J.~Electron Spectrosc.~Relat.~Phenonom.}
  \textbf{\bibinfo{volume}{76}}, \bibinfo{pages}{121} (\bibinfo{year}{1995}).

\bibitem{Gweon96}
G.-H.\ Gweon {\em et al.},
  \bibinfo{journal}{J.~Phys.-Condens.~Matter}
  \textbf{\bibinfo{volume}{8}}, \bibinfo{pages}{9923} (\bibinfo{year}{1996}).

\bibitem{Gweon01}
G.-H.\ Gweon {\em et al.},
  \bibinfo{journal}{J.~Electron
  Spectrosc.~Relat.~Phenonom.} \textbf{\bibinfo{volume}{117-118}},
  \bibinfo{pages}{481} (\bibinfo{year}{2001}).

\bibitem{Perfetti02}
L.\ Perfetti {\em et al.}, Phys.\ Rev.\ B {\bf 66}, 075107 (2002).

\bibitem{Loeser96}
A.\ G.\ Loeser {\em et al.}, Science \textbf{273}, 325 (1996);
H.\ Ding {\em et al.}, Nature (London) \textbf{382}, 51 (1996).

\bibitem{Damascelli02}
A.\ Damascelli, Z.-X.\ Shen, and Z.\ Hussain, Rev.\ Mod.\ Phys.\ (in press), cond-mat/0208504.

\bibitem[{\citenamefont{Lee et~al.}(1973)\citenamefont{Lee, Rice, and
  Anderson}}]{Lee73}
\bibinfo{author}{\bibfnamefont{P.~A.} \bibnamefont{Lee}},
  \bibinfo{author}{\bibfnamefont{T.~M.} \bibnamefont{Rice}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{P.~W.} \bibnamefont{Anderson}},
  \bibinfo{journal}{Phys.\ Rev.\ Lett.} \textbf{\bibinfo{volume}{31}},
  \bibinfo{pages}{462} (\bibinfo{year}{1973}).

\bibitem{Shannon00} N.\ Shannon and R.\ Joynt, Solid State Commun. {\bf 115}, 411 (2000).


\bibitem[{\citenamefont{Sadovskii}(1979)}]{Sadovskii79}
\bibinfo{author}{\bibfnamefont{M.~V.} \bibnamefont{Sadovskii}},
  \bibinfo{journal}{Zh.~Eksp.~Teor.~Fiz.} \textbf{\bibinfo{volume}{77}},
  \bibinfo{pages}{2070} (\bibinfo{year}{1979}), \bibinfo{note}{[Sov.~Phys.~JETP
  {\bf{50}}, 989 (1979)]}.

\bibitem[{\citenamefont{Tchernyshyov}(1999)}]{Tchernyshyov99}
\bibinfo{author}{\bibfnamefont{O.}~\bibnamefont{Tchernyshyov}},
  \bibinfo{journal}{Phys.\ Rev.\ B} \textbf{\bibinfo{volume}{59}},
  \bibinfo{pages}{1358} (\bibinfo{year}{1999}).

\bibitem[{\citenamefont{Millis and Monien}(2000)}]{Millis00b}
\bibinfo{author}{\bibfnamefont{A.~J.}~\bibnamefont{Millis}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{H.}~\bibnamefont{Monien}},
  \bibinfo{journal}{Phys.\ Rev.\ B} \textbf{\bibinfo{volume}{61}},
  \bibinfo{pages}{12496} (\bibinfo{year}{2000}).

\bibitem[{\citenamefont{Bartosch and Kopietz}(1999)}]{Bartosch99d}
\bibinfo{author}{\bibfnamefont{L.}~\bibnamefont{Bartosch}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{P.}~\bibnamefont{Kopietz}},
  \bibinfo{journal}{Phys.\ Rev.\ B} \textbf{\bibinfo{volume}{60}},
  \bibinfo{pages}{15488} (\bibinfo{year}{1999}).

\bibitem{Bartosch00d}
\bibinfo{author}{\bibfnamefont{L.}~\bibnamefont{Bartosch}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{P.}~\bibnamefont{Kopietz}},
  \bibinfo{journal}{Phys.\ Rev.\ B} \textbf{\bibinfo{volume}{62}},
  \bibinfo{pages}{R16223} (\bibinfo{year}{2000});
\bibinfo{author}{\bibfnamefont{L.}~\bibnamefont{Bartosch}},
  \bibinfo{journal}{Ann.\ Phys.\ (Leipzig)} \textbf{\bibinfo{volume}{10}},
  \bibinfo{pages}{799} (\bibinfo{year}{2001}).

%\bibitem[{\citenamefont{Bartosch}(2001)}]{Bartosch01b}
%\bibinfo{author}{\bibfnamefont{L.}~\bibnamefont{Bartosch}},
%  \bibinfo{journal}{Ann.\ Phys.\ (Leipzig)} \textbf{\bibinfo{volume}{10}},
%  \bibinfo{pages}{799} (\bibinfo{year}{2001}).

%\bibitem[{\citenamefont{Bartosch and Kopietz}(2000)}]{Bartosch00d}
%\bibinfo{author}{\bibfnamefont{L.}~\bibnamefont{Bartosch}} \bibnamefont{and}
%  \bibinfo{author}{\bibfnamefont{P.}~\bibnamefont{Kopietz}},
%  \bibinfo{journal}{Phys.\ Rev.\ B} \textbf{\bibinfo{volume}{62}},
%  \bibinfo{pages}{R16223} (\bibinfo{year}{2000}).

\bibitem[{\citenamefont{Monien}(2001)}]{Monien01}
\bibinfo{author}{\bibfnamefont{H.}~\bibnamefont{Monien}},
  \bibinfo{journal}{Phys.\ Rev.\ Lett.} \textbf{\bibinfo{volume}{87}},
  \bibinfo{pages}{126402} (\bibinfo{year}{2001}).

\bibitem[{\citenamefont{Halperin}(1965)}]{Halperin65}
\bibinfo{author}{\bibfnamefont{B.~I.} \bibnamefont{Halperin}},
  \bibinfo{journal}{Phys.\ Rev.} \textbf{\bibinfo{volume}{139}},
  \bibinfo{pages}{A104} (\bibinfo{year}{1965}).

\bibitem[{\citenamefont{Lifshits et~al.}(1988)\citenamefont{Lifshits,
  Gredeskul, and Pastur}}]{Lifshits88}
\bibinfo{author}{\bibfnamefont{I.~M.} \bibnamefont{Lifshits}},
  \bibinfo{author}{\bibfnamefont{S.~A.} \bibnamefont{Gredeskul}},
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{L.~A.}
  \bibnamefont{Pastur}}, \emph{\bibinfo{title}{Introduction to the Theory of
  Disordered Systems}} (\bibinfo{publisher}{Wiley, New York},
  \bibinfo{year}{1988}).

\bibitem[{\citenamefont{Kosztin et~al.}(1998)\citenamefont{Kosztin, Kos, Stone,
  and Leggett}}]{Kosztin98}
For a recent account of the subject, see
\bibinfo{author}{\bibfnamefont{I.} \bibnamefont{Kosztin}},
  \bibinfo{author}{\bibfnamefont{S.}~\bibnamefont{Kos}},
  \bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Stone}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{A.~J.} \bibnamefont{Leggett}},
  \bibinfo{journal}{Phys.\ Rev.\ B} \textbf{\bibinfo{volume}{58}},
  \bibinfo{pages}{9365} (\bibinfo{year}{1998}).

\bibitem[{\citenamefont{Scalapino et~al.}(1972)\citenamefont{Scalapino, Sears,
  and Ferrell}}]{Scalapino72}
\bibinfo{author}{\bibfnamefont{D.~J.} \bibnamefont{Scalapino}},
  \bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Sears}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{R.~A.} \bibnamefont{Ferrell}},
  \bibinfo{journal}{Phys.\ Rev.\ B} \textbf{\bibinfo{volume}{6}},
  \bibinfo{pages}{3409} (\bibinfo{year}{1972}).

%\bibitem[{\citenamefont{Negele and Orland}(1988)}]{Negele88}
%\bibinfo{author}{\bibfnamefont{J.~W.} \bibnamefont{Negele}} \bibnamefont{and}
%  \bibinfo{author}{\bibfnamefont{H.}~\bibnamefont{Orland}},
%  \emph{\bibinfo{title}{Quantum Many-Particle Physics}}
%  (\bibinfo{publisher}{Addison-Wesley, Redwood City}, \bibinfo{year}{1988}).

%\bibitem[{\citenamefont{Shankar}(2nd ed., 1994)}]{Shankar94}
%\bibinfo{author}{\bibfnamefont{R.}~\bibnamefont{Shankar}},
%  \emph{\bibinfo{title}{Principles of quantum mechanics}}
%  (\bibinfo{publisher}{Plenum Press, New York}, \bibinfo{year}{2nd ed., 1994}).

\bibitem[{\citenamefont{Gr{\"{u}}ner}(1994)}]{Gruener94}
\bibinfo{author}{\bibfnamefont{G.}~\bibnamefont{Gr{\"{u}}ner}},
  \emph{\bibinfo{title}{Density Waves in Solids}}
  (\bibinfo{publisher}{Addison-Wesley, Reading}, \bibinfo{year}{1994}).

\bibitem[{\citenamefont{Brazovskii and Dzyaloshinskii}(1976)}]{Brazovskii76}
\bibinfo{author}{\bibfnamefont{S.~A.} \bibnamefont{Brazovskii}}
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{I.~E.}
  \bibnamefont{Dzyaloshinskii}}, \bibinfo{journal}{Zh-Eksp.\ Teor.\ Fiz.}
  \textbf{\bibinfo{volume}{71}}, \bibinfo{pages}{2338} (\bibinfo{year}{1976}),
  \bibinfo{note}{[Sov. Phys. JETP {\bf{44}} (1976) 1233]}.
%
%\bibitem{McKenzie95a}
%R.\ H.\ McKenzie, Phys.\ Rev.\ B {\bf{52}}, 16428 (1995).
%
\bibitem{McKenzie96b}
R.\ H.\ McKenzie and D.\ Scarratt,
Phys.\ Rev.\ B {\bf{54}}, R12709 (1996).

\bibitem{Joynt99} For a different interpretation of the ARPES signal, see R.\ Joynt, Science \textbf{284}, 777 (1999).


\end{thebibliography}