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\begin{document} 
\title{A Short Review of Non Fermi Liquids}
\author{ Laura M. Robinson\\
	 \\
         Department of Physics\\ 
         University of Cincinnati \\
         Cincinnati, Ohio 45221\\[1em]}
 
\date{March 21, 1997}
\maketitle
\vspace{1.0in}

\begin{abstract}
%\Large
	The low temperature properties of ordinary metals became well known
with Landau's development of Fermi liquid theory in the late 1950s.
In some heavy fermion compounds, the predictions of this theory have been
extremely accurate.  In recent years, however, a new class of heavy fermion 
materials has become the subject of increased scrutiny because they 
possess unusual thermodynamic and magnetic properties that cannot be 
explained by conventional Fermi liquid theory.  These materials are known
as non Fermi liquids.  Although the mechanism that drives the difference 
in the behaviour of these seemingly similar heavy fermion systems is not 
fully understood, the low temperature physical properties of non Fermi liquids 
can be characterized and compared to those of conventional Fermi liquids.  
This paper attempts to classify the differences in these materials as 
well as provide some general information about non Fermi liquids.
\end{abstract}

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%\Large

%\tableofcontents
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\section{Fermi Liquids}

\indent

In order to better understand the low temperature properties of
non Fermi liquids (NFL), it is useful to grasp the essential nature of 
some heavy fermion systems that behave like ``ordinary'' Fermi liquids.  
Fermi liquid theory (FLT), originally introduced
by Landau in 1956, has formed the basis of our understanding of metallic
systems.  The basic premise of this theory is that a one-to-one 
correspondence exists between the excitations of a free Fermi gas 
and the interacting conduction electrons in a metallic system. \cite{refB} 
\cite{refD}  Another underlying idea in this theory is the existence of
a single energy scale.   This suggests that if the temperature and 
energy dependence of experimentally measured quantities, such as the 
specific heat, resistivity  and magnetic susceptibility, are consistent 
with the 
intrinsic energy scale of the electronic excitations, i.e. the Fermi 
energy, then the material behaves like a Fermi liquid.  \cite{refE}

So what are the physical properties of a heavy fermion system that is 
described by FLT?  Qualitatively, one expects that 
the electrical resistivity should increase with increasing temperature,
as is the case for ordinary metals like copper.  It turns out that this 
is not the case 
for heavy Fermi liquids.  To illustrate this behavior, consider $CeAl_{3}$. 
$CeAl_{3}$ provides an excellent example for illustrating the physical 
characteristics of these heavy fermion systems,
as it was one of the first discovered heavy electron compounds.  Figure 1 
shows the resistivity of $CeAl_{3}$ for both high and low (inset) temperatures.

\begin{figure}[htb]
\centerline{\psfig{figure=CeAl3rho.eps,height=2.5in}}
\caption[]{\em{$\rho$(T) of $CeAl_{3}$ between 1.5 and 300 K and below 0.1
K.  Data are from Ott et al. (high temperature) and Andres et al. (low 
temperature).  Taken from [3].}}
\end{figure}

Contrary to ordinary metals, $\rho$(T) has a minimum just below room
temperature (300 K) and increases with decreasing temperature, until about
35 K.  At this temperature, the resistivity reaches a maximum.  As 
the temperature continues to decrease, $\rho$(T) correspondingly decreases.
For extremely low temperatures, $\rho$(T) can be well approximated by \newline
\newline
\centerline{\[$\rho$(T) = $\rho_{0}$ + A $T^{2}$ \]} 
\newline
\newline 
where $\rho_{0}$ is the resistivity at zero temperature and A is a constant, 
both of which vary for different materials.  This model fits the data well 
up to roughly 0.3 K. \cite{refC}

In heavy fermion materials, the low temperature specific heat also behaves
unlike ordinary metals, for which it has a finite value at zero temperature 
and slightly increases linearly with increasing temperature.  
Figure 2 illustrates the contrasting low T behaviour of the specific heat for 
$CeAl_{3}$.  

\begin{figure}[htb]
\centerline{\psfig{figure=CeAl3C.eps,height=2.5in}}
\caption[]{\em{Low temperature specific heat C/T versus T for $CeAl_{3}$.
The data for T $>$ 1.5 K are from Ott et al., those below 0.3 K from 
Andres et al.  Taken from [3].}}
\end{figure}
Here, the specific heat increases with $\em{decreasing}$ temperature.  
It reaches a maximum somewhere between 0.3 and 1 K.  After the maximum,
the specific heat decreases and obtains a finite value at zero temperature
which is much larger than that of ordinary metals (2000 times greater
than copper, for example).  This is a result of the large effective
mass of these ``heavy'' fermion materials. \cite{refC}

\section{Non Fermi Liquids}
\indent

Now that the properties of a Fermi liquid have been described, the problem
of classifying non Fermi liquids begins.  Several different materials
exhibit NFL behaviour; most are cerium and uranium intermetallics 
that have been doped with a nonmagnetic material.  The Ce and U ions have
partially filled f-electron orbitals that interact with the conduction 
electrons in the materials, most certainly contributing to the NFL behaviour
observed in these intermetallics.  Some examples of NFLs are 
$U_{1-x}Th_{x}Pd_{2}Al_{3}$, $M_{1-x}U_{x}Pd_{3}$ (M=Sc, Y), and 
$UCu_{5-x}Pd_{x}$.  These materials differ from Fermi liquids in that they 
exhibit either logarithmic or weak power law divergences in some physical
characteristics, including the electrical resistivity and specific heat,
at low temperatures.\cite{refA}  In fact, it seems that NFLs are 
defined more by their low temperature behaviour than from an all-encompassing 
definition as was the case for Fermi liquids.  Recall that Fermi liquids were 
defined as materials displaying universal behaviour for energies less than the 
Fermi energy.  Analysis NFLs indicates that the scaling of electronic 
excitations in NFLs are more varied than predicted for Fermi liquids.  
In fact, this variable energy scale is believed to contribute to the 
NFL behaviour, and is also included in the working definition of NFLs.
\cite{refE}  

As previously mentioned, deviation from Fermi liquid has recently 
been observed in several f-electron intermetallics, including
$Y_{0.8}Th_{2-x}U_{x}Pd_{3}$.  In contrast to the quadratic temperature 
dependence of the resistivity of Fermi liquids at low temperatures, the 
low temperature resistivity for non Fermi liquids 
is given by \newline
\newline
\centerline{\[$\rho$(T) $\sim$  $\rho_{0}$ - A T \]}
\newline
\newline
where $\rho_{0}$ indicates the resistivity at zero temperature and 
A is a constant.
As in the case for Fermi liquids, these constants are material dependent, and
are related to the relevant energy scale for the material.
Figure 3 illustrates the low temperature behaviour of the resistivity of 
$Y_{0.8}Th_{2-x}U_{x}Pd_{3}$. 

\begin{figure}[htb]
\centerline{\psfig{figure=nflrho2.eps,height=2.5in}}
\caption[]{\em{Low temperature resistivity data (T $\le$ 20 K) plotted
as log(1 - $\Delta \rho$(T)/$\Delta$$\rho$(0) versus log T for the alloy
$Y_{0.8}Th_{2-x}U_{x}Pd_{3}$ $(0 \le x \le 0.2)$.  Solid lines represent
fits to a power law $\Delta \rho$(T)/$\Delta \rho$(0) = 1 - 
a$(T/T_{k})^{n}$ (n $\approx$ 1 $\pm$ 0.2), revealing 
non Fermi liquid behaviour. Taken from [1].}}
\end{figure}
Note that in this plot, the vertical axis has a different functional form
than the resistivity plotted for $CeAl_{3}$.  This does not mean that the
two plots are not related;  figure 3 simply illustrates the normalized
resistivity for $Y_{0.8}Th_{2-x}U_{x}Pd_{3}$.  Also note that the 
resistivity has been fitted to $T^{n}$.  In constrast to Fermi liquids 
where n is typically 2 (quadratic), the value of n for 
for the example show here is approximately 1.0 $\pm$ 0.2 (a similar 
result holds for other NFLs).  This result is also in agreement with 
the predicted linear temperature dependence of the resistivity.

The specific heat of non Fermi liquids also differs from that of a Fermi
liquid.  It is given functionally by \newline
\newline
\centerline{\[C $\sim$ $\gamma$ T log T ]\}
\newline
\begin{figure}[htb]
\centerline{\psfig{figure=nflcv1.eps,height=2.5in}}
\caption[]{\em{Low temperature specific heat C/T versus log T for 
various alloys which exhibit a T=0 logarthmic divergence, 
a non Fermi liquid behaviour.  Taken from [1].}}
\end{figure}
Figure 4 shows the low temperature dependence of the specific heat for
several NFLs, including $Y_{0.8}Th_{0.1}U_{0.1}Pd_{3}$ shown on a logarithmic
temperature scale.  Since these plots are linear, agreement is shown with
the functional form of C.  The specific heat diverges logarithmically for
low temperatures!  This behaviour is extremely different from that
of Fermi liquids, which attain a finite value for the specific heat at
low temperatures.

\section{Conclusions}
\indent

So why are NFL materials important?  As indicated above, a great deal of the 
interest in NFL behaviour simply results from the unusual temperature
dependence of the physical characteristics of NFLs.  NFL materials have 
also become increasingly more important as they display unconventional
superconductivity that has also been observed in some copper oxides.  While
these oxides differ from heavy fermion systems, they have strikingly similar
superconducting states (aside from the different observed critical 
temperatures.)  It is hoped that an understanding of the NFL systems will not
only aid in the understanding of the nature of the elecronic excitations
in these f-electron materials, but also in the understanding of this 
unconventional superconductivity. \cite{refA}

The mechanism that drives the difference between Fermi liquids and NFLs 
is not fully understood, but many theories exist.  Most agree that the
behaviour of NFLs is the result of several interactions.  One idea is that
many of the observed properties may be understood as a consequence of a broad
distribution of Kondo temperatures, although this is probably only part
of the explanation.

\begin{thebibliography}{99}
\bibitem{refA} M.B. Maple, M.C. de Andrade, J. Herrmann, Y. Dalichaouch,
D.A. Gajewski, C.L. Seaman, R. Chau, R. Movshovich, M.C. Aronson, and R.
Osborn,\ {\em J. Low Temp. Phys.} {\bf  99}, 223 (1995).
\bibitem{refB} E. Miranda, V. Dobrosavljevic, and G. Kotliar,\ {\em J. Phys.:
Condens. Matter} {\bf 8}, 9871, (1996).
\bibitem{refC} H.R. Ott,\ {\em Progress Low Temp. Phys.} {\bf XI}, 215 
(1987).
\bibitem{refD} M.C. Aronson, R. Osborn, R.A. Robinson, J.W. Lynn, R. Chau,
C.L. Seaman, M.B. Maple,\ {\em Phys. Rev. Lett.} {\bf  75}, 725 (1995).
\bibitem{refE} M.C. Aronson, M.B. Maple, R. Chau, A. Georges, A.M. Tsvelik, 
and R. Osborn,\ {\em J. Phys.: Condens. Matter} {\bf 8}, 9815, (1996).

\end{thebibliography}
\end{document}

