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4 pages, 1 figure
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quantum lattice model, strong-coupling, Falicov-Kimball model,
phase separation
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\documentstyle[aps,twocolumn,prl,epsf,amssymb,floats]{revtex} %with lines(3) '
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\begin{document}
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\title{Phase separation due to quantum mechanical correlations}
\author{James K. Freericks$^*$, Elliott H. Lieb$^{\dagger}$, and
Daniel Ueltschi$^\ddagger$}
\address{$^*$Department of Physics, Georgetown University, Washington, DC
20057, USA\\
$^\dagger$Departments of Mathematics and Physics, Princeton University, Princeton, NJ 08544,
USA\\
$^\ddagger$Department of Mathematics, University of California, Davis, CA 95616,
USA}
\date{\today}
\maketitle
\widetext
\begin{abstract}
Can phase separation be induced by strong electron correlations? We
present a theorem that affirmatively answers this question in the
Falicov-Kimball model away from half-filling, for any dimension. In the ground
state the itinerant electrons are spatially separated from
the classical particles.
\end{abstract}
\pacs{71.28+d, 71.30+h, and 71.10-Hf}
] %+++++
\narrowtext
%============================================================================
The Falicov-Kimball (FK) model~\cite{FK69}
can be viewed as a modification
of the Hubbard (H) model~\cite{H63} in which one species of electrons (say
spin down) has infinite mass. As such, its relation to the latter is
similar to the relation of the Ising model to the quantum Heisenberg
model~\cite{KL86}. Alternatively, it can be viewed as a model
of itinerant electrons and immobile ions. It possesses long range order at
low temperature in two or more dimensions at half filling, and this
checkerboard state (and higher-period
generalizations~\cite{GJL92,K94,K98,HK01}) remain to-date as the
only examples of crystallization
into a perfectly ordered structure whose periodicity is not that of the
underlying lattice.
In this paper, we report a theorem on the existence of phase separation
in the ground state of the FK model, away from half-filling and for large
repulsion between the particles. `Phase separation', or `segregation', means
that the system splits into two large domains, one being
occupied by the classical particles, and the other by the quantum particles.
In the language of the H model, this would mean segregation of spin up particles
from spin down particles --- resulting in a ferromagnetic state.
The question of whether strong interactions can drive quantum (electronic)
systems to phase separate was posed over ten years ago for the FK model
\cite{FF90} and the H model \cite{KE90}. This work
is a rigorous proof of this long-standing conjecture for the FK model.
Further discussion of the
relation between the FK and the H model is given later.
Phase separation is not new in classical
lattice models. It is present in the Ising model, and
in many other classical models. It also occurs in the FK model at half-filling
for some densities, as was proved in \cite{K98}. In these examples it is mainly a {\it
local} phenomenon --- interactions (or `effective interactions' in the case of
FK) tend to dislike boundaries, and the state with minimum boundary is phase
separated.
Away from half-filling, the
electrons of the FK model are in delocalized wave functions, and their energy cannot be
written as a sum of local terms. While we prove that the ground state energy is roughly
proportional to the boundary between occupied and empty sites, the mechanism is {\it
nonlocal} and genuinely quantum mechanical.
The FK Hamiltonian~\cite{FK69} is
\begin{equation}
H=-\sum_{{\bf x},{\bf y}\in {\Omega}} t({\bf x} - {\bf y}) c^{\dagger}_{\bf x}c_{\bf y}+
U\sum_{{\bf x}\in {\Omega}}
c^{\dagger}_{\bf x}c_{\bf x}w_{\bf x}.
\end{equation}
Here, $t({\bf x}-{\bf y})$ is the hopping coefficient between sites ${\bf x}$ and
${\bf y}$; it is translation
invariant, but may depend on the direction (this allows consideration of general Bravais
lattices). $c^{\dagger}_{\bf x}$
and $c_{\bf x}$ are creation and annihilation operators
for a spinless electron at site ${\bf x}$, and $w_{\bf x} = 1$ or 0 is a classical
variable
that denotes the presence or the absence of an ion at ${\bf x}$. (Spin degrees of freedom
have trivial behavior and are left aside here.)
$\Omega \subset {\mathbb Z}^d$ is
a finite $d$-dimensional lattice, and
$U$ is the on-site repulsion between the two species of particles. For any
given configuration $w = \{w_{\bf x}\}$ of
classical particles, the ground state for $N_{\rm e}$ electrons is determined by
diagonalizing a one-body operator given by the above Hamiltonian, and
filling in the lowest $N_{\rm e}$ states. The main question is to find which
configuration $w$,
with a given number of classical particles $N_{\rm c} = \sum_{\bf x} w_{\bf x}$,
minimizes the energy of the electrons.
Our theorem states upper and lower bounds for the energy of $N_{\rm e}$
electrons, for a given configuration $w$ of the classical particles. For
orientation, let us consider first
a configuration where the sites devoid of classical particles form a large,
`compact' region. The
expected energy of the electrons is a bulk
term that scales like the volume of this region, and a correction that
scales like its boundary. Our main result is a proof of this conjecture for
{\it all} configurations, not only `nice' ones.
We need some notation.
Let $\Lambda = \{ {\bf x} \in \Omega: w_{\bf x} = 0 \}$
denote the set of empty sites for the configuration $w$, and $\partial\Lambda$
its boundary, $\partial\Lambda = \{ {\bf x}\in\Lambda, {\rm dist}({\bf x},\Omega\setminus
\Lambda)=1 \}$. Their respective
number of sites are $|\Lambda|$ and $|\partial\Lambda|$. We
write $E(N_{\rm e},w)$ for the ground
state energy of $N_{\rm e}$ electrons in the configuration $w$.
An important quantity is $n=n_{\rm e}/(1-n_{\rm c})$, where
$n_{\rm e} = N_{\rm e}/|\Omega|$ and $n_{\rm c} = \sum_{\bf x} w_{\bf x}/|\Omega|$
are the densities for quantum and classical particles respectively.
It represents the
electronic density that would exist inside $\Lambda$, if all electrons
live inside the domain devoid of classical particles.
Let $e(n)$ be the usual kinetic energy per site for noninteracting electrons with
density $n$ in the thermodynamic limit (its expression is recalled below, see Eq.\
(\ref{eq: e_fermi})).
\vspace{1mm}
\noindent
{\bf Theorem.} {\it For all $\Lambda$ we have
upper and lower bounds,
$e(n) |\Lambda| + \alpha'(n) |\partial\Lambda| \ge E(N_e,w) \ge e(n) |\Lambda|
+ \alpha(n,U) |\partial\Lambda|$.
Here, $\alpha' (n)$ and $\alpha (n, U)$ are explicitly given positive
functions. For nearest-neighbor hoppings (i.e., $t({\bf x}) \neq 0$ if $|{\bf x}|=1$, $t({\bf
x})=0$ otherwise), $\alpha(n,U) = \alpha(n) -
\gamma(U)$, where $\alpha(n) = \alpha(1-n)$ is strictly positive for
$0 n_{\rm e}$ should also
be segregated. The central band that includes the line $n_{\rm e} + n_{\rm c}
= 1$ should be the host of numerous periodic phases and various coexistences between
periodic phases and empty or full phases. This is supported by numerical
simulations in 2D \cite{2d_numerics}.
In one dimension, the ground states are {\bf periodic}
\begin{itemize}
\item
when $(n_{\rm e},n_{\rm c})=(\frac12,\frac12)$ \cite{KL86};
\item
when $n_{\rm c}=1-n_{\rm e}$ (half-filling) and $n_{\rm c}=p/q$ is a rational
number (in an irreducible fraction). The periodicity is $q$ for $U$ sufficiently large
(depending on $q$) \cite{L92}.
\end{itemize}
For finite $U$, there is numerical evidence for {\bf coexistence} of two
periodic phases and free electrons
\begin{itemize}
\item
when $n_{\rm c}=1-n_{\rm e}$, $p/q