Download high-temperature superconductivity from short

FEATURE ARTICLE
HIGH-TEMPERATURE SUPERCONDUCTIVITY FROM
SHORT-RANGE REPULSION
BY
ANDRÉ-MARIE S. TREMBLAY
S
tandard solid-state physics books offer limited
help to understand much of the phenomenology of
high-temperature superconductors. Almost every
basic assumption behind the theories in these
books fails. For example, one expects the true elementary
excitations of a metal to have the same quantum numbers
as free electrons, in other words one expects the so-called
Fermi liquid theory to apply. That is not the case for the
cuprates. One instead finds that at half-filling they are
insulators and that, lightly-doped away from half-filling,
their metallic state at finite temperature is not a Fermi
liquid. Another tenet of solid state physics is that strong
Coulomb repulsion is detrimental to superconductivity.
Apparently not so for the cuprates since the
superconducting transition temperature is high despite the
strong Coulomb repulsion necessary to explain their
insulating behavior at half-filling.
The challenge, like in many other frontier fields of
physics, is to understand strong interactions. Much
ingenuity and hard work has been invested towards this
objective. The paper by Arun Paramekanti in this issue of
Physics in Canada explains in simple terms how strong
correlations lead to an insulator at half-filling. This
phenomenon is called the Mott transition. It also describes
some of the methods that have been developed to
understand the whole phase diagram.
Here I focus on two recently developed methods that
involve numerical computations, and on what we learn
from them. I outline mainly the contribution from our
group in Sherbrooke, so there is definitely some bias. I
begin by describing the model.
SUMMARY
Theoretical models of high temperature
superconductors cannot be treated by
perturbation theory. Here I briefly survey
some of the new theoretical methods that
have been developed for the Hubbard model
and the insights they have provided to
understand
the
Physics
of
strong
interactions
and
high-temperature
superconductors.
THE MODEL, WEAK AND STRONG
COUPLING
Simple models in physics can have surprisingly rich sets
of solutions that explain observed complex behavior with
minimal assumptions. The model proposed by John
Hubbard in 1963 [1] fits this category. Anderson [2]
proposed it for high-temperature superconductivity right
after its discovery in 1986. It contains two terms, one for
kinetic energy in a single band, and the other for shortrange Coulomb repulsion. The screened Coulomb
interaction is represented by the energy cost U of double
occupation of a single orbital. Without the Coulomb
repulsion, the Hamiltonian is diagonal in momentum
space, while with only the Coulomb interaction it is
diagonal in real space. When U is larger than the kinetic
energy bandwidth, we are in the strong coupling regime,
whereas we are in the weak coupling regime when the
kinetic energy dominates. In practice, the cuprates are
very near intermediate coupling where neither one or the
other term clearly dominates. This means that the exact
low energy eigenstates have components in a sizeable
fraction of the exponentially large Hilbert space, by
contrast with the weakly interacting case, for example,
where one can simply fill a Fermi sea and include
perturbative corrections.
Exact solutions to this model exist only in one-dimension
or in infinite dimension. The two-dimensional case is the
relevant one for cuprates.
WEAK COUPLING
Early progress towards explaining how superconductivity
can arise from purely repulsive interactions came from
weak coupling approaches. As explained by Paramekanti,
the first phenomenon that comes out simply from the
Hubbard
model
is
the
tendency
towards
antiferromagnetism. In regions where the system is not
antiferromagnetic, spin fluctuations can nevertheless be
present. The antiferromagnetic spin waves can then play
the role of phonons: the glue that binds Cooper pairs. One
important prediction of this approach [3-5], made first in
the context of organic and heavy fermions
superconductors in Ref. [3] by a group that included
Claude Bourbonnais in Sherbrooke, is that the order
parameter will have d-wave symmetry. The d-wave
A-M.S. Tremblay
<Andre-Marie.
Tremblay@
USherbrooke.ca>,
Département de
physique et RQMP,
Université de
Sherbrooke,
Sherbrooke, QC,
J1K 2R1
et membre du
programme Matériaux
quantiques, Institut
canadien de
recherches avancées,
Toronto, ON,
M5G 1Z8
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HIGH-TEMPERATURE SUPERCONDUCTIVITY ... (TREMBLAY)
symmetry was also predicted for the cuprates [6,7], where it is
indeed observed.
with dynamical mean-field theory, the subject of our next
section.
Weak coupling ideas can serve as a stepping stone to develop
non-perturbative approaches. The Two-Particle-Self-consistent
(TPSC) approach does not use Feynman diagrams [8,9]. Instead,
it enforces conservation laws, the Pauli principle and makes
one ansatz that allows double occupancy to be determined selfconsistently. The validity of this approach as an approximate
solution to the two-dimensional Hubbard model can be tested
by comparing with Quantum Monte Carlo calculations [8-10].
The great advantage of Monte Carlo is that it is exact within
statistical uncertainties that can be estimated. The disadvantage
is that since fermions are in antisymmetric states, a sign (±) has
to be attached to each Monte Carlo configuration. That
prevents the sampling to be efficient at low temperatures in
many cases. Nevertheless, there are some non-trivial regimes
where Monte Carlo can be used as a benchmark for
TPSC [10,11].
STRONG COUPLING
Within TPSC one other important theorem is satisfied: The
Mermin-Wagner theorem states that you cannot break a
continuous symmetry in two dimensions. Hence, the phase
transition that would appear in mean-field theory is replaced by
a crossover temperature below which the antiferromagnetic
correlation length ξ grows rapidly. When it becomes larger than
the thermal de Broglie wavelength, ξth, that you can find from
Δε ∼ υFΔk ~ υFξth-1 ~ kBT, then the electrons see the
background spin configuration as if it were almost ordered.
That leads to the opening of a pseudogap in the parts of the
Fermi surface that would become gapped if there were longrange order [9,12]. Neutron scattering has determined that, in
electron-doped cuprates, the pseudogap indeed opens up when
the condition ξ ∼ ξth is satisfied [13]. The Fermi velocity is
obtained from photoemission experiments, which are also in
excellent agreement with TPSC [14]. In this approach, one finds
a superconducting dome and the most recent calculations of
conductivity are promising [15]. In particular, experiment finds,
at the antiferromagnetic critical point and at larger doping, that
there is a correlation between Tc and the size of the linear term
in the resistivity [16,17]. The reason this approach works for
electron-doped systems, even though it is based on weak
coupling ideas extended to a non-perturbative regime, is that
for electron doping, the interaction strength U is about 6t
(where 8t is the bandwidth) which is below the value necessary
to drive a strong Mott transition. The system can still be an
insulator at half-filling above the Néel temperature because of
fluctuating antiferromagnetism in two dimensions [9]. Or, it is
also possible that U changes a bit with doping. Comparison of
cluster perturbation theory calculations with experimental
photoemission results have provided insight on the Fermi arc
structure observed in the pseudogap regime and suggested that
U is a bit smaller for electron-doped cuprates as compared with
hole-doped [18]. This was confirmed by more recent
calculations [19] based on first-principles methods combined
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The solution of the Hubbard model in infinite dimension was a
breakthrough that led to a new approach to the problem at
strong coupling. Georges and Kotliar [20] as well as Jarrell [21]
proposed that one could study the Hubbard model using an
approasch that has the same structure as the solution in infinite
dimension, namely a single interacting site in a self-consistent
bath of non-interacting electrons. The problem with the singlesite approach [22] is that high-temperature superconductors are
quasi two-dimensional and that actual observations of
electronic properties by photoemission have shown that the
self-energy is also momentum dependent. Hence, one cannot
completely neglect spatial correlations. A generalization was
thus proposed wherein a cluster of sites (instead of a single
site) is embedded in an infinite bath of non-interacting
electrons. There are two popular ways of doing this [23,24] that
become identical if the cluster is large enough. Generically one
calls these methods quantum cluster methods [25,26].
Some of the main results that have been found are as follows.
a) Finite-size studies confirm that there is a d-wave
superconducting transition at finite temperature [27,28].
b) The zero-temperature phase diagram contains
antiferromagnetism and d-wave superconductivity with a
tendency to coexist in the underdoped regime [29,30]. Such a
tendency is observed, but not to the extent suggested by the
calculations. It is difficult to study the stripe phase with these
methods because of the large clusters that would be necessary,
so one cannot rule out that stripe order would appear in this
model, as it does in some experiments.
c) Calculated spectral weights in the normal state are similar to
those observed experimentally [18,30].
d) Retardation effects that have been associated with the glue
in ordinary superconductors are also observed in the quantum
cluster calculations. The relevant frequencies are associated
with spin fluctuations, providing direct evidence for that
mechanism [31,32].
e) In the presence of a small orthorhombicity, very large
anisotropy in the transport coefficients is obtained if U is larger
than the critical U for the Mott transition [33,34].
f) If one does not allow long-range order and lets the normal
state survive at arbitrarily low temperature, one predicts that if
the interaction strength U is larger than the one needed for the
Mott transition, there appears a first-order transition between
two metallic phases with different compressibilities and spin
fluctuations [35,36]. That first-order transition ends at a critical
point at extremely low temperature when U is large. At
temperatures above that critical point, the scattering rate is
large. This reminds us of the phenomenology associated with
the quantum critical point discussed in the paper by Louis
Taillefer in this issue, except that, this time, long correlation
lengths are not necessary.
HIGH-TEMPERATURE SUPERCONDUCTIVITY ... (TREMBLAY)AA
WHAT HAVE WE LEARNED, AND SOME OF
THE OPEN QUESTIONS
Quantitative calculations that explain the many experiments
are what is needed before the community agrees that the hightemperature superconductivity problem is solved. Progress in
this direction has been steady and encouraging, as this
contribution and that of Paramekanti have tried to show. One
may need to refine the Hubbard model to explain every detail
of the experiments, but what is encouraging up to now is that
the hopping matrix elements suggested by band structure
calculations and a value of U of the order of the bandwidth
suffice to explain a lot of the phenomenology. It is noteworthy
that the same methods reproduce the main features of the zerotemperature phase diagram of layered BEDT organics [37,38].
This phase diagram [37] and that of the high Tc suggest that
antiferromagnetic fluctuations must be present for d-wave
superconductivity but that frustration by doping or by
exchange interaction must be large enough to prevent longrange antiferromagnetic order.
The pseudogap in electron-doped cuprates seems well
explained by precursors of antiferromagnetism [13,14], but the
question is still open for hole-doped cuprates. Although holeand electron-doped cuprates must be modeled with interaction
strengths U that are comparable, it could make a difference
whether U is smaller or larger than the critical U for the Mott
transition. In particular, a value of U larger than the critical U
for the Mott transition allows one to explain spectral weight
transfer as a function of doping, as observed in X-Ray
absorption spectroscopy [39]. It would also provide a way to
obtain large anisotropies (nematicity) in the pseudogap
regime [34] and large scattering rates above optimal doping, but
without the need for long magnetic correlation lengths [33]. It
will be necessary to perform calculations that allow long-range
antiferromagnetic order to find out whether the finite-doping
transition between two metals [35,36] merges with the
antiferromagnetic quantum critical point or is a different
phenomenon. In the hole-doped cuprates, antiferromagnetism
seems to end at a doping much smaller than the quantum
critical point extrapolated from the pseudogap temperature.
From a methodological point of view, including interactions
between particle-hole pairs (so-called vertex corrections) in
quantum cluster methods [40] would allow more reliable
calculations of transport properties. We also have to face the
fact that many experiments [41,42] suggest that to understand the
whole phase diagram of hole-doped cuprates, it may be
necesssary to include the effect of oxygens in the copperoxygen planes [43]. This means solving the three-band Hubbard
model, a very difficult task that may stimulate further
methodological improvements.
The difficulty associated with strong correlations in hightemperature superconductors has forced us to accept that
analytical methods cannot alone solve that problem. Much
remains to be done but there is clear hope that the solution is in
sight. And it is also clear that progress on that front has led and
will continue to lead to many insights on interesting materials
and perhaps even allow engineering heterostructures, as
discussed in the contribution of Cavaglia, or new materials
with properties we choose to design [44]. One can safely predict
that high-temperature superconductivity will continue to
stimulate developments in theoretical physics with far reaching
consequences.
ACKNOWLEDGEMENTS
I am indebted to many students, postdocs and collaborators
whom I warmly thank and to D. Sénéchal for reading the
manuscript and for collaboration on most of the work discussed
here. This work was partially supported by the Tier I Canada
Research Chair Program.
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