Research program, T. H. Hansson
Strongly correlated electron systems
In many materials, the strongly interacting electrons can be understood using Fermi-liquid
theory. The idea is that the main effect of interactions can be coded in a few phenomenological parameters, the most important one being the effective electron mass. Taking these so
called renormalizations into account, the electrons can be described as a collection of weakly
interacting quasi electrons and quasi holes. In other systems, like BCS superconductors, and
various magnetic systems, the picture is more complicated in that the ground states
spontaneously break various symmetries. In spite of such complications, there are in many
cases very good theoretical descriptions based on various weak coupling methods, mean field
approaches and renormalization group techniques. Typical examples are the Ginzburg-Landau
and BCS descriptions of superconductivity, and the spin wave theory of antiferromagnets. For
many systems, there are also very successful numerical schemes based on various mean field
approaches, such as Hartree-Fock and density functional techniques.
There are, however, classes of quantum systems that are not easily amenable to the
aforementioned methods. Intrinsically strongly coupled and quantum dominated systems
where there is no dominating classical (mean field) solution belong to this class. In certain
low dimensional examples, such as the Kondo effect, and the Schwinger and Luttinger
models, there are solutions based on specialized techniques such as bosonization and the
Bethe ansatz, but in general one has to resort to phenomenological models or brute force
numerical simulations, a typical example being the low energy physics of QCD.
Another set of systems is characterized by having many competing low energy states. Any
linear combination of these is a candidate for the quantum ground state, and small
perturbations can give rise to widely different phases. Technically one is faced with a very
complicated problem in degenerate perturbation theory. The archetypal example is the
quantum Hall effect, but certain classes of frustrated magnets, and presumably also the
cuprates of high Tc super-conductivity, fall into this general class. A fascinating aspect of
many of these systems, is that the elementary excitations can have unusual quantum numbers
and obey so called fractional statistics. This is established in two-dimensional systems in
strong magnetic fields (the quantum Hall system), but it is an exciting possibility that both
one-dimensional systems described by Luttinger liquid theory, and certain frustrated twodimensional systems with no magnetic field also have fractionalized charge.
The characterization of these so called “exotic” quantum liquids, and the study of their phase
diagrams, has been a very active area of research during the last decades. Interestingly, many
ideas originating in the study of different systems are converging, and one can now attempt a
systematic classification of the exotic quantum systems based on the new concepts of
“topological”i and “quantum”ii order.
Topological states of matter and fractionalized excitations
A topological state of matter is characterized by having properties that are protected against
small (and local) changes in the microscopic Hamiltonian. Typical such properties are values
of charges, number of ground states, and interference (or Berry) phases related to transport of
excitations around closed loops. Topological states differ from the commonly known states
resulting from spontaneous symmetry breakdown in that they cannot be described by a
Landau theory for a local order parameter. Instead the information about the protected
quantities is coded in a topological field theory, the prime example being the Chern-Simons
gauge theories describing the Abelian quantum Hall liquids. A particularly intriguing class of
topological states is the non-abelian QH liquids, that will be discussed later. Interestingly, the
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simplest of these states, the Moore-Read, or Pfaffian, stateiii, can be shown to be topologically
equivalent to a weakly coupled p-wave paired superconductoriv.
Up until fairly recently, it was believed that topological states are rather exotic beasts –
theoretically very interesting, but hard to find, and to study. Although it had been known for
some time that ordinary, weakly coupled, BCS superconductors coupled to electromagnetism
is an example of a topological statev, most researchers associated topological states primarily
with the QH liquids. There have been many attempts to detect other predicted topological
states, such as spin liquids and p-wave superconductors, but the experiments are still not
conclusive.
Today the view on these matters has changed dramatically, mainly due to the discovery of
topological insulators in both two and three dimensionsvi. These materials can be described by
band theory – i.e. by non-interacting electrons - but differ from ordinary band insulators by
the existence of gapless edge modes. The presence or absence of these modes is related to
topological quantum numbers characterizing the bulk state. In the two-dimensional case there
is a close analogy to the integer quantum Hall effect, while in three dimensions, the edge state
is a two-dimensional gapless electron gas. The spectrum of the electrons in this gas is of the
Dirac type, just as in graphene. This has many interesting consequences, for instance when it
comes to the quantization pattern seen in the Hall effect.
Since the topological insulators are three-dimensional materials, there are many interesting
possibilities to form devices by putting them in proximity to superconductors and magnetic
materials, as will be briefly discussed in a later section.
The topological band theory can be extended to paired states, i.e. to superconductors, so that
these can also be classified according to topology. In hindsight it is clear that both the integer
QH effect and the BCS superconductors show that there can be interesting topological
properties also in weakly interacting systems, but before the discovery of the topological
insulators, this was not widely appreciated.
A fascinating feature of superconductors with non-trivial band topology is the presence of
Majorna modes, which most easily can be thought of as half of a fermion (two of these modes
can be combined to a Dirac fermion). The Majorana particles are neutral, zero energy,
solutions to the BdG equations that describe the quasiparticles in superconductors. The
Majorana modes exist on the edges of the samples, and in the core of vortices, and it is these
exotic vortices that have non-abelian statistics. Not surprisingly, a lot of recent efforts has
been spent to get a better theoretical understanding of the physics of the Majorana modes.
Cold atoms – bosons, fermions and optical lattices
The continued interest in strongly correlated electron systems, is paralleled by developments
in the field of strongly correlated bosons – an area of research which has exploded since
Bose-Einstein condensation (BEC) was first observed in a cold atom gas in 1995.
The cold atomic gases provide an unprecedented laboratory for studying quantum effects.
Since both fermions and bosons can be trapped, and since one can simultaneously trap
different gases or different quantum states of the same gas, one can study a vast number of
mixtures of bosons and fermions. Also, there are techniques to tune the effective short-range
interactions between the neutral atoms, from strong to weak coupling, something that is not
possible in electron systems where the interaction is Coulombic. This opens for detailed
comparison with theories based on weak-coupling expansions, and also the possibility to trace
the development of a system from weak to strong coupling. Perhaps the most striking
example of this is the transition from a BCS state of cold fermionic atoms at weak coupling,
to a molecular BEC of tightly bound molecules at strong coupling.
Another very exciting area is that of optical lattices, where crossed laser beams are used to
form periodic traps for the cold atoms. By tuning the lattice constants, and the coupling
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strength between the atoms, one can artificially create systems that almost faithfully emulate
some of the models used in condensed matter physics. Of particular interest is the Hubbard
model that is believed to be at the heart of understanding the high Tc compounds.
There are various ways to put the BEC:s in rapid rotation and create an Abrikosov lattice of
vortices. At even faster rotation the lattice will melt, and ultimately the kinetic energy in the
rotating frame will quench, and the particles will all occupy the bose analogue to the lowest
Landau level for electrons. For these states there is a close analogue to quantum Hall physics,
and many of the methods and results can be carried over to the boson casevii. In particular one
can speculate on the possibility of realizing states with non-Abelian topological order.
Solid state quantum devices and quantum materials
This is not the place to attempt a general survey of the new and explosively developing field
of quantum information and quantum computing. Suffice to say that a central problem is to
actually build and understand components that satisfy the very strict criteria of phase
coherence, stability and reproducibility, which are required for any actual implementation.
Among the proposed schemes, solid-state devices based on Josephson junctions (JJ) have the
advantage of scalability and relative ease of production due to the advances in
nanofabrication. Another advantage is that they are relatively easy to manipulate, since the
information is carried by charges and currents that can be controlled by electronic circuitry.
This however also means that they are very susceptible to decoherence due to charge and flux
noise, and the present efforts in this field are directed towards controlling these sources of
error. Such control requires not only a good understanding of the individual quantum devices,
but of the whole quantum electronic circuit, including various biases, transmission lines,
filters etc.
A very tantalizing suggestion for how to overcome the problem of decoherence, is the
concept of topologically protected qubitsviii. The idea is to code the quantum information in
non-local, topological, degrees of freedom, which are insensitive to local perturbations such
as electric or magnetic impurities. Although such a scheme can in principle be achieved using
JJsix, the most widely studied proposal is based on the properties of the quasiparticles in nonabelian quantum Hall liquids. These particles obey non-Abelian statistics, which means that
when they move around each other, while all the time being far apart, the wave function can
change between distinct, but degenerate, quantum states that can be used to code the quantum
information. An obvious obstacle is that QH experiments are hard to perform since they
require very low temperatures, very clean samples and very high magnetic fields. Here the
topological isolators and superconductors might offer new and simpler possibilities, since
they are time-reversal invariant states that do not require magnetic fields. Typically, the
interesting effects instead occur because of strong spin-orbit couplings. In 2008, Fu and Kane
showed that a time-reversal invariant two-dimensional superconducting state supporting
vortices with Majorana fermions, can be formed at the interface between a topological
insulator, and an ordinary s-wave superconductorx. Later, various compound structures have
been investigated, and they provide a new, and potentially simpler, route to non-abelian states.
So even though the path towards a useful solid state based quantum computer might be long
and steep, and even end at an insurmountable wall, to follow it will for sure be very
intellectually rewarding, and those who do, are quite likely to find many technologically
important devices along the way.
A more conventional, yet very innovative, way to overcome the decoherence problem is to
use JJ qubits as artificial atoms in a on-chip microwave cavity – a system that is analogous to
the much studied case of a real atom in an optical cavity. Using such a set-up, the qubit can be
kept in a state which is much less susceptible to noise, than in the usual configurationsxi.
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At an even more applied level, the study of JJ:s in a microwave cavity opens up the
possibility of designing a new class of quantum metamaterials. A metamaterial is, roughly
speaking, a designed periodic structure with electromagnetic response of a type not found in
naturally occurring materials. It is however still a material, in the sense that its
electromagnetic response can be coded in bulk parameters independent of the lattice spacing.
Striking examples are the metamaterials with negative index of refraction that has been
reported both at microwave and optical wavelengthsxii. The term “quantum metamaterial” has
been proposed for metamaterials where the response cannot be understood using classical
physics. Pertinent examples are certain multiple-quantum-well structuresxiii, and the JJ-arrays
mentioned earlierxiv. These ideas are still in their infancy, and their practical usefulness has
still to be demonstrated.
Examples of specific projects
1. Excitons and exciton condensates in quantum Hall liquids
The methods we have developed over the last five year provide explicit candidate wave
functions for all abelian hierarchical QH states. These wave functions have the expected
topological properties, and are thus most likely to be in the correct universality class. In the
simplest cases they coincide with the celebrated, and numerically well tested, wave functions
proposed by Laughlin and Jain. It is, however, of principal importance to develop a general
scheme to improve on these “representative” wave functions, which in our approach are
written in terms of conformal blocks of various conformal field theories. Any such procedure
is severely restricted by the necessity to preserve the good topological properties of the
original wave functions. Together with my former student Maria Hermanns, and present
graduate students, I am developing such a scheme which is based on exciting excitons, i.e.
localized quasiparticle – quasihole pairs.
While exciting a few excitons can be considered as a correction to the original ground state,
exciting a finite density of them might drive the system through a quantum phase transition to
another state at the same filling fraction. With collaborators, I am exploring these
possibilities, and we have some encouraging preliminary results on the transition between the
non-abelian Moore-Read QH state, and the metallic Halperin-Lee-Read state.
2. Quantum Hall hierarchy states on closed manifolds, and QH viscosity.
So far, we have constructed wave functions in a disc geometry, which is the one closest to
the bar geometry most commonly used in experiments, and also closely connected to the
cylindrical geometry appropriate for Corbino-disc experiments. It is however of great
principal interest to construct the corresponding wave functions on closed geometries, the
simplest, and most important, being the sphere and the torus. The reason is two-fold. At a
practical level, most efficient numerical calculations are performed on closed geometries to
avoid the complications due to gapless edge modes. At a theoretical level, the wave functions
on closed geometries contain important topological information. Since the electrons in a QH
liquid can carry different amount of orbital spin, they will respond differently to spatial
curvature. This effect is reflected in a quantity called shift, which can be determined from the
spherical wave functions. On a torus, all QH wave functions are degenerate, and the ground
state degeneracy directly reflects the topological nature of the liquid.
We are presently adopting techniques known from string theory to directly calculate relevant
correlation functions in the spherical geometry and with a background magnetic field (which
is normally not present in the string theory context). The results so far are promising. We have
already established collaboration with the group of dr. Nicolas Regnault at Ecole Normale
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Supérieure in Paris, which is world leading in simulating QH wave functions, and with them
we plan to test some of our spherical states that are of particular experimental interest.
In a recent paper we have presented a generalization of the so-called Moore-Read conjecture
about the relationship between QH wave functions and conformal blocks. The essential new
point is that the conformal blocks should be considered as wave functions in a coherent state
basis, rather than in a position basis. This should allow us get arbitrary hierarchical wave
functions on any manifold, provided we can find the pertinent coherent states. We already
have preliminary results on the torus that are very encouraging.
Some year ago, N. Read stressed the importance of a non-dissipative component in the
viscosity tensor for the QH systemsxv. This QH viscosity is related to the shift on the sphere,
and thus to the orbital spin. On the torus, which has zero curvature, the QH viscosity can be
obtained from the response to modular transformations. It is a challenge to calculate the
viscosity of the hierarchical states, and hopefully our general methods for constructing wave
functions in different geometries will give some insight. I intend to spend time on this
problem in the coming years.
3. Non-Abelian QH states
The methods we developed for abelian hierarchical states can also be applied to construct
hierarchies of non-abelian states in both open and closed geometries. Another aspect of our
method is that it allows for the construction of both quasihole and quasielectron states. This is
not unimportant, since the latter are hard to find with other methods, even when the ground
state is known. In a recent publication, we have also shown how to construct explicit
candidate wave functions that are expected to be in the same universality class as the antiPfaffian, i.e. the particle-hole conjugate of the Moore-Read state. Using similar methods, one
can also construct particle-hole symmetric non-abelian states at filling fraction ½.
Testing these ideas requires rather heavy numerical simulations. We hope to do at least some
of this work in collaboration with the group at ENS and some other issues are already studied
in Prof. Jainendra Jain’s group at Penn State University.
4. Topological field theory for p-wave superconductors
I have recently started a new line of research aimed at finding topological field theories that
describe superconductors with non-trivial band topology. The simplest example is the chiral
p-wave paired state in two and three dimensions, but there are also the time-reversal invariant
interface states discussed above11, and the noncentrosymmetric superconductors discussed by
Fujimoto and Satoxvi. The starting point is the BF theories that describe s-wave paired states6,
which we have generalized by adding a novel type of topological term containing
fundamental Majorana fields. The theory has an extra fermionic gauge invariance, and is quite
interesting also from a purely quantum field theory point of view. We are presently writing a
first manuscript on this work, and I have already presented some preliminary results at a
workshop at Princeton University in spring 2011. We have so far mainly considered the twodimensional parity and time-reversal breaking case, but intend to generalize our description to
three dimensions, and to P and T symmetric cases.
5. Fractional charge in Luttinger liquids and at quantum Hall edges
Fractional quantum numbers, and in particular fractional electric charge, is one of the
hallmarks of exotic phases. As discussed above, these charges are deeply related to
topological properties of the ground states, and are typically rational fractions of the unit
charge e. An exception is the Luttinger liquid state of an one-dimensional electron system
where it has been claimed that the carriers have charge eg, where g is a non-universal
interaction parameter that determines certain critical exponentsxvii. In a 2010 paper by Prof.
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Jon-Magne Leinaas at Oslo University, dr. Mats Horsdal, now at the University of Leipzig,
and myself, we showed that the situation is more complicated. Our main conclusions
concerned the nature of fractional charges in gapless systems, but we also conjectured that the
relevant fractions expected to be observed would be e g , rather than eg. In a later
collaboration, with Prof. Eddy Ardonne at Nordita and Marianne Rypestøl at Oslo university,
we have shown that our conjecture was indeed correct, and in particular we have made a
theoretical and numerical study of the experiment proposed by Berg et al.xviii. A paper based
on these results is already in manuscript form, and we intend to follow it up by making
concrete proposals for experiments that could detect fractional charges on edges of QH bars.
We already have some ideas about what geometry would be optimal, but a detailed study of
parameter ranges, detector sensitivities etc. is still to be done. In this project we hope to
collaborate with experimentalists as well as with theorists specializing in transport in lowdimensional systems.
References
i
X. G. Wen, Advances in Physics, 44, 405 (1995).
X. G. Wen, Phys. Rev. B 65, 165113 (2002).
iii
G. Moore and N. Read, Nucl. Phys. B 360, 362 (1991).
iv
N. Read and D. Green, Phys. Rev. B 61, 10267 (2000).
v
T. H. Hansson, V. Oganesyan, and S.L. Sondhi, Ann. of Phys. 313, 497 (2004).
vi
M. Z. Hasan and C.L. Kane, Rev. Mod. Phys. 82, 3045 (2010).
vii
For reviews, see N.R. Cooper, Advances in Physics 57, 539 (2008); S. Viefers, J. Phys.:
Condens. matter 20 (2008) 123202; A.L. Fetter, Rev. Mod. Phys. 81, 647–691 (2009)
viii
C. Nayak et al., Rev. Mod. Phys. 80, 1083 (2008).
ix
L. B. Ioffe et al., Nature 415, 503 (2002).
x
L. Fu and C.L. Kane, Phys. Rev. Lett. 100, 096407 (2008).
xi
A. Wallraff et al., Nature 431, 162 - 167 (2004).
xii
For a review, see e.g. D. R. Smith, Science, 305, 788 (2004).
xiii
J. Plumridge et al., Solid State Comm.,146, 406, (2008).
xiv
A.L. Rakhmanov et al., Phys. Rev. B 77, 144507 (2008).
xv
N. Read, Phys. Rev. B 79, 045308 (2009).
xvi
M. Sato and S. Fujimoto, Phys. Rev. B 79, 094504 (2009).
xvii
Matthew P. A. Fisher and Leonid I. Glazman, arXiv:cond-mat:9610037; K.-V. Pham, M.
Gabay, and P. Lederer, Phys. Rev. B 61, 16 397 (2000).
xviii
E. Berg, Y. Oreg, E.-A. Kim, and F. von Oppen, Phys. Rev. Lett. 102, 236402 (2009).
ii
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