Research in Condensed matter theory at Fysikum
At present, our main activity is focused on strongly correlated electron systems, especially the Quantum Hall Effect, but we are also intersted in the physics of cold atoms. Together with the nano-physics group at the Royal institute of technology (KTH) we also work on problems related to various quantum electronics devices. At the end we list some special projects that we are currently working on.
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 phenomeno-logical constants, 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 T c 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 two-dimensional 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[1] and one can now attempt a systematic classification of the exotic quantum systems based on the new concepts of "topological"[2]and "quantum"[3] order.
1. See e.g . T. Senthil, et.al, Phys. Rev. B 70 , 144407 (2004).
2.
X. G. Wen, Advances in Physics, 44 , 405 (1995).
3.
X. G. Wen, Phys. Rev. B 65 , 165113 (2002).
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 form 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 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 T c 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 case(1). In particular one can speculate on the possibility of realizing states with non-Abelian topological order.
In one important aspect, the cold atoms are much harder to study than electrons. The very fact that they are neutral makes transport experiments very difficult, and so far the measurements are basically limited to observing the expanding clouds after having turned off the traps. It is an outstanding challenge to find other ways to probe the quantum nature of the cold atom states and, in the long perspective, to develop atomtronics.
(1) For a review, see e.g ., S. Viefers, J. Phys.: Condens. matter 20 (2008) 123202.
• The half filled Landau Level - from thin torus to bulk
In a series of papers, we have shown that the quantum Hall problem on a very thin cylinder, or torus, is exactly solvable, and we have given strong evidence to support the notion that these exact crystal like "TT"- states evolve adiabatically into the bulk quantum Hall liquids when the cylinder radius is increased. The QH hierarchy is manifest in our formulation, and it also strongly suggests a phase diagram that was proposed earlier based on an effective field theory approach[1]. We also established that in a particular thin limit, the low-energy sector of the half-filled lowest Landau level is that of free neutral one-dimensional fermions[2]. This has a striking resemblance to the composite fermion picture but with certain differences. Moving away from this limit we would generically expect the system to become a Luttinger liquid. This provides a precise hypothesis about the physical nature of the elusive ?=1/2 state, and we are at present exploring possible experimental consequences for instance in transport properties.
1. A.A. Kivelson, D.-H. Lee, and S.-C. Zhang, Phys. Rev. B 46 , 2223 (1992).
2.
E. J. Bergholtz and A. Karlhede, Phys. Rev. Lett. 94 , 026802 (2005).
• The quantum Hall hierarchy - quasihole condensates and microscopic theory
In another series of papers published during the last couple of years[1,2] we have developed methods based on conformal field theory (CFT) that allows us to construct wave functions for all hierarchical QH states that can be formed by condensation of quasielectrons only.
In collaboration with professor Viefers group at Oslo University, we are at present trying to extend this construction by finding a suitable CFT-description of quasihole condensation, which would allow us to generate the full hierarchy.
With the exception of the Laughlin states, no hierarchy QH state can be derived from a microscopic theory. We hope that the insights gained from the CFT-construction will allow for a derivation starting from a Chern-Simons-Landau-Ginzburg theory that is a proper representation of the microscopic physics.
1. T. H. Hansson, C.-C. Chang, J. K. Jain, and S. Viefers, Phys. Rev. B 76 , 075347 (2007);
2. E. J. Bergholtz, T. H. Hansson, M. Hermanns, and A. Karlhede, Phys. Rev. Lett. 99 , 256803 (2007)
• Non-Abelian QH states
With the methods referred to above - thin torus limits, and CFT approach - we can also address several important problems regarding the non-Abelian QH states. The thin limit approach naturally describes the quasiparticle excitations as domain walls between in-equivalent ground state[1]. Together with professor Eddy Ardonne at Nordita, we aim to explore this for more general QH states and in particular try to make a connection to the recent approach by Bernevig and Haldane to the non-Abelian states based on Jack polynomials[2].
A central part of our CFT approach to the QH hierarchy, was a novel way to construct quasi-electron, as opposed to quasihole, states. This approach will be generalized to the non-Abelian state, where no quasielectron wave functions are known. Using the methods we have developed, we can also construct completely new non-Abelian states
1.
E. J. Bergholtz, J. Kailasvuori, E. Wikberg, T. H. Hansson, and A. Karlhede, Phys. Rev. B 74 , 081308 (2006).
2.
B. Andrei Bernevig, F. D. M. Haldane, Fractional Quantum Hall States and Jack Polynomials arXiv:0707.3637.
• Rapidly rotating cold atoms
Although we have not published much in this field[1], we have kept a keen interest in it and are at present actively pursuing a project aimed at understanding possible topological phases in rapidly rotating BEC:s. There are strong reasons to believe that the non-Abelian states that were originally proposed in the context of QH physics, are much more likely to be observed in cold atom gases.
We plan to make a more systematic investigation of multi-component QH liquids built from both fermions and bosons. Particularly interesting are states with bosonic components with ? > 1, since several of the proposed non-Abelian states are in this range.
1. For an early paper, see however: S. Viefers,T. H. Hansson, and S. M. Reimann, Phys. Rev. A 62 , 053604 (2000)
• Monte Carlo simulation of a charged superconductor with disorder
The perhaps simplest, and certainly the most familiar, example of a topologically ordered state, is in fact that of an ordinary BCS superconductor provided the fluctuations of the electromagnetic field is properly taken into account. In work in collaboration with a group at Princeton University, we have studied the topological properties of such "charged" superconductors, especially in two dimensions[1]. One of the hallmarks of topologically ordered states, and thus the superconductor, is a ground state degeneracy depending on the topology of the underlying space, but independent of local variations in the Hamiltonian.
This degeneracy is split in a characteristic pattern due to certain tunneling processes that we, in collaboration with dr. Jack Lidmar at KTH, have studied numerically using a novel order parameter 19 . It is an important prediction of the general theory of topological order that this splitting pattern is insensitive to local impurities, and we plan to carry out simulations to put this claim to a test.
1. T. H. Hansson, V. Oganesyan, and S.L. Sondhi, Ann. of Phys. 313 , 497 (2004).
• Monte Carlo simulation of phase slips in a realistic model of a JJ-array
Josephson junction arrays are interesting not only from point of view of applications in quantum information devices, but also in their own right. In particular, they allow for detailed studies of the various quantum phase transitions that might occur. Central to this is the presence of vortices, or instantons, in the Euclidian description of the effective 2D, XY- like models. In collaboration with dr. Jack Lidmar at KTH, we use a method that we originally developed for the study of topological order in superconductors[1], to study flux tunneling through a periodic JJ-array. At the low temperatures relevant for the experiments (down to 50 mK) dissipation occurs via quantum phase slips where the phase of the superconducting order parameter suddenly unwinds by 2
1. A. Vestergren, J. Lidmar and T. H. Hansson, Europhys. Lett., 69 , 256 (2005).
• Josephson junction arrays, microwave cavities and quantum nanomaterials
In close collaboration with the KTH nanophysics group under professor David Haviland we study various configurations of JJ-arrays, both by themselves, and coupled to an on-chip microwave cavity. In particular we have investigated arrays with more complicated unit cells to engineer a gap in the plasmon spectrum. As a first result[1], we have found that, in a specific parameter regime, and frequency region, the impedance between the two input terminals becomes purely imaginary, while the mode spectrum develops a gap that persists when realistic boundary effects are taken into account. This opens for the possibility to study condensed matter like crystal physics using Josephson junction arrays, with the possibility to tune the band gaps by applying control fluxes. We hope for applications in metamaterials, photonic crystal physics and microwave electronics, such as filters and ultra fast switches.
1. C. Hutter et al , Josphson junction transmission lines as tunable artificial crystals, arXiv:0804.2099.