# Quantum Condensed Matter Physics

David Campbell, Antonio Castro Neto, Claudio Chamon, Anatoli Polkovnikov, Anders Sandvik

Theorists in quantum condensed-matter physics try to understand the behavior of complex systems of atoms and molecules. Their interests include high-temperature superconductivity, quantum phase transitions, spin systems, disorder, and other strongly correlated phenomena.

## Research

- Computational studies of quantum phase transitions
A continuous ground state phase transition occurring in a quantum-mechanical many-particle system as a function of some system parameter is referred to as a quantum phase transition. At the quantum-critical point separating two different types of ground states, the quantum fluctuations play a role analogous to thermal fluctuations in a phase transition occurring at nonzero temperature. An important aspect of these transitions is that the critical fluctuations and the associated scaling behavior of the quantum-critical point influences the system not only in the close vicinity of the ground-state critical point itself, but also in a wide finite-temperature region surrounding it. While many quantum phase transitions can be understood in terms of a mapping of the quantum mechanical problem onto a classical statistical-mechanics problem with an additional dimension (corresponding to time), recent attention has been focused on exotic transitions which fall outside the classical framework and may be important in strongly-correlated electronic systems such as the high-Tc cuprate superconductors. Prof. Sandvik’s group uses quantum Monte Carlo techniques to explore such transitions in model systems, primarily quantum spin systems. The purpose of this research is to find and characterize various quantum phase transitions in an un-biased (non-approximate) way, in order to provide benchmarks and guidance to developing theories. The influence of disorder (randomness) on the nature of quantum phase transitions is also studied.

- Electron fractionalization in graphene-like structures
Electron fractionalization is intimately related to topology. In one-dimensional systems, such as polyacetelene, fractionally charged states exist at domain walls between degenerate vacua. In two-dimensional systems, fractionalization exists in quantum Hall fluids, where time-reversal symmetry is broken by a large external magnetic field. Recently, there has been a tremendous effort in the search for examples of fractionalization in two-dimensional systems with time-reversal symmetry.

In this research work, we showed that fractionally charged topological excitations exist in tight-biding systems where time-reversal symmetry is respected. These systems are described, in the continuum approximation, by the Dirac equation in two space dimensions. The topological zero-modes are mathematically similar to fractional vortices in p-wave superconductors. They correspond to a twist in the phase in the mass of the Dirac fermions, akin to cosmic strings in particle physics. The quasiparticle excitations can carry irrational charge and irrational exchange statistics. These excitations can be deconfined at zero temperature, but when they are, the charge re-rationalizes to the value 1/2.

Publications:

Electron fractionalization in two-dimensional graphenelike structures

C.-Y. Hou, C. Chamon, and C. Mudry

Phys. Rev. Lett. 98, 186809 (2007),

arXiv:cond-mat/0609740Irrational vs. rational charge and statistics in two-dimensional quantum systems

C. Chamon, C.-Y. Hou, R. Jackiw, C. Mudry, S.-Y. Pi, and A. Schnyder,

arXiv:0707.0293Electron fractionalization for two-dimensional Dirac fermions

C. Chamon, C.-Y. Hou, R. Jackiw, C. Mudry, S.-Y. Pi, and G. Semenoff,

in preparation - Quantum Monte Carlo algorithms
Monte Carlo methods are powerful computational tools for studies of equilibrium properties of classical many-particle systems. Using a stochastic process for generating random configurations of the system degrees of freedom, such methods simulate thermal fluctuations, so that expectation values of physical observables of interest are directly obtained by averaging “measurements” on the configurations. In quantum mechanical systems (e.g., electrons in a metal or superconductor, localized electronic spins (magnetic moments) of a certain insulators, or atoms in a magnetic or optical trap), quantum fluctuations have to be taken into account as well, especially at low temperatures, and these pose a much greater challenge than thermal fluctuations alone. Several different quantum Monte Carlo techniques have been devised during the past three decades, but many challenges remain in developing efficient algorithms for reaching large system sizes and low temperatures, and extending the applicability to models that are currently intractable. Prof. Sandvik is the principal developer of a scheme known as Stochastic Series Expansion, which during the last few years has emerged as the quantum Monte Carlo method of choice for studies of several classes of spin and boson systems. Recently, the group has initiated a research program to develop algorithms for studying ground states of quantum spin systems in the so-called valence bond (singlet pair) basis. This approach shows great promise for studies of quantum phase transitions and may also be applicable to fermion and boson models.

- Research by Anatoli Polkovnikov
I am interested in understanding various properties of interacting many-particle systems, especially driven away from equilibrium. Particular topics include representation of quantum dynamics through classical trajectories: theory and applications, understanding non-equilibrium thermodynamics from microscopics, universal aspects of dissipation for nearly adiabatic evolution, dynamics near phase transitions. My research is closely tied to experiments in cold atom systems.

- Research by Antonio Castro Neto
Professor Castro Neto has broad interests in condensed matter theory research ranging from decoherence in quantum open systems (with applications to quantum computation and NEMS – Nanoelectromechanical systems), and quantum magnetism in ordered and disordered itinerant magnets, to high temperature superconductors (HTc). Some of the subjects studied by Prof. Castro Neto’s group are: the nature of inhomogeneous states (such as stripes) in HTc cuprate oxides; the interplay between quantum fluctuations, disorder, and dissipation in metallic magnets and the existence of Griffiths-McCoy singularities, responsible for singular behavior in the magnetic response of these systems at low temperatures (such as U and Ce intermetallics); the frustration of decoherence in open quantum systems; the interplay of dissipation and disorder in the physics quantum ferromagnets (such as UCu2Si(2-x)Ge(x)); the ferromagnetism and metal-insulator transitions in systems with very small electronic densities (such as Eu(1-x)Ca(x)B6); the study of strongly-correlated systems in finite geometries, such as in nano-structures; the renormalization group analysis of superconductivity in electron-boson systems. More recently, Prof. Castro Neto’s group, using field theoretical methods, has studied the electronic properties in two-dimensional (graphene) and three-dimensional (graphite) carbon based materials.