One of the principal aims of condensed matter physics is concerned with understanding the diverse phases of matter, ordered and disordered, and how phase transitions betweenthem occur. A milestone was the Landau-Ginzburg theory of phases and phase transitions, which relies on the existence of a local order parameter that vanishes on one side of the transition. For a long time it was believed that such paradigm is essentially universal (but for rare exceptions such as the Berezinskii-Kosterlitz-Thouless transition). In recent years, we have witnessed the dawn of a new class of systems that evade this paradigm and display a wide range of unusual and interesting properties. In particular, a new type of order devoid of any local order parameter has been discovered, where a topology-dependent number of degenerate liquid-like states appear, distinguished only by non-local measurements (topological order). The number of known systems that belong to this class is rapidly increasing. The fractional quantum Hall effect is arguably the most eminent example. Others include quantum dimer models, originally formulated in the context of high-temperature superconductivity, ice and vertex models.
The notion of quantum topological order has recently received renewed attention due to the possibility of exploiting it towards the implementation of quantum computing. Together with Prof. Claudio Chamon, we became interested in better understanding the nature of a quantum topologically ordered phase, and in investigating its robustness with respect to quantum as well as thermal fluctuations. By constructing discrete lattice models whose ground state wavefunctions can be computed exactly, we were able to study in detail a phase transition from a topologically ordered phase to a non-topologically ordered one, in the absence of a local order parameter both below and above the transition [1].
We also studied the effect of thermal fluctuations (at equilibrium), and we showed that quantum topological order is fragile in two dimensions, in the sense that its characteristic non-local order parameter -- the topological entropy -- vanishes identically in the thermodynamic limit, whenever the temperature is raised from zero [2]. This result helps understanding the limits of any practical use of topological order towards quantum computation.
We also showed how the topological entropy can be used as an order parameter in classical systems, and a well-defined notion of classical topological order can therefore be formulated, of which I provided examples in two-dimensions using appropriate hard-constrained systems [3].
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C. Castelnovo, and C. Chamon
A quantum topological phase transition at the microscopic level
arXiv:0707.2084 (2007) -
C. Castelnovo, and C. Chamon
Entanglement and topological entropy of the toric code at finite temperature
Physical Review B 76, 184442 (2007) -
C. Castelnovo, and C. Chamon
Topological order and topological entropy in classical systems
Physical Review B 76, 174416 (2007) -- selected for the November 2007 issue of Virtual Journal of Quantum Information.