Session P5: Entanglement Entropy in Condensed Matter Physics

Probing order beyond the Landau paradigm

For many years, it was thought that Landau's theory of symmetry breaking could describe essentially all phases and phase transitions. Then, in 1982, the limitations of Landau theory were exposed in a dramatic way with the discovery of the fractional quantum Hall (FQH) effect. The FQH states contain a new kind of order - known as ``topological order'' - that is fundamentally beyond the Landau paradigm. Topological order cannot be understood using symmetry breaking, order parameters, or long range order. This poses an interesting theoretical problem: these states must contain some kind of structure that is responsible for their unusual physical properties. But what is this structure and how can we probe it without order parameters? In my talk, I will describe recent progress in answering this question. I will show that topological order is intimately connected with nonlocal quantum entanglement. I will introduce a new quantity - called ``topological entropy'' - that measures precisely this nonlocal entanglement.   

Universal contributions to entanglement entropy at critical points in two spatial dimensions

The entanglement entropy of a pure quantum state of a bipartite system $A \cup B$ is defined as the von Neumann entropy of the reduced density matrix obtained by tracing over one of the two parts. Critical ground states of local Hamiltonians in one dimension have entanglement that diverges logarithmically in the subsystem size, with a universal coefficient that for conformally invariant critical points is related to the central charge of the conformal field theory. We find the entanglement entropy for a standard class of $z=2$ quantum critical points in two spatial dimensions with scale invariant ground state wave functions: in addition to a nonuniversal ``area law'' contribution proportional to the size of the $AB$ boundary, there is generically a universal logarithmically divergent correction. This logarithmic term is completely determined by the geometry of the partition into subsystems and the central charge of the field theory that describes the equal-time correlations of the critical wavefunction. Taken together with results on entanglement entropy in gapped, topologically ordered phases, these results indicate that even when the ``area law'' correctly predicts the leading behavior of entanglement, universal subleading terms can reflect important properties of a quantum many-body system.   

Entanglement entropy of quantum-critical spin chains with strong randomness

For disorder-free critical quantum spin chains, the entanglement of a segment of $N\gg 1$ spins with the remainder scales as $log_2 N$, with a prefactor fixed by the central charge of the associated conformal field theory. The mean entanglement entropy of quantum spin chains with randomness follows the same logarithmic scaling, and provides a universal critical entropy, which is equivalent to the central charge in the pure case. In my talk I will explore the origin and derivation of the universal entanglment entropy of the random spin-1/2 Heiseneberg model in the random-singlet phase, as well as that of the random spin-1 Heisenberg chain at the breakdown of its Haldane phase. The entanglement of these and related infinite-randomness fixed points makes it possible to speculate on possible extensions of the c-theorem of CFTs to the realm of systems with strong randomness.