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Multiplets in the Entanglement Spectrum

Ari Turner, University of Amsterdam

Entanglement (as in the Einstein, Podolsky, Rosen paradox) is a major distinctive feature of quantum mechanics. To demonstrate the existence of entanglement in the simplest way, entangling
the spins of photons, one has to be very careful in preparing the states, as the entanglement is fragile. However, entanglement is present in the ground state of every many body system.
In particular it plays an important role in a type of phase called a "topological phase." We will
illustrate this with the case of spin chains.

Often, spin chains do not have any long range order, because of quantum
mechanical fluctuations. Surprisingly, there can be phase transitions between two such phases, which suggests the existence of a hidden order (which has been called "topological order").

In this talk, I demonstrate that the entanglement spectrum can serve as an order
parameter for these unusual transitions. The central idea is to reduce a one
dimensional chain to a zero-dimensional imaginary system which describes
the entanglement and is called the "entanglement
Hamiltonian." One can then understand the phases of the original spin chain simply by
looking at the spectrum of the entanglement Hamiltonian, just as one deduces the
properties of atoms from their spectra.

The entanglement spectrum has also the potential for clarifying properties of
quantum phases without order in higher dimensions, as exhibited by topological
insulators.