What information can be determined about a state given
just the ground state wave function?
Quantum ground states, speaking intuitively, contain
fluctuations between many of the configurations one might want to understand.
The information about them can be organized by introducing an imaginary system,
dubbed the entanglement Hamiltonian.
What light does the dynamics of this Hamiltonian (a
precise version of the notion of "zero point motion") shed on the
I will start my discussion near critical points, where
Lorentz invariance often emerges and the entanglement Hamiltonian becomes
tractable, revealing that it exists in one dimension less than the real system.
One application is to the fluctuations of angular momentum in a spin chain.
The entanglement Hamiltonian is especially successful at
clarifying the properties of quantum phases without order, such as topological
insulators. In particular, spin chains often have no long range order due to
quantum fluctuations. Nevertheless there can be phase transitions between two
of these disordered states, suggesting the existence of a hidden order.
My main aim in the talk will be to demonstrate that the
entanglement spectrum can serve as an order parameter for these unusual transitions; this observation leads to a
classification of one-dimensional phases.
One can understand the phases of the actual system simply
by looking at the spectrum of the entanglement Hamiltonian, just as one deduces
the properties of atoms from their spectra.