KRUGER2	Entanglement Entropies of Fermions
Type	Theoretical
#students	1
Orientation	Several important connections have been established in recent years between concepts from quantum information and problems in many-body physics. One such connection is the use of entanglement entropies to understand how ground states of various quantum Hamiltonians show either criticality or topological order [1]. Quantum entanglement is the essential ingredient for building a quantum computer and is believed to play a crucial in the mechanism for high temperature superconductivity.
How	The simplest measures of quantum entanglement are the bipartite vonNeumann or Renyi entanglement entropies. These are calculated from a reduced density matrix after tracing over all degrees of freedom associated with a sub-volume of the system. While for bosons the entanglement entropies usually scale with the area of the interface between the sub-volumens, fermions exhibit an extra long-range entanglement [2]. This is the consequence of the large number of gapless particle-hole excitation around the Fermi surface [3]. One therefore expects a qualitative change in the entanglement scaling near quantum critical points where the Fermi reconstructs.
What	The aim of this project is to implement an algorithm to numerically calculate the first Renyi entanglement entropy for any given N-particle wave function. The code should ideally be written in C/C++. The first Renyi entropy is related to the square of the reduced density matrix. It can be obtained from the expectation value of a swap operator that exchanges particles between two copies of the system within the sub-volume [4]. This is most efficiently calculated by Monte-Carlo sampling. The student will test the code for non-interacting bosons and fermions and perform a careful finite-size scaling. If successful, this algorithm might be used to study the entanglement entropies of fermionic wave functions that describe the collapse of a Fermi liquid at a quantum critical point [5].
References	[1] J. Eisert, M. Cramer, and M.B. Plenio, Rev. Mod. Phys. 82, 277 (2010) [2] D. Gioev and I. Klich, Phys. Rev. Lett. 96, 100503 (2006) [3] B. Swingle, Phys. Rev. Lett. 105, 050502 (2010) [4] Y. Zhang, T. Grover, and A. Vishwanath, Phys. Rev. Lett. 107, 067202 (2011) [5] F. Kruger and J. Zaanen, Phys. Rev. B 78, 035104 (2008)
Supervisor	Dr Frank Kruger, f.kruger@ucl.ac.uk