Quantum Matter

The simulation of systems of anyons offers a significant challenge to the condensed matter physicist. These systems are presently of substantial theoretical and experimental interest due to their potential for universal quantum computation, but due to their non-trivial exchange statistics, the tools available for their study have been limited. In this talk, I will present a formalism whereby any existing tensor network algorithm may be adapted for use with both Abelian and non-Abelian anyons, culminating in our recent simulations of infinite 1-D chains of interacting anyons using the Multi-scale Entanglement Renormalisation Ansatz, or MERA, demonstrating that tensor network algorithms may be effectively employed in the study of anyonic systems.

Anderson localization emerges in quantum systems when randomised parameters cause the exponential suppression of motion. In this talk we will consider the localization phenomenon in the toric code, demonstrating its ability to sustain quantum information in a fault tolerant way. We show that an external magnetic field induces quantum walks of anyons, causing logical information to be destroyed in a time linear with the system size when even a single pair of anyons is present. However, by taking into account the disorder inherent in any physical realisation of the code, it is found that localization allows the memory to be stable in the presence of a finite anyon density. Enhancements to this effect are also considered using random lattices, and similar problems for anyons transported by thermal errors are considered.

More than forty years ago Nobel laureate P.W. Anderson studied the overlap between two nearby ground states. The result that the overlap tends to zero in the thermodynamics limit was catastrophic for those times. More recently the study of the overlap between ground states, i.e. the fidelity, led to the formulation of the so called fidelity approach to (quantum) phase transition (QPT). This new approach to QPT does not rely on the identification of order parameters or symmetry pattern; rathers it embodies the theory of phase transitions with an operational meaning in terms of measurements. Nowadays orthogonality of ground states is much less surprising. I will provide the general scaling behavior of the fidelity at regular and at critical points of the phase diagrams, Anderson's result being a particular case. These results are useful to many areas of theoretical physics. A related quantity extensively studied here, the fidelity susceptibility, is well known in various other contexts under different names. In metrology it is called quantum Fisher information; in the theory of adiabatic computation it represents the figure of merit for efficient computation; in yet another context it is known as (real part of) the Berry geometric tensor.

AdS/CFT has proven itself a powerful tool in extending our understanding of strongly coupled quantum theories. While studies of AdS/CFT have predominantly focused on tree level calculations, there has been growing interest in the loop effect recently. We studied the 1-loop correction to the gauge boundary-to-boundary correlator due to its coupling to a complex scalar field. In this talk, I would outline our main results, explain the Cutkosky rule in AdS space, and discuss an extra divergence we found in both real and imaginary part of the loop integral. I would then combine our analysis with the replica trick to demonstrate a possible application where one attempts to calculate the DC conductivity in a condensed matter system with random disorder and discuss the limitation and difficulties of our method in its current form.

Entanglement renormalization is a coarse-graining transformation for quantum lattice systems. It produces the multi-scale entanglement renormalization ansatz, a tensor network state used to represent ground states of strongly correlated systems in one and two spatial dimensions. In 1D, the MERA is known to reproduce the logarithmic violation of the boundary law for entanglement entropy, S(L)~log L, characteristic of critical ground states. In contrast, in 2D the MERA strictly obeys the entropic boundary law, S(L)~L, characteristic of gapped systems and a class of critical systems. Therefore a number of highly entangled 2D systems, such as free fermions with a 1D Fermi surface, Fermi liquids and spin Bose metals, which display a logarithmic violation of the boundary law, S(L)~L log L, cannot be described by a regular 2D MERA. It is well-known that at low energies, a many-body system may decouple into two or more independent degrees of freedom (e.g. spin-charge separation in 1D systems of electrons). In this talk I will explain how, in systems where low energy decoupling occurs, entanglement renormalization can be used to obtain an explicit decoupled description. The resulting tensor network state, the branching MERA, can reproduce a logarithmic violation of the boundary law in 2D and, as additional numeric evidence also suggests, might be a good ansatz for the highly entangled systems with a 1D Fermi (or Bose) surface mentioned above. In addition, after recalling that the MERA can be regarded as a specific (discrete) realization of the holographic principle, we will see that the branching MERA leads to exotic holographic geometries.

: In this talk I will review the common appearance of torsion in solids as well as some new developments. Torsion typically appears in condensed matter physics associated to topological defects known as dislocations. Now we are beginning to uncover new aspects of the coupling of torsion to materials. Recently, a dissipationless viscosity has been studied in the quantum Hall effect. I will connect this viscosity to a 2+1-d torsion Chern-Simons term and discuss possible thought experiments in which this could be measured. Additionally I will discuss a new topological defect in 3+1-d, the torsional monopole, which does not require a lattice deformation to exist. If present, torsional monopoles are likely to impact the behavior of materials with strong spin-orbit coupling such as topological insulators.

In this talk I will describe my recent work on the structure of entanglement in field theory from the point of view of mutual information. I will give some basic scaling intuition for the entanglement entropy and then describe how this intuition is better captured by the mutual information. I will also describe a proposal for twist operators that can be used to calculate the mutual information using the replica method. Finally, I will discuss the relevance of my results for holographic duality and entanglement based simulation methods for many body systems.