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1. Trapped Ions

This project is a collaboration with the Quantum Physics group at the University of Sydney. The motiviation behind this project is the ability to experimentally engineer 2-dimensional interactions with ion traps, . These systems can function as a 'quantum simulator', for example to probe quantum criticality or dynamical phase transitions.

The experiment can be modeled as a 2-dimenional Ising model with long-range interactions,

{$H = \sum_{ij} J_{ij} \sigma^z_i \sigma^z_j + \vec{B} \cdot \sum_i \vec\sigma_i $}

where {$J_{ij} = \frac{1}{|r_i - r_j|^\alpha}$} is a long-range power law interaction.

The aim of the project is to explore the capabilities of this system, groundstate phases, and dynamical phases.

Relevant previous work:

  • In 1 dimension, some work has been done on dynamical transitions,
  • The groundstate for 2 dimensions has been studied a bit. There is a mapping to a dimer model on a hexagonal

lattice. R. Moessner, S. L. Sondhi, and P. Chandra, “Two-dimensional periodic frustrated ising models in a transverse field,” Phys. Rev. Lett., vol. 84, pp. 4457–4460, May 2000. R. Moessner and S. L. Sondhi, “Ising models of quantum frustration,” Phys. Rev. B, vol. 63, p. 224401, May 2001. R. Moessner, S. L. Sondhi, and P. Chandra, “Phase diagram of the hexagonal lattice quantum dimer model,” Phys. Rev. B, vol. 64, p. 144416, Sep 2001.

  • Some iDMRG numerics for the groundstate phase diagram for infinite cylinders has been performed by Nariman, with probably enough data calculated to make a publication.

The idea behind an Honours project would be (1) assist with the completion of the groundstate phase diagram, and () start work on some dynamics.

2. Probing quantum phase transitions through fluctuations

In recent years, the growth of Quantum Information Science has inspired new ways of looking at condensed matter systems. One such viewpoint is the study of bipartite entanglement, which can be detected easily in DMRG. Beyond simple measures such as the von Neumann or Reyni entropies, the entanglement spectrum (ES) is very useful tool for probing symmetry properties. For example the nature of symmetry-proected topological phases is best described in terms of symmetry representations of the entanglement spectrum.

Another aspect of entanglement is bipartite fluctuations. This can be seen as a higher order effect of some order pr correlations in the system. Some theory work on this has been done by Kareyn Le Hur's group,, , and these ideas were used successfully in a study by Kjall, Zaletel et al

However it has received not much further attention, but probably should be one of the standard tools for MPS/tensor network calculations. So the idea is to test the method for some benchmark calculations and find some situations were it can be used effectively. We can also look at generalizations; in principle we can measure fluctuations of pretty much any quantity, so we could also look at energy fluctuations, or fluctuations associated a symmetry.

A good starting benchmark is the spin-1 chain, as this contains a variety of interesting quantum phase transitions. In particular, there is a still a minor controversey as to what happens in the dimer phase in the vicinity of the ferromagnet, that perhaps a stufy of the fluctuations would be able to resolve (or at least, provide some strong evidence).

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