Principal Investigator
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Project Title
| The Density Matrix Renormalization Group |
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Brief Description for General Publications
The Density Matrix Renormalization Group (DMRG) method is used to obtain solutions of the many-body Schroedinger equation for a lattice Hamiltonian. While the method is mostly limited to one-dimensional systems, it is very useful for strongly correlated electron systems where, due to the nature of the interactions, traditional techniques give very poor results. Recently the method was extended to calculate full non-equilibrium time evolution, making it the first general-purpose numerical method for calculating time evolution in quantum mechanical systems. In addition, 2004 saw the introduction of operator methods that allow, for example, systems to be studied at finite-temperature (with or without dissapative dynamics) with essentially the same generality and accuracy of ground-state calculations. These two breakthroughs provide the tools required to study thermodynamic and non-equilibrium properties of quantum systems, which is the new focus of this project. In particular, this method should be ideal for resolving one of the big mysteries of condensed matter physics, which is the finite-temperature phase diagram of Luttinger liquids. The Luttinger liquid is the canonical model for one-dimensional gapless quantum systems, and is characterized by low-lying collective bosonic excitations that carry separately the spin and the charge of the original electrons (spin-charge separation). However very little is known about Luttinger liquids away from zero temperature. Analytic calculations using the Bethe-Ansatz suggest that at low (but non-zero) temperature the specific heat has an exponential behaviour, signifying gapped excitations probably due to some sort of spin and charge density wave. This suggests that the T=0 point is a quantum critical point. At higher temperatures the behaviour changes again and spin-charge separation is aparantly recovered, but it is not clear why or how this happens. Finite-temperature DMRG should be an invaluable tool for investigating this behaviour. A second area of interest is to model optical lattices and nanostructures (such as quantum dots and point contacts). Current theories for optical lattices rely on zero-temperature approximations and generally give a poor resemblance to experimental results. The new DMRG methods described above should be able to model these structures directly and give an improved understanding of current and future experimental results. |