On the theory of weakly doped Mott insulators
(Dated: October 30, 2008)

 

During the years of 70's and 80's of last centuries, the quark confinement was of central interests to the high-energy
physics community although even today its mechanism remains unsolved. During that period, one of candidate is
to investigate the non-perturbative solution \instanton" which is responsible for the \vacuum tunneling" between
infinite degenerate classical vacua. This solution is a finite action solution of Euclidean Yang-Mills theory, so that the
boundary condition plays an important role that we then classify all such solutions by using the homotopy group. In
this manner, every solution with finite Euclidean action corresponds to a topological index which is an integer and
called \topological charge". Roughly, it is impossible for us to continuously deform a solution with a certain topological
charge into another solution with different charge. Surprisingly, instead of such non-singular mappings (solutions), it
is confirmed that at large distance physics of QCD (Quantum Chromodynamics), there are singular solutions (which
we then call \meron" configuration) that are relevant to the mechanism of quark confinement. Actually in Yang-Mills
theory, one instanton is composed of a meron-meron pair (likewise, anti-instanton, antimeron-antimeron pair), every
meron has a one half topological charge so that a single meron configuration reduces to a divergent action since this
variety of configurations consists of singularities in Euclidean space. As a toy model which is much easier than original
Yang-Mills theory, theorists focus on the 2-Dimensional O(3) model. Many similarities between them are found. For
example, they are both renormalizable, scale invariant, conformal invariant and asymptotically free. But we must pay
attention that actually they are different theories with different field theoretical structures. Especially their homotopic
mappings differ from each other: the Yang-Mills theory's physical space is the boundary of corresponding Euclidean
space, and internal space is embedded in a gauge group space (SU(2); SU(3) etc.); while in the O(3) model, the
boundary of physical space is the compactified 2-dim Euclidean space and the internal space is a 2-D sphere. In
spite of that, more investigations on the topological "non-perturbative" behavior of the O(3) model will help us for
understanding the meron configuration since the original Yang-Mills theory is quite formidable when non-perturbative
consideration is involved.

 

 Profoundly, recently the relevant research is also given rise to on the theory of weakly doped Mott insulators. This
theory is a mutual Chern-Simons gauge theory which is a coherent-path-integral formalism of the phase-string theory.
It is believed that in this theory the very non-trivial phase-string effect should be responsible for all quite uncon-
ventional properties of cuprates such as its transport, quasi-particle, etc. At half-filling, Ref.[4] gives us an explicit
low-energy effective field theory which possesses of dynamical O(3) symmetry in 3-D Euclidean space (corresponding
to SO(2; 1) relativistic Lorentz invariance in 2 + 1D Minkowski space).

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