Note: A New Concept of Phase Transition

Today I listened Prof. Xie Chen from Caltech giving her talk about her new framework of phase transition in some concrete topological ordered systems. I did not quite capture the detailed mathematics of various fancy models given by her, but I am quite impressed by the new concept of her about phase transition in condensed matter systems, typically in strongly correlated systems.

Xie’s concept basically goes like: if we are dealing with phase and phase transition problem in condensed matter systems, where correlations between individual degree of freedoms can be tuned relatively strong, the problem is hard to describe and solve if we keep the free fermion/boson concept of the problem. Indeed, there is no definite dispersions of electronic freedoms and the system will become gapless at the critical point where new low energy degree of freedom will emerge, such as spinons and holons in DQCP. Xie raised three different models, the Mott-superfluid transition in (2+1)D bosonic system, the symmetric-ferrormagnetic transtion in (1+1)D transverse field Ising model, and the superconductor-Kitaev chain transition in (1+1)D Majorana chain model. In all these models, the phase transition between two phases, tuned by a single dimensionless parameter (measures the competition between the two phases), can be described as a dual condensation of two competing degree of freedoms. One of these two freedoms will be condensing (gapless) in one phase, and correspondingly be gapped in another, vice versa. Here, she tried to use the word ‘condensation’ to describe the situation where adding one new such condensed object into the ground state of the phase will not change the ground state energy of the phase. In the Mott-superfluid transition in (2+1)D bosonic system, the bosons are gapped in the Mott inuslating phase, but are condensed (gapless) in the superfluid phase; wheras the vortex (the disorders in the bosonic order parameter field in the superfluid phase) are gapped in the superfluid phase, but are condensed (gapless) in the Mott insulating phase. At the crtical point, the two objetcts are all wanting to be condensed, thus seeking to a compromise in which they all becomes the low energy degree of freedoms at the critical point. So do the other two cases, where in the Ising model the two competitive objects are the Z2 charge and the domain wall, and in the Majorana chain model (which is actually equivalent to the Ising chain, by performing the Jordan-Wigner transformation) the two competitive objects are the Z2 flux and the Z2 flux plus one fermion.

This new concept of the condensation competition of two orders in two phases of the phase transition is quite important, in the meaning that it can rephrase thephase transtions even in free fermion/boson cases. This may help us to develop a totally new scheme of describing phase transition in many-body system with strongly correlation. I should keep following Xie’s work.