Tian Ma & Shouhong Wang, Topological Phase Transitions I: Quantum Phase Transitions, hal-01651908

The aim of the above paper is to provide a systematic theoretical study on quantum phase transitions associated with the Bose-Einstein condensates, the superfluidity and the superconductivity.

**I.** Based on current developments, we now know that there are only two types of phase transitions:

*dynamical phase transitions and**topological phase transitions (TPTs).*

The authors have developed the dynamic transition theory for dissipative systems; see [Ma & Wang, Phase Transition Dynamics, Springer-Verlag, 2013, 555pp.] and the references therein. This is a new notion of phase transitions, applicable to all dissipative systems, including nonlinear dissipative systems in statistical physics, fluid dynamics, atmospheric and oceanic sciences, biological and chemical systems etc.

A TPT refers to the transition in its topological structure in the physical space of the system, and quantum phase transitions (QPTs) belong to the category of TPTs. The notion of TPTs is originated from the pioneering work by J. Michael Kosterlitz and David J. Thouless (1972) where they identified a completely new type of phase transitions in two-dimensional systems where topological defects play a crucial role. With this work, together with F. Duncan M. Haldane, they received 2016 Nobel prize in physics.

There have been many attempts, but the basic theoretical understanding of TPTs is still largely open.

**II.** QPTs are TPTs for condensate systems, including the gaseous Bose-Einstein condensates, superconductivity, and superfluidity. The basic physical characteristics of a QPT are as follows:

- A QPT is a transition between quantum states, and the state quantities or the order parameters of the quantum system describing the transition should be the wave functions of the quantum states;
- the control parameters are non-thermal; and
- a QPT is a TPT of a condensate system, rather than a dynamical phase transition. Consequently, the state functions describing a QPT are in the wave function .

**III.** The field equations governing the condensation are determined by the principle of Hamiltonian dynamics (PHD) or equivalently by the principle of Lagrangian dynamics (PLD):

where is the control parameter. The associated phase transition equations are then the following *topological structure equations*:

- for particle number conserved systems,
where , , and represents the chemical potential;

- for non-conserved systems,

**IV.** Based on the above three characteristics and the topological structure equations (2), the wave function of the condensate is a function of the control parameter , i.e. . Then mathematically we say that the system undergoes a QPT at a critical if the topological structure of is different from that of .

**V.** We deduce from the topological structure equations (2) that the supercurrents of superconductivity and superfluidity are as follows:

VI. The supercurrents enjoy the divergence–free condition (incompressibility):

The authors have developed a geometric theory for incompressible flows to study the structure and its stability and transitions of incompressible fluid flows in the physical spaces. The complete account of this geometric theory is given in the authors’ research monograph [Ma & Wang, Geometric Theory of Incompressible Flows and Applications to Fluid Dynamics, AMS, 2005]. This geometric theory can then be directly applied to study the transitions of topological structure associated with the quantum phase transitions of BEC, superfluidity and superconductivity. We derive in particular the basic theory for different TPTs, leading to transparent physical pictures of various condensates.

**VII.** After careful examination of the formation of Cooper pairs in superconductivity, we derive the following microscopic mechanism for the Meissner effect:

- Below the critical temperature , an applied magnetic field induces spin Cooper pairs;
- only the Cooper pairs reversely parallel to the applied are stable, leading to their physical formation; and
- the total magnetic moment of all Cooper pairs with , together with the surface supercurrents, can cancel out the magnetism induced by the applied field in the superconductor, and resists the applied field from entering its body.

Also, this paper is the first one in the series of papers by the authors on TPTs, including the forthcoming papers for the following science problems:

- galactic spiral structures,
- electromagnetic eruptions on solar surface,
- boundary-layer separation of fluid flows, and
- interior separation of fluid flows.

*Tian Ma, Shouhong Wang*