Thermodynamic Properties Of Quantum Degenerate Fermi Gas With Spin-Orbit Coupling

Isothermal compressibility KT,which measures the fluidity of the liquid or the stiffness of the solids,is one of the most important physical quantities in thermodynamic physics.In ultracold atoms,this quantity has been widely explored in the measurement of the transition from normal gas to Bose-Einstein condensate(BEC),as well as the transition from superfluids to solid phases,such as supersolids and Mott insulators.The spin-orbit coupling(SOC)in degenerate Fermi gases can fundamentally change the fate of s-wave superfluid,giving rise to topological superfluids and associated Majorana zero modes.In this thesis,we mainly study the thermodynamic properties of Fermi degenerate gas in the presence of spin-orbit coupling and Zeeman field between in the free space and the optical lattice,including isothermal compression coefficient,pressure,pressure index,etc.Firstly,we introduce mean field theory for the quantum phase transition of the three-dimensional Fermi degenerate gases under the combined action of spin-orbit coupling and Zeeman field in the free space.It is known that the superfluid order parameter destroyed by a large Zeeman field can be restored by the SOC.With increasing strengths of the Zeeman field,there is a series of topological quantum phase transitions from a nontopological superfluid state with fully gapped fermionic spectrum to a topological superfluid state with four topologically protected Fermi points and then to a second topological superfluid state with only two Fermi points.In addition,We introduce that the SOC and the Zeeman field are counteracting,and that this competition tends to stabilize the uniform superfluid phase against the phase sepration.Secondly,we study the thermodynamic properties of the system,such as isothermal compression coefficient,pressure and the exponent of pressure.Starting from the Gibbs-Duhem equation,we show that KT comes from both the explicit contribution of chemical potential and implicit contribution of order parameter.In the Bardeen-Cooper-Schrieffer(BCS)limit,KT is determined by explicit compressibility,which is proportional to the density of states at the Fermi surface;while in the BEC regime it is determined by the implicit compressibility,which is inversely proportional to the scattering length.Between these two limits,we find a pronounced enhanced compressibility in the gapless Weyl phase regime,which can be regarded as a remanent shadow effect of phase separation.This enhanced compressibility also lead to anomalous exponent of pressure,which mainly characterizes the role of many-body interaction.Finally,motivated by the differences between the free space and the optical lattice,in the last part of the thesis,we examine thermodynamic properties in the spin-orbit-coupled degenerate Fermi gases in a square lattice model,in which the filling factor and particle-hole symmetry about half filling become an important influencing factor.With increasing particle density n,the compressibility decreases to a small magnitude,while n2KT may increase monotonically with increasing n from zero to half filling.In the strong-coupling regime,when the chemical potential and pairing strength are much larger than the tunneling strength and Zeeman field,this compressibility will approach n2KT=2/U,where U is the on-site interacting strength.We show that SOC and the Zeeman field can play opposite roles in the behavior of compressibilities.The combination of these two terms can give rise to pronouncedly enhanced isothermal compressibility in some proper parameter regime.The anomalous compressibilities also give rise to an anomaly in the exponent of pressure.In the lattice model,the enhanced peak can be found in both the fully gapped phase and gapless Weyl phases.