Using the t-J Hamiltonian of U(1) slave-boson symmetry, bose condensation is discussed by showing the occurrence of symmetry breaking in the hole doped high
Tc cuprates. The symmetry breaking is shown to take place with the d-wave hole pairing, but not with the s-wave hole pairing. Based on a derived supersymmetry Hamiltonian we find that there exists a possibility of supersymmetry conditions in association with the pairing order parameters of both spinon and holon.
We have numerically evaluated the energy of several kinds of wave functions considered to be candidate ground states of the two-dimensional t-J model at various hole densities. We searched a parameter space which includes d-wave and s-wave superconductivity and spin-density-wave ordering as well as the projected Fermi-liquid state. Coexistence of different orderings, such as the s+id state and d-wave spin-density-wave state, were found to be stable states. We find a phase diagram in the density-t/J plane which has coexistence of antiferromagnetism and superconductivity at very low hole concentrations and superconductivity up to rather high values of density---about 40%. At intermediate concentrations, the time-reversal symmetry-breaking s+id state is found.
Due to the interplay of multicomponent order parameters [e.g., a twisted bilayer superconductor with interlayer Josephson coupling or a frustrated $(n\ensuremath{\ge}3)$-band superconductor], a superconductor can possess a $\text{U}(1)\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ symmetry, corresponding to the superconducting
Tc and time-reversal symmetry breaking transition
TTRSB, respectively. It was then conjectured that in this class of Hamiltonians, there exists a vast parameter regime
O such that the system exhibits vestigial TRSB, i.e.,
TTRSB>Tc, while at the boundary $\ensuremath{\partial}\mathcal{O}$, the system possesses a single phase transition
TTRSB=Tc. In this paper, we provide evidence towards this conjecture by mathematically eliminating the possibility of a floating phase, i.e.,
TTRSB<Tc, for the strong coupling regime. More specifically, we prove that the correlation functions of
U(1) spins are bounded above by that of
Z2 spins for all temperatures and lattice structures (e.g.,
Zd for all
d). In particular, this guarantees the existence of high-
Tc TRSB (and consequently topological) superconductivity in a large class of Hamiltonians. Note that the same property can also be proven for a certain parameter regime $(\mathrm{\ensuremath{\Delta}}\ensuremath{\ge}4/5)$ of the generalized XY model on any lattice structure, despite belonging to an entirely distinct class of $\text{U}(1)\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ Hamiltonians.
In order to obtain a reasonable phase diagram involving hole-pair condensation for hole-doped cuprate systems, we propose a modified t-J Hamiltonian in which a Coulomb repulsion term between nearest neighbor holes is considered. We show from the investigation of free energy that holon pair condensation can occur as a result of symmetry breaking. In addition, we introduce a possibility of supersymmetry condition at
T=0K in the intermediate region between the antiferromagnetic phase and the superconducting phase.
The criteria for bosonization of Cooper pairs and novel Bose-liquid superconductivity and superfluidity in high-
Tc cuprates and other pseudogap matters are formulated by using the uncertainty principle and the boson mean field theory. We argue that the BCS-type s- or d-wave superconductivity occurring in the fermionic limit of Cooper pairs (which exist in ordinary metals and heavily overdoped cuprates with large Fermi energies) is not characteristic of underdoped to overdoped cuprates with low Fermi energies. The superconducting order parameter in high-
Tc cuprates and other pseudogap matters cannot be determined as the BCS-like (s- or d-wave) gap. We show that the unconventional superconductivity/superfluidity occurring in the bosonic limit of Cooper pairs exists in low Fermi energy systems where the bosonic Cooper pairs are formed at a pseudogap temperature T* above the uperconducting/superfluid transition temperature
Tc and then such bosons condenses into a Bose superfluid at
Tc. The pair condensation of attracting bosons occurs at
Tc and then their single particle condensation sets in at
Tc∗ lower than
Tc (in three dimensions (3D)) or at T=0 (in two dimensions (2D)). The coherent single particle and pair condensates of bosons exist as two distinct superfluid phases below
Tc. By solving the mean field equations for high-
Tc cuprates, the novel superconducting states (i.e., a vortex-like state existing below the temperature
Tv=Tc2D lower than T* but higher than
Tc=Tc3D and two superconducting phases below
Tc) and their properties characterized by the boson superfluid stiffness are determined and compared with the key experimental findings. The novel Bose-liquid superconductivity and superfluidity exists also in other exotic superconductors/superfluids, including quantum liquids, atomic Fermi gases and low-density nuclear matter.