We show that quantum and thermal fluctuations in spin-2 Bose-Einstein condensates lift the accidental degeneracy of the mean-field phase diagram. Fluctuations select the uniaxial (square biaxial) nematic state for scattering lengths a4>a2 (a4<a2). Paradoxically, the order is stronger at higher temperatures. For spin-2 87Rb and 23Na, a continuous Ising-type transition is predicted on raising the magnetic field, from a fluctuation stabilized uniaxial state to a field stabilized square biaxial order state. This is a promising experimental system to realize the "order-by-disorder" phenomenon.
In this thesis, we have investigated several fluctuation-driven phenomena in ultracold spinor Bose gases. In Bose-Einstein condensates of hyperfine spin-two (F=2) atoms, it is shown that zero-point quantum fluctuations completely lift the accidental continuous degeneracy in quantum spin nematic phases predicted by mean field analysis, and these fluctuations select out two distinct spin nematic states with higher symmetries. It is further shown that fluctuations can drive a novel type of coherent spin dynamics which is very sensitive to the variation of quantum fluctuations controlled by magnetic fields or potential depths in optical lattices. These results have indicated fundamental limitations of precision measurements based on mean field theories. In addition, fluctuation-driven coherent spin dynamics studied here is a promising tool to probe correlated fluctuations in many body systems. In another system -- a two-dimension superfluid of spin-one (F=1) Na²³ atoms -- we have investigated spin correlations associated with half quantum vortices. It is shown that when cold atoms become superfluid below a critical temperature a unique nonlocal topological order emerges simultaneously due to fluctuations in low dimensional systems. Our simulation have indicated that there exists a nonlocal softened pi-spin disclination structure associated with a half-quantum vortex although spin correlations are short ranged. We have also estimated fluctuation-dependent critical frequencies for half-quantum vortex nucleation in rotating optical traps. These results indicate that the strongly fluctuating ultracold spinor system is a promising candidate for studying topological orders that are the focus of many other fields.
Quantum phase transitions (QPTs) between uniaxial or biaxial spin nematic (SN) phases are investigated in biquadratic spin-1 and spin-2 XY infinite chains with the rhombic- and uniaxial-type single-ion anisotropies. Systematic discussions of distinctive singular behaviors are made to classify various types of QPT from one SN state to the other SN state in using the Wigner-Yanase skew information (WYSI), the bipartite entanglement entropy (BEE), and the quadrupole moments (QMs). For the spin-1 system with the three uniaxial SN phases, we find that a discontinuous QPT, signaled by discontinuous behaviors of all the considered WYSI, BEE, and QMs, occurs from the z-ferroquadrupole phase (FQP) to the x- or y-FQPs, while a continuous QPT occurs between the x- and y-FQPs. The central charge in the continuous QPT line is estimated as
c≃1 from the BEE. Compared to the spin-1 system, depending on a given strength of the uniaxial-type single-ion anisotropy, the spin-2 system undergoes four different types of QPTs between the two biaxial SN phases as the rhombic-type anisotropy varies: the quantum crossovers, connecting the two orthogonal biaxial SN states adiabatically without an explicit phase transition, the continuous and the discontinuous QPTs, and the SN to magnetic transitions via the antiferromagnetic phase (AFP). In a sharp contrast to the spin-1 system, for the transitions between the two biaxial SN phases, the discontinuous transition line is classified as a topological phase characterized by a doubly degenerate entanglement spectrum and a string order parameter defined by the Cartan generator of the
SO(5) symmetry group in spin-2 systems, while the continuous QPT is advocated by the central charge
c≃1. Whereas the QPT lines with
c≃1/2 indicate that the transition between the biaxial SN phase and the AFP belongs to the Ising universality class.
We show that the ground state of disordered rotor models with quadrupolar interactions can exhibit biaxial nematic ordering in the disorder-averaged sense. We present a mean-field analysis of the model and demonstrate that the biaxial phase is stable against small quantum fluctuations. We point out the possibility of experimental realization of such rotor models using ultracold spin-1 Bose atoms in a spin-dependent and disordered optical lattice in the limit of a large number of atoms per site and also suggest an imaging experiment to detect the biaxial nematicity in such systems.
We consider quantum phases of tightly confined spin-2 bosons in an external field under the presence of rotationally invariant interactions. Generalizing previous treatments, we show how this system can be mapped onto a quantum rotor model. Within the rotor framework, low-energy excitations about fragmented states, which cannot be accessed within standard Bogoliubov theory, can be obtained. In the spatially extended system in the thermodynamic limit there exists a mean field ground-state degeneracy between a family of nematic states for appropriate interaction parameters. It has been established that quantum fluctuations lift this degeneracy through the mechanism of order by disorder and select either a uniaxial or square-biaxial ground state. On the other hand, in the full quantum treatment of the analogous single-spatial-mode problem with finite-particle number, it is known that, due to symmetry-restoring fluctuations, there is a unique ground state across the entire nematic region of the phase diagram. Within the established rotor framework, we investigate the possible quantum phases under the presence of a quadratic Zeeman field, a problem which has previously received little attention. By investigating wave-function overlaps, we do not find any signatures of the order-by-disorder phenomenon which is present in the continuum case. Motivated by this, we consider an alternative external potential which breaks less symmetry than the quadratic Zeeman field. For this case, we do find the phenomenon of order by disorder in the fully quantum system. This is established within the rotor framework and with exact diagonalization.
We investigate quantum phase transitions and quantum coherence in infinite biquadratic spin-1 and -2 XY chains with rhombic single-ion anisotropy. All considered coherence measures such as the
l1 norm of coherence, the relative entropy of coherence, and the quantum Jensen-Shannon divergence, and the quantum mutual information show consistently that singular behaviors occur for the spin-1 system, which enables to identify quantum phase transitions. For the spin-2 system, the relative entropy of coherence and the quantum mutual information properly detect no singular behavior in the whole system parameter range, while the
l1 norm of coherence and the quantum Jensen-Shannon divergence show a conflicting singular behavior of their first-order derivatives. Examining local magnetic moments and spin quadrupole moments lead to the explicit identification of novel orderings of spin quadrupole moments with zero magnetic moments in the whole parameter space. We find the three uniaxial spin nematic quadrupole phases for the spin-1 system and the two biaxial spin nematic phases for the spin-2 system. For the spin-2 system, the two orthogonal biaxial spin nematic states are connected adiabatically without an explicit phase transition, which can be called quantum crossover. The quantum crossover region is estimated by using the quantum fidelity. Whereas for the spin-1 system, the two discontinuous quantum phase transitions occur between three distinct uniaxial spin nematic phases. We discuss the quantum coherence measures and the quantum mutual information in connection with the quantum phase transitions including the quantum crossover.