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arXiv:cond-mat/0702057 · Physical Review Letters

Nematic Order by Disorder in Spin-2 Bose-Einstein Condensates

Ari M. Turner, Ryan Barnett, Eugene Demler, Ashvin Vishwanath

Published May 11, 2007• 79 citations
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Abstract

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.

DOI
10.1103/physrevlett.98.190404

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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.
It is shown that zero point quantum fluctuations completely lift the accidental continuous degeneracy that is found in mean field analysis of quantum spin nematic phases of hyperfine spin-2 cold atoms. The result is two distinct ground states which have higher symmetries: a uniaxial spin nematic and a biaxial spin nematic with dihedral symmetry Dih4. There is a novel first-order quantum phase transition between the two phases as atomic scattering lengths are varied. We find that the ground state of 87Rb atoms should be a uniaxial spin nematic. We note that the energy barrier between the phases could be observable in dynamical experiments.
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.
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.
In this thesis, we investigate strongly correlated phenomena in quantum spin systems. In the first part of this work, we study geometrically frustrated antiferromagnets (AFMs). Generalizing the SU(2) Heisenberg Hamiltonian to Sp(N) symmetry, we obtain, in the large-N limit, the mean-field phase diagrams for the planar pyrochlore and cubic AFMs. We then use gauge theories to consider fluctuation effects about their respective mean-field configurations. We find, in addition to conventional Neel states, a plethora of novel magnetically disordered phases: two kinds of spin liquids, Z2 in 2+1D and U(1)in 3+1D, and several valence bond solids such as two and three-dimensional plaquette and columnar singlet states. We use the same approach to study the diamond lattice AFM which possesses extended classical ground state degeneracy. We demonstrate that quantum and entropic fluctuations lift this degeneracy in different ways. In the second part of the thesis, we study ultracold spinor atoms confined in optical lattices. We first demonstrate the feasibility of experimental realization of rotor models using ultracold spin-one Bose atoms in a spin-dependent and disordered optical lattice. We show that the ground state of such disordered rotor models with quadrupolar interactions can exhibit biaxial nematic ordering in the disorder-averaged sense, and suggest an imaging experiment to detect the biaxial nematicity in such systems. Finally, using variational wavefunction methods, we study the Mott phases and superfluid-insulator transition of spin-three bosons in an optical lattice with an anisotropic two dimensional optical trap. We chart out the phase diagrams for Mott states with n = 1 and n = 2 atoms per lattice site. We show that the long-range dipolar interaction stabilizes a state characterized by antiferromagnetic chains made of ferromagnetically aligned spins. We also obtain the mean-field phase boundary for the superfluid-insulator transition, and show that inside the superfluid phase and near the superfluid-insulator phase boundary, the system undergoes a first order antiferromagnetic-ferromagnetic spin ordering transition.
We investigate the effects of thermal and quantum fluctuations on the phase diagram of a spin-1 87Rb{}^{\text{87}}\text{Rb}87Rb Bose-Einstein condensate (BEC) under the quadratic Zeeman effect. Due to the large ratio of spin-independent to spin-dependent interactions of 87Rb{}^{\text{87}}\text{Rb}87Rb atoms, the effect of noncondensed atoms on the condensate is much more significant than that in scalar BECs. We find that the condensate and spontaneous magnetization emerge at different temperatures when the ground state is in the broken-axisymmetry phase. In this phase, a magnetized condensate induces spin coherence of noncondensed atoms in different magnetic sublevels, resulting in temperature-dependent magnetization of the noncondensate. We also examine the effect of quantum fluctuations on the order parameter at absolute zero and find that the ground-state phase diagram is significantly altered by quantum depletion.
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.