## Conventional BEC and "no-node" theorem

The ordinary ground state wavefunctions of bosons, including the superfluid 4He and most alkali boson BECs, obey the ``no-node'' theorem, or, more formally, Perron-Frobenus theorem. (See Feynmann's "Statistical Mechanics" textbook for an explanation of this important but often overlooked theorem.) It implies that their condensate wavefunctions are positive-definite and thus can only be the conventional s-wave like, since all unconventional symmetries (e.g., p, d-wave etc) necessarily require nodes.The "no-node" theorem is a very general statement, which applies to almost all well-known ground states of bosons, including the superfluid, Mott-insulating, density-wave, and super-solid ground states. It is also a very strong statement, which reduces the generally speaking complex-valued many-body wavefunctions to become positive-definite. This is why the ground state properties of bosons, such as 4He, can in principle be exactly simulated by the quantum Monte-Carlo method free of the sign problem. Furthermore, this statement implies that the ordinary ground states of bosons, including BEC and Mott-insulating states, cannot spontaneously break time-reversal (TR) symmetry, since TR transformation for the single component bosons is simply the operation of the complex conjugation.

## Unconventional BEC

In analogy to unconventional superconductivities, we proposed the new concept of unconventional BECs based on their symmetry properties [Ref. 1] . Consider a system with either a continuous or lattice rotational symmetry. If the condensate wavefunction Psi(r) belongs to a non-trivial representation of the rotation group, it is a UBEC, otherwise, it is conventional. UBECs must be nodal beyond the "no-node" theorem exhibiting non-trivial symmetries such as p-wave and d-wave, etc. Their condensation wavefunctions can be either real-valued exhibiting nodal lines, or, complex-valued exhibiting nodal points breaking time-reversal symmetry, spontaneously.Since 2006, we have been working on exploring novel UBEC states proposing to pump bosons into high-orbital bands (e.g. the 2nd band of p-orbitals) of optical lattices which are meta-stable excited states and thus refrain from the ``no-node'' constraint [Ref. 2] . Their condensate wavefunctions are complex-valued exhibiting the px \pm i py type symmetry breaking time-reversal symmetry spontaneously.

The preference of the complex condensates is due to the repulsive interaction: The complex condensates only have nodal points, while the real condensate px or py has nodal lines, hence, the complex ones are spatially more uniform to reduce interaction energy [Ref. 3] . In real space, the system exhibits a vortex-anti-vortex configuration. We also showed that even when the superfluid phase coherence is lost in the Mott-insulating state, the system still exhibit a staggered ordering of orbital angular momentum [Ref. 3] , and [Ref. 4] .

## Frustrated BEC and the four-coloring problem

We propose a novel four-coloring model which describes âfrustrated superfluidityâ of p-band bosons in the diamond optical lattice Ref. [5] . The superfluid phases of the condensate wave functions on the diamond-lattice bonds are mapped to four distinct colors at low temperatures. The fact that a macroscopic number of states satisfy the constraints that four differently colored bonds meet at the same site leads to an extensive degeneracy in the superfluid ground state at the classical level.We demonstrate that the phase of the superfluid wave function as well as the orbital angular momentum correlations exhibit a power-law decay in the degenerate manifold that is described by an emergent magnetostatic theory with three independent flux fields. Our results thus provide a novel example of critical superfluid phase with algebraic order in three dimensions. We further show that quantum fluctuations favor a NÃ©el ordering of orbital angular moments with broken sublattice symmetry through the order-by-disorder mechanism.

## Unconventional BEC with two-compnent bosons

In the context of Gross-Pitaevskii theory, we investigate the unconventional Bose-Einstein condensations in the two-species mixture with p-wave symmetry in the second band of a bipartite optical lattice Ref. [6] . An imaginary-time propagation method is developed to numerically determine the p-orbital condensation. Different from the single-species case, the two-species boson mixture exhibits two nonequivalent complex condensates in the intraspecies-interaction-dominating regime, exhibiting the vortex-antivortex lattice configuration in the charge and spin channels, respectively. When the interspecies interaction is tuned across the SU(2) invariant point, the system undergoes a quantum phase transition toward a checkerboardlike spin-density wave state with a real-valued condensate wave function. The influence of lattice asymmetry on the quantum phase transition is addressed.## Experimental tests

Our theory of UBEC with the $p$-wave symmetry has been tested by Hemmerich's group at Hamburg University in high-orbital bands of optical lattices (For details, please see their review paper**Kock et. al, J. Phys. B: At. Mol. Opt. Phys. 49, 042001(2016)**). Their 2nd band displays degenerate energy minima located at K_1=-K_1 and K_2=-K_2 modula reciprocal lattice vectors. Our analysis showed that the condensate wavefunctions should be \Psi_K1 \pm i \Psi_K2, whose real space distributions exhibit the vortex-antivortex lattice configuration. We have analyzed the quantum phase transition from the complex condensate to the real condensates \Psi_K1 and \Psi_K2 as tuning the lattice asymmetry [Ref. 3] .

Recently, Hemmerich's group has also performed the matter-wave interference experiments which directly show the phase difference \pm \pi/2 between \Psi_K1 and \Psi_K2 (

**Kock et al, PRL 114, 115301 (2015)**). The spirit of this experiment is very similar to the phase-sensitive experiments for the d-wave symmetry of high Tc cuprates, including the corner \pi-junction and the tri-crystal experiments.

## D-wave polariton-exciton condensation

In collaboration with Y. Yamamoto's experiment group at Stanford, we have provided theory support for explaining their observation of d-wave UBECs on the exciton-polariton lattice systems. Ref. [7]. ** References and talks **

**Congjun Wu,**"Unconventional Bose-Einstein Condensations Beyond the ``No-node'' Theorem" Mod. Phys. Lett. 23, 1 (2009). See pdf file .

**Congjun Wu**, "Atomic matter of nonzero-momentum Bose-Einstein condensation and orbital current order", Phys. Rev. A

**74**, 13607 (2006), see pdf file.

**Congjun Wu,**"Complex and real unconventional Bose-Einstein condensations in high orbital bands", Phys. Rev. A 84, 033635 (2011), See pdf file .

**Congjun Wu**, R. T. Scalettar, G. G. Batrouni, "Exotic phases of interacting p-band bosons" Phys. Rev. B 87, 224505 (2013) . See pdf file

**Congjun Wu,**

Four-coloring model and frustrated superfluidity in the diamond lattice

Phys. Rev. Lett. 112, 020601 (2014) , see pdf file and supplementary material .

**Congjun Wu,**"Unconventional Bose-Einstein Condensations of Two-species Bosons in the $p$-orbital Bands in Optical lattice",

Phys. Rev. A 93, 053623 (2016) . See pdf file .

**Congjun Wu,**Naoyuki Masumoto, Andreas LÃ¶ffler, Sven HÃ¶fling, Norio Kumada, Lukas Worschech, Alfred Forchel and Yoshihisa Yamamoto,

"Dynamical d-wave condensation of exciton<96>polaritons in a two-dimensional square-lattice potential",

Nature Physics 7, 681 (2011) See pdf file , and supplementary material .

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Last modified: July 15, 2007.