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 Néel 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
Four-coloring model and frustrated superfluidity in the diamond lattice
Phys. Rev. Lett. 112, 020601 (2014) ,
see pdf file
and supplementary material
.
Phys. Rev. A 93, 053623 (2016) .
See pdf file .
"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.