Crystal structure and the Bloch theorem play a fundamental role in
condensed matter physics.
Motivated by the recent development of quantum dynamics, including
both the cold-atom shaking lattice experiments and the
solid state laser-driven lattice experiments,
we define the concepts of dynamics crystal and ``space-time
group" .
This work provides a general framework for further studying physical
properties in this class of dynamic systems, for example, transport
and topological properties.
References and talks
"Dynamic crystal'' generalizes the standard concept of crystal to the
dynamic case.
Dynamic crystal exhibits the general intertwined space-time periodicities
in D+1 dimensions, including both the static crystal and the Floquet crystal
as special cases
[Ref. 1] .
In general, the space-time primitive unit cell is not a direct
product between spatial and temporal domains.
In the most general case, there may not even exist spatial translational
symmetry at any given time, nor temporal translational
symmetry at any spatial location.
Dynamic crystal
We also construct the Floquet-Bloch theorem for space-time crystals.
Each band eigenstate is characterized by a lattice momentum-energy vector
\kappa=(k,\omega_m).
Except a plane-wave phase factor, the eigenwavefunction takes the form of a
perodic function with the same periodicity of the space-time unit cell.
To describe the space-time crystalline symmetries, we propose a new mathematical
concept of ``space-time'' group.
The non-relativistic space-time symmetry is the direct product of the Euclidean
group in D-spatial dimensions and that along the time-direction E_D \otimes E_1.
We define ``space-time'' group as its discrete subgroup, and in general it is
space-time entangled.
It includes operations involving fractional translations along the time-direction.
Compared to space and magnetic groups, space-time group is augmented
by ``time-screw'' rotations and ``time-glide'' reflections involving
fractional translations along the time direction.
Space-time group
Similar to that space group classifies crystal, the classification
of ``space-time'' crystal is based on ``space-time'' group.
A complete classification in 1+1D shows that there are 13 space-time groups
in contrast to the 17 wallpaper space groups characterizing the 2D static crystals.
Due to the non-equivalence between spatial and temporal directions,
there are no square and hexagonal space-time crystal systems.
There exist two types of glide reflections: the time-glide
reflection g_x and ``glide-time-reversal'' g_t, which
are reflections with respect to the x-direction and t-direction
followed by a fractional translation along the t-direction and
x-direction, respectively.
The classifications of the space-time groups in higher dimensions are
generally complicated.
In 2+1D, we have classified 275 space-time groups.
The ``space-time'' group symmetries protect spectra degeneracies
of ``space-time'' crystals.
The Kramers-type degeneracy can arise from the glide time-reversal
symmetry without the half-integer spinor structure, which constrains
the winding number patterns of spectral dispersions.
In 2+1D, non-symmorphic space-time symmetries enforce spectral
degeneracies, leading to protected Floquet semi-metal states.
Phys. Rev. Lett. 120, 096401 (2018) .
See pdf file .
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Last modified: Oct 16, 2018.