# Route towards classical frustration and band flattening via optical lattice
distortion

Pil Saugmann  Department of Physics, Stockholm University, Se-106 91
Stockholm, Sweden  José Vargas  Institut für Laserphysik, Universität Hamburg,
22761 Hamburg, Germany  Zentrum für Optische Quantentechnologien and Institut
für Laser-Physik, Universität Hamburg, 22761 Hamburg, Germany  Yann Kiefer
Institut für Laserphysik, Universität Hamburg, 22761 Hamburg, Germany  Zentrum
für Optische Quantentechnologien and Institut für Laser-Physik, Universität
Hamburg, 22761 Hamburg, Germany  Max Hachman  Institut für Laserphysik,
Universität Hamburg, 22761 Hamburg, Germany  Zentrum für Optische
Quantentechnologien and Institut für Laser-Physik, Universität Hamburg, 22761
Hamburg, Germany  Raphael Eichberger  Institut für Laserphysik, Universität
Hamburg, 22761 Hamburg, Germany  Zentrum für Optische Quantentechnologien and
Institut für Laser-Physik, Universität Hamburg, 22761 Hamburg, Germany
Andreas Hemmerich  Institut für Laserphysik, Universität Hamburg, 22761
Hamburg, Germany  Zentrum für Optische Quantentechnologien and Institut für
Laser-Physik, Universität Hamburg, 22761 Hamburg, Germany  The Hamburg Center
for Ultrafast Imaging, Luruper Chaussee 149, 22761 Hamburg, Germany  Jonas
Larson  Department of Physics, Stockholm University, Se-106 91 Stockholm,
Sweden

###### Abstract

We propose and experimentally explore a method for realizing frustrated
lattice models using a Bose-Einstein condensate held in an optical square
lattice. A small lattice distortion opens up an energy gap such the lowest
band splits into two. Along the edge of the first Brillouin zone for both
bands a nearly flat energy-momentum dispersion is realized. For the excited
band a highly degenerate energy minimum arises. By loading ultracold atoms
into the excited band, a classically frustrated $XY$ model is formed,
describing rotors on a square lattice with competing nearest and next nearest
tunnelling couplings. Our experimental optical lattice provides a regime,
where a fully coherent Bose-Einstein condensate is observed, and a regime
where frustration is expected. If we adiabatically tune from the condensate
regime to the regime of frustration, the momentum spectra shows a complete
loss of coherence. Upon slowly tuning back to the condensate regime, coherence
is largely restored. Good agreement with model calculations is obtained.

###### pacs:

03.75.Lm, 67.85.Hj, 05.30.Rt

Introduction. – Classically frustrated systems are characterized by highly
degenerate ground states which may result in glassy or liquid like phases with
exotic properties frust1 ; spinglass ; spinliq . While extensively studied
during the last decades, due to the complexity of these systems defining their
properties is extremely challenging. Using the experimental platform of cold
atoms loaded into optical lattices, which has proven useful for controlled
simulations of complex systems review1 ; review2 , could open a new chapter
for systematic studies of frustration. The simplest example of a frustrated
system is that of spins on a triangular lattice with nearest neighbor anti-
ferromagnetic couplings. Such geometric frustration was recently
experimentally explored in a cold atom setting seng1 ; seng2  by making use of
spin degrees of freedom and engineered via external periodic driving.
Experimental studies of frustrated systems, using cold atoms in optical
lattices, are met with several difficulties. For example, without external
driving, atoms in the lowest Bloch band typically support tunneling couplings
with ferromagnetic order. Focused engineering of band structure degeneracies
must go well beyond the tight-binding approximation, such that these are not
lifted by higher order tunneling terms. Frustration may also appear on a
square lattice due to different mechanisms. In the Villain model villain ,
three of the tunneling terms are positive, while the fourth one, closing the
loop around a plaquette, is negative. Another possibility arises from the
competition between nearest and next nearest neighbor tunneling terms.

For classical rotors $s_{\bf i}=(\cos\phi_{\bf i},\sin\phi_{\bf i})$, the
frustrated $XY$ model is xy4a ; xy4b

$H_{XY}=J_{1}\sum_{\langle{\bf ij}\rangle}\cos(\phi_{\bf i}-\phi_{\bf
j})+J_{2}\sum_{\langle\\!\langle{\bf ij}\rangle\\!\rangle}\cos(\phi_{\bf
i}-\phi_{\bf j}),$ (1)

with positive $J_{1},\,J_{2}$ and the sums over nearest and next nearest
neighbor coupling terms, respectively. The above model supports ‘interaction
induced frustration’ rather than the aforementioned geometric frustration.
Despite four decades of intense research, the physics of the Villain and
frustrated $XY$ models is not fully clarified. Both models, sharing the same
universal features, support two symmetries; a continuous one, corresponding to
particle conservation, and a chiral Ising-like symmetry. Due to the Mermin-
Wagner theorem, the former cannot be spontaneously broken but nevertheless
manifests as a Kosterlitz-Thouless transition. It is believed, but still
debated xy3a ; xy3b ; xy3c ; xy3d ; xy3e ; xy3f ; xy3g , that upon lowering
the temperature the models pass through two nearby transitions. The phase
emerging between the transitions is argued to be a chiral spin liquid xy8 . At
zero temperature, and at the frustration point, $J_{1}=2J_{2}$, the ground
state is a liquid phase. The liquid phase may also survive beyond the
frustration point xy7 , even though a thorough analysis, based on a large
$N$-expansion, came to the conclusion that it exists only at the frustration
point xy6 . Spin wave theory suggests that for $J_{2}>2J_{1}$ anti-
ferromagnetic order is established in the two sublattices with a relative
‘collinear’ order xy4a ; xy4b , as a result of the order-by-disorder
phenomenon od . Needless to say, these speculations await any experimental
verification.

Figure 1: (Color online) (a) Conventional square lattice geometry with
equivalent $\mathcal{A}$ and $\mathcal{B}$ sites, obtained for $\Delta V=0$ in
Eq. (2). The lower panel shows the first (white square) and second (grey
triangles) BZs (dashed square denotes the reduced BZ for the distorted
lattice). (b) Bipartite lattice geometry obtained for $\Delta V\neq 0$ in Eq.
(2) with the corresponding first (white square) and second (grey triangles)
BZs in the lower panel. The two inequivalent high symmetry points $X_{\pm}$
are global minima of the second band. (c) The second band for $\Delta V\approx
0.063\,V_{0}$ and $V_{0}=10E_{\mathrm{R}}$ is plotted across the first BZ.

In this paper we propose, and experimentally realize, a surprisingly simple
realization of the frustrated $XY$ model (1) in terms of cold atoms loaded
into an optical lattice. The advantage of our method is that no external
driving (like lattice shaking or laser-induced tunneling) is required in order
to realize the desired anti-ferromagnetic tunneling couplings and strong
couplings beyond nearest neighbors. This becomes possible by considering meta-
stable states in higher Bloch bands. In particular, we consider bosonic atoms
loaded into the second band of a bipartite square lattice as described in Ref.
hemmerich . This lattice provides two classes of alternating potential wells,
denoted $\mathcal{A}$ and $\mathcal{B}$, with tunable relative depths $\Delta
V$. In the case $\Delta V=0$, a simple square lattice arises with a white
square denoting the first Brillouin zone (BZ) in Fig. 1(a), surrounded by four
grey triangles showing the second BZ. The first Bloch band shows constant
energy on the dashed rectangle in Fig. 1(a), dividing the BZ into two equally
sized areas. Tuning $\Delta V$ slightly away from zero has significant
consequences. The unit cell area increases by a factor of two, i.e. the size
of the first BZ shrinks by a factor two, such that it now is bordered by the
dashed rectangle (Fig. 1(b)). Hence, the lowest band of the $\Delta V=0$
lattice splits into two new bands, thus giving rise to the first and second
bands of the $\Delta V\neq 0$ lattice. If $\Delta V$ is kept close to zero,
each of these bands is nearly degenerate along the first BZ edge. Thus, the
second band provides an extended approximate minimum all along the edge of the
first BZ just like for the frustrated $XY$ model (Fig. 1(c)). At this stage
our experiment is not capable of directly confirming the frustration emerging
from such a quasi degeneracy. Nevertheless, we show in the following that
numerical simulations of the atomic momentum distributions in the frustrated
$XY$ model agree well with experimental time-of-flight measurements.

Model system. – We consider bosonic atoms within a two-dimensional bipartite
optical lattice. Along the transverse third direction the atoms are confined
by a harmonic trap. The two-dimensional lattice is formed by superimposing two
one-dimensional standing waves oscillating with a tunable relative phase
$\theta$. The optical potential reads

$\begin{array}[]{lll}V(x,y)&=&-V_{0}\,[\cos^{2}(kx)+\cos^{2}(ky)\\\
&&+2\cos(\theta)\cos(kx)\cos(ky)].\end{array}$ (2)

In the remainder of the paper, energies are given in terms of the recoil
energy $E_{\mathrm{R}}\equiv\hbar^{2}k^{2}/2m$, with $k$ the wave number and
$m$ the atomic mass. For $\theta=\pi/2$, a conventional separable square
lattice arises (cf. Fig. 1(a)). When $\theta\neq\pi/2$ (cf. Fig. 1(b)), the
last term in Eq. (2) distorts the lattice. This gives rise to a bipartite non-
separable lattice geometry, composed of two sub-lattices, with deep and
shallow sites $\mathcal{A}$ and $\mathcal{B}$, respectively. Their potential
energy difference is $\Delta V=-4V_{0}\cos(\theta)$. Due to this distortion,
the first BZ is decreased by a factor of two, and the first Bloch band of the
$\theta=\pi/2$ lattice splits into two new bands defining a new BZ. These
bands are separated by a gap, which for weak distortions scales as $\sim\Delta
V/V_{0}$ according to the ‘nearly free electron gas’ model ashcroft1976solid .
It is striking how flat the bands are along the BZ edge as compared to the
band gap. Reaching the regime of frustration requires a flatness characterized
by an energy spread which is below the energy associated with two-body contact
interaction between the atoms. Increasing the flatness, however, comes with
the price that the band width and the band gap are decreased. Decreasing the
band width slows down the relaxation toward the regime of frustration, while
decreasing the band gap increases the inter-band relaxation, i.e., the loss of
atoms from the second band via collisions metastable . Thus, there is a trade-
off in choosing the optimal configuration. For the rest of the paper we pick
parameters $V_{0}=10\,E_{\mathrm{R}}$ and $\Delta V=0.063\,V_{0}$, for which
we plot the second band in fig. 1(c). The flatness is not perfect; the band
width is about six times the energy spread along the BZ edge, while the gap is
roughly 230 times this spread.

To explore the properties of atoms residing solely in the second band, we
evoke a single-band approximation and derive an effective lattice model,

$\hat{H}_{1}=\sum_{{\bf i}\neq{\bf j}}(t_{\bf ij}\hat{a}_{\bf
i}^{\dagger}\hat{a}_{\bf j}+h.c.),$ (3)

where the matrix $t_{\bf ij}$ amounts to the tunneling amplitudes between
sites ${\bf i}$ and ${\bf j}$. These amplitudes can be chosen real. The above
lattice Hamiltonian becomes diagonal by Fourier-transforming it. Using this we
can identify $t_{\bf ij}$ as the Fourier coefficients of the dispersion
$E_{2}(q_{x},q_{y})$ for the second band of the full Hamiltonian. The relevant
numerically extracted amplitudes are presented in tab. 1. For the present
parameters, it is found that the system is close to the frustrated scenario,
i.e. after identifying $J_{1}$ and $J_{2}$ we have $J_{1}/J_{2}=1.43$ while
other tunneling terms are approximately zero in comparison. The ratio
$J_{1}/J_{2}$ does not equal the desired value 2, implying that the second
band is not fully flat along the BZ edge, as also seen in fig. 1(c). The final
step in mapping to the frustrated $XY$ model is accomplished by considering
the mean-field approximation in which the atomic operators are replaced by
their expectations, e.g. $\hat{a}\rightarrow\alpha_{\bf i}=\sqrt{n_{\bf
i}}e^{i\phi_{\bf i}}$ such that $n_{\bf i}$ is the (average) number of atoms
at site ${\bf i}$ and $\phi_{\bf i}$ is the intra-site condensate phase.

$t_{nm}$ |  0 |  1 |  2
---|---|---|---
0 | -13.97 | $1.4\cdot 10^{-3}$ | $-6\cdot 10^{-6}$
1 | $1.4\cdot 10^{-3}$ | $9.7\cdot 10^{-4}$ | $-4\cdot 10^{-6}$
2 | $-6\cdot 10^{-6}$ | $-4\cdot 10^{-6}$ | $-6\cdot 10^{-7}$
Table 1: Tunneling matrix for the parameters $V_{0}=10E_{\mathrm{R}}$ and
$\Delta V=0.063V_{0}$. The first diagonal term gives the onsite energy, the
terms $(0,1)$ and $(1,0)$ are the nearest neighbor tunneling rates $J_{1}$,
and $(1,1)$ the next nearest neighbor tunneling rate $J_{2}$ and so on.

We must further add one term accounting for atom-atom interaction and another
corresponding to the confining harmonic trap. Following the standard
assumptions, i.e. the local density approximation and onsite $s$-wave
scattering, we then derive the mean-field Hamiltonian

$H=H_{1}+\sum_{\bf i}\frac{\omega^{2}}{2}n_{\bf i}|{\bf
i}|^{2}+\frac{U}{2}\sum_{\bf i}n_{\bf i}^{2},$ (4)

where $\omega$ and $U$ are the dimensionless trap frequency and interaction
strength respectively. We can simulate the atomic relaxation that occurs in
the experiment (see below) by initializing a state $\psi({\bf i},0)$, defined
by some set of random variables $n_{\bf i}$ and $\phi_{\bf i}$, and then
evolving this state in imaginary time imagt . If the system would not be
frustrated, the initial lack of long range order rapidly disappears and a
phase-locking among the $\phi_{\bf i}$ will occur. For example, if the BZ edge
is not flat enough (with minima at the $X_{\pm}$ points), the global
condensate builds up a striped order characterized by a $\pi$-phase difference
between either the columns or rows of the lattice. However, if the BZ is
approximately flat, such order will not build up on any experimentally
relevant time scale. In Fig. 2 we show properties of the numerically extracted
condensate. Here, at a given time $t$, the condensate order parameter is
obtained from $\Psi(x,y)=\sum_{\bf i}\alpha_{\bf i}w_{\bf i}(x,y)$, where
$w_{\bf i}(x,y)$ is the Wannier function at site ${\bf i}$, which for the
figure has been taken as a real Gaussian. From $\Psi(x,y)$ we get the atomic
and momentum densities $\rho(x,y)$ ($=|\Psi(x,y)|^{2}$) and
$\tilde{\rho}(p_{x},p_{y})$ respectively. We note how the relaxation causes
the momentum distribution (c) to assemble along the BZ edge. The spatial
density (a) shows an envelope following the inverse shape of the trap with
seemingly complex local density fluctuations stemming from the frustration. A
manifestation of the frustration is especially evident in (b) where we plot
the condensate phase $\phi(x,y)=\mathrm{angle}[\Psi(x,y)]$. If we were to let
the condensate relax further, i.e. by propagating the simulation further, the
momentum distribution would get more and more concentrated around the
$X_{\pm}$ points, and domains of striped order would form in the spatial
condensate density. The sizes of these domains would grow with time to
eventually suppress the frustration completely.

Figure 2: (Color online) Numerical simulation of the atomic real space density
(a), its phase (b), and the momentum density profile (c). The relaxation is
mimicked by imaginary time-propagation for a dimensionless time $t_{f}=100$.
At this stage the condensate is still fairly uniformly distributed around the
BZ edge (c) The middle plot (b) shows how no evident order is established in
the condensate, which signals the presence of frustration. The system
parameters are chosen to represent the experimental ones; the dimensionless
trapping frequency $\omega=0.037$, $V_{0}=10E_{R}$, $\Delta V=0.063\,V_{0}$,
$U_{0}=0.011E_{R}$ and the number of atoms $N=10^{5}$.

Experimental demonstration. – A nearly pure spin-polarised Bose-Einstein
Condensate of $3\cdot 10^{5}$ rubidium (87Rb) atoms in the hyperfine state
$|F=2,m_{F}=2\rangle$ with a temperature of $30\,$nK is produced in a crossed
optical dipole trap (ODT). The atomic sample is adiabatically loaded into a
bipartite optical square lattice, which displays a double-well structure as
described in eq. (2). The optical lattice is formed by two mutually orthogonal
optical standing waves with a wavelength of $\lambda=1064\,$nm. The lattice
depth $V_{0}$ as well as the potential energy difference $\Delta V$ can be
actively tuned. The tuning of $\Delta V$ is realized via the control of the
time phase difference $\theta$ between the standing waves with a precision of
$\pi/300$ conforming to $\Delta V=-4V_{0}\cos(\theta)$. Rapid adjustment of
$\Delta V$ allows us to quench the atoms into higher Bloch bands. For
technical details see refs. fermionlattice ; bosonlattice .

Figure 3: Collapse and revival of coherence. Three momentum spectra are shown
in (a) for characteristic values of $\Delta V$ tuned according to the protocol
in (b). The first image at $\Delta V=0.439\,V_{0}$ shows coherent condensate
fractions at the $X_{\pm}$ points of the second band (cf. Fig. 1(b)). In the
second image at $\Delta V=0.063\,V_{0}$ coherence has completely vanished. The
third image, recorded after ramping back to $\Delta V=0.439\,V_{0}$, shows
partially restored coherence. The blue symbols in (c) and (d) show the
observed collapse and revival of coherent condensate fractions at the
$X_{\pm}$ points plotted against the ramp times $t_{\rm down}$ and $t_{\rm
up}$, respectively. The red symbols indicate the total population of the first
and second band. Error bars correspond to the standard deviation of the mean
for a set of eight measurements.

The experimental sequence is performed as follows (cf. Fig. 3): the atoms are
initially loaded into the lowest band of the lattice by adiabatically ramping
up the lattice depth $V_{0}$ to $10\,E_{\rm{R}}$ in $150\,{\rm ms}$ with fixed
$\Delta V=-1.236\,V_{0}$. In addition, we are able to reduce the temperature
of the atomic sample by simultaneously decreasing the potential depth of the
ODT. After a hold time of $100\,$ms, a quench protocol is applied to excite
the atoms into a metastable second Bloch band metastable  by rapidly tuning
$\Delta V$ to $0.439\,V_{0}$ in $t_{\rm exc}=0.1\,$ms. After a holding time of
$t_{\rm cond}=500\,$ms, the atoms are condensed at the two inequivalent energy
minima of the band, i.e. the $X_{\pm}$-points (cf. Fig. 3 (a) left).
Subsequently, the potential difference $\Delta V$ is ramped down during a
variable time $t_{{\rm down}}\in\\{0,100\\}\,$ms to $\Delta V=0.063\,V_{0}$.
After $t_{{\rm hold}}=0.02\,{\rm ms}$ the potential is ramped back to $\Delta
V=0.439\,V_{0}$ in a variable time $t_{up}\in\\{0,100\\}\,{\rm ms}$ for a
fixed $t_{{\rm down}}$ and the atoms are held in the lattice for another
$t_{{\rm hold}}=0.02\,{\rm ms}$. Finally, the lattice and trapping potentials
are instantaneously switched off followed by a ballistic expansion of
$22\,{\rm ms}$ and the momentum spectrum is recorded via an absorption image.
The temporal evolution of the coherent fraction, $\left(n_{\rm coh}-n_{\rm
incoh}\right)/\left(n_{\rm coh}+n_{\rm incoh}\right)$ of the population at the
$X_{\pm}$ points is observed. This observable is determined by counting the
atoms in a disk-shaped region of interest (ROI) around both $X_{\pm}$ points
$n_{\rm coh}$ and subtracting the atoms $n_{\rm incoh}$ enclosed by a ring-
shaped ROI with the same area of the disk-shaped ROIs metastable .

Figure 3(a) shows momentum spectra at $\Delta V=0.439\,V_{0}$ before the down-
ramp (left), at $\Delta V=0.063\,V_{0}$ after the $t_{\rm down}=60\,$ms down-
ramp and a short hold time $t_{\rm hold}=20\,\mu$s (centre), and after the
$t_{up}=20\,$ms up-ramp back to $\Delta V=0.439\,V_{0}$ followed by $t_{\rm
hold}=20\,\mu$s (right). At $\Delta V=0.439\,V_{0}$, the second band possesses
distinct energy minima at the $X_{\pm}$ points and a coherent condensate
fraction is observed as indicated by the clearly visible higher order Bragg
resonances. When the potential difference is ramped down to $\Delta
V=0.063V_{0}$ (cf. Fig. 3 (b)), the condensate fraction decreases. As
illustrated by the blue data points in Fig. 3(c), if the down-ramp time
$t_{{\rm down}}$ is sufficiently large ($>60\,$ms) to satisfy adiabaticity,
the condensate fraction practically vanishes and the atomic sample no longer
displays coherence. At $\Delta V=0.063V_{0}$, the energy-momentum dispersion
of the second band is almost degenerate along the edge of the first Brillouin
zone. The atoms are observed to spread over the entire edge of the second
Bloch band populating all available Bloch states. The associated momentum
distribution in the centre image of Fig. 3(a) shows a rectangular shape,
similar as in the theoretical prediction of Fig. 2(c). The loss of coherence
indicates that the condensate suffers from excessive fragmentation due to the
large number of nearly degenerate minimal energy states.

In order to exclude the possibility that the observed vanishing of coherence
is due to uncontrolled heating, we have added a second stage in the
experimental protocol in Fig. 3(b), where $\Delta V$ is ramped up again to the
initial value $\Delta V=0.439\,V_{0}$. If this ramp is processed
adiabatically, we observe a revival of the condensate fraction, as is seen in
the right image in Fig. 3(a). Figure 3(d), shows that in fact the recovery of
the condensed fraction at the $X_{\pm}$ points requires a sufficiently long
ramp time $t_{{\rm up}}$ to satisfy adiabaticity. Maximum coherence is found
for $t_{{\rm up}}$ around $20\,$ms. The total number of atoms in the first and
second band remains nearly constant throughout the measurement (red markers in
Fig. 3(c) and (d)). The observed recovery of the condensate fraction appears
incomplete, possibly due to residual band relaxation at $\Delta V=0.063V_{0}$
and associated heating processes. These findings indicate that the observed
loss of coherence, when tuning to $\Delta V=0.063V_{0}$, could be interpreted
as a signature of a glassy phase originating from massive degeneracy of the
second band. A similar protocol was used in Ref. Gre:02  to discriminate a
Mott insulation phase from a thermally incoherent phase.

Conclusion. – We proposed a new method, relying on a lattice distortion, for
realizing frustrated models in an optical lattice system. Our proposal does
not rely on any driving as in earlier schemes, but the desired anti-
ferromagnetic tunneling terms result naturally by considering the first
excited band. The method was experimentally demonstrated by loading a gas of
ultracold atoms to this second band, and the condensate coherence fraction was
measured upon tuning from a non-frustrated to a frustrated regime and then
back. In the frustrated regime no visible signatures of coherence were
observed, in agreement with numerical simulations of the frustrated system,
while coherence was reestablished after ramping back to the non-frustrated
regime. In a strict sense, this is no direct evidence of frustration, which
would require different experimental tools, e.g. single-site pair-correlation
measurements could give more insight into possible glass and liquid phases.
Nevertheless, our work is a first step realizing systematic experimental
studies of frustrated matter.

###### Acknowledgements.

We thank Axel Gagge and Themistoklis Mavrogordatos for helpful discussions. We
acknowledge financial support from the Knut and Alice Wallenberg foundation
and the Swedish research council (VR). We acknowledge support from the
Deutsche Forschungsgemeinschaft (DFG) through the collaborative research
center SFB 925 (Project No. 170620586, C1).

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