LETTERS
PUBLISHED ONLINE: 2 MAY 2016 | DOI: 10.1038/NPHYS3743
Quantum phase transitions with parity-symmetry
breaking and hysteresis
A. Trenkwalder
1
, G. Spagnolli
2
, G. Semeghini
2
, S. Coop
2,3
, M. Landini
1
, P. Castilho
1,4
, L. Pezzè
1,2,5
,
G. Modugno
2
, M. Inguscio
1,2
, A. Smerzi
1,2,5
and M. Fattori
1,2
*
Symmetry-breaking quantum phase transitions play a key role
in several condensed matter, cosmology and nuclear physics
theoretical models
1–3
. Its observation in real systems is often
hampered by finite temperatures and limited control of the
system parameters. In this work we report, for the first
time, the experimental observation of the full quantum phase
diagram across a transition where the spatial parity symmetry
is broken. Our system consists of an ultracold gas with
tunable attractive interactions trapped in a spatially symmetric
double-well potential. At a critical value of the interaction
strength, we observe a continuous quantum phase transition
where the gas spontaneously localizes in one well or the
other, thus breaking the underlying symmetry of the system.
Furthermore, we show the robustness of the asymmetric
state against controlled energy mismatch between the two
wells. This is the result of hysteresis associated with an
additional discontinuous quantum phase transition that we
fully characterize. Our results pave the way to the study
of quantum critical phenomena at finite temperature
4
, the
investigation of macroscopic quantum tunnelling of the order
parameter in the hysteretic regime and the production of
strongly quantum entangled states at critical points
5
.
Parity is a fundamental discrete symmetry of nature
6
conserved
by gravitational, electromagnetic and strong interactions
7
. It states
the invariance of a physical phenomenon under mirror reflection.
Our world is pervaded by robust discrete asymmetries, spanning
from the imbalance of matter and antimatter to the homo-chirality
of DNA of all living organisms
8
. Their origin and stability is
a subject of active debate. Quantum mechanics predicts that
asymmetric states can be the result of phase transitions occurring
at zero temperature, named in the literature as quantum phase
transitions (QPTs)
1,4
. The breaking of a discrete symmetry via a
QPT provides also asymmetric states that are particularly robust
against external perturbations. Indeed, the order parameter of a
continuous-symmetry-breaking QPT can freely (with no energy
cost) wander along the valley of a ‘mexican hat’ Ginzburg–Landau
potential (GLP) by coupling with gapless Goldstone modes
9
. In
contrast, the order parameter of discrete-symmetry-breaking QPTs
is governed by a one-dimensional double-well GLP
10
. The reduced
dimensionality suppresses Goldstone excitations, and the order
parameter can remain trapped at the bottom of one of the two wells.
This provides a robust hysteresis associated with a first-order QPT.
Evidence of parity-symmetry breaking has been reported in
relativistic heavy-ions collisions
11
and in engineered photonic
crystal fibres
12
. Observation of parity-symmetry breaking in a QPT
has been reported for neutral atoms coupled to a high-finesse optical
cavity
13
. However, this is a strongly dissipative system, with no direct
access to the symmetry-breaking mechanism necessary to study the
robustness of asymmetric states. In addition, previous theoretical
studies
14,15
have interpreted the puzzling spectral properties of a gas
of pyramidal molecules that date back to the 1950s (ref. 16), in terms
of the occurrence of a QPT with parity-symmetry breaking.
In the present work we report the observation of the full
phase diagram across a QPT where the spatial parity symmetry
is broken. Our system consists of ultracold atoms trapped in a
double-well potential
17,18
where the tunable strength of the attractive
interparticle interaction is the control parameter of the transition.
Additional control of the energy mismatch between the two wells
allows driving of discontinuous first-order QPTs in the non-
symmetric ordered part of the phase diagram and observation of
an associated hysteretic behaviour.
In our system, the atomic ground state depends on two
competing energy terms in the Hamiltonian H = H
a
+ gH
b
, where
H
a
=
dr
†
(r)[−(
¯
h
2
/2m)∇
2
+ V (r)](r) includes kinetic and
potential energy, and H
b
= (2π
¯
h
2
a
0
/m)
dr
†
(r)
†
(r)(r)(r)
accounts for contact interaction between the atoms. Here, (r)
is the many-body wavefunction,
†
(r) its hermitian conjugate (in
the following we consider normalization 〈
†
(r)(r)〉= 1), m the
atomic mass, a
0
is the Bohr radius,
¯
h the reduced Planck constant
and V (r) is a double-well trapping potential in the x direction
(see Fig. 1a) and a harmonic trap in the orthogonal plane. The
adimensional control parameter g = Na
s
/a
0
< 0 is the product of
the total number of atoms N and the scattering length a
s
< 0
characterizing the interatomic attractive interaction. The full many-
body Hamiltonian is invariant under x ↔−x mirror reflection.
This parity symmetry imposes a spatially symmetric ground state
for any value of the control parameter g . Because H
a
and H
b
do
not commute, the corresponding ground states are quite different.
H
a
is minimized by each atom equally spreading on both wells. A
finite energy gap, specified as the tunnelling energy J , separates
the ground and the first (antisymmetric) excited state of H
a
. J
can be tuned by controlling the height of the potential barrier
between the two spatial wells. In contrast, gH
b
is minimized by a
linear combination of two degenerate states, one having all atoms
localized in one well, the second with all atoms localized in the
other well. Thanks to the competition in the Hamiltonian between
the effective repulsion due to the kinetic energy and the attractive
interatomic interaction, the energy gap between the two low-lying
© Macmillan Publishers Limited . All rights reserved
1
Istituto Nazionale di Ottica-CNR, 50019 Sesto Fiorentino, Italy.
2
LENS European Laboratory for Nonlinear Spectroscopy, and Dipartimento di Fisica e
Astronomia, Università di Firenze, 50019 Sesto Fiorentino, Italy.
3
ICFO-Institut de Ciencies Fotoniques, Barcelona Institute of Science and Technology,
08860 Castelldefels (Barcelona), Spain.
4
Instituto de Física de São Carlos, Universidade de São Paulo, C.P. 369, 13560-970 São Carlos, São Paulo, Brazil.
5
Quantum Science and Technology in Arcetri, QSTAR, 50125 Firenze, Italy. *e-mail: fattori@lens.unifi.it
826 NATURE PHYSICS | VOL 12 | SEPTEMBER 2016 | www.nature.com/naturephysics