Equilibrium and Nonequilibrium Aspects of Phase Transitions in Quantum Physics

216 Pages·2018·5.214 MB·English

by  Ricardo Puebla

Springer Theses
Recognizing Outstanding Ph.D. Research
Ricardo Puebla

Equilibrium and
Nonequilibrium
Aspects of Phase
Transitions in
Quantum Physics
Springer Theses
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Ricardo Puebla
Equilibrium
and Nonequilibrium Aspects
of Phase Transitions
in Quantum Physics
Doctoral Thesis accepted by
ä
the Universit t Ulm, Ulm, Germany
123
Author Supervisor
Dr. RicardoPuebla Prof. Dr. MartinB. Plenio
Centrefor Theoretical Atomic, Institut für Theoretische Physik
Molecular andOptical Physics andIQST
Queen’s University Belfast UniversitätUlm
Belfast, UK Ulm, Germany
Tagder Promotion(date of the Ph.D. defense): 13.03.2018
ISSN 2190-5053 ISSN 2190-5061 (electronic)
SpringerTheses
ISBN978-3-030-00652-5 ISBN978-3-030-00653-2 (eBook)
https://doi.org/10.1007/978-3-030-00653-2
LibraryofCongressControlNumber:2018955173
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’
Supervisor s Foreword
Phasetransitionsareanexperienceofeverydaylifethathasfascinatedscientistsfor
centuries. The quest to understand them has led to many Nobel Prize-worthy
developments ranging from the renormalization group in statistical mechanics to
fundamentalaspectsofquantumfieldtheory.Freezingandboilingwaterconstitute
simple examples of first-order phase transitions for which the density, a first
derivative of the free energy, changes discontinuously. But these are by no means
the only phase transitions. Indeed, in many respects, more interesting are
second-orderphasetransitionssuchastheferromagneticphasetransition.Here,the
magnetization increases continuously from zero as the temperature is lowered
below the transition temperature, and it is the magnetic susceptibility, the second
derivative of the free energy, that changes discontinuously. Second-order phase
transitions are particularly interesting as they exhibit the remarkable phenomenon
of spontaneous symmetry breaking. When there is no external magnetic field and
the setting is therefore completely symmetric, the ferromagnetic material can
becomemagnetizedinoneoftwopossible directionswhicharechosenatrandom;
i.e., in each repetition of the experiment, there may be a new direction of magne-
tization. Although the physics was completely symmetric, the resulting state isnot
andwespeakofaspontaneouslybrokensymmetry.Themechanismofspontaneous
symmetry breaking underlies, for example, the theory of the Higgs boson in the
standard model.
In quantum mechanics, a second-order quantum phase transition in a quantum
systemdescribesanabruptchangeintheground-statepropertiesuponachangeofa
system parameter and separates quantum phases of matter, whose physical prop-
ertiescannotbeanalyticallyconnected.Accordingtothetraditionalunderstanding,
the emergence of phase transitions can be explained only by invoking the concept
ofthethermodynamiclimitofinfinitelymanysystemcomponents.Indeed,upuntil
recently, it has been widely assumed that a phase transition can only occur in this
limit. The reason for this assumption can be understood from the consideration
of the partition function. A partition function can be represented as a sum of
exponential functions, and the number of terms to be summed over is equal to the
dimension of the corresponding Hilbert space. As the sum of finite number of
v
vi Supervisor’sForeword
exponentialfunctions,whichareindividuallyanalyticfunctions,canneverbecome
non-analytic,thepossibilityofanynon-analyticityinthepartitionfunctionandthus
the possibility of phase transition only arise when this sum is over infinitely many
terms. This simple observation rules out the existence of a phase transition for
quantum systems where each system component has a finite Hilbert space
dimension, such as in the case of spin systems. In the folklore of the research
community that has been concentrated, not correctly as we will see, in the belief
that only infinitely extended systems can exhibit a phase transition.
Inhisdissertation,RicardoPueblashowsthatthisbeliefisnottrue.Indeed,ina
seminalcontribution,heshowsthatthisno-gotypeofprincipledoesnotapplytoa
system composed of unbounded quantum systems residing in an infinite dimen-
sionalHilbertspacesuchasaharmonicoscillator.Inthiscase,thepartitionfunction
is a sum over an infinite number of exponential functions and therefore the pos-
sibilityofnon-analyticbehaviorisnotnecessarilyruledoutbythefinitenumberof
systemcomponents.Buildingonthisinsight,hethenproceedstoconstructexplicit
examples of quantum phase transitions and critical phenomena in a minimal
quantum system, comprising a single two-level system and a bosonic mode. He
shows that in a suitable limit of the system’s parameters rather than in the con-
ventional thermodynamic limit in the particle number, this system exhibits all the
phenomenology of phase transitions. This novel notion, dubbed finite-component
size quantum phase transition, challenges the well-established idea of phase tran-
sitionswhichemergeduetoaninfinitenumberofdegreesoffreedom.Remarkably,
it is shown that by investigating and exploring systems featuring this novel phase
transition, which do not require large number of constituents, one can learn about
the critical traits of a truly quantum many-body system—typically more compli-
catedtorealizeexperimentally—inthesamemannerasonestudiessimplifiedIsing
modelstounderstandmorecomplexsystemswhichbelongtothesameuniversality
class. Ricardo does not only show that the ground-state properties exhibit a phase
transition but also extends these observations to time-dependent phenomena.
Indeed, when traversing a symmetry breaking second-order phase transition at a
finiterate,different partsofthesystemcannotinteractfastenoughandwillchoose
their symmetry broken phase at random and independent of each other. Hence,
different spatial regions may choose different symmetries, and the boundary
betweentwosuchregionsformsadomainwalloradefect.Inthetraditionalpicture
of phase transitions, remarkably, the number of these domains is a function of the
rate at which one traverses the phase transition with exponents that are, somewhat
surprisingly, determined by the equilibrium properties ofthe phase transition. This
is the celebrated Kibble–Zurek scaling, and it was by no means clear that this
phenomenology would also be realized in the finite system phase transition, but
Ricardo was able to show this too by assuming that quantum excitations play the
role of defects. Beyond the actual conceptual novelty, a detailed analysis of a
potential experimental realization unveils its feasibility based on dynamical
decoupling techniques to cope with different and realistic noise sources.
Experiments of this type are now underway with laser-cooled trapped ions.
Supervisor’sForeword vii
This thesis not only offers a detailed account for the aforementioned scientific
results, but also places them into context by reviewing previous findings and dis-
cussingprospectsdirectionsforthefuturework.Inaddition,theintroductionfirms
down the necessary general concepts for the understanding of the subsequent
chapters and discussion. This thesis is therefore an excellent read for those inter-
estedintherealmofcriticalphenomenaandKibble–Zurekphysicsaswellasinthe
novel notion offinite-component system quantum phase transitions.
Ulm, Germany Prof. Dr. Martin B. Plenio
July 2018
Abstract
Phase transitions are inherent to many-body systems and constitute a paradigmatic
example of an emergent phenomenon, which stems from an infinite number of
microscopic constituents. Different phases of matter, that are characterized by
having distinct macroscopic properties, are separated by critical values of certain
externalmagnitudesat whicha change of phase takes place, namelywhen aphase
transition occurs. As matter varies its collective properties in many and disparate
manners,phasetransitionsorcriticalphenomenaappearalsointhequantumrealm.
Inthisregard,ourunderstandingandthesubsequentexploitationofdistinctphases
of matter crucially rely on the comprehension of phase transitions and of their
underlying mechanisms. Continuous phase transitions deserve special attention as
they allow to investigate same physics across systems that are a priori utterly
distinct. This remarkable property is known as universality, and it provides a
powerfultoolfortheinspectionofcritical phenomenabyanalyzingsimplemodels
that reproduce the main traits of more complex systems. However, while these
notions are well established in a static or equilibrium case, much less it is known
about how universal features emerge or are inherited in nonequilibrium scenarios.
In this context, we find the Kibble–Zurek mechanism, a theory aiming to explain
defect formation promoted by finite-rate ramps across a phase transition. This
mechanism establishes a relation between universal equilibrium critical exponents
and nonequilibrium features of the dynamics involving a phase transition. In this
thesis, we analyze continuous phase transitions in different low-dimensional sys-
tems, addressing their equilibrium and nonequilibrium aspects, as well as paying
special attention not only to the Kibble–Zurek mechanism but also to other uni-
versal traits of the dynamics. Moreover, we show that a model with only two
constituent elements undergoes a phase transition in a suitable parameter limit
rather than in the conventional thermodynamic limit and thus challenging the
common notion about phase transitions.
The Kibble–Zurek mechanism evidenced that universal traits also apply to
nonequilibrium scenarios. However, since Kibble–Zurek mechanism is based on
properties only valid in the thermodynamic limit, their examination and ultimate
verification in finite-size systems must be done prudently. Here, we discuss how
ix
x Abstract
universality of phase transitions can be attained and extended by exploiting and
combining finite-size scaling theory with a Kibble–Zurek problem. This is illus-
trated in a Ginzburg–Landau model as well as in a realistic linear to zigzag phase
transition in a Coulomb crystal, where in addition an approximate method is pro-
posed to gain insight into the dynamics. Furthermore, a brief exploration of a
feasible quantum many-body system suggests that the quantum Kibble–Zurek
mechanism may be observed within current ion-trap technology.
A relevant part of this thesis has been devoted to investigate the quantum Rabi
model,whichdescribestheinteractionbetweenaqubitandasinglebosonicmode.
Despite its apparent simplicity, it is demonstrated that the quantum Rabi model
undergoes a quantum phase transition when the ratio of qubit over bosonic fre-
quencybecomesinfinite,whilekeepingfinitethenumberofsystemconstituents—a
novel manner of attaining criticality that we have dubbed finite-component system
phase transition. In this limit, the exact effective low-energy solution reveals a
spontaneous symmetrybreakingandtheemergence ofground-statesingularitiesat
the critical point. Furthermore, for finite, yet large, ratio values, an equivalent
formulation of finite-size scaling theory can be applied to this quantum phase
transition.ThisallowsustodisclosethatthequantumRabimodelfallsinthesame
universality class as that of the Dicke model. We extend our study of this critical
system to a nonequilibrium scenario, where among another findings, we show that
Kibble–Zurek physics can actually be observed in a zero-dimensional system,
contrary to the common belief prior this work. Hence, the quantum Rabi model
serves as an excellent test bed for the examination of different physics related to
quantum phase transitions. We complete our analysis by proposing a potential
experimentalimplementationofthequantumRabimodelusingasingletrappedion
toprobethedynamicsofasuperradiantquantumphasetransition,whileatthesame
time coping with realistic noise sources.