Tensor Networks in Condensed Matter [Elektronische Ressource] / Mikel Sanz Ruiz. Gutachter: Ignacio Cirac ; Alejandro Ibarra. Betreuer: Ignacio Cirac

Mikel Sanz

Sujets

• Mathematics

Informations

• Publié par: technische_universitat_munchen
• Publié le: 01 janvier 2011
• Nombre de lectures: 84
• Langue: Deutsch
• Poids de l'ouvrage: 10 Mo

Extrait

Technische Universitat Munchen
Max-Planck{Institut fur Quantenoptik
Tensor Networks in
Condensed Matter
Mikel Sanz Ruiz
Vollst andiger Abdruck der von der Fakult at fur Physik
der Technischen Universit at Munc hen
zur Erlangung des akademischen Grades eines
Doktors der Naturwissenschaften (Dr. rer. nat.)
genehmigten Dissertation.
Vorsitzender: Univ.-Prof. J. J. Finley, Ph. D.
Prufer der Dissertation: 1. Hon.-Prof. I. Cirac, Ph. D.
2. Univ.-Prof. Dr. A. Ibarra
Die Dissertation wurde am 21.03.2011
bei der Technischen Universit at Munc hen eingereicht
und durch die Fakult at fur Physik am 20.04.2011 angenommen.Nothing produced such odd results
as trying to get even.
Franklin P. Jones (1908-1980)Zusammenfassung
Diese Doktorarbeit ist dem Studium der Quantenvielteichentheorie gewidmet. Die
dazugeh orige Forschung wurde im Rahmen der Matrix Product States (MPS) und
deren Generalisierung in h oheren Dimensionen, Projected Entangled Pair States
(PEPS) durchgefuhrt. Im ersten Teil dieser Arbeit beschaftigen wir uns n aher
mit den mathematischen Eigenschaften solcher Tensornetzwerkzust ande. Die ersten
Kapitel behandeln eindimensionale Systeme. Dabei nutzen wir die Verbindungen
zwischen MPS und Quantenkan alen um mehrere neue Ergebnisse, wie zum Beispiel
die Quanten-Wielandt-Ungleichung oder die Konditionen fur die Konstruktion lokal-
invarianter Zust ande unter einer Symmetriegruppe zu beweisen. Kapitel 5-7 sind
h oherdimensionalen Systemen gewidmet. Hier de nieren wir die Bedingungen fur
die Herstellung invarianter PEPS . Im zweiten Teil der Arbeit gehen wir auf die
Anwendung in kondensierter Materie ein. Das achte Kapitel zeigt mehrere Meth-
oden quasi-l osbare Hamiltonoperatoren mit Zweik orperwechselwirkung und SU(2)-
Symmetrie zu konstruieren, w ahrend wir im neunten Kapitel darlegen, dass diese
Tensornetzwerkzust ande als Labor fur die Kondensierte-Materie-Theorie genutzt
werden k onnen: Zum Beispiel in der Charakterisierung der String-Order, der Ver-
allgemeinerung des Lieb-Schultz-Mattis-Theorems, dem Beweis neuer Theoreme die
versuchen die Quantenverschr ankung mit Magnetisierung oder mit Langstreckenin-
teraktion in Verbindung zu setzen und vielem mehr.
This thesis is devoted to the study of quantum many-body systems. This inves-
tigation is performed in the framework of Matrix Product States (MPS) and their
generalization to higher dimensions, Projected Entangled Pair States (PEPS). In
the rst part of the work, we discuss the mathematical properties of such tensor
network states in depth. In the rst chapters we deal with one-dimensional systems,
for which we use the connections between MPS and completely positive maps to
prove several new results, such as the quantum version of the Wielandt’s inequality
or the construction of locally invariant states under a symmetry group. Chapters 5-7
are dedicated to higher-dimensional systems, for which we provide the conditions to
construct invariant PEPS. The second part of the thesis is dedicated to applications
in condensed matter. In chapter 8 we provide several methods of constructing quasi-
solvable Hamiltonians with two-body interactions, while in chapter 9 we show that
these tensor network states can be used as a laboratory for theoretical condensed
matter in, for instance, the characterization of the string-order, the generalization
of the Lieb-Schultz-Mattis theorem, the demonstration of new theorems relating
entanglement to magnetisation or to long-range interactions, etc.
iiiAbstract
This Thesis contributes to the development of the theory of tensor network states
in many{body systems. The rst part is dedicated to improving our comprehension
of the mathematical properties of such states; in the second part, these properties
are employed to obtain results in condensed matter and quantum magnetism. The
cornerstone of these advances is the possibility of a local characterization of global
features. Properties such as the uniqueness of the ground state, the existence of
a non{vanishing spectral gap above this ground state, or the characterization of
symmetries are encoded in the tensor. This is based on the connection between
Matrix Product States and completely positive maps in one{dimensional systems.
For higher{dimensional networks, where such a connection does not exist, one can
attempt to transform the question into the former one{dimensional problem.
We prove a quantum generalization of the Wielandt’s inequality, well{known in
the context of classical channels and Markov chains. Thisy provides an
upper bound for the number of spins which must be gathered in order to nd a
gapped parent Hamiltonian which has the Matrix Product State as a unique ground
state. The bound, surprisingly, depends only on the tensor’s physical and bond
dimensions, and not on the explicit entries of the tensor. Many previous results on
Matrix Product States depended on the existence of this upper bound.
With this in hand, we provide a local characterization of the symmetries in
both Matrix Product States and Projected Entangled Pair States. As almost every
interesting Hamiltonian in condensed matter exhibits some kind of symmetry, to
be able to locally characterize these symmetries in the tensors is a key question.
Furthermore, we employ this characterization to systematically construct quasi{
solvable SU(2){invariant Hamiltonians with two{body interactions.
Finally, we apply the advances achieved in Matrix Product States’ and Pro-
jected Entangled Pair States’ theories to characterize the existence of string order
in one{dimensional systems (proposing as well an extension to higher dimensions),
to provide generalizations of existing theorems in the context of tensor networks,
and to exploit the simplicity of the structure of these states in order to make use of
them as a theoretical laboratory for condensed matter. An example of this would be
the proved relationship between fractional magnetization and entanglement, or the
proposed one which links long{range interacting Hamiltonians to the entanglement
of their ground states.
vContents
Introduction 1
1 Fundamentals of MPS 9
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2 Construction of Matrix Product States . . . . . . . . . . . . . . . . . 11
1.2.1 A constructive de nition . . . . . . . . . . . . . . . . . . . . . 11
1.3 Canonical form for Matrix Product States . . . . . . . . . . . . . . . 13
1.3.1 OBC{MPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3.2 PBC{MPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.4 Finitely correlated states . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.4.1 A bit about quantum channels . . . . . . . . . . . . . . . . . . 23
1.4.2 Finitely correlated states . . . . . . . . . . . . . . . . . . . . . 27
1.4.3 Reduced density matrix and expectation values . . . . . . . . 31
2 Injectivity 37
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.2 Injectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.3 Strong irreducibility . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.4 Classical channels and Wielandt’s inequality . . . . . . . . . . . . . . 45
2.5 Quantum Wielandt’s inequality . . . . . . . . . . . . . . . . . . . . . 48
2.5.1 Primitivity, full Kraus rank and strong irreducibility . . . . . 49
2.5.2 Quantum Wielandt’s inequality . . . . . . . . . . . . . . . . . 51
2.5.3 An application: zero-error capacity . . . . . . . . . . . . . . . 55
3 Parent Hamiltonians 57
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.2 De nition of Parent Hamiltonian . . . . . . . . . . . . . . . . . . . . 58
3.3 Uniqueness of the ground state . . . . . . . . . . . . . . . . . . . . . 62
3.4 Gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.5 Kinsfolk Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
viiCONTENTS
4 Symmetries in MPS 69
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.2 De nition and characterization . . . . . . . . . . . . . . . . . . . . . 70
4.3 Uniqueness of the construction method . . . . . . . . . . . . . . . . . 73
4.3.1 The case of SU(2) . . . . . . . . . . . . . . . . . . . . . . . . 75
4.4 Irreducibility implies injectivity . . . . . . . . . . . . . . . . . . . . . 76
5 Fundamentals of PEPS 79
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.2 Constructing Projected Entangled Pair States . . . . . . . . . . . . . 80
5.2.1 Regular 2D lattices . . . . . . . . . . . . . . . . . . . . . . . . 82
5.3 A canonical form for PEPS . . . . . . . . . . . . . . . . . . . . . . . . 84
5.3.1 Canonical form for MPS: improvement . . . . . . . . . . . . . 84
5.3.2 form for PEPS . . . . . . . . . . . . . . . . . . . . 87
6 Injectivity and parent Hamiltonians in PEPS 99
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.2 Injectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.3 Parent Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.3.1 De nition of parent Hamiltonian . . . . . . . . . . . . . . . . 103
6.3.2 Uniqueness of the ground state . . . . . . . . . . . .