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Tensor Network Methods in Many-Body Physics Andras Molnar

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Tensor Network Methods in Many-Body Physics Andras Molnar Tensor Network Methods in Many-body physics Andras Molnar Ludwig-Maximilians-Universit¨atM¨unchen Max-Planck-Institut f¨urQuanutenoptik M¨unchen2019 Tensor Network Methods in Many-body physics Andras Molnar Dissertation an der Fakult¨atf¨urPhysik der Ludwig{Maximilians{Universit¨at M¨unchen vorgelegt von Andras Molnar aus Budapest M¨unchen, den 25/02/2019 Erstgutachter: Prof. Dr. Jan von Delft Zweitgutachter: Prof. Dr. J. Ignacio Cirac Tag der m¨undlichen Pr¨ufung:3. Mai 2019 Abstract Strongly correlated systems exhibit phenomena { such as high-TC superconductivity or the fractional quantum Hall effect { that are not explicable by classical and semi-classical methods. Moreover, due to the exponential scaling of the associated Hilbert space, solving the proposed model Hamiltonians by brute-force numerical methods is bound to fail. Thus, it is important to develop novel numerical and analytical methods that can explain the physics in this regime. Tensor Network states are quantum many-body states that help to overcome some of these difficulties by defining a family of states that depend only on a small number of parameters. Their use is twofold: they are used as variational ansatzes in numerical algorithms as well as providing a framework to represent a large class of exactly solvable models that are believed to represent all possible phases of matter. The present thesis investigates mathematical properties of these states thus deepening the understanding of how and why Tensor Networks are suitable for the description of quantum many-body systems. It is believed that tensor networks can represent ground states of local Hamiltonians, but how good is this representation? This question is of fundamental importance as variational algorithms based on tensor networks can only perform well if any ground state can be approximated efficiently in such a way. While any state can be written as a tensor network state, the number of parameters needed for the description might be too large. This is not the case for one-dimensional systems: only a few parameters are required to have a good approximation of their ground states; that, in turn, allows for numerical algorithms based on tensor networks performing well. The situation in two dimensions is somewhat more complicated, but it is known that ground states of local Hamiltonians can be expressed as tensor networks with sub-exponentially many parameters. In the present thesis, we improve on these existing bounds strengthening the claim that the language of tensor networks is suitable to describe many-body systems. Another central question is how symmetries of the system such as translational in- variance, time-reversal symmetry or local unitary symmetry can be reflected in tensor networks. This question is important as systems appearing in nature might intrinsically possess certain symmetries; on one hand, understanding these symmetries simplifies the description of these systems. On the other hand, the presence of symmetries leads to the appearance of novel phases { symmetry-protected topological (SPT) order, { and tensor networks provide the right language to classify these phases. In one dimension and for vi Abstract certain classes of two-dimensional tensor networks (states generated by so-called injective tensors) it is well understood how symmetries of the state can be described. A general framework, however, has yet to be developed. In the present thesis, we contribute to the development of the theory in two ways. We first investigate the question for injective ten- sors, and generalize the existing proof for any geometry including the hyperbolic geometry used in the AdS/CFT correspondence. Second, we introduce a class of tensor network states that include previously known examples of states exhibiting SPT order. We show how symmetries are reflected in these states thus deepening the understanding of SPT order in two dimensions. Zusammenfassung Stark korrelierte Systeme zeigen Ph¨anomenewie Hochtemperatursupraleitung oder den Quanten-Hall-Effekt, die mit klassischen und semiklassischen Methoden nicht erkl¨arbar sind. Da die Dimension des zugrundeliegenden Hilbertraums exponentiell mit der Gr¨oße des Systems w¨achst, versagen viele der traditionellen Ans¨atzef¨urderartige Systeme. Es ist daher notwendig, neuartige numerische und analytische Methoden zu entwickeln, die die Physik in diesem Bereich erkl¨arenk¨onnen. Tensor-Netzwerkzust¨andek¨onnen diese Schwierigkeiten zum Teil ¨uberwinden, indem sie eine Familie von Zust¨andendefinieren, die nur von einer kleinen Anzahl von Param- etern abh¨angen. Diese Zust¨andetragen auf zwei Arten zur L¨osungdes Problems bei: Erstens werden sie als Variationsansatz in numerischen Algorithmen verwendet. Zweit- ens bieten sie einen analytischen Zugang zu einer großen Klasse genau l¨osbarerModelle, von denen angenommen wird, dass sie alle m¨oglichen Materiephasen repr¨asentieren. In der vorliegenden Arbeit werden mathematische Eigenschaften dieser Zust¨andeuntersucht, wodurch das Verst¨andnisdaf¨ur, wie und warum Tensor-Netzwerke f¨urdie Beschreibung von Quantensystemen geeignet sind, vertieft wird. Zun¨achst widmen wir uns der Frage, inwiefern Tensornetzwerke Grundzust¨andelokaler Hamiltonians darstellen k¨onnen. Diese Frage ist von grundlegender Bedeutung, da Varia- tionsalgorithmen, die auf Tensornetzwerken basieren, nur dann akkurate Ergebnisse liefern k¨onnen,wenn der Grundzustand nicht allzu weit von der zugrundeliegenden variationellen Mannigfaltigkeit entfernt ist. Zwar kann prinzipiell jeder Quantenzustand als Tensor- netzwerkstatus beschrieben werden. Jedoch ist die Anzahl der f¨urdie Beschreibung er- forderlichen Parameter m¨oglicherweise extrem groß. Dies ist bei eindimensionalen Sys- temen nicht der Fall: Nur wenige Parameter sind erforderlich, um eine gute N¨aherung ihrer Grundzust¨andezu erhalten. Aufgrund dieser theoretische Grundlage kann darauf vertraut werden, dass die Ergebnisse der tensornetzwerkbasierten Algorithmen akkurat sind. Die Situation in zwei Dimensionen ist komplizierter, aber es ist bekannt, dass Grundzust¨andelokaler Hamiltonians als Tensornetzwerke mit subexponentiell vielen Pa- rametern ausgedr¨uckt werden k¨onnen. In der vorliegenden Arbeit verbessern wir diese bestehenden Grenzen und verst¨arken die Behauptung, dass Tensornetzwerke geeignet ist, Vielteilchensysteme zu beschreiben. Eine weitere zentrale Frage ist, wie Symmetrien des Systems wie Translationsinvari- anz, Zeitumkehrsymmetrie oder lokale Symmetrie in Tensornetzwerken reflektiert werden k¨onnen. Das Verst¨andnisdieser Symmetrien vereinfacht einerseits die Beschreibung der viii Zusammenfassung Systeme, in denen diese Symmetrien auftreten. Auf der anderen Seite f¨uhrtdas Vorhan- densein von Symmetrien zum Entstehen neuer Phasen - sogenannter \symmetry protected topological phases" (SPT) -, und Tensornetzwerke liefern die richtige Beschreibung, um diese Phasen zu klassifizieren. In einer Dimension und f¨urbestimmte Klassen von zweidi- mensionalen Tensornetzwerken (Zust¨ande,die von sogenannten injektiven Tensoren erzeugt werden) ist es gut verstanden, wie Symmetrien des physikalischen System sich in ihrer Beschreibung als Tensornetzwerk widerspiegeln. Ein allgemeiner Rahmen muss jedoch noch entwickelt werden. In der vorliegenden Arbeit tragen wir auf zweierlei Weise zur Weiterentwicklung der Theorie bei. Wir untersuchen zun¨achst die Frage nach injektiven Tensoren und verallgemeinern den vorhandenen Beweis f¨urjede Geometrie, einschließlich der in der AdS / CFT-Korrespondenz verwendeten hyperbolischen Geometrie. Zweitens f¨uhrenwir eine Klasse von Tensornetzwerkzust¨andenein, die bereits bekannte Beispiele f¨urZust¨andemit SPT-Ordnung enthalten. Wir zeigen, wie sich Symmetrien in diesen Zust¨andenwiderspiegeln, wodurch das Verst¨andnisder SPT-Ordnung in zwei Dimensio- nen vertieft wird. List of publications First author publications: 1. Andras Molnar, Norbert Schuch, Frank Verstraete, and J. Ignacio Cirac. \Approx- imating Gibbs states of local Hamiltonians efficiently with projected entangled pair states". In: Phys. Rev. B 91.4, 045138 (4 Jan. 2015), p. 045138. arXiv: 1406.2973 2. Andras Molnar, Yimin Ge, Norbert Schuch, and J. Ignacio Cirac. \A generalization of the injectivity condition for Projected Entangled Pair States". In: J. Math. Phys. 59, 021902 (2018) 59.2, 021902 (June 22, 2017), p. 021902. arXiv: 1706.07329v1 3. Andras Molnar, Jos´eGarre-Rubio, David P´erez-Garc´ıa,Norbert Schuch, and J. Ig- nacio Cirac. \Normal projected entangled pair states generating the same state". In: New J. Phys. 20, 113017 (Nov. 2018), p. 113017. arXiv: 1804.04964 Non-first author publications: 1. Yimin Ge, Andr´asMoln´ar,and J. Ignacio Cirac. \Rapid Adiabatic Preparation of Injective Projected Entangled Pair States and Gibbs States". In: Phys. Rev. Lett. 116.8 (Feb. 2016), arXiv:1508.00570. arXiv: 1508.00570 2. Ilya Kull, Andras Molnar, Erez Zohar, and J. Ignacio Cirac. “Classification of Matrix Product States with a Local (Gauge) Symmetry". In: Annals of Physics, Volume 386, November 2017, Pages 199-241 386 (Aug. 1, 2017), pp. 199{241. arXiv: 1708.00362v2 3. G. Scarpa et al. \Computational complexity of PEPS zero testing". In: ArXiv e- prints (Feb. 22, 2018). arXiv: 1802.08214 4. David Sauerwein, Andras Molnar, J. Ignacio Cirac, and Barbara Kraus. \Matrix Product States: Entanglement, symmetries, and state transformations". In: arXiv e-prints, arXiv:1901.07448 (Jan. 2019), arXiv:1901.07448. arXiv: 1901.07448 x Table of contents Contents Abstract v Zusammenfassung vii List of publications
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