Fakultät für Physik, Technische Universität München
Max-Planck-Institut für Quantenoptik
Tensor network states for the description of quantum many-body systems
Thorsten Bernd Wahl
Vollständiger Abdruck der von der Fakultät für Physik der Technischen Universität München zur Erlangung des akademischen Grades eines
Doktors der Naturwissenschaften
Vorsitzender: Univ.-Prof. Dr. Rudolf Gross
Prüfer der Dissertation:
1. Hon.-Prof. J. Ignacio Cirac, Ph.D.
2. Univ.-Prof. Dr. Michael Knap
3. Univ.-Prof. Dr. Ulrich Schollwöck,
(nur schriftliche Beurteilung)
Die Dissertation wurde am 11. Mai 2015 bei der Technischen Universität München eingereicht und durch die Fakultät für Physik am 3. August 2015 angenommen.
This thesis is divided into two mainly independent parts: The first one elaborates on the classification of all Matrix Product States (MPS) with long range localizable entanglement, where the second is devoted to free fermionic and interacting chiral Projected Entangled Pair States (PEPS).
In the first part, we derive a criterion to determine when a translationally invariant MPS has long range localizable entanglement, which indicates that the corresponding state has some kind of non-local hidden order. We give examples fulfilling this criterion and eventually use it to obtain all such MPS with bond dimension 2 and 3 (which is the dimension of the virtual indices of the MPS tensor).
In the second part, we show that PEPS in two spatial dimensions can describe chiral topological states by explicitly constructing a family of such states with a non-trivial Chern number, thus defying previous arguments which seemed to rule out their very existence. They turn out to have power law decaying correlations, i.e., the local parent Hamiltonians for which they are ground states, are gapless. We show that such parent Hamiltonians lie at quantum phase transition points between different topological phases.
Furthermore, we also construct long range Hamiltonians with a flat energy spectrum for which those PEPS are unique ground states. The gap is robust against local perturbations, which allows us to define a Chern number for the PEPS. We demonstrate that free fermionic PEPS (GFPEPS) must necessarily be non-injective and have gapless parent Hamiltonians. Moreover, we provide numerical evidence that they can nevertheless approximate well the physical properties of Chern insulators with local Hamiltonians at arbitrary temperatures.
As for non-chiral topological PEPS, the non-trivial, topological properties can be traced down to the existence of a symmetry on the virtual level of the PEPS tensor that is used to build the state. Based on that symmetry, we construct string-like operators acting on the virtual modes that can be continuously deformed without changing the state. On the torus, the symmetry implies that the ground state space of the local parent Hamiltonian is two-fold degenerate. By adding a string wrapping around the torus one can change one of the ground states into the other. We use the special properties of PEPS to build the boundary theory and show how the symmetry results in the appearance of chiral modes and in a universal correction to the area law for the zero Rényi entropy.
Finally, we show that PEPS can also describe chiral topologically ordered phases. For that, we construct a simple PEPS for spin-1/2 particles in a two-dimensional lattice. We reveal a symmetry in the local projector of the PEPS that gives rise to the global topological character. We also extract characteristic quantities of the edge Conformal Field Theory using the bulk-boundary correspondence.
Diese Dissertation ist in zwei weitgehend unabhängige Teile gegliedert: Der erste beschreibt die Klassifizierung aller Matrixproduktzustände (MPS) mit langreichweitiger lokalisierbarer Verschrän-kung, während der zweite frei fermionischen und wechselwirkenden chiralen projizierten verschränk-ten Paarzuständen (PEPS) gewidmet ist.
Im ersten Teil leiten wir ein Kriterium her, mit dem sich bestimmen lässt, ob ein MPS lang-reichweitige lokalisierbare Verschränkung besitzt, was eine Art verborgene nichtlokale Ordnung des entsprechenden Zustandes bedeutet. Wir liefern Beispiele, die dieses Kriterium erfüllen, und verwenden es letztendlich um alle solchen MPS mit Bindungsdimension 2 und 3 (die Dimension der virtuellen Indizes des MPS-Tensors) zu erhalten.
Im zweiten Teil zeigen wir, dass in zwei Dimensionen PEPS chirale topologische Zustände beschreiben können, indem wir explizit eine Familie solcher Zustände mit nichttrivialer Chernzahl konstruieren, was bisherige Argumente widerlegt, welche deren Existenz auszuschließen schienen. Es stellt sich heraus, dass deren Korrelationen einem inversen Potenzgesetz folgend abfallen, d.h. die lokalen Parent Hamiltonians, für welche sie Grundzustände sind, haben keine Energielücke. Wir zeigen, dass solche Parent Hamiltonians auf Quantenphasenübergangspunkten zwischen verschiedenen topologischen Phasen liegen.
Außerdem konstruieren wir langreichweitige Hamiltonians mit einem flachen Energiespektrum, für welche diese PEPS nichtentartete Grundzustände sind. Die Energielücke ist stabil gegen lokale Störungen, was es uns erlaubt eine Chernzahl für diese PEPS zu definieren. Wir demonstrieren, dass frei fermionische PEPS (GFPEPS) notwendigerweise nichtinjektiv sein müssen und Parent Hamiltonians mit einer verschwindenden Energielücke besitzen. Darüberhinaus liefern wir numerische Belege dafür, dass sie dennoch die physikalischen Eigenschaften von Chernisolatoren mit lokalen Hamiltonians bei beliebigen Temperaturen hinreichend genau approximieren können.
Wie bei nichtchiralen topologischen PEPS können die nichttrivialen, topologischen Eigenschaften auf die Existenz einer Symmetrie im virtuellen Raum des PEPS-Tensors zurückgeführt werden, aus dem der Zustand aufgebaut wird. Auf dieser Symmetrie basierend konstruieren wir stringartige Operatoren, die auf die virtuellen Moden wirken und kontinuierlich verformt werden können ohne dass sich der Zustand dabei ändert. Auf dem Torus hat die Symmetrie eine zweifache Entartung des Grundzustandsraumes des lokalen Parent Hamiltonians zur Folge. Wenn man einen String einfügt, der den Torus umwickelt, kann man zwischen den beiden Grundzuständen wechseln. Wir verwenden die speziellen Eigenschaften von PEPS um die Randtheorie abzuleiten und weisen auf, wie die Symmetrie zu chiralen Moden und einer universellen Korrektur zum Flächengesetz der nullten Rényi-Entropie führt
Schließlich zeigen wir, dass PEPS auch chirale topologisch geordnete Systeme beschreiben können. Dazu konstruieren wir einfache PEPS von Spin-1/2-Teilchen auf einem zweidimensionalen Gitter. Wir enthüllen eine Symmetrie des lokalen PEPS-Projektors, die den globalen topologischen Charakter bedingt. Außerdem leiten wir durch Verwendung der Volumen-Rand-Entsprechung be-zeichnende Größen der Konformen Feldtheorie des Randes ab.
Shuo Yang, Thorsten B. Wahl, Hong-Hao Tu, Norbert Schuch, and J. Ignacio Cirac, “Chiral projected entangled-pair states with topological order”, Phys. Rev. Lett. 114, 106803 (2015).
Thorsten B. Wahl, Stefan T. Haßler, Hong-Hao Tu, J. Ignacio Cirac, and Norbert Schuch, “Symmetries and boundary theories for chiral projected entangled pair states”, Phys. Rev. B 90, 115133 (2014).
Thorsten B. Wahl, Hong-Hao Tu, Norbert Schuch, and J. Ignacio Cirac, “Projected entangled-pair states can describe chiral topological states”, Phys. Rev. Lett. 111, 236805 (2013).
Thorsten B. Wahl, David Pérez-García, and J. Ignacio Cirac, “Matrix Product States with long-range Localizable Entanglement”, Phys. Rev. A 86, 062314 (2012).
All publications have been incorporated into this thesis.
- 1 Introduction
2 Matrix Product States with Long Range Localizable Entanglement
- 2.1 Introduction to Matrix Product States
- 2.2 Introduction to Localizable Entanglement
- 2.3 A criterion for Matrix Product States with Long Range Localizable Entanglement
3 Chiral Topological Projected Entangled Pair States
- 3.1 Introduction to Projected Entangled Pair States
- 3.2 Introduction to Chiral Topological Systems
- 3.3 Free Fermionic Chiral Projected Entangled Pair States
- 3.4 Interacting Chiral Projected Entangled Pair States
- 4 Conclusion and Perspectives
- A Proof that LRLE for implies LRLE for
- B Properties of free fermionic chiral Projected Entangled Pair States
- C Numerical implementation of fermionic Projected Entangled Pair States on a cylinder
EŒOLEONARDO DA VINCI
The state of a system of classical particles is completely characterized by the particles’ positions and velocities. However, at the atomic length scale, the motion of particles can no longer be described by classical mechanics, but instead the rules of quantum mechanics need to be taken into consideration. In quantum mechanics, the motion of particles is dictated by the Schrödinger equation, and the particles’ positions and momenta no longer have fixed values, but follow a certain probability distribution. Since for many particles joint probabilities need to be taken into consideration, the complexity of the Schrödinger equation grows exponentially with the number of particles. This makes the exact description of quantum many-body systems of very few particles already in general intractable. However, many fascinating phenomena in Condensed Matter Physics emerge entirely due to the interplay of many quantum particles and cannot be explained without the laws of quantum mechanics. Examples include superconductivity, Bose-Einstein condensation [1, 2] and the Quantum Hall Effect . That is why efficient approximation methods to the many-particle problem have been devised.
Many of the above phenomena can be explained by assuming the particles to be effectively non-interacting, such that the Schrödinger equation can be decoupled into equations for the individual particles. The solution to the latter yields an approximate description of the quantum many body system under consideration. However, apart from obtaining inaccuracies via this method, it simply fails to predict the properties induced by strong interactions between the particles, like the Fractional Quantum Hall Effect [4, 5]. For this reason, other methods had to be conceived, which do not rely on the non-interacting picture, such as Quantum Monte Carlo simulations .
Throughout this thesis, we will consider systems defined on a lattice, where on the lattice sites either immobile spins reside or electrons that are allowed to hop. Although this is already a strong simplification, such systems are believed to accurately describe certain materials, and, most importantly, also host the above phenomena observed in continuous Condensed Matter systems. In second quantization, such systems are described by diagonalizing the many-body Hamiltonian, which is again exponentially hard in the number of particles. Therefore, the system sizes amenable to exact diagonalization are often too small in order to investigate the thermodynamic limit. The exponentially huge Hilbert space (the space the many-body Hamiltonian acts on) seems to pose strong restrictions on the system sizes that can be tackled precisely. However, it turns out that the ground states of most relevant Hamiltonians are not randomly distributed in the exponentially large Hilbert space, but instead live on a subspace that only grows polynomially with the system size. This seems to be in particular the case for Hamiltonians with a gap to the first excited state in the thermodynamic limit and local interactions, that is, interactions whose amplitudes decay at least exponentially with distance. The reason is that such Hamiltonians fulfill the so-called area law [7, 8, 9], which states that for their ground states (), the entanglement entropy of the reduced density matrix of some simply connected region grows only as its surface and not as its volume if the size of the region is increased. These points are illustrated in Fig. 1.1.
Even more excitingly, the above-mentioned relevant subspace of the Hilbert space can be parameterized by Tensor Network States [10, 11, 12, 13, 14, 15] (see Fig. 1.2). They can be constructed in such a way that they automatically fulfill the area law. Each component of the wave function can in principle be obtained by carrying out a contraction of a tensor network. The highest dimension of the contracted tensor indices is referred to as the bond dimension. By increasing the bond dimension, the desired ground state can be approximated with higher accuracy. It is believed that in arbitrary dimensions, the bond dimension and thus the number of parameters grows only polynomially with the system size if the TNS is required to have a certain overlap with the ground state wave function of a local Hamiltonian. This has been proven rigorously in one dimension . Apparently, nature does not occupy the exponentially large Hilbert space she has at her disposal, but contents herself with a polynomially large subspace that is in principle describable for us!
Another way to grasp this huge reduction in the complexity of the ground states of realistic Hamiltonians is to count their number of parameters: Consider for instance a Hamiltonian defined for a system of particles with spin . The dimension of the Hilbert space is . However, any physically relevant Hamiltonian has at most four-particle interactions. The number of possible four-particle interactions scales as . Hence, the number of possible Hamiltonians, and therefore also of possible ground states is polynomial in the system size and not exponential - even without the assumption of local interactions. However, to the best of the author’s knowledge, only for the ground states of local Hamiltonians a suitable parameterization of the corresponding polynomial subspace has been provided.
Let us now summarize important classes of TNS: In one dimension, Matrix Product States (MPS)  have been shown to provide an efficient approximation of realistic local Hamiltonians [8, 9, 16]. By efficient we mean that the number of parameters required to obtain a fixed accuracy of the approximation of any observable grows at most polynomially with the systems size. MPS can be obtained by sequential generation  and are the variational class underlying the extremely successful Density Matrix Renormalization Group (DMRG) [18, 19, 20] algorithms. Note that MPS not only allow for an efficient representation of the ground states of local Hamiltonians, but it has also been shown that the best MPS with a given bond dimension can be found in a time that is polynomial in the system size [21, 22]. Moreover, once the best MPS representation has been found, local observables and few-particle correlations can be computed efficiently. MPS can also be applied to critical systems (that is, gapless Hamiltonians); however, their polynomial scaling is worse in this case and other tensor network ansätze have been devised to remedy this problem. The first one was the Tree Tensor Network  (TTN), which has a real space renormalization group structure. However, the tensors on the highest level of an optimized TTN contain information about the local entanglement on its lowest level. The resulting reduction in computational efficiency could be overcome by the Multiscale Entanglement Renormalization Ansatz (MERA) , which removes short range entanglement. Critical systems in one dimension display a logarithmic violation of the area law, i.e., the entanglement entropy does not saturate as a function of the length of region (as is the case for gapped local Hamiltonians), but scales like the logarithm of its length. The TTN and the MERA also possess this logarithmic violation [23, 11].
The generalization of MPS to two and higher dimensions is denoted as Projected Entangled Pair States (PEPS) [13, 24, 25]. As opposed to Quantum Monte Carlo methods, they do not suffer from the so-called sign problem , i.e., they also allow for the description of frustrated and fermionic systems . However, PEPS still compete with other methods which are not that well suited for the thermodynamic limit of two dimensional systems, such as DMRG in two dimensions [28, 20]. The reason is that the scaling of the number of parameters required for a given overlap with the true ground state, although polynomial in the system size, is much worse then for MPS. Furthermore, there are some technical problems, which arise due to the fact that PEPS do not allow for an efficient calculation of observables (or even of the norm of the state) . This implies that approximate methods need to be applied in the contraction of PEPS, which are, however, usually well controlled [30, 31, 32]. In two dimensions, critical systems do not necessarily display logarithmic corrections to the area law. If they do, it would still be sensible to use PEPS to describe them, as PEPS probably also allow for an efficient approximation of critical local Hamiltonians as is the case for MPS in one dimension. Note that higher dimensional MERA  obey the area law and can in fact be represented efficiently by PEPS . However, for two dimensional critical systems it is possible to obtain a better (polynomial) scaling by employing the branching MERA [14, 15], which displays, depending on its branching parameter, logarithmic violations to the area law or even a volume law .
At finite temperature, the area law of the entanglement entropy for gapped local Hamiltonians is replaced by an area law of the mutual information . Tensor Network States can easily be generalized to mixed states. An obvious way is to define a purification of the mixed state one intends to represent . This can be achieved by doubling the physical dimension at each site (such that there is an additional physical index) and obtaining the mixed state by tracing out the extra degrees of freedom. Another possibility is to again double the physical dimension, but to consider the additional physical index as defined in the dual space (ket and bra). Then, one has a representation of mixed states by the same contraction scheme as for pure states. However, some care has to be taken to ensure that the constructed state is positive semidefinite. The simplest cases are Matrix Product Operators (MPO) [36, 37] and Projected Entangled Pair Operators (PEPO)  in one and higher dimensions, respectively. The two approaches have been compared in Ref. , where it has been shown that mixed states obtained by purifying MPS can be efficiently represented by MPOs, but not the other way around.
Most TNS can be defined such that they allow to read off properties of the wave function from the constituent tensors: Symmetries and topological properties of MPS and PEPS can be encoded locally in the corresponding tensors. Moreover, there is a standard procedure of constructing a parent Hamiltonian [10, 40] for any MPS or PEPS as a sum of local projectors which each annihilate the state. Ground states of this kind are called frustration free. For MPS it can be easily checked whether it is the unique ground state of its parent Hamiltonian. In one dimension, several such MPS with exact parent Hamiltonians of a simple form, such as the Majumdar-Ghosh state  and the AKLT model  (a special point of the spin-1 bilinear biquadratic Heisenberg model) have been found. For two dimensional systems determining whether the parent Hamiltonian has a unique ground state is a rather intricate issue.
The first part of this work is devoted to MPS with long range localizable entanglement (LRLE) [43, 44] and to chiral topological PEPS [45, 46, 47, 48]. Localizable entanglement plays an essential role in the field of quantum communication. It quantifies the amount of entanglement that can on average be localized between the spins at the ends of a spin chain by measuring all the remaining ones. If it is non-vanishing in the thermodynamic limit, the state is said to have LRLE. Such states can be used as ideal quantum repeaters , since they allow for the creation of entangled pairs over arbitrary long distances. Moreover, they appear at certain topological quantum phase transitions , i.e., LRLE can be used for their detection. A criterion will be presented which classifies all translationally invariant MPS with a fixed bond dimension, which possess LRLE. This is very relevant as it basically also classifies all ground states of local gapped Hamiltonians with LRLE, since the bond dimension of the corresponding MPS saturates as a function of the size of the system.
The major part of this thesis contains our contribution to the discovery and description of chiral topological PEPS [45, 47, 48]. In many cases, the interplay between the quantum particles of a topological state gives rise to macroscopic properties which are quantized and protected to disorder and perturbations: The field of topological systems emerged with the discovery of the Quantum Hall Effect in 1980 by von Klitzing , where it has been observed that for a sufficiently pure metallic sample at high magnetic fields, the transverse resistance (Hall resistance) displays sharp quantized plateaus if the magnetic field is varied, cf., Fig. 1.3. The Hall resistance is an integer multiple of , which is why the effect is also denoted as the Integer Quantum Hall Effect. In the meantime, the accuracy of the quantization of the Hall conductance has been measured to be around one in a billion . Moreover, a closer investigation for sufficiently pure samples revealed several smaller plateaus with conductances that are fractional multiples of , which is denoted as the Fractional Quantum Hall Effect discovered in 1982 . Due to the topological protection against perturbations and disorder, the following definition of a topological phase has been put forward: Two Hamiltonians are in the same topological phase if and only if they can be connected via a gapped path of Hamiltonians. The phase which contains the Hamiltonian in the atomic limit (i.e., where all sites of the lattice are decoupled) is also called the topologically trivial phase. The topological phases containing the Integer Quantum Hall states are labeled by the Chern number , which is up to a sign the Hall conductance expressed in units of .
Apart from systems that are intrinsically topological, i.e., that are stable to any small perturbation, the notion of symmetry protected topological phases  has been coined: Some states that possess a certain symmetry are topologically protected against any small perturbation that does not break the symmetry. However, an arbitrarily small perturbation that does break the symmetry can destroy the topological phase. More mathematically, a symmetry protected topological phase can be considered as the equivalence class of Hamiltonians that possess a certain symmetry and can be connected by a gapped path of Hamiltonians that obey the same symmetry.
Coming back to “genuine” topological systems, another distinction is usually made: They are divided into short range entangled and long range entangled topological systems . The latter are said to have topological order [58, 59]. Such states are characterized by a constant sub-leading correction to the area law, i.e., the entanglement entropy of a region grows like the size of its boundary minus a constant term, known as the topological entanglement entropy [60, 61]. A characteristic property of such systems is that the ground state degeneracy of the corresponding Hamiltonian depends on the topology of the manifold they are defined on. All other topological states are referred to as being short range entangled. They in particular include symmetry protected topological states, since, as elaborated above, they are connected to the trivial phase by small perturbations that break the corresponding symmetry. Another class of short range entangled states is the one of free fermionic systems, i.e., those fermionic systems that are described by a Hamiltonian that is quadratic in creation and annihilation operators. They can be diagonalized exactly by a computational cost that is polynomial in the system size, since there exist simple methods to write the Hamiltonian as a sum of terms that act only on single particle modes . Due to this great simplification, a complete characterization of free fermionic phases has been achieved [63, 64]. All free fermionic examples of “genuine” topological systems possess chiral edge modes at the boundary of the system, i.e., one or several quasi-particle modes that propagate in one direction along the boundary of the system and thus break time reversal symmetry. Those quasi-particle modes can be either fermionic modes, such as for the Integer Quantum Hall effect, or Majorana modes, such as in the case of topological superconductors.
Moving on to the interacting regime, which includes interacting electrons and spin systems, the picture is less clear and a variety of “genuine” topological systems has been found theoretically, and in the case of the Fractional Quantum Hall Effect also experimentally. They no longer have to possess chiral edge modes (or even gapless edge states). Prominent examples are Resonating Valence Bond states  proposed by Anderson for the description of high-temperature superconductivity, Levin-Wen  models and the toric code  (see Fig. 1.4). Being all topologically ordered, these examples have been studied intensively, since the topological degeneracy of the ground state subspace is stable to local perturbations and can thus be used as a quantum memory. Moreover, several systems with topological order have non-Abelian anyonic excitations, whose braiding might be used in order to carry out topologically protected quantum computations. It is thus a formidable task to find efficient numerical and analytic tools to describe those systems.
While for the above non-chiral examples of topologically ordered systems exact PEPS representations with low bond dimensions exist [68, 69, 70], no example of a PEPS has been known previously that represents a chiral topological system, not even in the free fermionic limit. This circumstance invoked arguments that seemed to rule out the existence of chiral PEPS. One of them is that at least free fermionic chiral PEPS could not exist, since the fact that they are generically unique frustration free ground states of local Hamiltonians allows for the construction of a complete set of localized Wannier wave functions [71, 72] - something that is incompatible with a non-vanishing Chern number . However, this work presents the discovery of the first family of free fermionic chiral PEPS , which was obtained independently from a similar example provided by Dubail and Read  at the same time. Hence, PEPS can be used for the description of chiral systems, as will be demonstrated by concrete numerical examples. Moreover, the properties of such chiral PEPS will be thoroughly investigated and eventually an example of an interacting chiral PEPS will be presented, whose construction is based on two copies of free fermionic chiral PEPS. This opens the doors to the description of general chiral topological systems by PEPS, from the numerical side, but also from the analytic side, via the construction of new interacting chiral models with exact PEPS ground states. This will enhance our understanding of interacting chiral topological systems and might eventually even help to classify them.
This thesis is organized as follows: Chapter 2 presents the description of states with long range localizable entanglement by MPS. Its first section, MPS will be introduced via the local projection of maximally entangled pairs that are located between neighboring sites. The graphical representation of MPS and the connection to the area law will be explained. Furthermore, it will be shown how symmetries on the physical level can be locally incorporated into the MPS tensor and also how it can be used to decide whether the corresponding parent Hamiltonian has a gap in the thermodynamic limit. In section 2.2, the notion of localizable entanglement will be introduced and its significance for quantum repeaters and the detection of quantum phase transitions. Our new findings are contained in section 2.3, where it will be shown how to determine if a given translationally invariant MPS has LRLE. Based on that condition, a criterion for all such MPS will be presented, which will in turn be used to parameterize all MPS with LRLE for bond dimension and explicitly.
Chapter 3 is devoted to chiral PEPS. In section 3.1 the construction of section 2.1 will be extended to two dimensional systems. This construction will be extended both to general fermionic systems and explicitly to free fermionic systems. Furthermore, a special emphasis will be made on how physical symmetries and topological properties can be encoded in the PEPS tensor. Section 3.2 contains a general introduction to chiral topological systems, first focusing on free fermionic topological systems and later providing tools from Conformal Field Theory [74, 75] for the description of interacting chiral fermionic phases. After establishing these concepts, the examples of chiral free fermionic PEPS we obtained will be presented in section 3.3. It will be shown that there are strong restrictions on free fermionic PEPS that support chiral edge modes - which in all generality lead to algebraically decaying correlations. However, they are nonetheless unique ground states of long-ranged gapped Hamiltonians, which will be demonstrated to be topologically protected against perturbations. The correspondence between the bulk energy spectrum and the spectrum of the boundary reduced density matrix will be conveyed (bulk-boundary correspondence ). A detailed treatment of the symmetries of chiral free fermionic PEPS will be given, which will eventually be used to fully classify all such PEPS with the smallest non-trivial bond dimension. Section 3.4 includes our results on chiral PEPS with topological order. Such a PEPS is obtained by applying a Gutzwiller projector on two copies of a free fermionic chiral PEPS. The resulting symmetries of the PEPS tensor will be presented and used to construct all ground states of the local frustration-free parent Hamiltonian. Furthermore, it will be explained how the bulk-boundary correspondence can be used to calculate the entanglement spectra  and how the later can be used to confirm the expected Conformal Field Theory of the model  (which characterizes the topological state). Our calculations showing that the gap of the transfer operator vanishes in the thermodynamic limit are presented. This implies that, again, correlations decay algebraically in real space.
Finally, chapter 4 concludes our results and gives an outlook on future research directions.
Chapter 2 and 3 are two mostly self-contained parts and can be read independently.
2 Matrix Product States with Long Range Localizable Entanglement
2.1 Introduction to Matrix Product States
As illustrated above, Matrix Product States (MPS) are well suited for the description of ground states of local gapped Hamiltonians in one dimension, as both fulfill the area law (this argument becomes rigorous if one refers to the area law of the Rényi entropy with parameter ). Historically, the first non-trivial example of an MPS was provided by Affleck, Kennedy, Lieb and Tasaki (AKLT)  in 1987 as the ground state of a certain parameter value of the bilinear biquadratic spin-1 Heisenberg Hamiltonian. However, they were formally introduced only in 1992 under the name of Finitely Correlated States , which are translationally invariant MPS. They captured broad interest only from 2007 on when it was realized  that their non-translationally invariant version is the variational class underlying the DMRG algorithm . Furthermore, it was observed that many statements about the whole MPS can be made (e.g., regarding the decay of correlations , symmetries , etc.) by studying the rank-3 tensors the MPS is composed of. In the meantime, MPS have been generalized to higher dimensional systems, resulting in Projected Entangled Pair States (PEPS), which are the subject of chapter 3, to critical systems violating the area law (Multiscale Entanglement Renormalization Ansatz ) and to continuous systems [82, 83] (both not discussed in this work).
This section is organized as follows: In the next subsection, the construction of MPS via pairs of maximally entangled spins is presented. The final form of MPS will be given, along with their graphical representation, which will be very useful to illustrate the contraction scheme of PEPS. Subsection 2.1.3 elaborates on the area law and the resulting efficient approximability of ground states of local Hamiltonians by MPS. In subsection 2.1.4 symmetries and, finally, in subsection 2.1.5 parent Hamiltonians for MPS will be considered.
We consider a chain of spin- particles. In order to construct an MPS for this system, we attach two virtual particles of spin to each site, one on the left and on one the right. Furthermore, we assume that they are in a product state of maximally entangled pairs connecting neighboring sites. E.g., the right virtual particle of site 1 and the left virtual particle of site 2 are in a maximally entangled state , where is the bond dimension (dimension of the Hilbert space of each virtual particle) and round brackets are used for vectors defined in the virtual Hilbert space.
In the case of open boundary conditions, the first site has only a right virtual particle and the -th site only a left virtual particle, see Fig. 2.1. For periodic boundary conditions, each site has two virtual particles, where the left one of the first site is maximally entangled with the right one of the -th, cf., Fig. 2.2. In both cases, we perform an arbitrary linear map at each site from the Hilbert space of the virtual particles to the Hilbert space of the corresponding physical particle. The map can be written in terms of a rank-3 tensor
where is the dimension of the physical Hilbert space for each site. In the case of open boundary conditions, the maps for and are different: For them, it is sufficient to use rank-2 tensors (i.e., matrices),
Hence, the resulting state for open boundary conditions is
where represents the matrix with entries . Hence, Eq. (2.3) is a product of matrices for each configuration , thus the name Matrix Product State. Note that is a row vector and a column vector. The other matrices can also be chosen to have other dimensions ( being a matrix) by using virtual particles of different spins . This case is included in the above construction if one sets . The graphical representation of Eq. (2.3) is given in Fig. 2.1.
On the other hand, for periodic boundary conditions, one can easily derive
with , as shown in Fig. 2.2. In Ref.  it has been proven that MPS with periodic boundary conditions can be represented by the same rank-3 tensor for each site, so the site index will be dropped in equations relating to periodic boundary conditions from now on.
It is obvious that an MPS with open boundary conditions is invariant under the replacement for a set of invertible matrices . For periodic boundary conditions, the MPS is invariant under for any invertible matrix . This gauge freedom can be used to derive canonical forms for MPS : Any MPS with open boundary conditions and bond dimension can be expressed as in Eq. (2.3), where the rank-3 tensors fulfill
where and is a diagonal strictly positive matrix with . For translationally invariant MPS, the rank-3 tensors can be gauged to a block form
The gauge given by Eqs. (2.7) to (2.9) is fixed in such a way that the completely positive map with (studied in more detail in subsection 2.1.5) has a unique fixed point of maximum absolute eigenvalue, which is . Note that an MPS with tensors as in Eq. (2.7) is a sum of MPS whose tensors are the diagonal blocks. In the limit of only those terms corresponding to the ’s of maximum magnitude contribute to the sum.
Using the completely positive map, one can easily show that the correlations of any MPS decay exponentially in real space : Let us consider the completely positive map of one block (we drop the index for clarity). It is equivalent to the so-called double tensor . It has as its highest eigenvalue and we denote by and the corresponding right and left eigenvector. For simplicity, let us assume that the eigenvalue of second highest magnitude, , is non-degenerate (the proof for the degenerate case is completely analogous). We call the respective right and left eigenvectors and . For large , the leading terms of are thus , where . The correlation of two single-site operators and is given by
where acts on site and on site . In the case of our MPS described by the tensor , we obtain for (with )
This demonstrates the exponential decay of all two-point correlations for the MPS corresponding to a single block in Eq. (2.7) and therefore also for the full MPS, which is a superposition of the MPSs corresponding to individual blocks.
In subsection 126.96.36.199, we will come back to the above approach in the context of PEPS, where cases with algebraically decaying correlation functions exist.
2.1.3 The area law and efficient approximability
Why are MPS useful for the description of one-dimensional spin systems? The reason is that ground states of local gapped Hamiltonians obey the area law , which constrains such ground states to occupy only a subspace of the full Hilbert space that grows polynomially with the system size . As MPS (and also PEPS) by construction fulfill the area law, their cost to describe such ground states grows only polynomially with the system size, too. Let us settle these statements more concretely for the one dimensional case (MPS): In one dimension, the area law demands that for the reduced density matrix of a local gapped Hamiltonian the entanglement entropy saturates as a function of (for ). The entanglement entropy  (also known as the von Neumann entropy) of a density matrix is defined as
It is the limiting case of the Rényi entropy 
Hence, the area law states for arbitrary . That MPS fulfill the area law can be seen by considering the Schmidt decomposition of a region of length with the rest
where lives in region , refers to the complementary region and . If one decomposes the wave functions described by Eqs. (2.3) and (2.4) into a connected region of sites and its complement, i.e., a basis for is , it becomes obvious that there can be at most Schmidt coefficients . Realizing that the eigenvalues of are the Schmidt coefficients squared, the von Neumann entropy of (2.14) reads
since the entropy is maximized for . Hence, the von Neumann entropy is bounded by times the number of bonds that are cut when tracing out all but consecutive sites (cf. Figs. 2.1 and 2.2). By the same token, this can be shown for PEPS.
The argument that both gapped ground states of local Hamiltonians and MPS fulfill the area law is of course by far not sufficient to prove that they can be used to efficiently approximate the ground states of those Hamiltonians. The rigorous proof can be found in Ref.  and shows that for any local one-dimensional Hamiltonian acting on sites whose ground state violates the area law at most logarithmically, an MPS approximation can be found with overlap with polynomial in for fixed . Therefore, also critical systems can be efficiently approximated by MPS, where MERA, however, scales better as a function of the system length.
Let us add that in Ref.  it has been shown that MPS do not only allow for an efficient representation of ground states of gapped local Hamiltonians, but that this representation can also be found numerically in a time that is polynomial in the system size.
The results of Refs. [86, 81] shall be restated here (for a related approach, see Ref. ): Let us consider translationally invariant states with local symmetries with a symmetry group , which is represented by the unitary matrices acting on the physical degree of freedom, that is,
where the phases form a one dimensional representation of .
Let us now consider separately the two cases of discrete and continuous . If is discrete and is an MPS obeying that symmetry, its tensor fulfills
with acting only on the virtual degrees of freedom and being a matrix permuting the blocks with index as in Eq. (2.7). is given by (with arbitrary phases ).
2.1.5 Injectivity and parent Hamiltonians
For each translational invariant MPS a Hamiltonian can be found that acts on sites and has the MPS as a ground state. The question is whether this ground state is unique. To tackle this problem, the concept of injectivity of MPS has been introduced [80, 10]: Consider the map
with being the tensor of a translationally invariant MPS in the canonical form (2.7). An MPS for which the map is injective, but not for any is said to be -injective or simply injective. In this case, is denoted as the injectivity length of the MPS. If and one has only one block in Eq. (2.7), -injectivity implies that is injective for any (e.g., for this is shown by inserting such that is full rank into Eq. (2.19)). In words, an -injective MPS has the property that by blocking sites the whole virtual space is accessible by acting on the physical space of the block (or, conversely, the virtual space is fully mapped to the physical space).
Injectivity turns out to be a generic property of MPS, i.e., the tensors which correspond to non-injective MPS have Haar measure zero. For injective MPS it has been demonstrated  that there is only one block in the canonical form Eq. (2.7) and that the reduced density operator has rank for . For injective MPS,
Quite related to injectivity is the completely positive map defined via
Let us consider how to construct the so-called parent Hamiltonian of an MPS, which is frustration free and has the MPS as its ground state. By frustration free it is meant that the Hamiltonian is a sum of projectors (possibly with prefactors) that each act only on a finite range of sites. E.g., a parent Hamiltonian with sums of projectors acting non-trivially on sites can be obtained by calculating the reduced density matrix of the MPS and setting
denotes the kernel (the null space) of a matrix and represents its support, i.e., the span of its column vectors. The reduced density matrix can be calculated via
For any MPS with injectivity length , there is a gapped parent Hamiltonian which is a sum of projectors acting on sites and has the MPS as its unique ground state. Note that for such an MPS it is also possible to construct a Hamiltonian with a continuous spectrum and the MPS as its unique ground state, denoted as the uncle Hamiltonian .
2.2 Introduction to Localizable Entanglement
Soon after a theoretical framework for MPS had been established , they were used to detect certain kinds symmetries and orders that may be present in a system. One of them is string order , which is a non-local quantity defined as the expectation value of a string of unitaries multiplied with local operators acting on the end sites of the string. Within the set of MPS it was shown to be related to the existence of a local symmetry in the system. Another such quantity, which appears to be truly non-local in general, is the localizable entanglement (LE) . It quantifies the amount of entanglement that can on average be generated between the two spins at the ends of a spin chain by measuring all the remaining ones. It can be lower bounded by two-point connected correlation functions (this has been shown rigorously for qubits in Ref.  and extended to spin-1 systems in Ref. ).
LE is not only of importance as a non-local order parameter, but also as a resource for quantum communication, because if it is close to 1 for a long spin chain, one can create with high probability highly entangled states over large distances. The limiting case is known as long range localizable entanglement (LRLE), which means that the LE is finite even for an infinitely long spin chain. This scenario corresponds to an ideal quantum repeater , which is a device to create maximally entangled pairs over large distances. LRLE is also relevant for Condensed Matter Physics, as, like string order, it appears at certain quantum phase transitions  and can therefore be used for their detection.
We will proceed by defining LE and revealing its long range behavior at certain quantum phase transitions.
2.2.2 Definition and relation to quantum phase transitions111The remainder of this chapter is a slight modification of the content of Ref. , copyright American Physical Society.
The LE is defined as the maximum average of the entanglement that can be generated between two spins of a spin chain by measuring the remaining ones . Let denote the density matrix of the original state. With probability the outcome of a measurement will be and the system will be in the corresponding two-particle state . Hence, the LE is given by
where is the class of allowed measurements and an entanglement measure.
Let us consider the slightly more general case of a chain of spin- particles along with two auxiliary particles of spin at each of the boundaries. The particles of the actual chain are the ones to be measured (measurement outcomes ), and the class of allowed measurements is the set of local projective von Neumann measurements, where the same measurement is carried out on each party (in particular, we exclude adaptive strategies). Therefore, the maximization of the average entanglement is performed by choosing the optimal physical basis .
An example of a state with LRLE is the ground state of the AKLT model  defined for a spin-1 chain with periodic boundary conditions; its Hamiltonian is given by
As mentioned above, it has a unique ground state that is an MPS given by the matrices , and . If one measures the spins in the basis , the LE is non-vanishing in the thermodynamic limit. In Ref.  a deformation has been introduced into the AKLT Hamiltonian via a parameter , such that for the AKLT Hamiltonian is recovered. If one varies , LRLE is observed only at , signaling a quantum phase transition at this point. What is even more important is that the correlation length is finite for any , i.e., it does not allow for the detection of this quantum phase transition (note that correlations provide only a lower bound on the LE [91, 92]). However, it turns out that this particular quantum phase transition can also be detected by the emergence of string order at . An example of a state that has LRLE but no string order will be provided below.
2.3 A criterion for Matrix Product States with Long Range Localizable Entanglement
In this section a necessary and sufficient criterion will be derived for all translationally invariant MPS with LRLE. In subsection 2.3.1, we will derive the LE of an MPS and show how it can be simplified to the case of the spins at the ends of the spin chain being qubits. Thereafter, in subsection 2.3.2, a criterion will be derived which allows to check for an MPS of arbitrary bond dimension based on a single tensor (i.e., without having to contract them), whether it possesses LRLE. This criterion will be used in subsection 2.3.3 to parameterize all MPS with bond dimension 2 and 3 that display LRLE.
2.3.1 Localizable entanglement of an MPS
The question to be answered in this section is for which translationally invariant MPS a finite amount of entanglement can be localized between the two ancillas in the limit . We assume the state of the system to be translationally invariant apart from boundary effects; for this reason, the rank-three tensors corresponding to the spin- particles are taken equal. Those consist of complex matrices () and can be assumed to be in the canonical form (2.7). As mentioned at the end of section 2.1.2, the MPS decomposes into a sum of MPS, where is the number of blocks in Eq. (2.7). Thus, we will only consider one block, which we simply call , in the following. The results to be derived will apply to the other blocks, too. The maps and satisfy (cf. Eqs. (2.8) and (2.9))
with a diagonal matrix , cf. Eqs. (2.8) and (2.9). Our goal is to find necessary and sufficient conditions on the matrices to give rise to LRLE for some matrices of the auxiliary particles. The latter can be chosen at will and are denoted by , where is the Hilbert space dimension of the individual auxiliary spins. The initial MPS is therefore
where the Hilbert space vectors of the auxiliary particles are denoted by round brackets, cf. Fig. 2.3.
Subsequent to a measurement , the initial state of the system reduces with probability
to , where (excluding cases with )
is the normalized two-particle state after the measurement. In appendix A it is shown that any MPS with LRLE for also has LRLE for (the converse is obvious), and also that w.l.o.g. we can take and as isometries. We can thus choose and the concurrence  as the measure of entanglement, , where is the matrix with coefficients . Dropping the superscripts and , the LE reads
This optimization problem is in general hard, since the sum needs to be evaluated for large before optimizing over the physical basis . Its maximization succeeded only in special cases, like for a chain of spins without auxiliary ones and measurement on all spins but those at the borders . In this case, , which implies that there is LRLE for . In the following, we slightly generalize this family of states and adapt them to our system that includes the auxiliary spins at the boundary:
Block structure of unitaries.
Consider an MPS described by the matrices
where is a permutation matrix. The are unitaries () and . We can realize that this MPS has LRLE for by choosing + (where are basis vectors of the auxiliary qubits and the basis vectors of the basis in which the matrices are given) and . Then, due to and therefore , Eq. (2.30) is finite in the thermodynamic limit. The fact that there is also LRLE for will become clear below.
In the above example, the matrices exhibit a block structure of unitaries. One may think that all states with LRLE need to have this property in some basis . Interestingly, this is not the case as shown by the following counterexample for :
For the MPS with
and it is a simple exercise to show that (2.30) remains finite in the limit . The matrices fulfill . However, , and thus they are not of the form in any basis . Note that the MPS can numerically be verified to be injective , and thus it is the unique ground state of a local translationally invariant gapped Hamiltonian. Moreover, it can be easily shown (analytically) that it is invariant under the -symmetry generated by the transformation , , .
2.3.2 Derivation of the criterion
The question arises of how one can check whether a given MPS has LRLE without having to resort to evaluating Eq. (2.30) for large numerically. In the following, a necessary and sufficient criterion will be derived, which allows to decide this based on the matrices directly. We first rewrite (2.30) by inserting to the right of a projector on the subspace spanned by the row vectors of , which can be written as , where is an isometry with . Thus, Eq. (2.30) reads now
Using the SVD of , which is (neglecting the indices ), along with the inequality of arithmetic and geometric means, one obtains for the factors of Eq. (2.33)
since . The inequality becomes an equality if and only if , i.e.,
is proportional to a unitary. The intuitive fact that all factors but the last one of Eq. (2.33) have to be exactly 1 to get LRLE turns out to be correct:
Formally, this lemma reads
which is what we want to show in the following. Thus, we assume that for the sum
there is a minimum integer such that
After building blocks of terms in Eq. (2.33), each of them can be upper bounded by
which shows that the LE must fulfill
i.e., there is no LRLE. ∎
Therefore, a necessary condition for any MPS to give rise to LRLE is that there exists an isometry , such that for a certain basis
denoting some unitary. Redefining shows that one can require the RHS of (2.41) to be the identity. After multiplying this from the right by , one obtains
which can also be written as
It follows that Eq. (2.43) is a necessary criterion to have LRLE. Whether it is also sufficient depends on if can always been chosen such that (2.33) is finite, which is confirmed by the following lemma: