Symmetry reduction induced by anyon condensation: a tensor network approach

Topological ordered phases are related to changes in the properties of their quasi-particle excitations (anyons). We study these relations in the framework of projected entanglement pair states (\textsf{PEPS}) and show how condensing and confining anyons reduces a local gauge symmetry to a global on-site symmetry. We also study the action of this global symmetry over the quasiparticle excitations. As a byproduct, we observe that this symmetry reduction effect can be applied to one-dimensional systems as well, and brings about appealing physical interpretations on the classification of phases with symmetries using matrix product states (\textsf{MPS}). The case of $\mathbb{Z}_2$ on-site symmetry is studied in detail.Comment: 21+5 pages, 15+3 figures. Introduction and conclusions enlarged, references and figure added, minor typos corrected, appendix about dyons adde

Mathematical open problems in Projected Entangled Pair States

Projected Entangled Pair States (PEPS) are used in practice as an efficient parametrization of the set of ground states of quantum many body systems. The aim of this paper is to present, for a broad mathematical audience, some mathematical questions about PEPS.Comment: Notes associated to the Santal\'o Lecture 2017, Universidad Complutense de Madrid (UCM), minor typos correcte

Simetrías en estados de redes tensoriales topológicas: clasificación, construcción y detección

Tesis inédita de la Universidad Complutense de Madrid, Facultad de Ciencias Matemáticas, Departamento de Análisis y Matemática Aplicada, leída el 10/01/2019The exotic properties which appear in the quantum setting, mainly manifested in strongly-correlated systems, oer potential applications in future technologies. For instance, high-precision measurementsor the new paradigm of a quantum computer. One of the most prominent features of quantum physics is entanglement: the correlationsbetween the parties of a system that cannot be described classically. This property is believed to be the one endowing quantum mechanics its complexity. Therefore, characterizing the entanglement properties of strongly-correlated systems plays a fundamental role for condensed-matter physics. However, this complexity comes hand in hand with a challenge: the number of parameters needed to describe a system grows exponentially with the number of parties in the system. This challengelies at the heart of the mathematical description of quantum mechanics. Such a situation happens naturally in many-body systems and in particular, in condensed-matter physics where the relevant physics appears when considering large systems. Then, how can we deal with this diculty? Thekey observation here is that realistic physical systems are the ones whose parties interact locally and this restricts the entanglement pattern in the low-energy sector (zero temperature). So thequestion is shifted to: Is there a framework that captures states with such entanglement pattern? The answer is yes: tensor network states...Las inusuales propiedades que surgen en la mecánica cuantica, manifestadas especialmente en sistemas fuertemente correlacionados, ofrecen aplicaciones potenciales a nuevas tecnologías. Por ejemplo, mediciones de alta precision o la creacion un nuevo modelo de computacion (el ordenador cuantico). Una de las propiedades mas destacadas de la física cuantica es el entrelazamiento: las correlaciones entre las partes de un sistema que no se pueden describir clasicamente. Se cree que esta propiedad es la que dota a la mecanica cuantica de su complejidad. Por lo tanto, caracterizar el entrelazamiento de sistemas fuertemente correlacionados desempeña un papel fundamental en la física de la materia condensada. Pero esta complejidad viene unida a una dificultad: el numero de parámetros necesario para describir un sistema crece exponencialmente con el numero de partes en el sistema. Este reto esta en el corazon de la descripcion matematica de la mecánica cuantica. Esto ocurre de forma natural en sistemas de muchos cuerpos y en particular, en física de la materia condensada, donde la física relevante aparece cuando el tamaño del sistema es grande. Entonces, ¿como podemos lidiar con esta dificultad? La observación clave es que los sistemas físicos reales son aquellos en los que las partes interaccionan localmente y esto restringe el patron de entrelazamiento en el sector de baja energía de los sistemas. Por lo que ahora la cuestion es: ¿hay un formalismo que capture los estados con ese tipo de patrón de entrelazamiento? La respuesta es si: los estados de redes tensoriales...Depto. de Análisis Matemático y Matemática AplicadaFac. de Ciencias MatemáticasTRUEunpu

String order parameters for symmetry fractionalization in an enriched toric code

We study a simple model of symmetry-enriched topological order obtained by decorating a toric code model with lower-dimensional symmetry-protected topological states. We show that the symmetry fractionalization in this model can be characterized by string order parameters, and that these signatures are robust under the effects of external fields and interactions, up to the phase transition point. This extends the recent proposal of [New Journal of Physics 21, 113016 (2019)] beyond the setting of fixed-point tensor network states, and solidifies string order parameters as a useful tool to characterize and detect symmetry fractionalization. In addition to this, we observe how the condensation of an anyon that fractionalizes a symmetry forces that symmetry to spontaneously break, and we give a proof of this in the framework of projected entangled pair states. This phenomenon leads to a notable change in the phase diagram of the toric code in parallel magnetic fields.Comment: Changes in title, astract and introdution. 11+7 pages, 7+3 figures, 2 table

Subsystem symmetry enriched topological order in three dimensions

We introduce a model of three-dimensional (3D) topological order enriched by planar subsystem symmetries. The model is constructed starting from the 3D toric code, whose ground state can be viewed as an equal-weight superposition of two-dimensional (2D) membrane coverings. We then decorate those membranes with 2D cluster states possessing symmetry-protected topological order under line-like subsystem symmetries. This endows the decorated model with planar subsystem symmetries under which the loop-like excitations of the toric code fractionalize, resulting in an extensive degeneracy per unit length of the excitation. We also show that the value of the topological entanglement entropy is larger than that of the toric code for certain bipartitions due to the subsystem symmetry enrichment. Our model can be obtained by gauging the global symmetry of a short-range entangled model which has symmetry-protected topological order coming from an interplay of global and subsystem symmetries. We study the non-trivial action of the symmetries on boundary of this model, uncovering a mixed boundary anomaly between global and subsystem symmetries. To further study this interplay, we consider gauging several different subgroups of the total symmetry. The resulting network of models, which includes models with fracton topological order, showcases more of the possible types of subsystem symmetry enrichment that can occur in 3D.Comment: 21 pages. v2: Published version. Updated Section IV with new figure and tabl

Normal projected entangled pair states generating the same state

Tensor networks are generated by a set of small rank tensors and define many-body quantum states in a succinct form. The corresponding map is not one-to-one: different sets of tensors may generate the very same state. A fundamental question in the study of tensor networks naturally arises: what is then the relation between those sets? The answer to this question in one dimensional setups has found several applications, like the characterization of local and global symmetries, the classification of phases of matter and unitary evolutions, or the determination of the fixed points of renormalization procedures. Here we answer this question for projected entangled-pair states (PEPS) in any dimension and lattice geometry, as long as the tensors generating the states are normal, which constitute an important and generic class

Brca1 Alternative Splicing Landscape In Breast Tissue Samples

Background: BRCA1 is a key protein in cell network, involved in DNA repair pathways and cell cycle. Recently, the ENIGMA consortium has reported a high number of alternative splicing (AS) events at this locus in blood-derived samples. However, BRCA1 splicing pattern in breast tissue samples is unknown. Here, we provide an accurate description of BRCA1 splicing events distribution in breast tissue samples. Methods: BRCA1 splicing events were scanned in 70 breast tumor samples, 4 breast samples from healthy individuals and in 72 blood-derived samples by capillary electrophoresis (capillary EP). Molecular subtype was identified in all tumor samples. Splicing events were considered predominant if their relative expression level was at least the 10% of the full-length reference signal. Results: 54 BRCA1 AS events were identified, 27 of them were annotated as predominant in at least one sample. Delta 5q, Delta 13, Delta 9, Delta 5 and del 1aA were significantly more frequently annotated as predominant in breast tumor samples than in blood-derived samples. Predominant splicing events were, on average, more frequent in tumor samples than in normal breast tissue samples (P = 0.010). Similarly, likely inactivating splicing events (PTC-NMDs, Non-Coding, Delta 5 and Delta 18) were more frequently annotated as predominant in tumor than in normal breast samples (P = 0.020), whereas there were no significant differences for other splicing events (No-Fs) frequency distribution between tumor and normal breast samples (P = 0.689). Conclusions: Our results complement recent findings by the ENIGMA consortium, demonstrating that BRCA1 AS, despite its tremendous complexity, is similar in breast and blood samples, with no evidences for tissue specific AS events. Further on, we conclude that somatic inactivation of BRCA1 through spliciogenic mutations is, at best, a rare mechanism in breast carcinogenesis, albeit our data detects an excess of likely inactivating AS events in breast tumor samples