We present an algebraic framework for interacting extended quantum systems to study complex phenomena characterized by the coexistence and competition of different states of matter. We start by showing how to connect different (spin-particle-gauge) languages by means of exact mappings (isomorphisms) that we name dictionaries and prove a fundamental theorem establishing when two arbitrary languages can be connected. These mappings serve to unravel symmetries which are hidden in one representation but become manifest in another. In addition, we establish a formal link between seemingly unrelated physical phenomena by changing the language of our model description. This link leads to the idea of universality or equivalence. Moreover, we introduce the novel concept of emergent symmetry as another symmetry guiding principle. By introducing the notion of hierarchical languages, we determine the quantum phase diagram of lattice models (previously unsolved) and unveil hidden order parameters to explore new states of matter. Hierarchical languages also constitute an essential tool to provide a unified description of phases which compete and coexist. Overall, our framework provides a simple and systematic methodology to predict and discover new kinds of orders. Another aspect exploited by the present formalism is the relation between condensed matter and lattice gauge theories through quantum link models. We conclude by discussing applications of these dictionaries to the area of quantum information and computation with emphasis in building new models of computation and quantum programming languages.
We present an algebraic framework for interacting extended quantum systems to study complex phenomena characterized by the coexistence and competition of different states of matter. We start by showing how to connect different (spin-particle-gauge) languages by means of exact mappings (isomorphisms) that we name dictionaries and prove a fundamental theorem establishing when two arbitrary languages can be connected. These mappings serve to unravel symmetries which are hidden in one representation but become manifest in another. In addition, we establish a formal link between seemingly unrelated physical phenomena by changing the language of our model description. This link leads to the idea of universality or equivalence. Moreover, we introduce the novel concept of emergent symmetry as another symmetry guiding principle. By introducing the notion of hierarchical languages, we determine the quantum phase diagram of lattice models (previously unsolved) and unveil hidden order parameters to explore new states of matter. Hierarchical languages also constitute an essential tool to provide a unified description of phases which compete and coexist. Overall, our framework provides a simple and systematic methodology to predict and discover new kinds of orders. Another aspect exploited by the present formalism is the relation between condensed matter and lattice gauge theories through quantum link models. We conclude by discussing applications of these dictionaries to the area of quantum information and computation with emphasis in building new models of computation and quantum programming languages.
We introduce an algebraic framework for interacting quantum systems that enables studying complex phenomena, characterized by the coexistence and competition of various broken symmetry states of matter. The approach unveils the hidden unity behind seemingly unrelated physical phenomena, thus establishing exact connections between them. This leads to the fundamental concept of {\it universality} of physical phenomena, a general concept not restricted to the domain of critical behavior. Key to our framework is the concept of {\it languages} and the construction of {\it dictionaries} relating them.
We introduce an algebraic framework for interacting quantum systems that enables studying complex phenomena, characterised by the coexistence and competition of various broken symmetry states of matter. The approach unveils the hidden unity behind seemingly unrelated physical phenomena, thus establishing exact connections between them. This leads to the fundamental concept of universality of physical phenomena, a general concept not restricted to the domain of critical behaviour. Key to our framework is the concept of languages and the construction of dictionaries relating them.
Quantum phases of matter are resources for notions of quantum computation. In this work, we establish a new link between concepts of quantum information theory and condensed matter physics by presenting a unified understanding of symmetry-protected topological (SPT) order protected by subsystem symmetries and its relation to measurement-based quantum computation (MBQC). The key unifying ingredient is the concept of quantum cellular automata (QCA) which we use to define subsystem symmetries acting on rigid lower-dimensional lines or fractals on a 2D lattice. Notably, both types of symmetries are treated equivalently in our framework. We show that states within a non-trivial SPT phase protected by these symmetries are indicated by the presence of the same QCA in a tensor network representation of the state, thereby characterizing the structure of entanglement that is uniformly present throughout these phases. By also formulating schemes of MBQC based on these QCA, we are able to prove that most of the phases we construct are computationally universal phases of matter, in which every state is a resource for universal MBQC. Interestingly, our approach allows us to construct computational phases which have practical advantages over previous examples, including a computational speedup. The significance of the approach stems from constructing novel computationally universal phases of matter and showcasing the power of tensor networks and quantum information theory in classifying subsystem SPT order.
Quantum phases of matter are resources for notions of quantum computation. In this work, we establish a new link between concepts of quantum information theory and condensed matter physics by presenting a unified understanding of symmetry-protected topological (SPT) order protected by subsystem symmetries and its relation to measurement-based quantum computation (MBQC). The key unifying ingredient is the concept of quantum cellular automata (QCA) which we use to define subsystem symmetries acting on rigid lower-dimensional lines or fractals on a 2D lattice. Notably, both types of symmetries are treated equivalently in our framework. We show that states within a non-trivial SPT phase protected by these symmetries are indicated by the presence of the same QCA in a tensor network representation of the state, thereby characterizing the structure of entanglement that is uniformly present throughout these phases. By also formulating schemes of MBQC based on these QCA, we are able to prove that most of the phases we construct are computationally universal phases of matter, in which every state is a resource for universal MBQC. Interestingly, our approach allows us to construct computational phases which have practical advantages over previous examples, including a computational speedup. The significance of the approach stems from constructing novel computationally universal phases of matter and showcasing the power of tensor networks and quantum information theory in classifying subsystem SPT order.
An algebraic theory of dualities is developed based on the notion of bond algebras. It deals with classical and quantum dualities in a unified fashion explaining the precise connection between quantum dualities and the low temperature (strong-coupling)/high temperature (weak-coupling) dualities of classical statistical mechanics (or (Euclidean) path integrals). Its range of applications includes discrete lattice, continuum field and gauge theories. Dualities are revealed to be local, structure-preserving mappings between model-specific bond algebras that can be implemented as unitary transformations, or partial isometries if gauge symmetries are involved. This characterization permits us to search systematically for dualities and self-dualities in quantum models of arbitrary system size, dimensionality and complexity, and any classical model admitting a transfer matrix or operator representation. In particular, special dualities such as exact dimensional reduction, emergent and gauge-reducing dualities that solve gauge constraints can be easily understood in terms of mappings of bond algebras. As a new example, we show that the ℤ2 Higgs model is dual to the extended toric code model in any number of dimensions. Non-local transformations such as dual variables and Jordan–Wigner dictionaries are algorithmically derived from the local mappings of bond algebras. This permits us to establish a precise connection between quantum dual and classical disorder variables. Our bond-algebraic approach goes beyond the standard approach to classical dualities, and could help resolve the long-standing problem of obtaining duality transformations for lattice non-Abelian models. As an illustration, we present new dualities in any spatial dimension for the quantum Heisenberg model. Finally, we discuss various applications including location of phase boundaries, spectral behavior and, notably, we show how bond-algebraic dualities help constrain and realize fermionization in an arbitrary number of spatial dimensions.
We propose a unified mathematical framework for classifying phases of matter. The framework is based on different types of combinatorial structures with a notion of locality called lattices. A tensor lattice is a local prescription that associates tensor networks to those lattices. Different lattices are related by local operations called moves. Those local operations define consistency conditions for the tensors of the tensor network, the solutions to which yield exactly solvable models for all kinds of phases. We implement the framework to obtain models for symmetry-breaking and topological phases in up to three space-time dimensions, their boundaries, defects, domain walls and symmetries, as well as their anyons for 2+1-dimensional systems. We also deliver ideas of how other kinds of phases, like SPT/SET, fermionic, free-fermionic, chiral, and critical phases, can be described within our framework. We also define another structure called contracted tensor lattices which generalize tensor lattices: The former associate tensors instead of tensor networks to lattices, and the consistency conditions for those tensors are defined by another kind of local operation called gluings. Using this generalization, our framework also covers mathematical structures like axiomatic (non-fully extended or defective) TQFTs, that do not directly describe phases on a microscopic physical level, but formalize certain aspects of potential phases, like the anyon statistics of 2+1-dimensional phases. We also introduce the very powerful concept of (contracted) tensor lattice mapping, unifying a lots of different operations, such as stacking, anyon fusion, anyon condensation, equivalence of different fixed point models, taking the Drinfel'd centre, trivial defects or interpreting a bosonic model as a fermionic model.