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arXiv:cond-mat/0207076 · International Journal of Modern Physics B

HIDDEN UNITY IN THE QUANTUM DESCRIPTION OF MATTER

Gerardo Ortíz, C. D. BATISTA

Published Nov 10, 2003• 1 citation
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Abstract

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.

DOI
10.1142/s0217979203020521

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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.
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.
Exactly solvable quantum systems represent a fascinating and critically important class of models in theoretical physics, offering exact solutions to non-trivial problems that often shed light on the behavior of more complex, intractable systems. The remarkable feature of these systems lies not only in their solvability but also in the underlying mathematical structures that facilitate such solutions. This paper explores the concept of a universal algebraic structure that underpins exactly solvable quantum systems. We delve into how diverse models, ranging from one-dimensional spin chains to many-body systems and field theories, exhibit common algebraic symmetries, such as Lie algebras, affine Lie algebras, and quantum groups. These algebraic frameworks provide a unified language for understanding integrability, the existence of an infinite number of conserved quantities, and the ability to construct exact eigenstates. We review the historical development of these ideas, from the Bethe ansatz to the development of the inverse scattering method and quantum groups. Our discussion highlights how these mathematical structures are not merely tools for calculation but fundamental descriptors of the physical reality of these systems, offering profound insights into non-perturbative quantum phenomena. The identification of a universal algebraic architecture suggests a deeper coherence in the landscape of quantum mechanics, paving the way for new discoveries and classifications of quantum models.
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.
Human languages employ constructions that tacitly assume specific properties of the limited range of phenomena they evolved to describe. These assumed properties are true features of that limited context, but may not be general or precise properties of all the physical situations allowed by fundamental physics. In brief, human languages contain `excess baggage' that must be qualified, discarded, or otherwise reformed to give a clear account in the context of fundamental physics of even the everyday phenomena that the languages evolved to describe. The surest route to clarity is to express the constructions of human languages in the language of fundamental physical theory, not the other way around. These ideas are illustrated by an analysis of the verb `to happen' and the word `reality' in special relativity and the modern quantum mechanics of closed systems.
An SU(4) model of high-temperature superconductivity and antiferromagnetism has recently been proposed. The SO(5) group employed by Zhang [Science 275, 1089 (1997)] is embedded in this SU(4) as a subgroup, suggesting a connection between our SU(4) model and the Zhang SO(5) model. In order to understand the relationship between the the two models, we have used generalized coherent states to analyze the nature of the SO(5) subgroup. By constructing coherent-state energy surfaces, we demonstrate explicitly that the $\mathrm{SU}(4)\ensuremath{\supset}\mathrm{SO}(5)$ symmetry can be interpreted as a critical dynamical symmetry interpolating between superconducting and antiferromagnetic phases, and that this critical dynamical symmetry has many similarities to critical dynamical symmetries identified previously in other fields of physics. More generally, we demonstrate with this example that the mathematical techniques associated with generalized coherent states may have powerful applications in condensed-matter physics because they provide a clear connection between microscopic many-body theories and their broken-symmetry approximate solutions. In addition, these methods may be interpreted as defining the most general Bogoliubov transformation subject to a Lie group symmetry constraint, thus providing a mathematical connection between algebraic formulations and the language of quasiparticle theory. Finally, we suggest that the identification of the SO(5) symmetry as a critical dynamical symmetry implies deep algebraic connections between high-temperature superconductors and seemingly unrelated phenomena in other field of physics.
Human languages employ constructions that tacitly assume specific properties of the limited range of phenomena they evolved to describe. These assumed properties are true features of that limited context, but may not be general or precise properties of all the physical situations allowed by fundamental physics. In brief, human languages contain `excess baggage' that must be qualified, discarded, or otherwise reformed to give a clear account in the context of fundamental physics of even the everyday phenomena that the languages evolved to describe. The surest route to clarity is to express the constructions of human languages in the language of fundamental physical theory, not the other way around. These ideas are illustrated by an analysis of the verb `to happen' and the word `reality' in special relativity and the modern quantum mechanics of closed systems.