Quantum Field Theory of Many-Body Systems: From the Origin of Sound to an Origin of Light and Electrons
DOI: 10.1063/1.2155760
During the past two decades, a quiet but persistent paradigm shift in the quantum theory of solids has been steadily brewing. The field is currently in flux, and the uncertainty as to how it will shake down is palpable. In the old school, band theory—together with Lev Landau’s Fermi-liquid theory, which subsumes electron interactions into effective parameters—readily accounts for the basic difference among simple metals, insulators, and semiconductors. In metals, residual interactions that drive instabilities toward, for example, superconductivity or magnetism can be treated with Landau’s theory of phase transitions. The central message in Xiao-Gang Wen’s remarkably original new book, Quantum Field Theory of Many-Body Systems: From the Origin of Sound to an Origin of Light and Electrons , is that an exciting and rich world lies beyond Landau’s theories. Belying its familiar main title, the book takes its readers on an exotic tour to the outer reaches of modern many-body theory and seeks to distill the essential ingredients of a possible new paradigm.
Perhaps by necessity within a book this ambitious, the experimental physics fueling the theoretical exploration is largely absent. But such an absence ought not to be taken as dismissal of the pressing—even urgent—need for new ways to understand strongly interacting, many-electron systems. In fact, the experimental picture is quite clear: Many crystalline solids, in which the electrons at the Fermi energy come from partially filled atomic d or f shells, are nonconformist “bad actors.” They stubbornly refuse to fit into the standard framework of band and Landau theory. High-temperature superconductors are the best-studied example but appear to be only the tip of the iceberg of such complex crystalline solids. Constructing a new framework, however, has been exceedingly challenging, with experts disagreeing on even its rough outline.
Wen has arguably been one of the most creative theorists seeking to supplant the existing foundations of solid-state theory. His creative approach finds its roots in the fractional quantum Hall effect, a phenomenon discovered in the early 1980s. When electrons are confined to two dimensions, cooled to cryogenic temperatures, and subjected to intense magnetic fields, they can condense into a novel quantum-liquid state. Like ordinary liquids, these fractional quantum Hall fluids are featureless. But hidden within, the electrons are undergoing a remarkably intricate sequence of dance patterns: They swirl around one another in a dizzying yet coherent fashion. Wen recognized that these dance patterns could be fruitfully characterized in terms of a special kind of hidden order, which he christened “topological order.” If confined to the surface of a torus (surely a gedanken experiment!), the ground state of the quantum Hall fluid is multiply degenerate: The ground states are locally indistinguishable and differ only in the nature of the nonlocal entanglement between electrons encircling the whole system. Such topologically ordered fluids generically support particle-like excitations that carry fractional quantum numbers: charge e/3 for the celebrated Laughlin quasiparticles.
Before venturing into such exotica, one finds that the early chapters of Wen’s book are, by and large, devoted to standard topics: the path-integral formulation of quantum mechanics, weakly interacting boson systems, and weakly interacting fermion systems, with the latter including Landau’s two pillars. But the author’s unique perspective is amply evident on almost every page. For example, Wen offers a new hydrodynamic approach to the Fermi liquid, and the Berry phase plays a rather central role throughout. Those early chapters could serve as a textbook to augment a standard graduate course in many-body theory.
In chapter 6, lattice gauge theory is demystified. Gauge symmetry is right-fully exposed as not being a symmetry at all but rather just a theoretical construct employing a many-to-one labeling of quantum states. That chapter serves as a point of departure from standard theory, with the latter part of the book devoted to Wen’s vision of an emerging new paradigm. Central to his vision is the notion of topological order and a more subtle, and apparently less well-defined, concept of quantum order. Exotic spin-liquid quantum states, which arise in toy models of quantum magnetism, are concrete examples. Despite their aesthetic appeal, the relevance of these exciting new theoretical developments to the experimental puzzles presented by complex, strongly correlated electronic crystals is currently unclear—although it would seem that the potential relevance is quite significant.
Throughout the book, Wen’s prose is informal yet strikingly clear and incisive. The message is buoyant and optimistic. Each section has a highlighted bullet or two encapsulating a key snippet of philosophy. Problems are scattered throughout. One minor stylistic quibble: Quantum field theories vacillate between 1+2 and 2+1 spacetime dimensions—perhaps a case of quantum dyslexia.
In the final chapter, Wen turns his attention to particle physics. He argues that it should be possible to obtain the standard model of particle physics—nonabelian gauge interactions, the photon, and Dirac fermions—starting with a simple lattice model of bosons with short-range interactions. Because fermions are intrinsically nonlocal (fermion operators anticommute at arbitrary large spatial separations), Wen maintains that they cannot be the input of any truly fundamental theory but, rather, must emerge from a more basic, entirely local theory of bosons. Frankly, I find his point of view appealing, but I am quite certain that such feelings are not shared by most of my particle-theory colleagues—including, presumably, Wen’s doctoral-thesis supervisor Edward Witten, the modern sage of string theory.
As a whole, Quantum Field Theory of Many-Body Systems is an inspirational and forward-looking book exploring the mysteries and never-ending wonders of many-particle quantum mechanics. One senses an approaching sea change in our understanding of complex electronic solids. Both for ambitious graduate students and for gray-haired veterans, Wen’s book offers a refreshing new look at the mysterious quantum world.
More about the Authors
Matthew Fisher. Kavli Institute of Theoretical Physics, University of California, Santa Barbara, US .
