Thomas C. Spencer | |
---|---|

Born | December 24, 1946 77) | (age

Education | AB, University of California, Berkeley PhD, New York University |

Employer | Institute for Advanced Study |

Title | Professor |

Spouse | Bridget Murphy |

Awards | Henri Poincaré Prize (2015) Dannie Heineman Prize for Mathematical Physics (1991) |

**Thomas C. Spencer** (born December 24, 1946) is an American mathematical physicist, known in particular for important contributions to constructive quantum field theory, statistical mechanics, and spectral theory of random operators. He is an emeritus faculty member at the Institute for Advanced Study.^{ [1] }

Spencer earned his doctorate in 1972 from New York University with a dissertation titled *Perturbation of the Po2 Quantum Field Hamiltonian* written under the direction of James Glimm. Since 1986, he has been a faculty member in the School of Mathematics at the Institute for Advanced Study.^{[ citation needed ]}

- Together with James Glimm and Arthur Jaffe he invented the cluster expansion approach to quantum field theory that is widely used in constructive field theory.
^{ [2] } - Together with Jürg Fröhlich and Barry Simon, he invented the approach of the infrared bound, which has now become a classical tool to derive phase transitions in various models of statistical mechanics.
^{ [3] } - Together with Jürg Fröhlich, he devised a 'multi-scale analysis' to provide, for the first time, mathematical proofs of: the Kosterlitz–Thouless transition,
^{ [4] }the phase transition in the one-dimensional ferromagnetic Ising model with interactions^{ [5] }and Anderson localization in arbitrary dimension.^{ [6] } - Together with David Brydges, he proved that the scaling limit of the self-avoiding walk in dimension greater or equal than 5 is Gaussian, with variance growing linearly in time.
^{ [7] }To achieve this result, they invented the technique of the lace expansion that since then has had wide application in probability on graphs.^{ [8] }

Spencer is a member of the United States National Academy of Sciences,^{ [1] } and the recipient of the Dannie Heineman Prize for Mathematical Physics (joint with Jürg Fröhlich, "*For their joint work in providing rigorous mathematical solutions to some outstanding problems in statistical mechanics and field theory.*").^{ [9] }^{ [10] }

The **Ising model**, named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that represent magnetic dipole moments of atomic "spins" that can be in one of two states. The spins are arranged in a graph, usually a lattice, allowing each spin to interact with its neighbors. Neighboring spins that agree have a lower energy than those that disagree; the system tends to the lowest energy but heat disturbs this tendency, thus creating the possibility of different structural phases. The model allows the identification of phase transitions as a simplified model of reality. The two-dimensional square-lattice Ising model is one of the simplest statistical models to show a phase transition.

The **classical XY model** is a lattice model of statistical mechanics. In general, the XY model can be seen as a specialization of Stanley's *n*-vector model for *n* = 2.

The **Berezinskii–Kosterlitz–Thouless** (**BKT**) **transition** is a phase transition of the two-dimensional (2-D) XY model in statistical physics. It is a transition from bound vortex-antivortex pairs at low temperatures to unpaired vortices and anti-vortices at some critical temperature. The transition is named for condensed matter physicists Vadim Berezinskii, John M. Kosterlitz and David J. Thouless. BKT transitions can be found in several 2-D systems in condensed matter physics that are approximated by the XY model, including Josephson junction arrays and thin disordered superconducting granular films. More recently, the term has been applied by the 2-D superconductor insulator transition community to the pinning of Cooper pairs in the insulating regime, due to similarities with the original vortex BKT transition.

**Critical exponents** describe the behavior of physical quantities near continuous phase transitions. It is believed, though not proven, that they are universal, i.e. they do not depend on the details of the physical system, but only on some of its general features. For instance, for ferromagnetic systems, the critical exponents depend only on:

In quantum field theory and statistical mechanics, the **Hohenberg–Mermin–Wagner theorem** or **Mermin–Wagner theorem** states that continuous symmetries cannot be spontaneously broken at finite temperature in systems with sufficiently short-range interactions in dimensions *d* ≤ 2. Intuitively, this means that long-range fluctuations can be created with little energy cost, and since they increase the entropy, they are favored.

**Barry Malcolm McCoy** is an American physicist, known for his contributions to classical statistical mechanics, integrable models and conformal field theories.

**Tai Tsun Wu** is a Chinese-born American physicist and writer well known for his contributions to high-energy nuclear physics and statistical mechanics.

**Jürg Martin Fröhlich** is a Swiss mathematician and theoretical physicist. He is best known for introducing rigorous techniques for the analysis of statistical mechanics models, in particular continuous symmetry breaking, and for pioneering the study of topological phases of matter using low-energy effective field theories.

In statistical mechanics, the **Griffiths inequality**, sometimes also called **Griffiths–Kelly–Sherman inequality** or **GKS inequality**, named after Robert B. Griffiths, is a correlation inequality for ferromagnetic spin systems. Informally, it says that in ferromagnetic spin systems, if the 'a-priori distribution' of the spin is invariant under spin flipping, the correlation of any monomial of the spins is non-negative; and the two point correlation of two monomial of the spins is non-negative.

The **Kibble–Zurek mechanism** (**KZM**) describes the non-equilibrium dynamics and the formation of topological defects in a system which is driven through a continuous phase transition at finite rate. It is named after Tom W. B. Kibble, who pioneered the study of domain structure formation through cosmological phase transitions in the early universe, and Wojciech H. Zurek, who related the number of defects it creates to the critical exponents of the transition and to its rate—to how quickly the critical point is traversed.

**Quantum simulators** permit the study of a quantum system in a programmable fashion. In this instance, simulators are special purpose devices designed to provide insight about specific physics problems. Quantum simulators may be contrasted with generally programmable "digital" quantum computers, which would be capable of solving a wider class of quantum problems.

**Antti Kupiainen** is a Finnish mathematical physicist.

**Jorge V. José** is a Mexican/American physicist born in Mexico City. Currently the James H. Rudy Distinguished Professor of Physics at Indiana University. He has made seminal contributions to research in a variety of disciplines, including condensed matter physics, nonlinear dynamics, quantum chaos, biological physics, computational neuroscience and lately precision psychiatry. His pioneering work on the two-dimensional x-y model has been exceedingly influential in many areas of physics and has garnered many citations. He edited the book on the “40 Years of Berezinskii-Kosterlitz-Thouless Theory”, on two-dimensional topological phase transitions in 2013. Three years later KT were awarded the 2016 Nobel Physics Prize.

**David Chandos Brydges** is a mathematical physicist.

**Tetsuji Miwa** is a Japanese mathematician, specializing in mathematical physics.

**Gian Michele Graf** is a Swiss mathematical physicist.

The **KTHNY-theory** describes the melting of crystals in two dimensions (2D). The name is derived from the initials of the surnames of John Michael Kosterlitz, David J. Thouless, Bertrand Halperin, David R. Nelson, and A. Peter Young, who developed the theory in the 1970s. It is, beside the Ising model in 2D and the XY model in 2D, one of the few theories, which can be solved analytically and which predicts a phase transition at a temperature .

The **quantum clock model** is a quantum lattice model. It is a generalisation of the transverse-field Ising model. It is defined on a lattice with states on each site. The Hamiltonian of this model is

**Giovanni Felder** is a Swiss mathematical physicist and mathematician, working at ETH Zurich. He specializes in algebraic and geometric properties of integrable models of statistical mechanics and quantum field theory.

**Krzysztof Gawędzki** was a Polish mathematical physicist, a graduate of the University of Warsaw and professor at the École normale supérieure de Lyon. He was primarily known for his research on quantum field theory and statistical physics. In 2022, he shared the Dannie Heineman Prize for Mathematical Physics with Antti Kupiainen.

- 1 2 IAS website
- ↑ Glimm, J; Jaffe, A; Spencer, T (1974). "The Wightman axioms and particle structure in the quantum field model".
*Ann. of Math*.**100**(3): 585–632. doi:10.2307/1970959. JSTOR 1970959. - ↑ Fröhlich, J.; Simon, B.; Spencer, T. (1976). "Infrared bounds, phase transitions and continuous symmetry breaking".
*Comm. Math. Phys*.**50**(1): 79–95. Bibcode:1976CMaPh..50...79F. CiteSeerX 10.1.1.211.1865 . doi:10.1007/bf01608557. S2CID 16501561. - ↑ Fröhlich, J.; Spencer, T. (1981). "The Kosterlitz–Thouless transition in two-dimensional abelian spin systems and the Coulomb gas".
*Comm. Math. Phys*.**81**(4): 527–602. Bibcode:1981CMaPh..81..527F. doi:10.1007/bf01208273. S2CID 73555642. - ↑ Fröhlich, J.; Spencer, T. (1982). "The phase transition in the one-dimensional Ising model with 1/
*r*^{2}interaction energy".*Comm. Math. Phys*.**84**(1): 87–101. Bibcode:1982CMaPh..84...87F. doi:10.1007/BF01208373. S2CID 122722140. - ↑ Fröhlich, J.; Spencer, T. (1983). "Absence of diffusion in the Anderson tight binding model for large disorder or low energy".
*Comm. Math. Phys*.**88**(2): 151–184. Bibcode:1983CMaPh..88..151F. doi:10.1007/bf01209475. S2CID 121435596. - ↑ Brydges, D.; Spencer, T. (1985). "Self-avoiding walk in 5 or more dimensions".
*Comm. Math. Phys*.**97**(1–2): 125–148. Bibcode:1985CMaPh..97..125B. doi:10.1007/bf01206182. S2CID 189832287. - ↑ Slade, G. (2006).
*The lace expansion and its applications*. Lecture Notes in Mathematics. Vol. 1879. Springer. ISBN 9783540311898. - ↑ APS website
- ↑ 1991 Dannie Heineman Prize for Mathematical Physics Recipient, American Physical Society. Accessed June 24, 2011

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