In many-body physics, most commonly within condensed-matter physics, a gapped Hamiltonian is a Hamiltonian for an infinitely large many-body system where there is a finite energy gap separating the (possibly degenerate) ground space from the first excited states. A Hamiltonian that is not gapped is called gapless.
The property of being gapped or gapless is formally defined through a sequence of Hamiltonians on finite lattices in the thermodynamic limit. [1] [ unreliable source? ]
An example is the BCS Hamiltonian in the theory of superconductivity.
In quantum many-body systems, ground states of gapped Hamiltonians have exponential decay of correlations. [2] [3] [4]
In quantum field theory, a continuum limit of many-body physics, a gapped Hamiltonian induces a mass gap.
A dynamical billiard is a dynamical system in which a particle alternates between free motion and specular reflections from a boundary. When the particle hits the boundary it reflects from it without loss of speed. Billiards are Hamiltonian idealizations of the game of billiards, but where the region contained by the boundary can have shapes other than rectangular and even be multidimensional. Dynamical billiards may also be studied on non-Euclidean geometries; indeed, the first studies of billiards established their ergodic motion on surfaces of constant negative curvature. The study of billiards which are kept out of a region, rather than being kept in a region, is known as outer billiard theory.
Quantum Monte Carlo encompasses a large family of computational methods whose common aim is the study of complex quantum systems. One of the major goals of these approaches is to provide a reliable solution of the quantum many-body problem. The diverse flavors of quantum Monte Carlo approaches all share the common use of the Monte Carlo method to handle the multi-dimensional integrals that arise in the different formulations of the many-body problem.
In quantum field theory and statistical mechanics, the 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.
Adiabatic quantum computation (AQC) is a form of quantum computing which relies on the adiabatic theorem to do calculations and is closely related to quantum annealing.
The numerical renormalization group (NRG) is a technique devised by Kenneth Wilson to solve certain many-body problems where quantum impurity physics plays a key role.
In computational complexity theory, QMA, which stands for Quantum Merlin Arthur, is the set of languages for which, when a string is in the language, there is a polynomial-size quantum proof that convinces a polynomial time quantum verifier of this fact with high probability. Moreover, when the string is not in the language, every polynomial-size quantum state is rejected by the verifier with high probability.
Vladimir E. Korepin is a professor at the C. N. Yang Institute of Theoretical Physics of the Stony Brook University. Korepin made research contributions in several areas of mathematics and physics.
Patrick Hayden is a physicist and computer scientist active in the fields of quantum information theory and quantum computing. He is currently a professor in the Stanford University physics department and a distinguished research chair at the Perimeter Institute for Theoretical Physics. Prior to that he held a Canada Research Chair in the physics of information at McGill University. He received a B.Sc. (1998) from McGill University and won a Rhodes Scholarship to study for a D.Phil. (2001) at the University of Oxford under the supervision of Artur Ekert. In 2007 he was awarded the Sloan Research Fellowship in Computer Science. He was a Canadian Mathematical Society Public Lecturer in 2008 and received a Simons Investigator Award in 2014.
In quantum many-body physics, topological degeneracy is a phenomenon in which the ground state of a gapped many-body Hamiltonian becomes degenerate in the limit of large system size such that the degeneracy cannot be lifted by any local perturbations.
Symmetry-protected topological (SPT) order is a kind of order in zero-temperature quantum-mechanical states of matter that have a symmetry and a finite energy gap.
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.
The Lieb–Robinson bound is a theoretical upper limit on the speed at which information can propagate in non-relativistic quantum systems. It demonstrates that information cannot travel instantaneously in quantum theory, even when the relativity limits of the speed of light are ignored. The existence of such a finite speed was discovered mathematically by Elliott H. Lieb and Derek W. Robinson in 1972. It turns the locality properties of physical systems into the existence of, and upper bound for this speed. The bound is now known as the Lieb–Robinson bound and the speed is known as the Lieb–Robinson velocity. This velocity is always finite but not universal, depending on the details of the system under consideration. For finite-range, e.g. nearest-neighbor, interactions, this velocity is a constant independent of the distance travelled. In long-range interacting systems, this velocity remains finite, but it can increase with the distance travelled.
A fracton is an emergent topological quasiparticle excitation which is immobile when in isolation. Many theoretical systems have been proposed in which fractons exist as elementary excitations. Such systems are known as fracton models. Fractons have been identified in various CSS codes as well as in symmetric tensor gauge theories.
Many-body localization (MBL) is a dynamical phenomenon occurring in isolated many-body quantum systems. It is characterized by the system failing to reach thermal equilibrium, and retaining a memory of its initial condition in local observables for infinite times.
In quantum mechanics, the spectral gap of a system is the energy difference between its ground state and its first excited state. The mass gap is the spectral gap between the vacuum and the lightest particle. A Hamiltonian with a spectral gap is called a gapped Hamiltonian, and those that do not are called gapless.
Hamiltonian simulation is a problem in quantum information science that attempts to find the computational complexity and quantum algorithms needed for simulating quantum systems. Hamiltonian simulation is a problem that demands algorithms which implement the evolution of a quantum state efficiently. The Hamiltonian simulation problem was proposed by Richard Feynman in 1982, where he proposed a quantum computer as a possible solution since the simulation of general Hamiltonians seem to grow exponentially with respect to the system size.
Gian Michele Graf is a Swiss mathematical physicist.
Roberto Longo is an Italian mathematician, specializing in operator algebras and quantum field theory.
Klaus Fredenhagen is a German theoretical physicist who works on the mathematical foundations of quantum field theory.
Stability of matter refers to the problem of showing rigorously that a large number of charged quantum particles can coexist and form macroscopic objects, like ordinary matter. The first proof was provided by Freeman Dyson and Andrew Lenard in 1967–1968, but a shorter and more conceptual proof was found later by Elliott Lieb and Walter Thirring in 1975.