The **topological entanglement entropy**^{ [1] }^{ [2] }^{ [3] } or *topological entropy*, usually denoted by *γ*, is a number characterizing many-body states that possess topological order.

A non-zero topological entanglement entropy reflects the presence of long range quantum entanglements in a many-body quantum state. So the topological entanglement entropy links topological order with pattern of long range quantum entanglements.

Given a topologically ordered state, the topological entropy can be extracted from the asymptotic behavior of the Von Neumann entropy measuring the quantum entanglement between a spatial block and the rest of the system. The entanglement entropy of a simply connected region of boundary length *L*, within an infinite two-dimensional topologically ordered state, has the following form for large *L*:

*-γ* is the topological entanglement entropy.

The topological entanglement entropy is equal to the logarithm of the total quantum dimension of the quasiparticle excitations of the state.

For example, the simplest fractional quantum Hall states, the Laughlin states at filling fraction 1/*m*, have *γ* = ½log(*m*). The *Z*_{2} fractionalized states, such as topologically ordered states of *Z*_{2} spin-liquid, quantum dimer models on non-bipartite lattices, and Kitaev's toric code state, are characterized *γ* = log(2).

In physics, **topological order** is a kind of order in the zero-temperature phase of matter. Macroscopically, topological order is defined and described by robust ground state degeneracy and quantized non-Abelian geometric phases of degenerate ground states. Microscopically, topological orders correspond to patterns of long-range quantum entanglement. States with different topological orders cannot change into each other without a phase transition.

A **spin ice** is a magnetic substance that does not have a single minimal-energy state. It has magnetic moments (i.e. "spin") as elementary degrees of freedom which are subject to frustrated interactions. By their nature, these interactions prevent the moments from exhibiting a periodic pattern in their orientation down to a temperature much below the energy scale set by the said interactions. Spin ices show low-temperature properties, residual entropy in particular, closely related to those of common crystalline water ice. The most prominent compounds with such properties are dysprosium titanate (Dy_{2}Ti_{2}O_{7}) and holmium titanate (Ho_{2}Ti_{2}O_{7}). The orientation of the magnetic moments in spin ice resembles the positional organization of hydrogen atoms (more accurately, ionized hydrogen, or protons) in conventional water ice (see figure 1).

A **topological quantum computer** is a theoretical quantum computer that employs two-dimensional quasiparticles called anyons, whose world lines pass around one another to form braids in a three-dimensional spacetime. These braids form the logic gates that make up the computer. The advantage of a quantum computer based on quantum braids over using trapped quantum particles is that the former is much more stable. Small, cumulative perturbations can cause quantum states to decohere and introduce errors in the computation, but such small perturbations do not change the braids' topological properties. This is like the effort required to cut a string and reattach the ends to form a different braid, as opposed to a ball bumping into a wall. Alexei Kitaev proposed topological quantum computation in 1997. While the elements of a topological quantum computer originate in a purely mathematical realm, experiments in fractional quantum Hall systems indicate these elements may be created in the real world using semiconductors made of gallium arsenide at a temperature of near absolute zero and subjected to strong magnetic fields.

**Oleg Sushkov** is a professor at the University of New South Wales and a leader in the field of high temperature super-conductors. Educated in Russia in quantum mechanics and nuclear physics, he now teaches in Australia.

**Jozef T. Devreese** is a Belgian scientist, with a long career in condensed matter physics. He is Professor Emeritus of Theoretical Physics at the Universiteit Antwerpen.

In condensed matter physics, a **string-net** is an extended object whose collective behavior has been proposed as a physical mechanism for topological order by Michael A. Levin and Xiao-Gang Wen. A particular string-net model may involve only closed loops; or networks of oriented, labeled strings obeying branching rules given by some gauge group; or still more general networks.

**Quantum dimer models** were introduced to model the physics of resonating valence bond (RVB) states in lattice spin systems. The only degrees of freedom retained from the motivating spin systems are the valence bonds, represented as dimers which live on the lattice bonds. In typical dimer models, the dimers do not overlap.

**Xiao-Gang Wen** is a Chinese-American physicist. He is a Cecil and Ida Green Professor of Physics at the Massachusetts Institute of Technology and Distinguished Visiting Research Chair at the Perimeter Institute for Theoretical Physics. His expertise is in condensed matter theory in strongly correlated electronic systems. In Oct. 2016, he was awarded the Oliver E. Buckley Condensed Matter Prize.

**Subir Sachdev** is Herchel Smith Professor of Physics at Harvard University specializing in condensed matter. He was elected to the U.S. National Academy of Sciences in 2014, and received the Lars Onsager Prize from the American Physical Society and the Dirac Medal from the ICTP in 2018.

**Yambo** is a computer software package for studying many-body theory aspects of solids and molecule systems. It calculates the excited state properties of physical systems from first principles, e.g., from quantum mechanics law without the use of empirical data. It is an open-source software released under the GNU General Public License (GPL). However the main development repository is provate and only a subset of the features available in the private repository are cloned into the public repository and thus distributed.

The ** SP formula** for the dephasing rate of a particle that moves in a fluctuating environment unifies various results that have been obtained, notably in condensed matter physics, with regard to the motion of electrons in a metal. The general case requires to take into account not only the temporal correlations but also the spatial correlations of the environmental fluctuations. These can be characterized by the spectral form factor , while the motion of the particle is characterized by its power spectrum . Consequently, at finite temperature the expression for the dephasing rate takes the following form that involves

The **toric code** is a topological quantum error correcting code, and an example of a stabilizer code, defined on a two-dimensional spin lattice It is the simplest and most well studied of the quantum double models. It is also the simplest example of topological order—*Z*_{2} topological order (first studied in the context of *Z*_{2} spin liquid in 1991). The toric code can also be considered to be a *Z*_{2} lattice gauge theory in a particular limit. It was introduced by Alexei Kitaev.

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.

**Tilman Esslinger** is a German experimental physicist. He is Professor at ETH Zurich, Switzerland, and works in the field of ultracold quantum gases and optical lattices.

**Scissors Modes** are collective excitations in which two particle systems move with respect to each other conserving their shape. For the first time they were predicted to occur in deformed atomic nuclei by N. LoIudice and F. Palumbo, who used a semiclassical Two Rotor Model, whose solution required a realization of the O(4) algebra that was not known in mathematics. In this model protons and neutrons were assumed to form two interacting rotors to be identified with the blades of scissors. Their relative motion (Fig.1) generates a magnetic dipole moment whose coupling with the electromagnetic field provides the signature of the mode.

**Quantum illumination** is a paradigm for target detection that employs quantum entanglement between a signal electromagnetic mode and an idler electromagnetic mode, as well as joint measurement of these modes. The signal mode is propagated toward a region of space, and it is either lost or reflected, depending on whether a target is absent or present, respectively. In principle, quantum illumination can be beneficial even if the original entanglement is completely destroyed by a lossy and noisy environment.

**James (Jim) P. Eisenstein** is the Frank J. Roshek Professor of Physics and Applied Physics at the physics department of California Institute of Technology.

**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.

Many-body localization (MBL) is a dynamical phenomenon which leads to the breakdown of equilibrium statistical mechanics in isolated many-body systems. Such systems never reach local thermal equilibrium, and retain local memory of their initial conditions for infinite times. One can still define a notion of phase structure in these out-of-equilibrium systems. Strikingly, MBL can even enable new kinds of exotic orders that are disallowed in thermal equilibrium --- a phenomenon that goes by the name of **localization protected quantum order** (LPQO) or **eigenstate order**

- ↑ Hamma, Alioscia; Ionicioiu, Radu; Zanardi, Paolo (28 March 2004). "Ground state entanglement and geometric entropy in the Kitaev model".
*Physics Letters A*.**337**(1–2). arXiv: quant-ph/0406202 . - ↑ Kitaev, Alexei; Preskill, John (24 March 2006). "Topological Entanglement Entropy".
*Physical Review Letters*.**96**(11): 110404. arXiv: hep-th/0510092 . doi:10.1103/physrevlett.96.110404. ISSN 0031-9007. PMID 16605802. - ↑ Levin, Michael; Wen, Xiao-Gang (24 March 2006). "Detecting Topological Order in a Ground State Wave Function".
*Physical Review Letters*.**96**(11): 110405. arXiv: cond-mat/0510613 . doi:10.1103/physrevlett.96.110405. ISSN 0031-9007. PMID 16605803.

- Haque, Masudul; Zozulya, Oleksandr; Schoutens, Kareljan (6 February 2007). "Entanglement Entropy in Fermionic Laughlin States".
*Physical Review Letters*.**98**(6): 060401. arXiv: cond-mat/0609263 . doi:10.1103/physrevlett.98.060401. ISSN 0031-9007. - Furukawa, Shunsuke; Misguich, Grégoire (5 June 2007). "Topological entanglement entropy in the quantum dimer model on the triangular lattice".
*Physical Review B*.**75**(21): 214407. arXiv: cond-mat/0612227 . doi:10.1103/physrevb.75.214407. ISSN 1098-0121.

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