Curvature Renormalization Group Method information
In theoretical physics, the curvature renormalization group (CRG) method is an analytical approach to determine the phase boundaries and the critical behavior of topological systems. Topological phases are phases of matter that appear in certain quantum mechanical systems at zero temperature because of a robust degeneracy in the ground-state wave function. They are called topological because they can be described by different (discrete) values of a nonlocal topological invariant. This is to contrast with non-topological phases of matter (e.g. ferromagnetism) that can be described by different values of a local order parameter. States with different values of the topological invariant cannot change into each other without a phase transition. The topological invariant is constructed from a curvature function that can be calculated from the bulk Hamiltonian of the system. At the phase transition, the curvature function diverges, and the topological invariant correspondingly jumps abruptly from one value to another. The CRG method works by detecting the divergence in the curvature function, and thus determining the boundaries between different topological phases. Furthermore, from the divergence of the curvature function, it extracts scaling laws that describe the critical behavior, i.e. how different quantities (such as susceptibility or correlation length) behave as the topological phase transition is approached. The CRG method has been successfully applied to a variety of static, periodically driven, weakly and strongly interacting systems to classify the nature of the corresponding topological phase transitions.[1][2][3][4][5][6][7][8][9][10]
^Chen, W. (2016). "Scaling theory of topological phase transitions". J. Phys.: Condens. Matter. 28 (2): 055601. arXiv:1505.05345. Bibcode:2016JPCM...28e5601C. doi:10.1088/0953-8984/28/5/055601. PMID 26790004. S2CID 26562531.
^Chen, Wei; Sigrist, Manfred; Schnyder, Andreas P (2016-09-14). "Scaling theory of Z topological invariants". Journal of Physics: Condensed Matter. 28 (36): 365501. arXiv:1604.07662. doi:10.1088/0953-8984/28/36/365501. ISSN 0953-8984. PMID 27400801. S2CID 46854459.
^Kourtis, Stefanos; Neupert, Titus; Mudry, Christopher; Sigrist, Manfred; Chen, Wei (2017-11-10). "Weyl-type topological phase transitions in fractional quantum Hall like systems". Physical Review B. 96 (20): 205117. arXiv:1708.04244. Bibcode:2017PhRvB..96t5117K. doi:10.1103/PhysRevB.96.205117. ISSN 2469-9950. S2CID 118933016.
^Chen, Wei; Schnyder, Andreas P. (July 2019). "Universality classes of topological phase transitions with higher-order band crossing". New Journal of Physics. 21 (7): 073003. arXiv:1901.11468. Bibcode:2019NJPh...21g3003C. doi:10.1088/1367-2630/ab2a2d. ISSN 1367-2630. S2CID 119057056.
^Chen, Wei (2018-03-14). "Weakly interacting topological insulators: Quantum criticality and the renormalization group approach". Physical Review B. 97 (11): 115130. arXiv:1801.00697. Bibcode:2018PhRvB..97k5130C. doi:10.1103/PhysRevB.97.115130. ISSN 2469-9950. S2CID 119078563.
^Molignini, Paolo; Chen, Wei; Chitra, R. (2018-09-17). "Universal quantum criticality in static and Floquet-Majorana chains". Physical Review B. 98 (12): 125129. arXiv:1805.09698. Bibcode:2018PhRvB..98l5129M. doi:10.1103/PhysRevB.98.125129. ISSN 2469-9950. S2CID 62836882.
^Chen, Wei; Sigrist, Manfred (2019-03-15), Luo, Huixia (ed.), "Topological Phase Transitions: Criticality, Universality, and Renormalization Group Approach", Advanced Topological Insulators, John Wiley & Sons, Inc., pp. 239–280, doi:10.1002/9781119407317.ch7, ISBN 978-1-119-40731-7, S2CID 126525389
In theoretical physics, the curvaturerenormalizationgroup (CRG) method is an analytical approach to determine the phase boundaries and the critical behavior...
connect different renormalization constants. The first gauge theory quantized was quantum electrodynamics (QED). The first methods developed for this...
property of space and time or four-dimensional spacetime. In particular, the curvature of spacetime is directly related to the energy and momentum of whatever...
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the curvature and covariant derivatives, the Feynman rules for QCD can be derived through path integral methods. The techniques for renormalization of...
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curvature tensor. The concept of Newton's gravity: "two masses attract each other" replaced by the geometrical argument: "mass transform curvatures of...
theory of gravitons due to an outstanding mathematical problem with renormalization in general relativity. In string theory, believed by some to be a consistent...
renormalization works very well in practice, Feynman was never entirely comfortable with its mathematical validity, even referring to renormalization...
development of renormalizationgroup in curved space, the discovery of asymptotic conformal invariance, and a full description of curvature-induced phase...
scaling behaviour as they approach criticality—can be shown, via renormalizationgroup theory, to share the same fundamental dynamics. For instance, the...
involve quantitative behavioral finance, phase field models, and renormalizationmethods in differential equations. His initial research focused on rigorous...
Fourier methods are applicable are complex-valued, and possibly vector-valued. Still further generalization is possible to functions on groups, which,...
a meaningful physical theory. At low energies, the logic of the renormalizationgroup tells us that, despite the unknown choices of these infinitely many...
language of differential forms, specifically as an adjoint bundle-valued curvature 2-form (note that fibers of the adjoint bundle are the su(3) Lie algebra);...
Beta function describing the change of coupling constant under the renormalizationgroup flow γ 1. Dirac matrix 2. One of the two conformal superghost...
a regularization to be applied, to obtain the renormalized amplitudes. In order for the renormalization to be meaningful, coherent and consistent, the...
problem of constructing models of cosmological inflation) occurs: renormalization theory predicts that scalar fields should acquire large masses. The...
functional theory and variational Monte Carlo and 1992 density matrix renormalizationgroup (DMRG).[citation needed] In 2014, variational principles were part...
quantum field theorists before the development in the 1970s of the renormalizationgroup, a mathematical formalism for scale transformations that provides...
called the metric tensor. The Einstein field equations describe how this curvature is produced. Newtonian gravitation is now superseded by Einstein's theory...
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