Generalised on renormalization group

อารีฟ บุญเสือ

Syed Muntasir Mamun

N C K SC K SC Recent advances at the intersection of quantum information geometry and machine-learning approaches to the renormalisation group suggest a fertile reinterpretation of the Similarity Renormalization Group (SRG). The continuous unitary transformations that drive a Hamiltonian toward band-diagonal form may be viewed as geodesics on a manifold of quantum states (or of density operators) equipped with a monotone metric, or equivalently as successive layers of a deep network that progressively disentangle degrees of freedom. This independent study develops that correspondence. It reviews the information-geometric structures relevant to quantum field theories and many-body Hamiltonians, examines neural-network renormalisation-group (NN-RG) architectures, and maps the SRG flow—originally clarified in the pedagogical setting of Szpigel & Perry (2000)—onto both geometric and learning-theoretic languages. The resulting perspective yields analytic insights into the optimality of particular generators and practical algorithms for approximate Hamiltonian diagonalisation that combine SRG ideas with modern differentiable programming. Error-scaling diagnostics originating in the two-dimensional laboratory are re-expressed as statements about geodesic deviation and about generalisation error in the associated learning problem. Also available on Academia.edu: https://www.academia.edu/attachments/133749950/download_file?s=portfolio

Heber R. da Cruz, Robin B. Stinchcombe

The authors elucidate the violation of the Harris criterion that often occurs in real-space renormalization group (RSRG) treatments of quenched random systems. The system studied is the bond-diluted q-state Potts model on the diamond Berker lattice, since it allows either sign for the pure specific heat exponent alpha which is calculated exactly and satisfies a generalised hyperscaling relation involving the fractal dimension. It is argued that it is necessary to allow the bond probability distribution to evolve, under length scaling, from its initial binary form. The width of the distribution (excluding the weight of the missing bonds) is found to play a crucial role and is used as an additional parameter. These conclusions are confirmed by actual calculation for this system of the RG flow in a generalised parameter space, in which the random fixed point is located. The associated fixed-point distributions are discussed, and it is demonstrated that the generalised view allows the Harris criterion to be satisfied.

Mich�le Irac-Astaud, B. Jouvet

Kai-Yan Michael

Raf Dekeyser, G. Kamieniarz

It is possible to generalise in several ways the sefconsistency conditions inherent in the mean field approximation. One can then obtain a substantial improvement on the estimates of critical parameters through rather simple calculations; also critical exponents can be obtained through a combination with the mean-field renormalization group method. These techniques can also be applied to quantummechanical systems.