Phase transitions and dimensional reduction

Leo P. Kadanoff

Emilio Cobanera

Paul Fendley, Oleg Tchernyshyov

Yuan Yang, Zheng-Zhi Sun, Shi-Ju Ran +1 more

Identifying quantum phases and phase transitions is key to understanding complex phenomena in statistical physics. In this work, we propose an unconventional strategy to access quantum phases and phase transitions by visualization based on the distribution of ground states in Hilbert space. By mapping the quantum states in Hilbert space onto a two-dimensional feature space using an unsupervised machine learning method, distinct phases can be directly specified and quantum phase transitions can be well identified. Our proposal is benchmarked on gapped, critical, and topological phases in several strongly correlated spin systems. As this proposal directly learns quantum phases and phase transitions from the distributions of the quantum states, it does not require priori knowledge of order parameters of physical systems, which thus indicates a perceptual route to identify quantum phases and phase transitions particularly in complex systems by visualization through learning.

Mehran Kardar A, Bruce McClain, Cyrus C. Taylor

We consider phase transitions in the presence of random magnetic fields, with long-range correlations such that $〈h(0)h(r)〉\ensuremath{\sim}\frac{1}{{r}^{d\ensuremath{-}\ensuremath{\sigma}}}$. The upper critical dimension is $6+\ensuremath{\sigma}$. A superspace renormalization-group calculation is carried out to order ${\ensuremath{\epsilon}}^{2}$, where $\ensuremath{\epsilon}=6+\ensuremath{\sigma}\ensuremath{-}d$. To order $\ensuremath{\epsilon}$, the critical properties are that of a pure system in $d\ensuremath{-}2\ensuremath{-}\ensuremath{\sigma}$ dimensions. This dimensional reduction is also justified by physical arguments concerning the lower critical dimensionality, and the hyperscaling relationship. However, it fails at order ${\ensuremath{\epsilon}}^{2}$.

Michael N. Barber