We present a measure of the dynamical critical behavior: the ``multiscale dynamical exponent (MDE),'' ${\mathit{z}}^{\mathit{m}\mathit{d}}$ . Using dynamical renormalization-group concepts we study the relaxation times, ${\mathrm{\ensuremath{\tau}}}_{\mathrm{\ensuremath{\Lambda}}}$, of a family of space scales defined by \ensuremath{\Lambda}=1/k. Assuming dynamical universality we argue, and verify numerically, that ${\mathit{z}}^{\mathit{m}\mathit{d}}$ has the same value as the usually defined z. We measure ${\mathit{z}}^{\mathit{m}\mathit{d}}$ in the two-dimensional Ising model with the Metropolis and cluster dynamics and find ${\mathit{z}}_{\mathrm{met}}^{\mathit{m}\mathit{d}}$ =2.1\ifmmode\pm\else\textpm\fi{}0.1 and ${\mathit{z}}_{\mathrm{wolff}}^{\mathit{m}\mathit{d}}$ =0\ifmmode\pm\else\textpm\fi{}0.15, respectively. We note that in our approach ${\mathit{z}}^{\mathit{m}\mathit{d}}$ is measured using a single temperature and a single lattice size. In addition, in the Metropolis case we present a method which helps to overcome critical slowing down in the dynamical measurements themselves.We present a measure of the dynamical critical behavior: the ``multiscale dynamical exponent (MDE),'' ${\mathit{z}}^{\mathit{m}\mathit{d}}$. Using dynamical renormalization-group concepts we study the relaxation times, ${\mathrm{\ensuremath{\tau}}}_{\mathrm{\ensuremath{\Lambda}}}$, of a family of space scales defined by \ensuremath{\Lambda}=1/k. Assuming dynamical universality we argue, and verify numerically, that ${\mathit{z}}^{\mathit{m}\mathit{d}}$ has the same value as the usually defined z. We measure ${\mathit{z}}^{\mathit{m}\mathit{d}}$ in the two-dimensional Ising model with the Metropolis and cluster dynamics and find ${\mathit{z}}_{\mathrm{met}}^{\mathit{m}\mathit{d}}$=2.1\ifmmode\pm\else\textpm\fi{}0.1 and ${\mathit{z}}_{\mathrm{wolff}}^{\mathit{m}\mathit{d}}$=0\ifmmode\pm\else\textpm\fi{}0.15, respectively. We note that in our approach ${\mathit{z}}^{\mathit{m}\mathit{d}}$ is measured using a single temperature and a single lattice size. In addition, in the Metropolis case we present a method which helps to overcome critical slowing down in the dynamical measurements themselves.

arXiv:hep-lat/9411074Physical review. B, Condensed matterMar 1, 19951 citation

Dynamical scaling

Springer eBooksJan 25, 20060 citations

A short introduction to Generalized Hydrodynamics

Fabian H. L. Eßler

These are notes based on lectures given at the 2021 summer school on Fundamental Problems in Statistical Physics XV. Their purpose is to give a very brief introduction to Generalized Hydrodynamics, which provides a description of the large scale structure of the dynamics in quantum integrable models. The notes are not meant to be comprehensive or provide an overview of all relevant literature, but rather give an exposition of the key ideas for non-experts, using a simple fermionic tight-binding model as the main example.These are notes based on lectures given at the 2021 summer school on Fundamental Problems in Statistical Physics XV. Their purpose is to give a very brief introduction to Generalized Hydrodynamics, which provides a description of the large scale structure of the dynamics in quantum integrable models. The notes are not meant to be comprehensive or provide an overview of all relevant literature, but rather give an exposition of the key ideas for non-experts, using a simple fermionic tight-binding model as the main example.

arXiv:2306.17072Physica A Statistical Mechanics and its ApplicationsMay 20, 202253 citations

From dynamical scaling to local scale-invariance: a tutorial

Malte Henkel

arXiv:1610.06122The European Physical Journal Special TopicsApr 1, 201717 citations

2D CFTs - Large Charge is not enough

Thiago R. Araujo, Omar Celikbas, Domenico Orlando +1 more

In this note we study two-dimensional CFTs at large global charge. Since the large-charge sector decouples from the dynamics, it does not control the dynamics and an EFT construction that works in higher-dimensional theories fails. It is however possible to use large charge in a double-scaling limit when another controlling parameter is present. We find some general features of the spectrum of models that admit an NLSM description in a WKB approximation and use the large-charge sector of the solvable $SU(2)_k$ WZW model to argue the regimes of applicability of both the large-Q expansion and the double-scaling limit.In this note we study two-dimensional CFTs at large global charge. Since the large-charge sector decouples from the dynamics, it does not control the dynamics and an EFT construction that works in higher-dimensional theories fails. It is however possible to use large charge in a double-scaling limit when another controlling parameter is present. We find some general features of the spectrum of models that admit an NLSM description in a WKB approximation and use the large-charge sector of the solvable $SU(2)_k$ WZW model to argue the regimes of applicability of both the large-Q expansion and the double-scaling limit.

arXiv:2112.03286Dec 6, 20211 citation

The Dynamical Cluster Approximation

Mark Jarrell

Bulletin of the American Physical SocietyMar 21, 20051 citation

We present a measure of the dynamical critical behavior: the ``multiscale dynamical exponent (MDE),''

{\mathit{z}}^{\mathit{m}\mathit{d}}

. Using dynamical renormalization-group concepts we study the relaxation times, ${\mathrm{\ensuremath{\tau}}}_{\mathrm{\ensuremath{\Lambda}}}$, of a family of space scales defined by \ensuremath{\Lambda}=1/k. Assuming dynamical universality we argue, and verify numerically, that

{\mathit{z}}^{\mathit{m}\mathit{d}}

has the same value as the usually defined z. We measure

{\mathit{z}}^{\mathit{m}\mathit{d}}

in the two-dimensional Ising model with the Metropolis and cluster dynamics and find

{\mathit{z}}_{\mathrm{met}}^{\mathit{m}\mathit{d}}

=2.1\ifmmode\pm\else\textpm\fi{}0.1 and

{\mathit{z}}_{\mathrm{wolff}}^{\mathit{m}\mathit{d}}

=0\ifmmode\pm\else\textpm\fi{}0.15, respectively. We note that in our approach

{\mathit{z}}^{\mathit{m}\mathit{d}}

is measured using a single temperature and a single lattice size. In addition, in the Metropolis case we present a method which helps to overcome critical slowing down in the dynamical measurements themselves.

SU(2)_k

WZW model to argue the regimes of applicability of both the large-Q expansion and the double-scaling limit.