Topological Aspects of Dense Matter: Lattice Studies

Topological fluctuations change their nature in the different phases of strong interactions, and the interrelation of topology, chiral symmetry and confinement at high temperature has been investigated in many lattice studies. This review is devoted to the much less explored subject of topology in dense matter. After a short overview of the status at zero density, which will serve as a baseline for the discussion, we will present lattice results for baryon rich matter, which, due to technical difficulties, has been mostly studied in two-color QCD, and for matter with isospin and chiral imbalances. In some cases, a coherent pattern emerges, and in particular the topological susceptibility seems suppressed at high temperature for baryon and isospin rich matter. However, at low temperatures the topological aspects of dense matter remain not completely clear and call for further studies.

Lattice study of QCD at finite chiral density: topology and confinement

In this paper we study the properties of QCD at nonzero chiral density $$\rho _5$$ ρ 5 , which is introduced through chiral chemical potential $$\mu _5$$ μ 5 . The study is performed within lattice simulation of QCD with dynamical rooted staggered fermions. We first check that $$\rho _5$$ ρ 5 is generated at nonzero $$\mu _5$$ μ 5 and in the chiral limit observe $$\rho _5 \sim \varLambda _{QCD}^2 \mu _5$$ ρ 5 ∼ Λ QCD 2 μ 5 . We also test the possible connection between confinement and topological fluctuations. To this end, we measured the topological susceptibility $$\chi _{\mathrm{top}}$$ χ top and string tension $$\sigma $$ σ for various values of $$\mu _5$$ μ 5 . We observed that string tension grows with $$\mu _5$$ μ 5 . It seems that topological susceptibility also rises with $$\mu _5$$ μ 5 , but to state this more reliably the uncertainties should be reduced. We believe that our results indicate possible connection between topological fluctuations and the strength of confinement.

Form invariance, topological fluctuations and mass gap of Yang–Mills theory

In order to have a new perspective on the long-standing problem of the mass gap in Yang–Mills theory, we study in this paper the quantum Yang–Mills theory in the presence of topologically nontrivial backgrounds. The topologically stable gauge fields are constrained by the form invariance condition and the topological properties. Obeying these constraints, the known classical solutions to the Yang–Mills equation in the three- and four-dimensional Euclidean spaces are recovered, and the other allowed configurations form the nontrivial topological fluctuations at quantum level. Together, they constitute the background configurations, upon which the quantum Yang–Mills theory can be constructed. We demonstrate that the theory mimics the Higgs mechanism in a certain limit and develops a mass gap at semiclassical level on a flat space with finite size or on a sphere.

Replays of spatial memories suppress topological fluctuations in cognitive map

The spiking activity of the hippocampal place cells plays a key role in producing and sustaining an internalized representation of the ambient space—a cognitive map. These cells do not only exhibit location-specific spiking during navigation, but also may rapidly replay the navigated routs through endogenous dynamics of the hippocampal network. Physiologically, such reactivations are viewed as manifestations of “memory replays” that help to learn new information and to consolidate previously acquired memories by reinforcing synapses in the parahippocampal networks. Below we propose a computational model of these processes that allows assessing the effect of replays on acquiring a robust topological map of the environment and demonstrate that replays may play a key role in stabilizing the hippocampal representation of space.

Non-Gaussian behavior of the internet topological fluctuations

The traditional Gibrat's hypotheses were once used to model the topological fluctuations of Internet. Although it seems to reproduce the scaling relation of Internet's degree distribution, the detailed micro-dynamics have never been empirically validated. Here, we analyze the distribution of degree growth rates of the Internet for various time scales. We find that in contrast to the traditional Gibrat's assumptions, none of the degree growth rates are normally distributed, but behaves as an exponential decrease on its body and a power-law decay on its tail. Moreover, the observed growth rate distribution turns out independent of the initial degree when the time interval enlarges to a year. Our observations do not consist with the traditional Gibrat law model and suggest a more complex fluctuation mechanism underlying the evolution of Internet.