Strongly Coupled Anisotropic Gauge Theories and Holography

We initiate a nonperturbative study of anisotropic, nonconformal, and confining gauge theories that are holographically realized in gravity by generic Einstein-axion-dilaton systems. In the vacuum, our solutions describe renormalization group flows from a conformal field theory in the UV to generic scaling solutions in the IR with generic hyperscaling violation and dynamical exponents $\theta$ and $z$. We formulate a generalization of the holographic $c$ theorem to the anisotropic case. At finite temperature, we discover that the anisotropic deformation reduces the confinement-deconfinement phase transition temperature suggesting a possible alternative explanation of inverse magnetic catalysis solely based on anisotropy. We also study transport and diffusion properties in anisotropic theories and observe, in particular, that the butterfly velocity that characterizes both diffusion and growth of chaos transverse to the anisotropic direction saturates a constant value in the IR which can exceed the bound given by the conformal value.

Figure 1

Free energy as a function of $T$ for different values of the anisotropy parameter $a/j=0$, 1, 3 (black, blue, and red curves, respectively). The parameters $\sigma =\sqrt{2/3}+1/10$, $\gamma =1/5$, $\mathrm{\Delta }=3$ were chosen such that the undeformed theory is confining. The horizontal axis corresponds to the confined state while all other branches correspond to deconfined phases. The insets show details of an additional phase transition for large $a$, as discussed in the text.

Figure 2

Phase diagram of the system in the $a\text{−}T$ plane. We observe two phases, confined and “plasma I,” for $a/j<2.08$. For larger $a/j$, there exist three phases, confined, “plasma I,” and “plasma II.” The blue and red curves indicate lines of first order transitions.

Figure 3

The viscosity over entropy density ratio for several values of $\theta$ and $z$. Increase of the scaling parameter $z$ leads to lower values of the ratio (solid lines), as well as a decrease of the value of $\theta$ (dashed lines).

Figure 4

Butterfly velocities $v_{B\perp }$ (solid lines) and $v_{B\parallel }$ (dashed lines). In the longitudinal direction, the information diffuses slower with increasing anisotropy, with vanishing velocity in the IR. Perturbations in the transverse plane can propagate at a faster rate, with a new upper bound attained in the IR.