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It is widely expected that at sufficiently high temperatures order is always lost, e.g. magnets loose their ferromagnetic properties. We pose the question of whether this is always the case in the context of quantum field theory in $d$ space dimensions. More concretely, one can ask whether there exist critical points (CFTs) which break some global symmetry at arbitrary finite temperature. The most familiar CFTs do not exhibit symmetry breaking at finite temperature, and moreover, in the context of the AdS/CFT correspondence, critical points at finite temperature are described by an uncharged black brane which obeys a no-hair theorem. Yet, we show that there exist CFTs which have some of their internal symmetries broken at arbitrary finite temperature. Our main example is a vector model which we study both in the epsilon expansion and arbitrary rank as well as the large rank limit (and arbitrary dimension). The large rank limit of the vector model displays a conformal manifold, a moduli space of vacua, and a deformed moduli space of vacua at finite temperature. The appropriate Nambu-Goldstone bosons including the dilaton-like particle are identified. Using these tools we establish symmetry breaking at finite temperature for finite small $epsilon$. We also prove that a large class of other fixed points, which describe some of the most common quantum magnets, indeed behave as expected and do not break any global symmetry at finite temperature. We discuss some of the consequences of finite temperature symmetry breaking for the spectrum of local operators. Finally, we propose a class of fixed points which appear to be possible candidates for finite temperature symmetry breaking in $d=2$.
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In this work we explore the possibility of spontaneous breaking of global symmetries at all nonzero temperatures for conformal field theories (CFTs) in $D = 4$ space-time dimensions. We show that such a symmetry-breaking indeed occurs in certain families of non-supersymmetric large $N$ gauge theories at a planar limit. We also show that this phenomenon is accompanied by the system remaining in a persistent Brout-Englert-Higgs (BEH) phase at any temperature. These analyses are motivated by the work done in arXiv:2005.03676 where symmetry-breaking was observed in all thermal states for certain CFTs in fractional dimensions. In our case, the theories demonstrating the above features have gauge groups which are specific products of $SO(N)$ in one family and $SU(N)$ in the other. Working in a perturbative regime at the $Nrightarrowinfty$ limit, we show that the beta functions in these theories yield circles of fixed points in the space of couplings. We explicitly check this structure up to two loops and then present a proof of its survival under all loop corrections. We show that under certain conditions, an interval on this circle of fixed points demonstrates both the spontaneous breaking of a global symmetry as well as a persistent BEH phase at all nonzero temperatures. The broken global symmetry is $mathbb{Z}_2$ in one family of theories and $U(1)$ in the other. The corresponding order parameters are expectation values of the determinants of bifundamental scalar fields in these theories. We characterize these symmetries as baryon-like symmetries in the respective models.
Holographic entanglement entropy and the first law of thermodynamics are believed to decode the gravity theory in the bulk. In particular, assuming the Ryu-Takayanagi (RT)cite{ryu-takayanagi} formula holds for ball-shaped regions on the boundary around CFT vacuum states impliescite{Nonlinear-Faulkner} a bulk gravity theory equivalent to Einstein gravity through second-order perturbations. In this paper, we show that the same assumptions can also give rise to second-order Lovelock gravity. Specifically, we generalize the procedure in cite{Nonlinear-Faulkner} to show that the arguments there also hold for Lovelock gravity by proving through second-order perturbation theory, the entropy calculated using the Wald formulacite{Wald_noether} in Lovelock also obeys an area law (at least up to second order). Since the equations for second-order perturbations of Lovelock gravity are different in general from the second-order perturbation of the Einstein-Hilbert action, our work shows that the holographic area law cannot determine a unique bulk theory even for second-order perturbations assuming only RT on ball-shaped regions. It is anticipated that RT on all subregions is expected to encode the full non-linear Einstein equations on asymptotically AdS spacetimes.
The loss of criticality in the form of weak first-order transitions or the end of the conformal window in gauge theories can be described as the merging of two fixed points that move to complex values of the couplings. When the complex fixed points are close to the real axis, the system typically exhibits walking behavior with Miransky (or Berezinsky-Kosterlitz-Thouless) scaling. We present a novel realization of these phenomena at strong coupling by means of the gauge/gravity duality, and give evidence for the conjectured existence of complex conformal field theories at the fixed points.
Using the recently developed fractional Virasoro algebra cite{la_nave_fractional_2019}, we construct a class of nonlocal CFTs with OPEs of the form $T_k(z)Phi(w) sim frac{ h_gamma Phi}{(z-w)^{1+gamma}}+frac{partial_w^gamma Phi}{z-w},$ and $T_k(z)T_k(w) sim frac{ c_kZ_gamma}{(z-w)^{3gamma+1}}+frac{(1+gamma ) T_k(w)}{(z-w)^{1+gamma}}+frac{partial^gamma_w T_k}{z-w}$ which naturally results in a central charge, $c_k$, that is state-dependent, with $k$ indexing a particular grading. Our work indicates that only those theories which are nonlocal have state-dependent central charges, regardless of the pseudo-differential operator content of their action. All others, including certain fractional Laplacian theories, can be mapped onto an equivalent local one using a suitable covering/field redefinition. In addition, we discuss various perturbative implications of deformations of fractional CFTs that realize a fractional Virasoro algebra through the lense of a degree/state-dependent refinement of the 2 dimensional C-theorem.
We propose a roadmap for bootstrapping conformal field theories (CFTs) described by gauge theories in dimensions $d>2$. In particular, we provide a simple and workable answer to the question of how to detect the gauge group in the bootstrap calculation. Our recipe is based on the notion of emph{decoupling operator}, which has a simple (gauge) group theoretical origin, and is reminiscent of the null operator of $2d$ Wess-Zumino-Witten CFTs in higher dimensions. Using the decoupling operator we can efficiently detect the rank (i.e. color number) of gauge groups, e.g., by imposing gap conditions in the CFT spectrum. We also discuss the physics of the equation of motion, which has interesting consequences in the CFT spectrum as well. As an application of our recipes, we study a prototypical critical gauge theory, namely the scalar QED which has a $U(1)$ gauge field interacting with critical bosons. We show that the scalar QED can be solved by conformal bootstrap, namely we have obtained its kinks and islands in both $d=3$ and $d=2+epsilon$ dimensions.
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