Strong coupling effective Higgs potential and a first order thermal phase transition from AdS/CFT duality

TL;DR

Using the AdS/CFT correspondence, the paper analyzes a strongly coupled ${N}=2$ gauge theory with fundamental matter by mapping the Higgs branch to instanton moduli on D7-branes and computing the finite-$T$ and finite-$\mu$ effective potential $V(Q)$. The isospin chemical potential induces a quadratic instability $V(Q)\sim -c\,\mu^2 Q^2$, signaling Bose-Einstein condensation, while at $\mu=0$ and finite $T$ a first-order thermal phase transition occurs, with the Higgs VEV $Q$ acting as the order parameter and a topology change of D7 embeddings at a critical ratio $m/b$ (with $b\sim T$) driving the transition. The results provide a nonperturbative, geometric picture of Higgs-branch dynamics in flavored holographic theories and demonstrate how strong coupling reshapes finite-temperature symmetry breaking. They also suggest extensions to other SUSY-breaking perturbations and to baryon-number chemical potential, where holography may offer complementary insights beyond lattice methods.

Abstract

We use AdS/CFT duality to study the thermodynamics of a strongly coupled N=2 supersymmetric large Nc SU(Nc) gauge theory with Nf =2 fundamental hypermultiplets. At finite temperature T and isospin chemical potential mu, a potential on the Higgs branch is generated, corresponding to a potential on the moduli space of instantons in the AdS description. For mu =0, there is a known first order phase transition around a critical temperature Tc. We find that the Higgs VEV is a suitable order parameter for this transition; for T>Tc, the theory is driven to a non-trivial point on the Higgs branch. For non-zero mu and T=0, the Higgs potential is unbounded from below, leading to an instability of the field theory due to Bose-Einstein condensation.

Figures (5)

• Figure 1: Brane embeddings in the AdS Schwarzschild background for different values of the quark mass. In the plot we have set $b=1$.
• Figure 2: Potential $V(Q)$ as a function of the instanton size / Higgs VEV $Q$ for various values of the quark mass $m$ (we set $b=1$ here); in fig. a) we sample the whole spectrum from $m=0$ to $m\to \infty$, in fig. b) we vary $m$ close to the phase transition region. The potential for $m \to \infty$ is flat and coincides with the horizontal axis.
• Figure 3: Position of the minimum of the potential $Q_0$ versus the bare quark mass $m_q$, zoom of the critical region.
• Figure 4: Position of the endpoint $y_{min}$ of the embedded brane on the horizon versus the potential minimum $Q_0$. Region I corresponds to solutions with $0\le$$m_q$$< 0.91$. In this range, for each $m_q$ there is only one regular D7 embedding, reaching the horizon. In the phase transition region ($0.913 \lesssim m_q \lesssim 0.926$), there are three embeddings for each $m_q$ (see the text for details). Of the two that reach the horizon one lies on the straight line in region II, the other in region III (dashed line).${}^8$ The solution that does not touch the horizon has $y_{min}=Q_0=0$, thus lying at the origin IV (we use a dot to describe this kind of solutions). Finally for large enough quark masses ($m_q>0.93$), there is again only one regular embedding for each $m_q$, which does not reach the horizon. These solutions all accumulate in the origin IV.
• Figure 5: Solutions in critical region: There are three condensate values which give different regular solutions for the each quark mass. $m_q=0.918$ in the plot with condensate values $c_1=-0.0165$, $c_2=-0.0172$, $c_3=-0.0265$.