Engineering holographic phase diagrams

By introducing interacting scalar fields, we tried to engineer physically motivated holographic phase diagrams which may be interesting in the context of various known condensed matter systems. We introduce an additional scalar field in the bulk which provides a tunable parameter in the boundary theory. By exploiting the way the tuning parameter changes the effective masses of the bulk interacting scalar fields, desired phase diagrams can be engineered for the boundary order parameters dual to those scalar fields. We give a few examples of generating phase diagrams with phase boundaries which are strikingly similar to the known quantum phases at low temperature such as the superconducting phases. However, the important difference is that all the phases we have discussed are characterized by neutral order parameters. At the end, we discuss if there exists any emerging scaling symmetry associated with a quantum critical point hidden under the dome in this phase diagram.

Figure 1

The density plot for the expectation value of $O_{\varphi }$ vs the rescaled temperature $T/{\mu }_q$ and the rescaled doping parameter (the source) $J_{\varphi }/({\mu }_q{)}^{{\mathrm{\Delta }}_{\varphi }^{-}}$, both by appropriate power in the chemical potential ${\mu }_q$. The dotted line is where $O_{\varphi }$ vanishes. Note that $O_{\varphi }$ is nonvanishing in general. And the relation between $O_{\varphi }$ and $J_{\varphi }$ is similar to the “equation of state” of the doping matter dual to scalar field $\varphi$.

Figure 2

The phase diagram of model I with positive doping parameter. It is the density plot for phase 1(blue) and phase 2 (orange) with coupling functions $-F_2(\varphi )={\varphi }^2-5{\varphi }^4/24\cong F_1(\varphi )$.

Figure 3

The free energy density difference between the ordered and normal phases. The rescaled temperature is chosen as $(T/{\mu }_q)\simeq 0.014\times {10}^{-3}$. The dashed black line stands for the baseline of the free energy of the normal phase with only the $\varphi$ condensate. The blue and orange lines correspond to the free energy difference of phase 1 and phase 2, respectively.

Figure 4

The phase diagram of model II around a natural quantum critical point. It is the density plot phase 1 (blue) and phase 2 (orange) with coupling functions $F_1(\varphi )=\varphi (\varphi +2)$, $F_2(\varphi )={\varphi }^2/2$. The green parts are the overlap region.

Figure 5

The energy difference between the ordered and normal phases in Fig. 4 at $(T/{\mu }_q)\times 10^3\simeq 0.028$. The dashed black line stands for the baseline of the free energy of the normal phase with only the $\varphi$ condensate. The blue and orange lines correspond to the free energy difference of phase 1 and phase 2, respectively.

Figure 6

The density plot of the order parameter $O_{\mathrm{\Psi }}$ on the $\{{\mu }_q,T\}$ plane, which is equivalent to Fig. 9 of a single charged scalar field model in Appendix pp2. The transparent middle region has vanishing order parameter and is the normal phase, where the equation of state ${\tilde{n}}_q({\tilde{\mu }}_q)$ in Eq. (13) is independent of the temperature. There are two condensed phases: the right-hand side one has a positive charge density $n_q$, while the left-hand side one has a negative $n_q$. The dashed lines indicate the scaling trajectories. Note that this toy model has scaling symmetry even in the condensed phases. In this figure, we have set $g_F=1$, $L=1$ for convenience.

Figure 7

Left: the 3D schematic diagram of Fig. 2 in terms of $\{{\mu }_q,{\tilde{J}}_{\varphi },T\}$, where the blue and orange regions correspond to ordered phase 1 and phase 2. Three light green surfaces indicate parameter constraints of constant $r_h$. Right: Phase diagram in terms of $\{T,{\tilde{J}}_{\varphi }\}$ at a fixed ${\mu }_q/T$ cross-profile, which corresponds to fixing the parameter $q$ in (5). The dashed line indicates the scalings, and the solid gray line indicates the parameter constraint of constant $r_h$.

Figure 8

Left: the 3D schematic diagram of Fig. 4 in terms of $\{{\mu }_q,{\tilde{J}}_{\varphi },T\}$, where the blue, orange and green regions correspond to ordered phase 1, phase 2 and the overlap phase. Three light green surfaces indicate the parameter constraint of constant $r_h$. Right: Phase diagram in terms of $\{T,{\tilde{J}}_{\varphi }\}$ at a fixed ${\mu }_q/T$ cross profile, which corresponds to fixing the parameter $q$ in (5). The dashed line indicates the scalings, and the solid gray line indicates the parameter constraint of constant $r_h$.

Figure 9

The plot of expectation value $O_{\mathrm{\Psi }}$ vs the temperature $T$, at fixed chemical potential ${\mu }_q$. Both of $O_{\mathrm{\Psi }}$ and $T$ are rescaled by the appropriate power of chemical potential ${\mu }_q$. The following parameters ate used: $J_{\mathrm{\Psi }}=0$, $m_{\mathrm{\Psi }}^2{L}^2=-2.1$, ${\lambda }_{\mathrm{\Psi }}=1$ and $q_{\mathrm{\Psi }}=0.5$.