Quantum oscillations and black hole ringing

We show that strongly coupled field theories with holographic gravity duals at finite charge density and low temperatures can undergo de Haas–van Alphen quantum oscillations as a function of an external magnetic field. Exhibiting this effect requires computation of the one-loop contribution of charged bulk fermions to the free energy. The one-loop calculation is performed using a formula expressing determinants in black hole backgrounds as sums over quasinormal modes. At zero temperature, the periodic nonanalyticities in the magnetic susceptibility as a function of the inverse magnetic field depend on the low energy scaling behavior of fermionic operators in the field theory, and are found to be softer than in weakly coupled theories. We also obtain numerical and WKB results for the quasinormal modes of charged bosons in dyonic black hole backgrounds, finding evidence for nontrivial periodic behavior as a function of the magnetic field.

Figure 1

The zero temperature magnetic susceptibility for bosons as a function of the magnetic field. The expression plotted has been made dimensionless by dividing by the sample area and multiplying by the boson mass $m$.

Figure 2

The zero temperature magnetic susceptibility to leading order at large $N$ as a function of the magnetic field. The expression plotted has been made dimensionless by dividing by the sample area and multiplying by the chemical potential $\mu$.

Figure 3

Motion of the quasinormal frequencies closest to the real axis as $k_{\perp }/\mu$ is varied from $-1$ to $+1$, according to (61). The temperature is $T=0.005\mu$. The other constants are taken to have values $q=1$, $\gamma =\sqrt{12}$, $\nu =1/3$, $\theta =\pi /6$, $h={\mu }^{1/3}$.

Figure 4

Left: Contributions of the lowest few quasinormal modes to the magnetic susceptibility, according to (53), as a function of $k_{\perp }/\mu$. The darker line is the pole nearest the real axis. Right: The total magnetic susceptibility due to the lowest 50 modes. The temperature is $T=0.005\mu$. The constants have the same values as in Fig. 3. The vertical axis is proportional to $\tilde{\chi }$ of (68).

Figure 5

The magnetic susceptibility at $T=0$ as a function of $k_{\perp }/\mu$. The constants have the same values as in Fig. 3. The vertical axis is proportional to $\tilde{\chi }$ of (68).

Figure 6

Left: Quasinormal modes at temperature $T/\mu =0.075$. Right: Quasinormal modes at zero temperature $T=0$. The plots show the lower half $z/\mu$ frequency plane, and bright spots denote quasinormal modes. Both plots have $q=0$, $m^2=0$, $\gamma =1$ and no magnetic field. The plots were generated as density plots of $|\det A^{\prime }(z)/\det A(z)|$. The finite temperature plot has $N=200$ while the zero temperature plot has $N=500$. The discreteness of the poles on the imaginary axis at $T=0$ is an artifact of finite $N$. As $N\to \infty$ the poles coalesce and form a branch cut.

Figure 7

The location of the quasinormal mode closest to the real axis as a function of charge. The temperature is $T/\mu =0.05$ and $B=0$. The charge ranges from $q=0$ to $q=4.3$. The mass is $m^2=0$ while $\gamma =1$. At the upper limit of $q$ the mode crosses into the upper half plane, indicating the onset of a superconducting instability.

Figure 8

Quasinormal modes at temperature $T/\mu =0.075$. Charges from left to right: $q=1$, $q=2$ and $q=4$. The plots show the lower half frequency plane $z/\mu$, and bright spots denote quasinormal modes. Both plots have $m^2=0$, $\gamma =1$ and no magnetic field.

Figure 9

Emergence of quasinormal modes from a branch cut as a function of magnetic field and at zero temperature. All plots have $q=4$, $\gamma =1$, $m^2=10$ and $\ell =10$. The plots show the lower half frequency plane $z/\mu$, and bright spots denote quasinormal modes. From top left $k/\mu \equiv \sqrt{2\ell |qB|}/\mu \approx 0.283,0.632,0.894,1.26,1.79,2.53$. The closely spaced modes are a branch cut that has been discretized into poles by the finite value of $N=500$ in (80).