Figure 1

The motion of poles of the Green’s functions (72) of spinors (left) and scalars (right) in the complex frequency plane. For illustration purposes, we have chosen parameters and rescaled $|\omega |(\to |\omega {|}^{\#}\mathrm{\text{with small}}\#)$ to give a better global picture. The poles are exponentially spaced along a straight line (dotted line) with angle ${\theta }_c$ given by (83). There are infinitely many poles, only a few of which are shown. Left plot: The black dashed line crossing the origin corresponds to the value of ${\theta }_c$ at $k=k_o$ [see (68)]: the boundary of the oscillatory region. As $k\to k_o$, most of the poles approach the branch point $\omega =0$ except for a finite number of them which become quasiparticle poles for the Fermi surfaces at larger values of $k$. The color dashed lines in the right half indicate the motion of poles on another sheet of the complex frequency plane at smaller values of $k$ [see the end of Sec. 3c for the choice of branch cut in the $\omega$-plane]. Right plot: the two dashed lines correspond to $k=0$ (upper one) and $k=k_o$ (lower one). Again most of the poles approach the branch point $\omega =0$ as $k\to k_o$. These plots are only to be trusted near $\omega =0$.Reuse & Permissions

Figure 2

A geometric illustration that poles of the spinor Green’s function never appear in the upper-half $\omega$-plane of the physical sheet, for two choices of ${\nu }_{k_F}<\frac{1}{2}$. Depicted here is the ${\omega }^{2{\nu }_{k_F}}$ covering space on which the Green’s function (88), with the $\omega /v_F$ term neglected, is single valued. The shaded region is the image of the upper-half $\omega$-plane of the physical sheet. The pole lies on the line $2{\nu }_{k_F}{\theta }_c=-{\gamma }_{k_F}$ for $k_{\perp }>0$ and on $2{\nu }_{k_F}{\theta }_c=\pi -{\gamma }_{k_F}$ for $k_{\perp }<0$, which are indicated by the purple solid line in the figure. The triangle formed by dashed arrows and solid lines in the upper left quadrant gives the geometric illustration for the equation $\pi -{\gamma }_k=\arg (e^{2\pi i{\nu }_k}-e^{-2\pi q{e}_d})$ [following from the first equation of (60)], which makes it manifest that for $k_{\perp }<0$ the pole lies outside the shaded region. In contrast, for a scalar one needs to reverse the direction of the horizontal dashed line and the pole lies inside the shaded region. Similarly, the triangle in the lower right quadrant gives the illustration for $-{\gamma }_k=\arg (-e^{2\pi i{\nu }_k}+e^{-2\pi q{e}_d})$, which is relevant for $k_{\perp }>0$. We also indicated the angles ${\alpha }_{\pm }$, which will be introduced and discussed in detail around (104).Reuse & Permissions

Figure 3

Examples of the motion of the pole for a spinor as $k_{\perp }$ is varied (arrows indicating the directions of increasing $k_{\perp }$). Left plot: ${\nu }_{k_F}<\frac{1}{2}$, for which the pole moves in a straight line. The plot shows an example where the pole moves to another sheet of the Riemann plane for $k_{\perp }>0$ (i.e., ${\alpha }_{+}>\frac{\pi }{2}$); the ${\alpha }_{\pm }$ indicated there are introduced in (104). Right plot: ${\nu }_{k_F}>\frac{1}{2}$ for which the dispersion (real part of the pole) is linear.Reuse & Permissions

Figure 4

Spinor Green’s function $G_2$ (as defined in Appendix ) at $k=0.918$ as a function of $\omega$, computed numerically. We have chosen parameters $r_{*}=R=g_F=q=1$, $m=0$, and $d=3$ for which the Fermi momentum is $k_F=0.91853$. The real and imaginary parts are shown in blue and orange dotted curves, respectively. Also shown is (88) (solid lines) with $h_1$, $h_2$ computed numerically using the method of Appendix  and ${\nu }_k$ given by (56).Reuse & Permissions

Figure 5

The values of $k_F$ as a function of $q$ for the Green’s function $G_2$ are shown by solid lines for $m=-0.4$, 0, 0.4. In this plot and ones below, we use units where $R=1$, $r_{*}=1$, $g_F=1$, and $d=3$. The oscillatory region, where ${\nu }_k=\frac{1}{\sqrt{6}}\sqrt{k^2+m^2-\frac{q^2}{2}}$ is imaginary, is shaded. From (a20) in Appendix , $G_1(k)=G_2(-k)$, so $k_F$ for $G_1$ can be read from these plots by reflection through the vertical $k=0$ axis. The $m=-0.4$ plot corresponds to alternative quantization for $m=0.4$ following from Eq. (a26). For convenience, we have included in each plot the values of $k_F$ for the alternative quantization using the dotted lines. Thus, the first ($m=-0.4$) and the third plot ($m=-0.4$) in fact contain identical information; they are related by taking $k\to -k$ and exchanging the dotted and solid lines. Also, as discussed after (a29), for $m=0$ the alternative quantization is equivalent to the original one. This is reflected in the middle plot in the fact that the dotted lines and solid lines are completely symmetric. All plots are symmetric with respect to $q$, $k\to -q$, $-k$ as a result of Eq. (a24).Reuse & Permissions

Figure 6

The “phase diagram”: Shown here are contour plots of the exponent ${\nu }_{k_F}$ evaluated on the primary Fermi surface (the one with the largest $k_F$), as a function of $m$ and $q$, for each of the two components of the spinor operator. The top and bottom rows differ only in the function used to shade the region ${\nu }_{k_F}\in [0,\frac{1}{2}]$. The top row is shaded according to ${\alpha }_{-}$, while the bottom row is shaded according to ${\alpha }_{+}$. The ${\alpha }_{\pm }$ were introduced in (104) and discussed in detail around there. The darker region corresponds to larger values of the angles. Both angles are zero at the line of ${\nu }_{k_F}=\frac{1}{2}$ and increase with a decreasing ${\nu }_{k_F}$. When an angle exceeds $\frac{\pi }{2}$, the corresponding pole moves into another Riemann sheet, the regions for which are indicated in the plots. As anticipated in the discussion below (104), ${\alpha }_{-}$ becomes $O(1)$ only for small $q$, which ${\alpha }_{+}$ becomes small only for ${\nu }_{k_F}$ close to $\frac{1}{2}$. We also indicated the region where there exists an oscillatory region for momentum satisfying (68). This region lies above the dashed lines. It is clear from the plots that for $m>0$, the region which allows a Fermi surface always lies inside the region which allows the oscillatory region. It is also interesting to note that in the left plot the dashed line in fact meets with the line for ${\nu }_{k_F}=\frac{1}{2}$ at $m=-\frac{1}{2}$ (not shown in figure). This happens for $d=3$ only. For general $d$ dimension, the dashed line intersects with $m=-\frac{1}{2}$ at ${\nu }_{k_F}=\frac{1}{2(d-2)}$.Reuse & Permissions

Figure 7

The Fermi velocity of the primary Fermi surface of various components as a function of ${\nu }_{k_F}>\frac{1}{2}$. Dotted lines are for $G_2$. Various values of $m$ are indicated.Reuse & Permissions

Figure 8

$h_1$ and $h_2/|c(k_F)|$ coefficients in (88) for the primary Fermi surface of various components as a function of ${\nu }_{k_F}$. Various values of $m$ are indicated. In the $h_2$ plot for $G_2$ at $m=-0.4$: there is a zero of $c(k_F)$ at $\nu =m{R}_2\approx .16$, at which $h_2$ changes sign. We explain the (lack of) significance of this phenomenon in Appendix . For convenience, we also plot $|c(k_F)|$ separately in Fig. 9 below.Reuse & Permissions

Figure 9

Real and imaginary part of $c(k_F)$ as a function of ${\nu }_{k_F}$ for $G_1$ at $m=0$. The plot for $G_2$ is very similar. Note that due to the Gamma function prefactor in (55), the real part diverges at half integers.Reuse & Permissions

Figure 10

The potential for the scalar field defined by (b3) for $m^2=-3/2$, $-2$, $-9/4$. In terms of the tortoise coordinate $s$ the horizon is located at $s=-\infty$ and the boundary at $s=0$. The oscillatory region is associated with the continuum for $s\to -\infty$ and “Fermi” surfaces are bound states in the potential well to the right of this continuum. Note the behavior of the potential close to the boundary is $V(s)\sim (2+m^2)/s^2$.Reuse & Permissions

Figure 11

A zero-energy Schrödinger potential which is equivalent to the spinor bound state problem (b4). The three plots are for $m=-1/4$, 0, $+1/4$, respectively, for varying values of the parameter $k/{\mu }_q$. For $m=-1/4$ we have rescaled $s_F\to s_F/|s_{*}|$, $V_F\to V_F{s}_{*}^2$ so that for all values of ${\mu }_q/k$ the potential can be drawn on the same interval $-1<s_F<0$.Reuse & Permissions

Figure 12

The parameter region where there exist Fermi surfaces for large $k_F$, $m$, $q$ in the WKB approximation, for different field theory space-time dimensions $d=3$, 4, 6, 50. For a given dimension the upper and lower boundary lines are defined by the Fermi surface moving into the oscillatory region (lower) and the nonexistence of classical orbits (upper.) See Fig. 13 for an alternative way to state this in terms of the radii of the two turning points. The extreme point to the right of the allowed region is $P_d=(1/\sqrt{3},(d-2)/\sqrt{3d(d-1)})$ and the point at the lower left corner is $Q_d=(0,(d-2)/\sqrt{d(d-1)})$.Reuse & Permissions

Figure 13

The WKB allowed region for $d=3$ showing contours of fixed $\nu$ and $m$ (above) and fixed $q$ (below) based on the WKB quantization condition (b19) for $n=0$. Upper plot: The lowest horizontal contour is $\nu =0$, increasing towards $\nu \to \infty$ at the upper boundary. The vertical contours are for fixed $m$ with $m=0$ lying on the $k_F/{\mu }_q$ axis. The contours in this plot demonstrate that the point $P_3$ controls the asymptotic slope of the fixed $\nu$ contours in Fig. 6. Lower plot: The contours of fixed $q$ move towards the upper boundary with increasing $q$. Note that these contours end on the lower boundary (the oscillatory region). In particular, if we fix $q$ and increase $m$, following along the contours in this plot, we see that $k_F$ decreases until eventually the bound state enters the oscillatory region.Reuse & Permissions

Figure 14

The Fermi velocity in the limit $q$, $k_F$, $m$ large for $d=3$. The limiting velocity is determined by the radius $r_k$ (which is in turn determined by $mR/q{g}_F$) using formula (c27). For $mR/q{g}_F=0$ the wave function goes towards the boundary ($r_k\to \infty$) and $v_F\to 1$. This is consistent with the asymptotic behavior of $v_F$ in Fig. 7, since these plots are for fixed $m$ and increasing $q$.Reuse & Permissions

Figure 15

A: The phase and amplitude of $c(k_F)$ as a function of $\nu$ on the primary Fermi surface of $G_2$ for $m=0.4$. Note that the phase jumps by $\pi$ when $c(k_F)$ has a zero. B: As defined in Eq. (88), the sign of $h_2$ (for $m=0.4$, $\alpha =2$) jumps at $\nu =m{R}_2\approx 0.16$. In spite of the singularity in $h_2/|c(k_F)|$ visible in the lower right panel of Fig. 8, $|h_2|$ as a function of $\nu$ is completely smooth.Reuse & Permissions