Theory of metal-insulator transitions in graphite under high magnetic field

Graphite under high magnetic field exhibits consecutive metal-insulator (MI) transitions as well as reentrant insulator-metal (IM) transitions in the quasiquantum limit at low temperature. In this paper, we identify the low-$T$ insulating phases as excitonic insulators with spin nematic orderings. We first point out that graphite under the relevant field regime is in the charge neutrality region, where electron and hole densities compensate each other. Based on this observation, we introduce interacting electron models with electron pocket(s) and hole pocket(s) and enumerate possible umklapp scattering processes allowed under the charge neutrality. Employing effective boson theories for the electron models and renormalization group analyses for the boson theories, we show that there exist critical interaction strengths above which the umklapp processes become relevant and the system enters excitonic insulator phases with long-range order of spin superconducting phase fields (“spin nematic excitonic insulator”). We argue that when a pair of electron and hole pockets gets smaller in size, a quantum fluctuation of the spin superconducting phase becomes larger and destabilizes the excitonic insulator phases, resulting in the reentrant IM transitions. We also show that an odd-parity excitonic pairing between the electron and hole pockets reconstructs surface chiral Fermi arc states of electron and hole into a 2-dimensional helical surface state with a gapless Dirac cone. We discuss field and temperature dependencies of in-plane resistance by surface transport via these surface states.

Figure 1

Theoretical phase diagram for graphite under high magnetic field. The phase diagram is obtained from the RG equations, Eqs. (42, 43, 44) for $H<H_0$ and Eqs. (67, 68, 69) for $H_0<H<H_1$. “SNEI-I” and “SNEI-II” stand for two distinct spin nematic excitonic insulator phases (strong-coupling phase). For $H<H_0$, the electronic state near the Fermi level comprises two electron pockets ($n=0$ LL with $\uparrow$ spin and $\downarrow$ spin) and two hole pockets ($n=-1$ LL with $\uparrow$ spin and $\downarrow$ spin). At $H=H_0$, the outer two pockets ($n=0$ LL with $\uparrow$ spin and $n=-1$ LL with $\downarrow$ spin) leave the Fermi level. For $H_0<H<H_1$, the electronic state has one electron pocket ($n=0$ LL with $\downarrow$) and one hole pocket ($n=-1$ LL with $\uparrow$). We choose $H_0=50\mathrm{T}$ and $H_1=120\mathrm{T}$. For a detailed parameter set of the RG equations, see Appendix pp4. Our theory may not be able to predict much about a transition between SNEI-I and SNEI-II phases (the shaded area around $H=H_0$); see the discussion in Sec. 10b. $T=0$ metal-insulator transition at $H=H_{c,1}$ and insulator-metal transition at $H=H_{c,2}$ are the quantum phase transition with the dynamical exponent $z=1$.

Figure 2

Schematic picture of electronic states of graphite under high field ($H<H_0$). Solid/dotted lines describe Fermi surfaces of two electron/hole pockets in both bulk and edge regions. Two electron/hole pockets in the bulk region are terminated by electron/hole-type surface chiral Fermi arc states in edge regions, respectively. Namely, $E_{0,\sigma }(k_z,y_j)/E_{-1,\sigma }(k_z,y_j)$ goes higher/lower in energy, when $y_j$ goes from the bulk region to the edge regions (see Appendix pp1).

Figure 3

Schematic pictures of one of the umklapp scatterings, $H_{\mathrm{u},2}$. As in Fig. 2, the vertical axis denotes the momentum along the field direction ($k_z$), while the horizontal axis denotes the chain index $y_j=k_j{l}^2$ with $k_j\equiv 2\pi j/L_x[j=1,2,...,L_x{L}_y/\left(2\pi l^2\right)]$. The two-particle scatterings with solid/dotted arrows are the exchange processes ($m=n$) of the first/fourth terms in Eq. (11) with $(0,\uparrow )\equiv 1$, $(0,\downarrow )\equiv 2$, $(-1,\uparrow )\equiv 3$, and $(-1,\downarrow )\equiv 4$.

Figure 4

Schematic pictures of one of the interpocket scatterings, $H_{\mathrm{b},2}$. They are the exchange processes ($m=n$) of the first two terms in Eq. (14) with $(0,\uparrow )\equiv 1$, $(0,\downarrow )\equiv 2$, $(-1,\uparrow )\equiv 3$, and $(-1,\downarrow )\equiv 4$. As in Fig. 2, the vertical axis denotes the momentum along the field direction ($k_z$), while the horizontal axis denotes the chain index $y_j=k_j{l}^2$ with $k_j\equiv 2\pi j/L_x[j=1,2,...,L_x{L}_y/\left(2\pi l^2\right)]$.

Figure 5

Theoretical calculation results of the optical conductivity ${\sigma }_{zz}\left(\omega \right)$ in the SNEI-I phase ($H=40$, 45, $49.5\mathrm{T}$). Inset: ${\sigma }_{zz}\left(\omega \right)$ in the SNEI-II phase ($H=55\mathrm{T}$). We use the same parameter sets as in Fig. 1. For details, see Appendix pp4. Unlike its appearance in the figures, the delta function at $\omega ={\omega }_{*}$ is the most prominent in amplitude, while the continuum spectrum is much less significant. The renormalized gap ${\omega }_{*}$ is on the order of $\sqrt{E_{\mathrm{int}}{E}_{\mathrm{bw}}}$, where $E_{\mathrm{int}}$ is an interaction energy scale, $E_{\mathrm{int}}\sim e^2/\left(\epsilon l\right)$, and $E_{\mathrm{bw}}$ is a bandwidth energy scale (see Appendix pp4).

Figure 6

Left: Schematic pictures of two-particle umklapp scatterings that are allowed in the two-pocket model under the charge neutrality condition, $H_{\mathrm{u},1}^{\prime }$ and $H_{\mathrm{u},2}^{\prime }$. They are direct ($j=n$) and exchange ($m=n$) processes of Eq. (57), respectively. Middle: Two-particle intrapocket scatterings $H_{\mathrm{d},1}^{\prime }$: exchange processes ($m=n$) of Eq. (59). Right: Two-particle interpocket scatterings $H_{\mathrm{b},2}^{\prime }$: exchange processes ($m=n$) of Eq. (58). As in Fig. 2, the vertical axis denotes the momentum along the field direction ($k_z$), while the horizontal axis denotes the chain index $y_j=k_j{l}^2$ with $k_j\equiv 2\pi j/L_x[j=1,2,...,L_x{L}_y/\left(2\pi l^2\right)]$.

Figure 7

Renormalization group flow at $T=0$ in the two-dimensional parameter space subtended by $n_{\left(2\right)}$ and $p_{\left(2\right)}={\overline{p}}_{\left(2\right)}$. Weak/strong-coupling phases stand for normal metal phase/spin nematic excitonic insulator (SNEI-II) phase, respectively. Quantum criticality of the quantum phase transition between these two is controlled by a fixed point named “FP1”. The scaling dimension of the relevant parameter at FP1, ${\nu }_2$, is given in Eq. (77).

Figure 8

(a) Single-particle electronic states in normal metal phase (two-pocket model). The electron pocket (blue curve) is formed by the $n=0$ LL with $\downarrow$ spin, and the hole pocket (yellow curve) is by the $n=-1$ LL with $\uparrow$ spin. (b) Single-particle electronic states with the excitonic pairing. (c) Single-particle electronic states in the vacuum region.

Figure 9

Schematic pictures of (a) side surfaces (gray area) with the two-dimensional helical surface state with a gapless Dirac cone, and (b) top surface (gray area) with the two-dimensional Chalker-Coddington network model.

Figure 10

Schematic picture of energetically degenerate SSH end states within the bulk excitonic band gap (blue dotted line). In a generic situation, the degeneracy is lifted by an electrostatic potential (black solid curve). An associated spatial gradient of the end-state eigenenergy with respect to $y$ leads to a chiral electric current along the $-x$ direction.

Figure 11

Schematic picture of a possible RG phase diagram in the presence of a relevant perturbation (denoted by $X$) around the decoupled Luttinger liquid (LL) fixed point (denoted by “FP0”). In the presence of such relevant perturbation, the LL fixed point is unstable; the normal metal phase is characterized by a new stable fixed point (denoted by “FP3”). The horizontal axis ($Y$) denotes the umklapp and interpocket scattering terms that drive the system into the excitonic insulator (EI) phases. FP2 represents a stable fixed point characterizing the EI phases. The critical properties of the metal-insulator (MI) and reentrant insulator-metal (IM) transitions are characterized by a new saddle fixed point (denoted by “FP4”) instead of by the FP1. In this schematic picture, we assume that the fixed point for the EI phases (FP2) is locally stable against the small perturbation $X$.