Phase diagram structure of topological Mott transition for zero-gap semiconductors beyond conventional Landau-Ginzburg-Wilson scenario

Moyuru Kurita, Youhei Yamaji, and Masatoshi Imada

Phys. Rev. B 88, 115143 – Published 27 September 2013

Abstract

We show that a wide class of unconventional quantum criticality emerges when orbital currents cause quantum phase transitions from zero-gap semiconductors such as Dirac fermions to a topological insulator or a Chern insulator. Changes in Fermi-surface topology concomitant with [SU(2) or time-reversal] symmetry breakings generate quantum critical lines (QCLs) even beyond the quantum critical point. This QCL running at temperature $T = 0$ separates two distinct topological phases. This is in contrast to the simple termination of the finite-temperature critical line at the quantum critical point without any extension of it at $T = 0$ . Topology change causes the unconventionality beyond the concept of simple spontaneous symmetry breaking assumed in the conventional Landau-Ginzburg-Wilson scenario. The unconventional universality implied by mean-field critical exponents $β > 1 / 2$ and $δ < 3$ is protected by the existence of the quantum critical line. It emerges for several specific lattice models including the honeycomb, kagome, diamond, and pyrochlore lattices. We also clarify phase diagrams of the topological phases in these lattices at finite temperatures.

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Received 2 May 2012

DOI:https://doi.org/10.1103/PhysRevB.88.115143

Authors & Affiliations

Moyuru Kurita, Youhei Yamaji, and Masatoshi Imada

Department of Applied Physics, University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan and CREST, JST, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan

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Vol. 88, Iss. 11 — 15 September 2013

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Figure 1
(a) Honeycomb, (b) kagome, and (c) pyrochlore lattice structures. (d) Schematics showing how degeneracy at a band crossing point is lifted. The loop current in this momentum space is equivalent to that in the real space shown in (g). Spin-orbital-current configurations on a hexagon (e) and a tetrahedron (f): Along each nearest-neighbor bond $(r_{i}, r_{j})$ , opposite spins on the directions $\pm b_{i j} \times d_{i j}$ flow in the opposite directions as illustrated in (g). Here, we define $d_{i j} = r_{i} - r_{j}$ and $b_{i j} = (r_{i} + r_{j}) / 2 - o_{i j}$ with $o_{i j}$ being the gravity center of the hexagon (e) [tetrahedron (f)]. In (g), all the vectors schematically illustrate only the correct directions while the lengths are not correct.Reuse & Permissions
Figure 2
(a)–(c) Examples for vectors $r_{i}$ , $r_{j}$ , $o_{i j}$ , $b_{i j}$ , and $d_{i j}$ for the honeycomb, kagome, and pyrochlore lattice. (d)–(f) Examples of symmetric operations applied to a bond $(i, j)$ .Reuse & Permissions
Figure 3
(a) Schematic phase diagram in the plane of $a$ proportional to interaction $V_{c} - V$ and the field $λ$ conjugate to the order parameter $ζ$ . Free-energy structures in each phase representing topological insulators (TIs) [Chern insulators (CIs)]; TMI and semimetal (SM) are also shown. (b) Phase diagram of orbital-current phase induced by Coulomb repulsion in the space of temperature $T$ , $λ$ , and interaction $V$ obtained by Hartree-Fock calculation. In the (pink) shaded area at $λ = 0$ , spontaneous orbital current flows and the solid blue line illustrates its critical line. The dashed blue line shows a crossover between the non-LGW and LGW regions characterized by $ζ \propto {| a |}^{n / (d - n)}$ , and $ζ \propto {| a |}^{1 / 2}$ , respectively (Ref. 26). The white circle at $V = V_{α c}$ and $λ = T = 0$ is the unconventional quantum critical point (QCP) of the TMT from which the white topological critical line (semimetal) at $T = 0$ and $λ = 0$ starts unlike the conventional QCP.Reuse & Permissions
Figure 4
The band structure of $H = H_{0}$ on the honeycomb lattice. Six Dirac cones shown in the figure are linked to either $K$ or $K^{'}$ by reciprocal-lattice vectors.Reuse & Permissions
Figure 5
(a) $V_{2}$ dependence of the order parameter $ζ$ at $T = 0$ . $ζ$ grows linearly at $T = 0$ for small $V_{2} - V_{2 c}$ . The dashed line shows Eq. (55). (b) $V_{2}$ dependence of the order parameter $ζ$ for several temperatures. $ζ$ grows with square root of $V_{2} - V_{2 c} (T)$ at finite temperatures. (c) The square (red) points show $V_{2 c} (T)$ vs $T$ . They constitute a line which separates the ordered and normal states. The solid black line is the theoretical critical line defined by Eq. (59). The cross points linked by a dashed line indicate the crossover between Ising-like and quantumlike areas. The dotted (green) line shows a theoretical expression of the crossover defined by Eq. (64).Reuse & Permissions
Figure 6
(a) The logarithm of $ζ$ and $V_{2} - V_{2 c} (T)$ for several temperatures. The slope of the dashed line is given by $β = \frac{1}{2}$ . It justifies that $ζ$ grows with square root of $V_{2} - V_{2 c} (T)$ for $T > 0$ . The solid (red) line is Eq. (61). The crossover value of $V_{2}$ is exemplified for the case of $T = 0.02$ by the intersection of the dotted and solid lines. (b) The logarithm of $ζ$ and $V_{2} - V_{2 c} (T = 0)$ showing how Ising-like behaviors reach to the quantum like behavior.Reuse & Permissions
Figure 7
(a) $λ$ dependence of $ζ$ for several values of $V_{2}$ and $T$ . We note that $V_{2 c} (T = 0) ≃ 1.18 t$ and $V_{2 c} (T = 0.04) ≃ 1.25 t$ for the honeycomb lattice. (b) Logarithmic plot of data points shown in (a). The slope of the data in (b) are scaled by $1 / δ$ and can be used to determine $δ$ . The dotted lines are the slopes expected from the values of $δ$ written nearby the lines. This shows the presence of three different criticality; $δ = 1$ for $V_{2} < V_{2 c} (T = 0)$ and $T = 0$ , $δ = 2$ for $V_{2} = V_{2 c} (T = 0)$ , and $T = 0$ and $δ = 3$ for $T > 0$ .Reuse & Permissions
Figure 8
Three-dimensional phase diagram of the honeycomb lattice. In the (pink) shaded surface at $λ = 0$ , the time-reversal symmetry is spontaneously broken and the solid (blue) line at $T > 0$ shows its critical line. This critical line follows the Ising universality. The dashed blue line shows a crossover between the quantum and Ising regions characterized by $ζ \propto | V_{2} - V_{2 c} |$ , and $ζ \propto | V_{2} - V_{2 c} |^{1 / 2}$ . The white circle at $V_{2} ≃ 1.18 t$ and $λ = T = 0$ is the MQCP.Reuse & Permissions
Figure 9
The band structure of the kagome lattice. The highest occupied band at $2 / 3$ filling touches quadratically to the lowest unoccupied flat band at $Γ$ point.Reuse & Permissions
Figure 10
(a) $V_{1}$ dependence of $ζ$ at $T = 0$ with theoretical curve described by Eq. (80). (b) $λ$ dependence of $ζ$ at $T = 0$ and $V_{1} = 0$ with theoretical curve described by Eq. (81). $Λ = 1.03$ is used in both cases. Both figures show good agreement of numerical calculations and analytical expressions. (c) $V_{1}$ dependence of $ζ$ for several different choices of temperatures on the kagome lattice. (d) Logarithm of data points at $T = 0.01$ and the line with slope $1 / δ = 1 / 2$ , which indicate that the criticality belongs to the Ising universality at finite temperatures. (e) $λ$ dependence of $ζ$ for $T = 0.01$ and $V_{1} = V_{1 c} (T = 0.01) ≃ 0.467 t$ . (f) Logarithm of data points in (a) plotted together with dashed line whose slope is $1 / δ = 1 / 3$ (Ising universality).Reuse & Permissions
Figure 11
(a) Critical lines of kagome lattice with $t_{3} = t / 12$ and $t_{3} = 0$ . Solid (blue) line is the analytical expression Eq. (88) with $Λ = 1.03$ . The universality is the Ising type at finite temperatures. Though the numerical results and theoretical expression slightly differ, we can see their qualitative correspondence. In this case, $T = V_{1} = 0$ is the MQCP, which is a unique feature of $(n, d) = (2, 2)$ case. (b) Phase diagram of kagome lattice in the parameter space of $V_{1}$ , $λ$ , and $T$ .Reuse & Permissions
Figure 12
(a) $V_{1}$ dependence of $ζ$ at $T = 0$ on the pyrochlore lattice. (b) Logarithmic plot of (a) with the dashed line indicating the slope $β = 2$ . An unconventional criticality expected from the singular form of the free-energy expansion, Eq. (101), is indeed confirmed here as $ζ \propto {(V_{1} - V_{1 c})}^{2}$ , namely $β = 2$ .Reuse & Permissions
Figure 13
(a) $λ$ dependence of $ζ$ at $V_{1} = 2.62 t$ and $T = 0$ . (b) $λ$ dependence of $ζ$ at $V_{1} = 2.0$ and $T = 0$ . (c) Logarithmic plot of (a) and (b) with lines of slope given by $1 / δ$ .Reuse & Permissions
Figure 14
$V_{1}$ dependence of $ζ$ for several values of $T$ on the pyrochlore lattice. $ζ$ grows with ${(V_{1} - V_{1 c})}^{2}$ at $T = 0$ . At finite temperature, however, we can see that the transition is of first order, which was not observed in the other lattices.Reuse & Permissions
Figure 15
Schematic picture of unit cell of TMI+CDW phase. Black and green sites stand for charge-rich and charge-poor sites, respectively. Arrows indicate vectors $v_{i j}$ in Eq. (105).Reuse & Permissions
Figure 16
Phase diagram of pyrochlore lattice. When $λ = 0$ , the system undergoes a transition from SM for small $V_{1}$ to TMI (dark orange line) at the MQCP, $V_{1} = V_{1 c} ≃ 2.62 t$ at $T = 0$ (white circle). The MQCP shows a universality similar to Fig. 3 beyond the conventional LGW scheme. When $λ < 0$ , SMs for small $V_{1}$ make a first-order transition (the light blue surface) to a topological semiconductor coexisting with charge order (TS+CDW). The TMI coexiting with charge order (TMI+CDW) at $λ \leq 0, T = 0$ crosses over to TS+CDW at finite temperatures. It terminates at a QCL (dark blue). For $λ > 0$ , a topological insulator with a gap is stabilized for all $V_{1} \geq 0$ at $T = 0$ . At $T > 0$ , through a light green surface, $ζ$ jumps. This surface separates a gaslike and liquidlike topological semiconductor (TS) and terminates on the Ising critical (bold green) line, which further terminates at the MQCP. The MQCP also terminates two QCLs at $T = 0$ , one along $λ = 0, V < V_{1 c}$ (bold white line) and the other (the dark blue line) representing that between TS+CDW and SM.Reuse & Permissions
Figure 17
(a),(b) Gap enhancement due to the electron interaction in the honeycomb lattice. (c),(d) Gap enhancement due to the electron interaction in the pyrochlore lattice. In each case, significant enhancement of the gap is observed even when the interaction does not exceed the critical value for $λ = 0$ . (e) Interaction dependence of the gap in the pyrochlore lattice with $λ < 0$ . In this case, the external field does not bring the system to the topological insulator, and the phase transition due to the electron interaction is observed.Reuse & Permissions
Figure 18
Classification of quantum phase transitions. Three circles, “LGW,” “Fermi,” and “ $Z$ , $Z_{2}$ ,” stand for the symmetry breaking, topological changes of the Fermi surface, and topological changes of the integer index characterizing the system, respectively. Lifshitz transitions are typical examples of quantum phase transitions involving the topological changes in the Fermi surface only, denoted as “Lifshitz.” Disorders and resultant Anderson localizations offer examples of changes in topological invariants without involving Fermi-surface topology changes and symmetry breakings, denoted as “Anderson” (Ref. 52). Overlaps among these three circles offer various qunatum phase transitions. For example, “AF in KMH” stands for the antiferromagnetic transitions in the Kane-Mele-Hubbard model (Ref. 53). Band inversion transitions between topologically trivial and nontrivial insulators are denoted by “band inversion” (Ref. 24). The unconventional universality focused in the present paper appears within the overlap of “LGW” and “Fermi.” This class of quantum phase transitions contains the BCS and AF transitions in the honeycomb Hubbard model, denoted as “AF in HH.” The TMT has all three of these characters (LGW, Fermi, and $Z$ , $Z_{2}$ ).Reuse & Permissions

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