Classification of gapless ${\mathbb{Z}}_2$ spin liquids in three-dimensional Kitaev models

Frustrated quantum magnets can harbor unconventional spin-liquid ground states in which the elementary magnetic moments fractionalize into new emergent degrees of freedom. While the fractionalization of quantum numbers is one of the recurring themes in modern condensed matter physics, it often remains a challenge to devise a controlled analytical framework tracking this phenomenon. A notable exception is the exactly solvable Kitaev model, in which spin degrees of freedom fractionalize into Majorana fermions and a ${\mathbb{Z}}_2$ gauge field. Here, we discuss the physics of fractionalization in three-dimensional Kitaev models and demonstrate that the itinerant Majorana fermions generically form a (semi)metal which, depending on the underlying lattice structure, exhibits Majorana Fermi surfaces, nodal lines, or topologically protected Weyl nodes. We show that the nature of these Majorana metals can be deduced from an elementary symmetry analysis of the projective time-reversal and inversion symmetries for a given lattice. This allows us to comprehensively classify the gapless spin liquids of Kitaev models for the most elementary tricoordinated lattices in three dimensions. We further expand this classification by addressing the effects of time-reversal symmetry breaking and additional interactions.

Figure 1

Example of a minimal volume arising for the (10,3)b hyperhoneycomb lattice, for which four elementary loops of length 10 are needed to enclose a closed volume. We provide illustrations of such minimal volumes for all lattices in Appendix pp1.

Figure 2

(a) Mirror line for the honeycomb lattice (6,3). (b) One of the three mirror planes for (8,3)b. The other two are obtained by $120 {}^{\circ }$ rotations around the $\widehat{z}$ direction. The combination determines the eigenvalues of all the fundamental loop operators. (c) One of the mirror planes for (8,3)n. Another is obtained by a $90 {}^{\circ }$ rotation around $\widehat{z}$. A third mirror plane (not shown) lies in the $xy$ plane. These determine the eigenvalues of all but one loop operator per unit cell.

Figure 3

Vison gap obtained for the smallest vison loop as a function of system size. The dotted line marks the extrapolation of the gap for infinite system size, and the gray bar denotes the error of the extrapolation. Details on the vison loops can be found in Appendix pp2.

Figure 4

Visualization of the A and B sublattices (in white and blue, respectively) for the honeycomb, square, and hyperoctagon lattices. While the sublattices of the honeycomb lattice have the same translation vectors as the original lattice, the same is not true for the square and hyperoctagon lattices, which lead to a finite value for ${\mathbf{k}}_0$.

Figure 5

The distinct features of the (8,3)a lattice [shown in (a)] are corotating spirals, while the (8,3)b lattice [shown in (b)] has counter-rotating spirals. The two different rotation directions are indicated by orange and blue, respectively.

Figure 6

(a) Unit cell and translation vectors for the Kitaev model on the (8,3)a lattice. (b) Phase diagram for (8,3)a. The parameter regions shaded darker orange have topological Fermi surfaces, while the lighter orange regions have trivial Fermi surfaces. (c) Visualization of the four Majorana Fermi surfaces for isotropic Kitaev couplings.

Figure 7

Left-hand side: Brillouin zone with Majorana Fermi surfaces and high-symmetry points for (8,3)a. Right-hand side: Energy dispersion along the high-symmetry lines. The Weyl points are indicated at the band crossings between the green and pink bands between $K$ and $\mathrm{\Gamma }$, as well as between $A$ and $H$.

Figure 8

(a) Unit cell and translation vectors, (b) phase diagram, and (c) Brillouin zone and position of Weyl points for isotropic couplings for the lattice (8,3)b. Yellow/red denote Weyl points with negative/positive charge.

Figure 9

Left-hand side: At the isotropic point, the Weyl points are located on the ${120}^{\circ }$ rotation invariant line, marked in blue. The right-hand side shows the energy dispersion along this high-symmetry line.

Figure 10

The left-hand side shows the Brillouin zone with the position of the Weyl points, the right-hand side shows the corresponding Chern number. The positions of the Weyl points, as well as the three example planes (defined by ${\mathbf{k}}_3=0,\pm \frac{1}{4}$), are indicated as a guide to the eye.

Figure 11

(a) Evolution of Weyl points for varying the coupling constants $0\le J_z\lesssim 0.43$ and $J_x=J_y=(1-J_z)/2$ for (8,3)b. (b) Corresponding Fermi arcs. The Fermi arcs touch and reconnect at $J_z\approx 0.201$. (c) Surface Brillouin zone for the 001 surface.

Figure 12

(a) Evolution of the Weyl nodes for (8,3)b in presence of a magnetic field for varying $\kappa$ from $0,...,0.25$. (b) Corresponding Fermi arc evolution. (c) Visualization of the surface Brillouin zone for open boundary conditions in the 001 direction.

Figure 13

(a) Unit cell and translation vectors for the lattice (8,3)c. (b) Phase diagram of the Kitaev model on (8,3)c. Around the isotropic point, there is a gapless phase with line nodes. (c) The line nodes (marked in blue) are located precisely at the edge of the Brillouin zone for isotropic couplings.

Figure 14

The left-hand side shows the Brillouin zone with high-symmetry points, and the gapless modes indicated in blue for (8,3)c. The right-hand side shows the energy dispersion along high-symmetry lines for isotropic couplings (colored) and anisotropic couplings $J_x=\frac{3}{8}$ and $J_z=J_y=(1-J_x)/2=\frac{5}{16}$ (gray). The zero-energy surface immediately gaps completely when departing from the isotropic point.

Figure 15

(a) Evolution of the Weyl nodes for (8,3)c in presence of a magnetic field for varying $\kappa$ from $0,...,0.5$. (b) Corresponding Fermi arc plot for $\kappa =0.25$. (c) Visualization of the surface Brillouin zone for open boundary conditions in the 100 direction.

Figure 16

(a) Unit cell and translation vectors for the (8,3)n lattice. (b) The Kitaev model on the (8,3)n lattice has no gapless phase. The blue line indicates the phase transition between the two distinct gapped phases, and the red dot marks the isotropic point. (c) Phase diagram for $\kappa =0.05$. For finite $\kappa$, the Dirac nodes split into pairs of oppositely charged Weyl nodes, and the phase transition line evolves to an entire phase of a gapless Weyl spin liquid, marked in light orange.

Figure 17

Brillouin zone with high-symmetry points and energy dispersion along the corresponding high-symmetry lines for (8,3)n. The spectrum at the isotropic point is fully gapped.

Figure 18

Brillouin zone with a path cutting through the Dirac/Weyl nodes appearing in the energy dispersion for (8,3)n for $\kappa =0$ (gray lines) and $\kappa =0.05$ (colored lines), respectively.

Figure 19

(a) Unit cell and translation vectors for the (9,3)a lattice. (b) The phase diagram is symmetric for $J_x\leftrightarrow J_y$. (c) Brillouin zone with position of the gapless Weyl nodes: red denotes a source, yellow a sink, and black a neutral combination of several Weyl nodes.

Figure 20

A deformed version of the (9,3)a lattice can be obtained by coupling honeycomb layers via mid-bond sites. The eight elementary plaquettes of length nine per unit cell are marked by the gray transparent polygons. A visualization of the plaquettes in the undeformed (9,3)a lattice can be found in Appendix pp2.

Figure 21

Possible flux assignments for the (9,3)a lattice for the eight elementary loops shown in Fig. 20.

Figure 22

(a) Visualization of the Kitaev couplings, unit cell, and translation vectors for the (10,3)a lattice. (b) Phase diagram for (10,3)a; the gapless region is shaded orange and the gapped blue. The parameter regions shaded darker orange have topological Fermi surfaces, while the lighter orange regions have trivial Fermi surfaces. (c) Visualization of the gapless modes for isotropic couplings.

Figure 23

Brillouin zone with high-symmetry points (left) and energy dispersion along the corresponding high-symmetry lines for lattice (10,3)a (right). Gray shaded curves indicate energy dispersion upon time-reversal symmetry breaking with magnetic field strength $\kappa$.

Figure 24

Deformation of the Fermi surface of the Majorana metal for the (10,3)a lattice when breaking time-reversal symmetry. Plots are shown for varying $\kappa$ parametrizing the magnetic field strength in Eq. (9).

Figure 25

View on (10,3)b along (a) (1,0,0). and (b) (0,1,10). View on (10,3)c along (c) (1,0,0) and (d) along (1,10,1).

Figure 26

(a) Visualization of the Kitaev couplings, unit cell, and translation vectors for the (10,3)b lattice. (b) Phase diagram for (10,3)b. (c) At the isotropic point, the gapless modes form a ring in the $k_x+k_y=0$ plane, indicated in gray.

Figure 27

The energy dispersion of the Kitaev model on the (10,3)b hyperhoneycomb lattice for various values of $\kappa$ (parametrizing the effective magnetic field) along certain high-symmetry lines indicated in the Brillouin zone on the right-hand side. The gray hexagon indicates the plane $k_x=-k_y$ on which the line of gapless mode (black line) is located. Figure adapted from Ref. [21].

Figure 28

(a) Evolution of the Weyl nodes for (10,3)b in the presence of a magnetic field for varying $\kappa$ from $0,...,\infty$. (b) Corresponding Fermi arc evolution. (c) Visualization of the surface Brillouin zone for open boundary conditions in the 100 direction. Figure adapted from Ref. [21].

Figure 29

(a) Visualization of the Kitaev couplings, the unit cell, and the translation vectors for the lattice (10,3)c. (b) Phase diagram for (10,3)c. (c) Visualization of the gapless nodal line in the $k_z=0$ plane, indicated in gray.

Figure 30

Brillouin zone with high-symmetry points and energy dispersion along the corresponding high-symmetry lines for (10,3)c.

Figure 31

Illustration of the 12 Fermi pockets enclosing a Weyl node (indicated by the yellow and red dots) and the energy dispersion upon breaking of time-reversal symmetry for (10,3)c.

Figure 32

(Left) Visualization of the surface Brillouin zone for the 001 direction of lattice (10,3)a. (Right) Gapless surface modes illustrated in the surface Brillouin zone. Aside from two puddles arising from the projection of the Majorana Fermi surfaces, two Fermi arcs (indicated by the blue line) form between the projected Weyl nodes (indicated by the red and yellow dots).

Figure 33

Deformation of the Majorana Fermi surface of the (10,3)a lattice when changing the $J_z$ coupling constant from 0 to $J_z=\frac{1}{2}$ where the system becomes gapped. Red (yellow) dots denote the position of Weyl points of positive (negative) chirality. The black dots are pairs of Weyl points of opposite chirality. For $J_z>\frac{1}{3}$ the Weyl points start to split and annihilate at $J_z=\sqrt{2}-1$.

Figure 34

Deformation of the (topological) Majorana Fermi surfaces of lattice (8,3)a when changing $J_z$. Red (yellow) dots denote the position of Weyl points of positive (negative) chirality. The black dots are pairs of Weyl points of opposite chirality. Around $J_z\approx 0.31$, each Weyl point splits into three. At the isotropic point, two pairs of Weyl points with opposite charges are annihilated while one is created at the touching points of the Fermi surfaces. This results in topologically trivial Fermi surfaces for $J_z>\frac{1}{3}$.

Figure 35

Effect of time-reversal symmetry breaking on (a) the (10,3)a and (b) the (8,3)a lattice, with the time-reversal invariant system on the left and the time-reversal symmetry breaking system $\left(\kappa =J_K/10\right)$ on the right, respectively. Breaking time-reversal symmetry destroys the perfect nesting condition with wave vector ${\mathbf{k}}_0$, marked by the arrow.

Figure 36

The lattices (a) (10,3)a, (b) (10,3)b, and (c) (8,3)b can be embedded in a network of edge-sharing octahedra, the elementary building block of all candidate materials realizing bond-directional Kitaev-type exchange [10]. Lattice (10,3)b is realized in the iridate $\beta \text{−}{\text{Li}}_{\text{2}}{\text{IrO}}_{\text{3}}$, while the other two lattice structures still await a material realization in the form of a spin-orbit entangled Mott insulator.

Figure 37

The (8,3)a lattice viewed along the (a) crystallographic $\widehat{c}$ axis and (b) crystallographic $\widehat{a}$ axis.

Figure 38

The (8,3)b lattice viewed along the (a) crystallographic $\widehat{c}$ axis and (b) crystallographic $\widehat{a}$ axis.

Figure 39

The (8,3)c lattice viewed along the (a) crystallographic $\widehat{c}$ axis and (b) crystallographic $\widehat{a}$ axis.

Figure 40

The (8,3)n lattice viewed along (a) $\widehat{a}+\widehat{b}$ axis and (b) $\widehat{c}$ axis.

Figure 41

The (9,3)a lattice viewed along the (a) crystallographic $\widehat{c}$ axis and (b) crystallographic $\widehat{b}$ axis.

Figure 42

The (10,3)a lattice viewed along the (a) fourfold screw rotation axis $\widehat{a}$, (b) twofold rotation axis $\widehat{a}+\widehat{b}$, and (c) threefold rotation axis $\widehat{a}+\widehat{b}+\widehat{c}$.

Figure 43

The (10,3)b lattice viewed along the (a) crystallographic $\widehat{c}$ axis and (b) crystallographic $\widehat{b}$ axis.

Figure 44

The (10,3)c lattice viewed along the (a) crystallographic $\widehat{c}$ axis and (b) crystallographic $\widehat{b}$ axis.

Figure 45

Loop operators of the lattice (8,3)a forming a volume constraint.

Figure 46

Vison excitation threading two plaquettes of length 8 in the lattice (8,3)a, as depicted on the left. In addition, it threads several plaquettes of length 14, two examples of which are depicted on the right shaded in magenta. The flipped bond operator is pictured in red.

Figure 47

Loop operators of the lattice (8,3)b forming a volume constraint. The loop operators in the bottom row are related to those in the top row by lattice translation vectors.

Figure 48

Vison excitation threading four plaquettes in the lattice (8,3)b, two of length 8 shaded in yellow and two of length 12 shaded in magenta. The flipped bond operator is pictured in red.

Figure 49

(Left) Fermi arc surface states in the [001]-surface Brillouin zone of lattice (8,3)b for a slab geometry 512 layers thick. (Right) Probability densities as functions of slab layer of the zero-energy states with momentum corresponding to one of the crossing points of the Fermi arcs.

Figure 50

Loop operators of the lattice (8,3)c forming two unique volume constraints in (a) and (b).

Figure 51

Vison excitation threading four plaquettes of length 8 (shaded in yellow) in the lattice (8,3)c. The flipped bond operator is pictured in red.

Figure 52

Loop operators of the lattice (8,3)n forming four unique volume constraints in (a), (b), (c), and (d).

Figure 53

Vison excitation threading two plaquettes of length 8 (yellow) and two plaquettes of length 10 (magenta) in the lattice (8,3)n. The flipped bond operator is pictured in red.

Figure 54

The lattice (9,3)a has eight loops of length 9 per unit cell, labeled by 1,$...,$ 8 in (a), that are subject to two volume constraints shown in (a) and (b).

Figure 55

Vison excitation threading four plaquettes of length 9 in the lattice (9,3)a. The flipped bond operator is pictured in red.

Figure 56

Loop operators of the lattice (10,3)a forming a volume constraint.

Figure 57

Vison excitation threading 10 plaquettes of length 10 in the lattice (10,3)a. The flipped bond operator is pictured in red.

Figure 58

Loop operators of the lattice (10,3)b forming a volume constraint.

Figure 59

Vison excitation threading six plaquettes of length 10 in the lattice (10,3)b. The flipped bond operator is pictured in red.

Figure 60

Loop operators of the lattice (10,3)c forming a volume constraint.

Figure 61

Vison excitation threading three plaquettes of length 10 (yellow) in the lattice (10,3)c. In addition, 11 plaquettes of length 12 are excited; two examples of such plaquettes (shaded in magenta) are shown in the second row. The flipped bond operator is pictured in red.

Figure 62

Visualization of the Kitaev couplings, the unit cell, and the translation vectors for the lattice (10,3)c with enlarged unit cell.