Phenomenology of bond and flux orders in kagome metals

Despite much experimental and theoretical work, the nature of the charge order in the kagome metals belonging to the family of materials ${A\text{V}}_3{\mathrm{Sb}}_5$ ($A=\text{Cs,Rb,K}$) remains controversial. A crucial ingredient for the identification of the ordering in these materials is their response to external perturbations, such as strain or magnetic fields. To this end, we provide a comprehensive symmetry classification of the possible charge orders in kagome materials with a $2\times 2$ increase of the unit cell. Motivated by the experimental reports of time-reversal symmetry breaking and rotational anisotropy, we consider the interdependence of flux and bond orders. Deriving the relevant Landau free energy for possible orders, we study the effect of symmetry-breaking perturbations such as strain and magnetic fields. Our results thus provide a road map for future tests of these intricate orders.

Figure 1

Schematic of the $2\times 2$ enlarged unit cell of the kagome lattice with definitions of the elements of $C_{6v}^{\prime \prime \prime }$. We define the three translation operations $t_i$, mirror reflections ${\sigma }_v,{\sigma }_d$, and rotations $C_n$ with $n=2,3,6$. Sites and bonds are indicated in gray, while plaquettes are indicated by a black cross. There are 12 sites, 24 bonds, and 12 plaquettes within the $2\times 2$ enlarged unit cell (green shading).

Figure 2

Schematics of the bond order $F_1,F_3$ and flux orders $F_2^{\prime },F_4^{\prime }$. For the bond orders, solid (dashed) lines indicate positive (negative) values of the order parameter. Colors indicate the components ${\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2,{\mathrm{\Delta }}_3$ of the bond order parameter. For the flux order parameter, the arrows indicate the direction of the current. Colors indicate the components ${\mathrm{\Delta }}_1^{\prime },{\mathrm{\Delta }}_2^{\prime },{\mathrm{\Delta }}_3^{\prime }$ of the flux order parameter.

Figure 3

(a) Phase diagram without a third-order term applicable to the free energies ${\mathcal{F}}_{32},{\mathcal{F}}_{34},{\mathcal{F}}_{42},{\mathcal{F}}_{44}$. (b) Phase diagram with a third-order term applicable to the free energies ${\mathcal{F}}_{12},{\mathcal{F}}_{14},{\mathcal{F}}_{22},{\mathcal{F}}_{24}$. The color indicates the angle $\varphi =arctan\frac{{\sum }_i{\mathrm{\Delta }}_i^{\prime 2}}{{\sum }_i{\mathrm{\Delta }}_i^2}$, while the intensity denotes ${\sum }_i({\mathrm{\Delta }}_i^2+{\mathrm{\Delta }}_i^{\prime 2})$. Solid lines indicate first-order phase transitions and dashed lines indicate second-order transitions. Hatching denotes an anisotropic solution of the order parameter. TRS is broken when ${\mathrm{\Delta }}^{\prime }>0$. The Landau free-energy coefficients are chosen to be $\alpha ={\alpha }^{\prime }={\lambda }_1={\lambda }_2={\mu }_i={\beta }_1=-{\beta }_2=1$, ${\lambda }_{i>2}=0$.

Figure 4

(a) Phase diagram in the presence of a magnetic field without a third-order term applicable to the free energy ${\mathcal{F}}_{34}$. (b) Phase diagram in the presence of a magnetic field with a third-order term applicable to the free energy ${\mathcal{F}}_{12}$. The color indicates the angle $\varphi =arctan\frac{{\sum }_i{\mathrm{\Delta }}_i^{\prime 2}}{{\sum }_i{\mathrm{\Delta }}_i^2}$, while the intensity denotes ${\sum }_i({\mathrm{\Delta }}_i^2+{\mathrm{\Delta }}_i^{\prime 2})$. Solid lines indicate first-order phase transitions, and dashed lines indicate second-order transitions. Hatching denotes an anisotropic solution of the order parameter. TRS is broken everywhere due to the applied field. The Landau free-energy coefficients are chosen to be $\alpha ={\alpha }^{\prime }={\lambda }_1={\lambda }_2={\mu }_i={\beta }_1=-{\beta }_2=1$, ${\lambda }_{i>2}=0$.

Figure 5

Temperature derivative of the order-parameter amplitude ${\mathrm{\Delta }}^2+{\mathrm{\Delta }}^{\prime 2}$ for the free energy (D1) with a term $d{\mathrm{\Delta }}^3$. Large values of the derivative (yellow) indicate first-order phase transitions. (a) In the case $d=0$, there is a first-order transition for a finite range of $T_c-T_c^{\prime }$. The red curves show the analytical results (D6) and (D8). (b) In the case $d>0$, there is a first-order phase transition for all $T_c>T_c^{\prime }$.

Figure 6

We plot the six order parameters and the phase boundaries between the different solutions. The blue line shows the boundary between solution 1 and solution 2 ($\sim T^{*}$), while the green line shows the boundary between solution 3 and solution 4 ($\sim T_c^{\prime }$). In between the blue and green lines, there is a crossover from solution 2 to solution 3. The yellow lines show the onset of $\mathrm{\Delta }$ order ($\sim T_c$). The wedge between the blue and green lines is where the solution is anisotropic.

Figure 7

We plot the six order parameters for a set of different magnetic fields. The order parameters are computed at a fixed temperature as a function of $T_c-T_c^{\prime }$. For $B=0$ we see the transitions between the four different solutions described in Appendix pp5. The region where the solution is most stable (at the left of the phase diagram) becomes larger as the $B$-field increases.

Figure 8

We plot the order parameter of the anisotropy ${\sum }_{i,j}[{({\mathrm{\Delta }}_i^2-{\mathrm{\Delta }}_j^2)}^2+{({\mathrm{\Delta }}_i^{\prime 2}-{\mathrm{\Delta }}_j^{\prime 2})}^2]$ for a set of different magnetic fields. We fix the temperature $T=0.2$. We plot the anisotropy at a fixed temperature as a function of $T_c-T_c^{\prime }$. There is a range of $T_c-T_c^{\prime }$ where the magnetic field significantly enhances the anisotropy.

Figure 9

Increase of the anisotropy ${\sum }_{i,j}[{({\mathrm{\Delta }}_i^2-{\mathrm{\Delta }}_j^2)}^2+{({\mathrm{\Delta }}_i^{\prime 2}-{\mathrm{\Delta }}_j^{\prime 2})}^2]$ as a function of the applied field. We fix $T_c-T_c^{\prime }=0.5$ and $T=0.2$.

Figure 10

Increase of the anisotropy ${\sum }_{i,j}[{({\mathrm{\Delta }}_i^2-{\mathrm{\Delta }}_j^2)}^2+{({\mathrm{\Delta }}_i^{\prime 2}-{\mathrm{\Delta }}_j^{\prime 2})}^2]$ as a function of the applied strain for two different values of the magnetic field. We fix $T=0.55$. In the case $T_c>T_c^{\prime }$ the application of the magnetic field increases the anisotropy, consistent with the experimental results from Ref. [24].