Quantum Heisenberg model on a sawtooth-chain lattice: rotation-invariant Green’s function method

Abstract

We apply the rotation-invariant Green’s function method (RGM) to study the spin $S=1/2$ Heisenberg model on a one-dimensional sawtooth lattice, which has two nonequivalent sites in the unit cell. We check the RGM predictions for observable quantities by comparison with the exact-diagonalization and finite-temperature-Lanczos calculations. We discuss the thermodynamic and dynamic properties of this model in relation to the mineral atacamite Cu$_2$Cl(OH)$_3$ complementing the RGM outcomes by results of other approaches.

Graphic abstract

Appendix: Numerical solution of the self-consistent equations

Fig. 8

Numerical solution of Eq. (4.1) using (A.1) (solid) and (A.5) (dashed), see the text of appendix: (upper panel) ${\tilde{\alpha }}_{ij}$; (middle panel) $\rho$, $\alpha _1$, $\alpha _2$; (lower panel) achieved values of the objective functions

Fig. 9

Values of correlators $\vert c_{ij}\vert$ as they follow by numerical solution of Eq. (4.1) using (A.1) (solid) and (A.5) (dashed). Gray straight lines correspond to high-temperature asymptotes

Fig. 10

RGM results for thermodynamic quantities per site, (upper panel) e(T), (middle panel) c(T), (lower panel) s(T), cf. Fig. 3, as they follow by numerical solution of Eq. (4.1) using (A.1) (solid) and (A.5) (dashed)

Fig. 11

RGM results for $S_q$ at various temperatures, cf. Fig. 6, as they follow by numerical solution of Eq. (4.1) using (A.1) (solid) and (A.5) (dashed)

In this appendix, we discuss how to solve numerically the self-consistent equations reported in Sect. 4. To this end, we consider a six-dimensional space of values $\xi _1\equiv {\tilde{\alpha }}_{10}$, $\xi _2\equiv {\tilde{\alpha }}_{01}$, $\xi _3\equiv {\tilde{\alpha }}_{20}$, $\xi _4\equiv {\tilde{\alpha }}_{11}$, $\xi _5\equiv {\tilde{\alpha }}_{02}$, and $\xi _6\equiv \rho$, see Eq. (4.1). We introduce the (nonnegative) objective function [62]

$$\begin{aligned} {\mathfrak {F}}(\xi _1\!,\!\ldots \!,\!\xi _6)\!=\! & {} \!\left[ \!\xi _1\xi _6\!-\!\Xi _1(\!\xi _1\!,\!\ldots \!,\!\xi _6\!)\!\right] ^2 +\!\left[ \!\xi _2\!-\!\Xi _2(\!\xi _1\!,\!\ldots \!,\!\xi _6\!)\!\right] ^2 \nonumber \\&+\left[ \!\xi _3\xi _6\!-\!\Xi _3(\!\xi _1\!,\!\ldots \!,\!\xi _6\!)\!\right] ^2 \!+\!\left[ \!\xi _4\!-\!\Xi _4(\!\xi _1\!,\!\ldots \!,\!\xi _6\!)\!\right] ^2 \nonumber \\&+\left[ \!\xi _5\!-\!\Xi _5(\!\xi _1\!,\!\ldots \!,\!\xi _6\!)\!\right] ^2 \!+\!\left[ \Xi _6(\xi _1\!,\!\ldots \!,\!\xi _6)\right] ^2 \!\ge \!0\nonumber \\ \end{aligned}$$

(A.1)

with

$$\begin{aligned} \Xi _1(\xi _1\!,\!\ldots \!,\!\xi _6)=\frac{1}{2\pi }\!\int \limits _{-\pi }^{\pi }\!\!\!\mathrm{d}q \mathrm{e}^{\mathrm{i}q}\langle S_{q1}^-S_{q1}^+ \rangle _{{\tilde{\alpha }},\rho }, \nonumber \\ \vdots \nonumber \\ \Xi _{5}(\xi _1\!,\!\ldots \!,\!\xi _6)=\frac{1}{2\pi }\!\int \limits _{-\pi }^{\pi }\!\!\!\mathrm{d}q \mathrm{e}^{\mathrm{i}q}\langle S_{q2}^-S_{q2}^+ \rangle _{{\tilde{\alpha }},\rho }, \nonumber \\ \Xi _6(\xi _1\!,\!\ldots \!,\!\xi _6)=\frac{1}{2\pi }\!\!\int \limits _{-\pi }^{\pi }\!\!\!\mathrm{d}q\langle S_{q1}^- S_{q1}^+\rangle _{{\tilde{\alpha }},\rho }\! \!-\!\!\frac{1}{2\pi }\!\!\int \limits _{-\pi }^{\pi }\!\!\!\mathrm{d}q \langle S_{q2}^-S_{q2}^+ \rangle _{{\tilde{\alpha }},\rho },\nonumber \\ \end{aligned}$$

(A.2)

see Eqs. (4.1), (4.2). The objective function ${\mathfrak {F}}(\xi _1,\ldots ,\xi _6)$ can be evaluated according to Eqs. (A.1) and (A.2) for any point in the six-dimensional space $(\xi _1,\ldots ,\xi _6)$ presuming that $\Xi _i(\xi _1,\ldots ,\xi _6)$, $i=1,\ldots ,6$ exist. Obviously, ${\mathfrak {F}}$ defined in Eq. (A.1) vanishes at the point $(\xi _1^*,\ldots ,\xi _6^*)$ which corresponds to the solution of Eq. (4.1), i.e., ${\mathfrak {F}}(\xi _1^*,\ldots ,\xi _6^*)=0$. Importantly, the set of equations at hand [Eq. (4.1)] does not have a unique solution, i.e., we are interested in the physical one only, which should be discriminated from irrelevant ones.

We begin with a sufficiently high temperature T (e.g., $T=50$ or $T=100$), when the correlation functions can be calculated using the asymptotic high-temperature values $c_{10}=-J_1/(8T)$, $c_{01}=-J_2/(8T)$, $c_{20}=J_1^2/(32T^2)$, $c_{11}=J_1J_2/(32T^2)$, $c_{20}=J_2^2/(32T^2)$ and $\rho =1$; these initial values are denoted as $\xi _1^{(0)},\ldots ,\xi _6^{(0)}$ and ${\mathfrak {F}}(\xi _1^{(0)},\ldots ,\xi _6^{(0)})\ne 0$. However, if $(\xi _1^{(0)},\ldots ,\xi _6^{(0)})$ is quite close to $(\xi _1^{*},\ldots ,\xi _6^{*})$ the objective function ${\mathfrak {F}}(\xi _1,\ldots ,\xi _6)$ has (approximately) the form of a paraboloid in the seven-dimensional space around $(\xi _1^{(0)},\ldots ,\xi _6^{(0)})$, i.e.,

$$\begin{aligned} {\mathfrak {F}}(\xi _1,\ldots ,\xi _6)\!\approx & {} \!{{{\mathcal {F}}}}(\xi _1,\ldots ,\xi _6); \nonumber \\ {{{\mathcal {F}}}}(\xi _1,\ldots ,\xi _6)\!= & {} \!\sum _{i=1}^6\sum _{j=i}^6C_{ij}\left( \xi _i-\xi _i^*\right) \left( \xi _j-\xi _j^*\right) \!. \end{aligned}$$

(A.3)

Considering close points with $\xi _1^{\pm }\!=\!\xi _1^{(0)}\!\pm \!\delta \xi _1/p$, ..., $\xi _6^{\pm }\!=\!\xi _6^{(0)}\!\pm \!\delta \xi _6/p$, where $\delta \xi _1,\ldots ,\delta \xi _5$ are (small) differences of the asymptotic high-temperature values at T and, e.g., at 1.01T, $\delta \xi _6$ is, e.g., 0.001, whereas p is, e.g., 600, and using ${{{\mathcal {F}}}}(\xi _1,\ldots ,\xi _6) \approx {\mathfrak {F}}(\xi _1,\ldots ,\xi _6)$, we determine the coefficients $C_{ij}$ in Eq. (A.3). Furthermore, we obtain the prediction for $\xi _1^*,\ldots ,\xi _6^*$ from Eq. (A.3). The objective function ${\mathfrak {F}}$, in general, does not vanish at the determined point $(\xi _1^*,\ldots ,\xi _6^*)$ (since ${\mathfrak {F}}$ and ${{{\mathcal {F}}}}$, in general, are only approximately equal). We declare this point as the initial one, i.e., $\xi _1^*\rightarrow \xi _1^{(0)},\ldots ,\xi _6^*\rightarrow \xi _6^{(0)}$, and repeat calculations. While seeking for the new coefficients $C_{ij}$ and the new prediction for $\xi _1^*,\ldots ,\xi _6^*$ we decrease $\delta \xi _1/p$, ..., $\delta \xi _6/p$ by factor 2. We repeat calculations (e.g., 10 times), evaluating the value of the objective function ${\mathfrak {F}}$ at this temperature T, see the solid line in the lower panel of Fig. 8, and its small values allow us to conclude that we have found the solution of Eq. (4.1) at the temperature T.

Next step is to decrease the temperature: $T\rightarrow T-\Delta T$; $\Delta T$ varies from 0.01 to $0.000\,01$, see below. We do not use the asymptotic high-temperature values any more. Instead, we use the determined values $\xi _1^*,\ldots ,\xi _6^*$ at the temperature T as the initial values $\xi _1^{(0)},\ldots ,\xi _6^{(0)}$ for the lower temperature $T-\Delta T$. Furthermore, $\delta \xi _i$ now are the differences of $\xi _i^{(0)}$ at $T-\Delta T$ and $\xi _i^{(0)}$ at T. This way we proceed approaching extremely low temperatures; simultaneously we observe the objective function ${\mathfrak {F}}$ which should be small enough. As can be seen in the lower panel of Fig. 8, ${\mathfrak {F}}$ is as small as $10^{-40}\ldots 10^{-50}$ (solid curve) and thus evidences that we have found the solution of Eq. (4.1).

Few comments on the explained scheme of numerical solution of the self-consistent equations are in order here. First, the described procedure, which is based on the assumption about a paraboloid for the objective function (A.3), requires a reasonable amount of time on personal computer that is obviously an advantage in comparison with the numerical solution described in Ref.  [62] (seeking for the minimum of the objective function within a cuboid).

Second, at certain low temperature the described scheme may fail. What is the reason for that? We observed that it may occur because $f_{-}$ [see Eq. (3.10)] becomes negative at the points of the six-dimensional space which are used to determine the coefficients $C_{ij}$ in Eq. (A.3). Sometimes, this obstacle can be overcome by a change of the specific parameter values employed in the described scheme. Here we have arrived at the third comment, which regards a jump behavior of the solid curve in the lower panel of Fig. 8. The jumps are related to a change of the step $\Delta T$: If the described scheme fails at some temperature, one may decrease $\Delta T$ or increase p etc. and this may allow to proceed further decreasing the temperature. For example, we set $\Delta T=0.01$ at high temperatures, but $\Delta T=0.001$ while approaching $T=1$, $\Delta T=0.000\,1$ for $T=1\ldots 0.1$ and $\Delta T=0.000\,01$ below $T=0.1$.

Fourth, it is worth noting that the second and the fourth equations (4.1) may be replaced by the physically equivalent ones

$$\begin{aligned} {\tilde{\alpha }}_{01}=\frac{1}{2\pi }\int \limits _{-\pi }^{\pi }\mathrm{d}q \langle S_{q1}^-S^+_{q2}\rangle _{{\tilde{\alpha }},\rho }, \nonumber \\ {\tilde{\alpha }}_{11}=\frac{1}{2\pi }\int \limits _{-\pi }^{\pi }\mathrm{d}q \mathrm{e}^{\mathrm{i}q} \langle S_{q1}^-S^+_{q2}\rangle _{{\tilde{\alpha }},\rho }, \end{aligned}$$

(A.4)

respectively. This simply corresponds to another possible choice of $c_{01}=\langle S_{j,1}^-S_{j,2}^+\rangle$ and $c_{11}=\langle S_{j,1}^-S_{j+1,2}^+\rangle$, see Fig. 1 and Eqs. (2.2) and (3.7). In these cases, however, the explained numerical scheme for the resulting set of self-consistent equations fails at higher temperatures and therefore they were discarded.

Finally, for the set $J_1=3.294$, $J_2=1$ we have detected the following problem. At high temperatures ${\tilde{\alpha }}_{02}$ is the smallest quantity and can be hardly controlled by the objective function (A.1). Therefore $c_{02}$ does not follow (4.3) in the temperature range where other four correlators, $c_{10}$, $c_{01}$, $c_{20}$, and $c_{11}$, are quite close their high-temperature asymptotes (4.3). However, the correct high-temperature behavior of all correlators is inherent in the self-consistent Eq. (4.1): Fixing $c_{02}$ by the relation $c_{02}=2c_{01}^2$ which holds at high temperatures and utilizing the described numerical scheme now in a five-dimensional space we achieve the values of the objective function as small as $10^{-20} \ldots 10^{-30}$.

Yet another comment is worth mentioning. In the beginning of appendix we introduce a six-dimensional space defining $\xi _1,\ldots ,\xi _6$ and then apply the explained numerical scheme. However, another choice of the coordinates which describe the points of a six-dimensional space is also possible. More specifically, we may consider another six-dimensional space of values, $\xi _1\equiv \rho {\tilde{\alpha }}_{10}$, $\xi _2\equiv {\tilde{\alpha }}_{01}$, $\xi _3\equiv \rho {\tilde{\alpha }}_{20}$, $\xi _4\equiv {\tilde{\alpha }}_{11}$, $\xi _5\equiv {\tilde{\alpha }}_{02}$, and $\xi _6\equiv \rho$, and use the following objective functions:

$$\begin{aligned} {\mathfrak {F}}(\xi _1\!,\!\ldots \!,\!\xi _6) \!=\! \sum _{i=1}^{5}\!\left[ \!\xi _i\!-\!\Xi _i(\!\xi _1\!,\!\ldots \!,\!\xi _6\!)\!\right] ^2 \!+\!\left[ \Xi _6(\xi _1\!,\!\ldots \!,\!\xi _6)\right] ^2, \nonumber \\ \end{aligned}$$

(A.5)

where $\Xi _i(\xi _i,\ldots ,\xi _6)$, $i=1,\ldots ,6$ are defined in Eq. (A.2), see Eqs. (4.1), (4.2). As can be seen in Figs. 8 and 9, dashed curves, in such a case the high-temperature asymptote for $c_{02}$ is reproduced better and this correlator has smaller values. However, the values of the objective function are much larger (dashed curve in the lower panel of Fig. 8). Thermodynamic quantities along with the static structure factor as they follow by numerical solution of Eq. (4.1) using (A.1) and (A.5) are compared in Figs. 10 and 11 (solid versus dashed curves). Although the results are different in detail (and correspond to a big difference of the objective function values) they look qualitatively quite similar.

Summarizing this appendix, we emphasize that a numerical solution of the self-consistent equations emerging after the Kondo–Yamaji approximation is an important ingredient of the RGM approach. While in the previous studies this issue has not been discussed in great detail, in the case of nonequivalent sites in the unit cell, when the number of equations increases, a controlled solving of this set of equations is vitally necessary to make possible a successful application of the RGM approach.