Spin-half Heisenberg antiferromagnet on a symmetric sawtooth chain: rotation-invariant Green’s functions and high-temperature series

Abstract

We apply the rotation-invariant Green’s function method to study the finite-temperature properties of a $S{=}1/2$ sawtooth-chain (also called $\Delta$-chain) antiferromagnetic Heisenberg model at the fully frustrated point when the exchange couplings along the straight-line and zig–zag paths are equal. We also use 13 terms of high-temperature expansion series and interpolation methods to get thermodynamic quantities for this model. We check the obtained predictions for observable quantities by comparison with numerics for finite systems. Although our work refers to a one-dimensional case, the utilized methods work in higher dimensions too and are applicable for examining other frustrated quantum spin lattice systems at finite temperatures.

Graphic Abstract

Appendices

Appendix A: Brief illustration of the finite-temperature Lanczos method (FTLM)

In this appendix, we provide the basics of the FTLM for convenience, see Refs. [91,92,93,94]. Within the FTLM scheme, the sum over an orthonormal basis in the partition function Z is replaced by a much smaller sum over R random vectors (in the present study, we take $R=50$ for $N=24,28,32$ and $R=20$ for $N=36$), that is

$$\begin{aligned} Z\approx \sum _{\gamma =1}^{\Gamma } \frac{\dim (\mathcal{{H}}(\gamma ))}{R} \sum _{\nu =1}^{R}\sum _{n=1}^{N_\textrm{L}} \textrm{e}^{-\frac{\epsilon _n^{(\nu )}}{T}} \left| \langle n(\nu )\vert \nu \rangle \right| ^2, \end{aligned}$$

(A.1)

where $\vert \nu \rangle$ labels random vectors for each symmetry-related orthogonal subspace $\mathcal{{H}}(\gamma )$ of the Hilbert space with $\gamma$ labeling the respective symmetry. The exponential of the Hamiltonian H in Eq. (A.1) is approximated by its spectral representation in a Krylov space spanned by the $N_\textrm{L}$ Lanczos vectors starting from the respective random vector $\vert \nu \rangle$, where $\vert n(\nu )\rangle$ is the nth eigenvector of H in this Krylov space with the energy $\epsilon _n^{(\nu )}$. To perform the symmetry-decomposed numerical Lanczos calculations, we use J. Schulenburg’s spinpack code [95, 96].

Appendix B: HTE series for the $S=1/2$ $J_1-J_2$ sawtooth-chain Heisenberg model

In this appendix, we report HTE series for the specific heat (per site) and the uniform susceptibility (per site), see Eq. (4.1), for a more general $S=1/2$ sawtooth-chain Heisenberg model with the exchange couplings along the straight line $J_1$ and along the zig–zag path $J_2$, see Fig. 1. We rewrite the coefficients $d_i$ and $c_i$, $i=2,\ldots ,13$ in Eq. (4.1) as follows:

$$\begin{aligned} d_i=\sum _{j=0}^i d_{i,j}J_1^{i-j}J_2^{j}; \;\;\; c_i=\sum _{j=0}^{i-1} c_{i,j}J_1^{i-j}J_2^{j}. \end{aligned}$$

(B.1)

The coefficients $d_{i,j}$, $j=0,\ldots ,i$ are as follows:

$$\begin{aligned} d_{2,j}= & {} \frac{3}{32},0,\frac{3}{16}, \nonumber \\ d_{3,j}= & {} \frac{3}{64},0,\frac{{-}9}{64},\frac{3}{32}, \nonumber \\ d_{4,j}= & {} \frac{{-}15}{512},0,\frac{{-}15}{128},\frac{{-}3}{64},\frac{{-}15}{256}, \nonumber \\ d_{5,j}= & {} \frac{{-}15}{512},0,\frac{15}{512},\frac{{-}5}{128},\frac{25}{256},\frac{{-}15}{256}, \nonumber \\ d_{6,j}= & {} \frac{21}{8\,192},0,\frac{291}{4\,096},\frac{13}{2\,048},\frac{261}{8\,192},\frac{63}{1\,024},\frac{21}{4\,096}, \nonumber \\ d_{7,j}= & {} \frac{917}{81\,920},0,\frac{777}{81\,920},\frac{77}{4\,096}, \nonumber \\{} & {} \quad \frac{{-}4\,739}{81\,920},\frac{217}{8\,192},\frac{{-}2\,611}{81\,920},\frac{917}{40\,960}, \nonumber \\ d_{8,j}= & {} \frac{1\,417}{655\,360},0,\frac{{-}2\,317}{81\,920},\frac{27}{8\,192},\frac{{-}3\,545}{98\,304}, \nonumber \\{} & {} \quad \frac{{-}1\,757}{61\,440},\frac{119}{15\,360},\frac{{-}4\,793}{122\,880},\frac{1\,417}{327\,680}, \nonumber \\ d_{9,j}= & {} \frac{{-}4\,303}{1\,376\,256},0,\frac{{-}4\,053}{327\,680},\frac{{-}375}{57\,344},\frac{6\,357}{286\,720}, \nonumber \\{} & {} \quad \frac{{-}24\,579}{1\,146\,880},\frac{3\,503}{114\,688},\frac{{-}3\,873}{1\,146\,880},\frac{2\,613}{1\,146\,880},\frac{{-}4\,303}{688\,128}, \nonumber \\ d_{10,j}= & {} \frac{{-}334\,433}{220\,200\,960},0,\frac{167\,591}{22\,020\,096},\frac{{-}35\,111}{11\,010\,048}, \nonumber \\{} & {} \quad \frac{370\,365}{14\,680\,064},\frac{16\,043}{1\,835\,008},\frac{8\,905}{22\,020\,096},\frac{303\,755}{11\,010\,048}, \nonumber \\{} & {} \quad \frac{{-}196\,571}{22\,020\,096},\frac{92\,629}{5\,505\,024},\frac{{-}334\,433}{110\,100\,480}, \nonumber \\ d_{11,j}= & {} \frac{37\,543}{62\,914\,560},0,\frac{9\,098\,771}{1\,321\,205\,760},\frac{169829}{110100480}, \nonumber \\{} & {} \quad \frac{{-}2\,735\,029}{792\,723\,456},\frac{1\,681\,801}{132\,120\,576},\frac{{-}16\,820\,155}{792\,723\,456},\frac{6\,456\,659}{990\,904\,320}, \nonumber \\{} & {} \quad \frac{{-}272\,833}{49\,545\,216},\frac{{-}330\,539}{66\,060\,288},\frac{2\,603\,711}{660\,602\,880},\frac{37\,543}{31\,457\,280}, \nonumber \\ d_{12,j}= & {} \frac{3\,987\,607}{6\,341\,787\,648},0,\frac{{-}3\,926\,113}{4\,404\,019\,200},\frac{31\,687\,379}{19\,818\,086\,400}, \nonumber \\{} & {} \quad \frac{{-}333\,299\,077}{26\,424\,115\,200},\frac{{-}72\,097}{157\,286\,400},\frac{{-}119\,844\,841}{19\,818\,086\,400}, \nonumber \\{} & {} \quad \frac{{-}20\,513\,567}{1\,321\,205\,760},\frac{46\,078\,849}{5\,872\,025\,600},\frac{{-}10\,265\,929}{707\,788\,800}, \nonumber \\{} & {} \quad \frac{9\,148\,231}{3\,303\,014\,400},\frac{{-}274\,151}{51\,609\,600},\frac{3\,987\,607}{3\,170\,893\,824}, \nonumber \\ d_{13,j}= & {} \frac{{-}1\,925\,339}{83\,047\,219\,200},0,\frac{{-}681\,805\,033}{249\,141\,657\,600},\frac{{-}263\,393}{2\,422\,210\,560}, \nonumber \\{} & {} \quad \frac{{-}609\,068\,681}{290\,665\,267\,200},\frac{{-}452\,833\,147}{79\,272\,345\,600},\frac{109\,267\,717}{10\,380\,902\,400}, \nonumber \\{} & {} \quad \frac{{-}585\,023\,153}{79\,272\,345\,600},\frac{12\,470\,300\,791}{1\,743\,991\,603\,200},\frac{2\,984\,548\,619}{871\,995\,801\,600}, \nonumber \\{} & {} \quad \frac{{-}242\,563\,763}{83\,047\,219\,200},\frac{66\,616\,537}{15\,571\,353\,600},\frac{{-}61\,211\,267}{20\,761\,804\,800}, \nonumber \\{} & {} \quad \frac{{-}1\,925\,339}{41\,523\,609\,600}. \end{aligned}$$

(B.2)

The coefficients $c_{i,j}$, $j=0,\ldots ,i-1$ are as follows:

$$\begin{aligned} c_{2,j}= & {} \frac{{-}1}{16},\frac{{-}1}{8}, \nonumber \\ c_{3,j}= & {} 0,\frac{1}{16},0, \nonumber \\ c_{4,j}= & {} \frac{1}{192},0,\frac{1}{256},\frac{1}{96}, \nonumber \\ c_{5,j}= & {} \frac{5}{3\,072},\frac{{-}1}{192},\frac{{-}1}{3\,072},\frac{{-}23}{1\,536},\frac{5}{1\,536}, \nonumber \\ c_{6,j}= & {} \frac{{-}7}{10\,240},\frac{{-}5}{3\,072},\frac{{-}29}{12\,288},\frac{17}{6\,144},\frac{{-}49}{12\,288},\frac{{-}7}{5\,120}, \nonumber \\ c_{7,j}= & {} \frac{{-}133}{245\,760},\frac{7}{10\,240},\frac{{-}29}{245\,760},\frac{1\,141}{368\,640}, \nonumber \\{} & {} \quad \frac{59}{122\,880},\frac{9}{2\,560},\frac{{-}133}{122\,880}, \nonumber \\ c_{8,j}&{=}&\frac{1}{32\,256},\frac{133}{245\,760},\frac{485}{589\,824},\frac{{-}43}{1\,474\,560}, \nonumber \\{} & {} \quad \frac{1\,393}{983\,040},\frac{{-}161}{92\,160},\frac{5\,863}{2\,949\,120},\frac{1}{16\,128}, \nonumber \\ c_{9,j}&{=}&\frac{1\,269}{9\,175\,040},\frac{{-}1}{32\,256},\frac{2\,623}{11\,796\,480},\frac{{-}1\,847}{1\,966\,080}, \nonumber \\{} & {} \quad \frac{{-}281}{1\,835\,008},\frac{{-}2\,657}{2\,949\,120}, \frac{{-}54\,223}{82\,575\,360},\frac{{-}23\,629}{41\,287\,680},\frac{1\,269}{4\,587\,520}, \nonumber \\ c_{10,j}= & {} \frac{3\,737}{148\,635\,648},\frac{{-}1\,269}{9\,175\,040},\frac{{-}73\,531}{330\,301\,440},\frac{{-}4\,399}{20\,643\,840}, \nonumber \\{} & {} \quad \frac{{-}47\,869}{82\,575\,360},\frac{1\,457}{2\,949\,120},\frac{{-}72\,833}{99\,090\,432}, \nonumber \\{} & {} \quad \frac{107\,419}{165\,150\,720},\frac{{-}34\,337}{47\,185\,920},\frac{3\,737}{74\,317\,824}, \nonumber \\ c_{11,j}&{=}&\frac{{-}339\,691}{11\,890\,851\,840},\frac{{-}3\,737}{148\,635\,648},\frac{{-}1\,488\,731}{11\,890\,851\,840}, \nonumber \\{} & {} \quad \frac{470\,969}{1\,981\,808\,640},\frac{{-}87\,187}{3\,963\,617\,280},\frac{179\,867}{412\,876\,800},\frac{858\,749}{2\,972\,712\,960}, \nonumber \\{} & {} \quad \frac{3\,757}{70\,778\,880},\frac{152\,969}{339\,738\,624}, \frac{{-}22\,843}{2\,972\,712\,960},\frac{{-}339\,691}{5\,945\,425\,920}, \nonumber \\ c_{12,j}= & {} \frac{{-}1\,428\,209}{108\,999\,475\,200},\frac{339\,691}{11\,890\,851\,840},\frac{9\,716\,173}{237\,817\,036\,800}, \nonumber \\{} & {} \quad \frac{14\,512\,039}{118\,908\,518\,400},\frac{7\,500\,233}{33\,973\,862\,400},\frac{{-}828\,713}{8\,493\,465\,600}, \nonumber \\{} & {} \quad \frac{795\,319}{2\,264\,924\,160},\frac{{-}3\,465\,593}{11\,890\,851\,840},\frac{9\,934\,111}{39\,636\,172\,800}, \nonumber \\{} & {} \quad \frac{{-}152\,533}{1\,061\,683\,200},\frac{3\,379\,349}{15\,854\,469\,120},\frac{{-}1\,428\,209}{54\,499\,737\,600}, \nonumber \\ c_{13,j}= & {} \frac{18\,710\,029}{4\,484\,549\,836\,800},\frac{1\,428\,209}{108\,999\,475\,200}, \nonumber \\{} & {} \quad \frac{6\,694\,733}{135\,895\,449\,600},\frac{{-}670\,989\,853}{15\,695\,924\,428\,800},\frac{5\,017\,897}{99\,656\,663\,040}, \nonumber \\{} & {} \quad \frac{{-}482\,107\,547}{2\,615\,987\,404\,800},\frac{{-}390\,798\,299}{4\,484\,549\,836\,800},\frac{{-}21\,335\,483}{348\,798\,320\,640}, \nonumber \\{} & {} \quad \frac{{-}529\,232\,611}{2\,092\,789\,923\,840},\frac{71\,879\,767}{784\,796\,221\,440},\frac{{-}741\,118\,447}{3\,487\,983\,206\,400}, \nonumber \\{} & {} \quad \frac{40\,887\,437}{747\,424\,972\,800},\frac{18\,710\,029}{2\,242\,274\,918\,400}.\nonumber \\ \end{aligned}$$

(B.3)

After setting $J_1=J_2=1$ in Eqs. (B.1)–(B.3), one arrives at Eq. (4.1) for the $S=1/2$ symmetric sawtooth-chain Heisenberg antiferromagnet (2.1).

Appendix C: Some intermediate results of the entropy-method interpolation

Fig. 8

Entropy method results based on the HTE series up to 13th order for (from top to bottom) c(e), T(e), and c(T). Here, we report the outcomes for nine Padé approximants [u, d] in Eq. (4.3), namely, [2, 11], [3, 10], [4, 9], [5, 8], [6, 7], [7, 6], [8, 5], [9, 4], and [10, 3]

Following the steps described in Sect. 4, we obtain the (approximate) entropy given in Eq. (4.4) and then the specific heat c(e) and the temperature T(e):

$$\begin{aligned} c(e)=-\frac{{s^\prime }^2}{s^{\prime \prime }}, \;\;\; T(e)=\frac{1}{s^\prime }; \end{aligned}$$

(C.1)

here, the prime denotes the derivative with respect to e. Equation (C.1) is a parametric representation of the temperature dependence of the specific heat c(T). The resulting c(T) curves obtained by entropy method based on the HTE series up to 10th order are smooth. However, c(T) curves obtained using the HTE series of 11th, 12th, and 13th orders are inadequate; see, e.g., Fig. 8. While the c(T) profile based on [2, 11], [3, 10], [9, 4], and [10, 3] in Eq. (4.3) is smooth (these curves are among the ones reported in Fig. 7), c(T) based on [4, 9], [5, 8], [6, 7], [7, 6], and [8, 5] in Eq. (4.3) abruptly falls to zero at certain temperatures (such curves are not shown in Fig. 7). The reason for that can be traced back to the Padé approximants in Eq. (4.3): $Q_d(e)$ may become zero at certain $e=e_d^\star$, $Q_d(e_d^\star )=0$, but $P_u(e)$ remains finite at this value of $e=e_d^\star$, $P_u(e_d^\star )\ne 0$; see Table 1. Previously, such Padé approximants were declared as unphysical and discarded. However, as can be seen from Fig. 8, it may be sufficient to discard from further consideration only a small region around $e=e_d^\star$, while other values of e are applicable for further manipulations to get c(T).

Table 1 Roots of polynomials $P_u(e)$ and $Q_d(e)$ [see Eq. (4.3)] denoted as $e^\star _{u}$ and $e^\star _{d}$, respectively

It might be worth noting that the entropy method yields a physical result even for $Q_d(e^\star _d)=0$, $e_0\le e^\star _d\le 0$, if $P_u(e^\star _u)=0$, $e^\star _u=e^\star _d$. We notice that in our calculations, poles $e^\star _d$ and zeros $e^\star _u$ may be close but not equal, see the second column in Table 1, resulting in nonapplicability of such a Padé approximant around $e=e_d^\star$; see the two upper panels in Fig. 8. Since s(e) and G(e), Eqs. (4.3) and (4.4), are expected to be smooth, the Padé approximants [4, 9], [5, 8], [6, 7], [7, 6], and [8, 5] have “defects” (a defect is the name given to an extraneous pole and a nearby zero; see Ref. [97]). The nearby zero of numerator and denominator may be regarded as canceling approximately; this is how to put the defects in the proper perspective; see Ref. [97].