c Here we will examine how the Lieb-Schultz-Mattis argument (Theorem 6.3 in p. 162) applies to the Majumdar-Ghosh model. First try twisting one of the dimer states as $\left|\Phi^{\prime}\right\rangle=\hat{U}_{\mathrm{LSM}}\left|\Phi_{\text {dimer }}^{\text {odd }}\right\rangle$. We see from Lemma 6.4 that the state $\left|\Phi^{\prime}\right\rangle$ has energy expectation value close to $E_{\mathrm{GS}}$. Compute the overlap $\left\langle\Phi_{\text {dimer }}^{\text {odd }} \mid \Phi^{\prime}\right\rangle$, especially its limiting value as $L \uparrow \infty$. Why doesn't Lemma 6.2 apply? Next consider the translation invariant ground state $\left|\Phi_{\text {dimer }}^{+}\right\rangle=\left|\Phi_{\text {dimer }}^{\text {odd }}\right\rangle+\left|\Phi_{\text {dimer }}^{\text {even }}\right\rangle$, and examine the nature of its twist $\left|\Phi^{\prime \prime}\right\rangle=\hat{U}_{\mathrm{LSM}}\left|\Phi_{\text {dimer }}^{+}\right\rangle$. In particular how does the state $\left|\Phi^{\prime \prime}\right\rangle$ look like?