Abstract
In this preparatory chapter, we define quantum spin systems, and discuss basic features of the ground states of the Heisenberg models. These basic results will be useful throughout the book. After studying this chapter the reader may start with any of the parts I, II, or III. After reviewing basic properties of a single quantum mechanical spin in Sect. 2.1, we introduce general quantum spin systems and fix notations in Sect. 2.2. We review in some detail the notion of time-reversal in Sect. 2.3. Then in Sects. 2.4 and 2.5, we discuss the ferromagnetic and antiferromagnetic Heisenberg models, respectively, placing main emphasis on general properties of their ground states. The Marshall-Lieb-Mattis theorem (Theorem 2.2 in p. 39) for the antiferromagnetic model plays a central role throughout the book.
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- 1.
To be rigorous this relations should be \(\hat{\varvec{S}}^2=S(S+1)\hat{1}\) with \(\hat{1}\) being the identity operator. We usually omit \(\hat{1}\).
- 2.
In magnetic systems, spin angular momenta come mostly from electron spins, which have spin quantum number 1/2. In some atoms, however, electrons in certain orbits are coupled according to the Hund rule to form a single effective spin with higher S.
- 3.
If the reader is new to quantum spins, it is a good idea to try examining, for the \(S=1/2\) case, what are the states where the spin is pointing in the positive 1 or 2 directions.
- 4.
To be very precise, the rotation operators are representations of elements of SU(2).
- 5.
One can take arbitrary products of \(\hat{U}^{(\alpha )}_\theta \) with various \(\alpha =1,2,3\) and \(\theta \).
- 6.
This corresponds to the mathematical fact that SU(2) is “twice larger” than the three dimensional rotation group SO(3). See Appendix A.5.
- 7.
For general S, the corresponding state \(|\psi _{\theta ,\varphi }\rangle =e^{-i\varphi \hat{S}^{(3)}}e^{-i\theta \hat{S}^{(2)}}|\psi ^S\rangle \) is known as the spin coherent state. See, e.g., Chap. 6 of [4].
- 8.
For a deeper reason for the appearance of the phase factor, see, e.g., Sect. 6.5 (in particular p. 85) of [4].
- 9.
More precisely, one defines a representation \(\rho \) by \(\rho (e)=\mathsf {I}\), \(\rho (a)=\mathsf {R}^{(1)}_\pi \), \(\rho (b)=\mathsf {R}^{(2)}_\pi \), and \(\rho (c)=\mathsf {R}^{(3)}_\pi \).
- 10.
We learned this elegant derivation from Akinori Tanaka.
- 11.
More precisely, one defines a representation \(\rho \) by \(\rho (e)=\hat{1}\), \(\rho (a)=\hat{u}_1\), \(\rho (b)=\hat{u}_2\), and \(\rho (c)=\hat{u}_3\).
- 12.
- 13.
The identity \(e^{-i\pi \hat{S}^{(\alpha )}}=-2i\hat{S}^{(\alpha )}\) (which is valid only for \(S=1/2\)) can be derived from (2.1.26). But, by noting that \(e^{-i\pi \hat{S}^{(3)}}|\psi ^{\pm 1/2}\rangle =e^{\mp i\pi /2}|\psi ^{\pm 1/2}\rangle =\mp i|\psi ^{\pm 1/2}\rangle =-2i\hat{S}^{(3)}|\psi ^{\pm 1/2}\rangle \), the identity for \(\alpha =3\) readily follows. Other cases then follow by symmetry. Similarly the identity \(e^{-i\pi \hat{S}^{(\alpha )}}=\hat{1}-2(\hat{S}^{(\alpha )})^2\) for \(S=1\) can be derived from (S.3) in p. 494, but can also be derived in a similar (simpler) manner.
- 14.
See Footnote 2 in p. 14.
- 15.
Probably the notation \(|\psi ^{\sigma }\rangle _x\) is more logical (see Appendix A.1), but we use this compact notation.
- 16.
One can take arbitrary products of \(\hat{U}^{(\alpha )}_\theta \) with various \(\alpha =1,2,3\) and \(\theta \).
- 17.
Note that \(|\!\uparrow \rangle _1|\!\downarrow \rangle _2=\sqrt{2}(\frac{1}{2}+\hat{S}^{(3)}_1)(\frac{1}{2}-\hat{S}^{(3)}_2)|\varPhi _{0,0}\rangle \).
- 18.
In (classical) mechanics the angular momentum \(\varvec{J}\) of a particle at \(\varvec{r}\) with velocity \(\varvec{v}\) is given by \(\varvec{J}=\varvec{r}\times \varvec{v}\).
- 19.
The product of a unitary operator and the complex-conjugation map, such as \(\hat{\varTheta }\), is called an antiunitary operator. See Appendix A.4.3 for more details.
- 20.
This relation can be interpreted as \(\hat{K}^\dagger =\hat{K}\). See Appendix A.4.3.
- 21.
Let \(\langle \varphi |\) and \(\hat{A}\) be an arbitrary bra state and a linear operator. Then there exists a bra state \(\langle \xi |\) such that \(\langle \varphi |(\hat{A}|\psi \rangle )=\langle \xi |\psi \rangle \) for any \(|\psi \rangle \). We therefore write \(\langle \xi |=\langle \varphi |\hat{A}\). But note, e.g., that there are no \(\langle \varphi |\) and \(\langle \xi |\) such that \(\langle \varphi |(\hat{K}|\psi \rangle )=\langle \xi |\psi \rangle \) for any \(|\psi \rangle \).
- 22.
One can also write \(\hat{\varTheta }^{-1}\) as \(\hat{\varTheta }^\dagger \). See Appendix A.4.3.
- 23.
The linearity is a consequence of the general fact that the product of two antilinear operators is linear. See Appendix A.4.3.
- 24.
There is indeed a deep reason behind this observation. Define a map \(\varGamma _\mathrm{tr}\) by \(\varGamma _\mathrm{tr}(\hat{S}^{(\alpha )}_x)=-\hat{S}^{(\alpha )}_x\) for any \(x\in \varLambda \) and \(\alpha =1,2,3\), \(\varGamma _\mathrm{tr}(\hat{A}^\dagger )=\varGamma _\mathrm{tr}(\hat{A})^\dagger \), \(\varGamma _\mathrm{tr}(\alpha \hat{A}+\beta \hat{B})=\alpha ^*\varGamma _\mathrm{tr}(\hat{A})+\beta ^*\varGamma _\mathrm{tr}(\hat{B})\), and \(\varGamma _\mathrm{tr}(\hat{A}\hat{B})=\varGamma _\mathrm{tr}(\hat{A})\varGamma _\mathrm{tr}(\hat{B})\) for any operators \(\hat{A},\hat{B}\) and \(\alpha ,\beta \in \mathbb {C}\). Such a map is called an antilinear \(*\)-automorphism. Then a general result known as Wigner’s theorem guarantees that there exists an antiunitary operator \(\hat{\varTheta }\) such that \(\varGamma _\mathrm{tr}(\hat{A})=\hat{\varTheta }^{-1}\hat{A}\hat{\varTheta }\) for any \(\hat{A}\). See Appendix A.6.
- 25.
This conclusion is intuitively understood if we assume that the magnetic field is generated by an electromagnet. Then the time-reversal changes the direction of the electric current, and hence reverses the magnetic field.
- 26.
In the language of graph theory, \(\varLambda \) is the set of vertices and \({\mathscr {B}}\) is the set of edges.
- 27.
To be precise this can be true only in three or higher dimensions if the system is macroscopically large. In one or two dimensions, it is known from the Hohenberg-Mermin-Wagner theorem (Theorem 4.24 in p. 124) that the ferromagnetic Heisenberg model does not exhibit ferromagnetic order at nonzero temperatures.
- 28.
A lattice \((\varLambda ,{\mathscr {B}})\) is connected if for any \(x,y\in \varLambda \) such that \(x\ne y\), there exists a finite sequence \(z_1, z_2,\ldots , z_n\in \varLambda \) with the properties \(z_1=x\), \(z_n=y\), and \(\{z_j,z_{j+1}\}\in {\mathscr {B}}\) for \(j=1,\ldots ,n-1\).
- 29.
This fact follow also from the Perron–Frobenius theorem (Theorem A.18 in p. 475).
- 30.
Here we are in fact minimizing the classical Hamiltonian \(H=\sum _{\{x,y\}\in {\mathscr {B}}}\overrightarrow{S}_x\cdot \overrightarrow{S}_y\). If the lattice is non-bipartite, even the classical problem of minimization becomes nontrivial. This is known as the problem of frustration.
- 31.
In the classical limit with \(S\gg 1\), one may neglect the \(O(|{\mathscr {B}}|S)\) term to find that \(\hat{H}|\varPhi _{\text {N}\acute{\mathrm{{e}}}\text {el}}\rangle \simeq -|{\mathscr {B}}|S^2\,|\varPhi _{\text {N}\acute{\mathrm{{e}}}\text {el}}\rangle \), and \(|\varPhi _{\text {N}\acute{\mathrm{{e}}}\text {el}}\rangle \) is a (near) ground state.
- 32.
See Footnote 28 in p. 33.
- 33.
This is standard. That \((\hat{\varvec{S}}_\mathrm{tot})^2|\varPhi _\mathrm {GS}\rangle =0\) means \(0=\langle \varPhi _\mathrm {GS}|(\hat{\varvec{S}}_\mathrm{tot})^2|\varPhi _\mathrm {GS}\rangle =\Vert \hat{S}^{(1)}_\mathrm{tot}|\varPhi _\mathrm {GS}\rangle \Vert ^2+\Vert \hat{S}^{(2)}_\mathrm{tot}|\varPhi _\mathrm {GS}\rangle \Vert ^2+\Vert \hat{S}^{(3)}_\mathrm{tot}|\varPhi _\mathrm {GS}\rangle \Vert ^2\), which shows \(\Vert \hat{S}^{(\alpha )}_\mathrm{tot}|\varPhi _\mathrm {GS}\rangle \Vert =0\).
- 34.
In general we say that a state \(|\varPhi \rangle \) is SU(2) invariant if it holds for any \(\alpha \) and \(\theta \) that \(\hat{U}^{(\alpha )}_\theta |\varPhi \rangle =c|\varPhi \rangle \) with some constant \(c\in \mathbb {C}\) (which may depend on \(\alpha \) and \(\theta \)) with \(|c|=1\).
- 35.
The reader new to this kind of proof is suggested to first consider the case with \(S=1/2\).
- 36.
This precisely corresponds to the successive operation of \(\hat{S}_{x_j}^+\hat{S}_{y_j}^-\) with \(j=1,\ldots ,n\) to \(|\psi ^\sigma \rangle \).
- 37.
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Tasaki, H. (2020). Basics of Quantum Spin Systems. In: Physics and Mathematics of Quantum Many-Body Systems. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-41265-4_2
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