Rigorous analysis of the effects of electron–phonon interactions on magnetic properties in the one-electron Kondo lattice model

Abstract

The Kondo lattice model (KLM) is a typical model describing heavy fermion systems. In this paper, we consider the interaction of phonons with the system described by the one-electron KLM. Magnetic properties of the ground state of this model are revealed in a rigorous form. Furthermore, we derive the effective Hamiltonian in the strong coupling limit ($J\rightarrow \infty$) for the strength of the spin-exchange interaction J; we examine the magnetic properties of the ground state of the effective Hamiltonian and prove that the Aizenman–Lieb theorem concerning the magnetization holds for the effective Hamiltonian at finite temperatures. Generalizing the obtained results, we clarify a mechanism for the stability of magnetic properties of the ground state in the one-electron KLM system.

A Brief overview of the definition of stability classes

The theory of stability classes in many-electron systems is originated in [13]. A more mathematically refined construction of the theory is presented in [16]. The essence of the idea is as follows.

Let $\mathfrak {H}_1$ be a Hilbert space and $\mathfrak {H}_0$ be its closed subspace. Suppose that the total spin operators $\varvec{S}_\mathrm{tot}$ act on the two Hilbert spaces, and furthermore that the two Hilbert spaces $\mathfrak {H}_0$ and $\mathfrak {H}_1$ are subspaces of the $M=0$-subspace: $\ker (S^{(3)}_\mathrm{tot})$.

Let $P_{1, 0}$ be the orthogonal projection from $\mathfrak {H}_1$ to $\mathfrak {H}_0$. Moreover, let $\mathfrak {P}_0$ and $\mathfrak {P}_1$ be Hilbert cones in $\mathfrak {H}_0$ and $\mathfrak {H}_1$, respectively. Now, suppose that

$$\begin{aligned} P_{1, 0}\mathfrak {P}_1=\mathfrak {P}_0 \end{aligned}$$

(A.1)

holds. Assume that $\psi _0\in \mathfrak {P}_0$ is strictly positive and has total spin $S_0$: ${\varvec{S}}^2_\mathrm{tot}\psi _0=S_0(S_0+1) \psi _0$. Next, assume that a vector $\psi _1$ in $\mathfrak {H}_1$ is strictly positive with respect to $\mathfrak {P}_1$ and has total spin $S_1$. With these settings, we claim that

$$\begin{aligned} S_0=S_1 \end{aligned}$$

(A.2)

holds. Indeed, because $P_{1, 0}\psi _1\ne 0$ and $P_{1, 0}\psi _1\ge 0$ w.r.t. $\mathfrak {P}_0$ hold due to (A.1), we get the positive overlap property: $\langle \psi _0|P_{1, 0} \psi _1\rangle >0$, which impels that

$$\begin{aligned} S_0(S_0+1) \langle \psi _0|P_{1, 0} \psi _1\rangle =\langle {\varvec{S}}^2_\mathrm{tot}\psi _0|P_{1, 0}\psi _1\rangle =\langle \psi _0|P_{1, 0} {\varvec{S}}^2_\mathrm{tot}\psi _1\rangle =S_1(S_1+1) \langle \psi _0|P_{1, 0} \psi _1\rangle . \end{aligned}$$

(A.3)

Hence, we conclude (A.2).

Let ${\mathscr {C}}(\mathfrak {P}_0)$ be the set of pairs $(\mathfrak {H}, \mathfrak {P})$ of a Hilbert space and a Hilbert cone satisfying the following conditions:

• $\mathfrak {P}$ is a Hilbert cone in $\mathfrak {H}$;

• $\mathfrak {H}_0$ is a subspace of $\mathfrak {H}$;

• a similar relation to (A.1): $P \mathfrak {P}=\mathfrak {P}_0$ is fulfilled, where P is the orthogonal projection from $\mathfrak {H}$ to $\mathfrak {H}_0$.

If we choose arbitrarily a pair $(\mathfrak {H}, \mathfrak {P})\in {\mathscr {C}}(\mathfrak {P}_0)$, and assume that a vector $\psi \in \mathfrak {H}$ is strictly positive with respect to $\mathfrak {P}$ and has total spin S, then we conclude $S=S_0$ using the positive overalp property as in the previous discussion.

Now, suppose we are given a Hamiltonian H acting on $\mathfrak {H}$. Assume then that this Hamiltonian satisfies the following conditions:

• (i) H commutes with the total spin operators $S_\mathrm{tot}^{(j)}\ (j=1, 2, 3)$.

• (ii) The heat semigroup $\{e^{-\beta H}\}$ is ergodic w.r.t. $\mathfrak {P}$.

Theorem 2.10 shows that the ground state $\psi _g$ of H is strictly positive with respect to $\mathfrak {P}$, and consequently, we know that $\psi _g$ has total spin $S_0$ due to the positive overlap property.

Now, let ${\mathscr {A}}_{\mathfrak {P}_0}(\mathfrak {H}, \mathfrak {P})$ be the set of all Hamiltonians satisfying the two conditions (i) and (ii). Then the ground state of any Hamiltonian belonging to ${\mathscr {A}}_{\mathfrak {P}_0}(\mathfrak {H}, \mathfrak {P})$ has total spin $S_0$. The $\mathfrak {P}_0$-stability class is defined by

$$\begin{aligned} {\mathscr {A}}_{\mathfrak {P}_0}=\bigcup _{(\mathfrak {H}, \mathfrak {P}) \in {\mathscr {C}}(\mathfrak {P}_0)} {\mathscr {A}}_{\mathfrak {P}_0}(\mathfrak {H}, \mathfrak {P}). \end{aligned}$$

(A.4)

From the construction, we see that the ground states of all Hamiltonians belonging to ${\mathscr {A}}_{\mathfrak {P}_0}$ have total spin $S_0$.

This theory describes the stability of magnetic properties of ground states in many-electron systems. To a given Hamiltonian $H_1$, let $H_2$ be a Hamiltonian obtained by adding complicated interaction terms with the environment typified by phonons. In general, the analysis of $H_2$ becomes more difficult, but if we find that $H_1$ and $H_2$ both belong to the same stability class ${\mathscr {A}}_{\mathfrak {P}_0}$, we conclude that the ground states of these Hamiltonians have the same total spin $S_0$.

Define

$$\begin{aligned} \mathfrak {P}_{\mathrm{NT}}={\mathrm{coni}}\Bigg (\bigg \{\left| {\varvec{\sigma }}_x\right\rangle _\mathrm{NT} : {\varvec{\sigma }}\in {\mathcal {S}}_{\varLambda }, \sum _{y\in \varLambda \setminus \{x\}} \sigma _y=0, x\in \varLambda \bigg \}\Bigg ). \end{aligned}$$

(A.5)

Then, $\mathfrak {P}_{\mathrm{NT}}$ is a Hilbert cone in the Hilbert space $Q \bigwedge ^{|\varLambda |-1} \big (\ell ^2(\varLambda )\oplus \ell ^2(\varLambda )\big ) \cap \ker (S_\mathrm{tot}^{(3)})$ that emerged in defining the NT system,⁷ The $\mathfrak {P}_\mathrm{NT}$-stability class constructed from $\mathfrak {P}_\mathrm{NT}$ is called the Nagaoka–Thouless stability class. Theorem 1.12 and other fundamental properties of the NT stability class are discussed in detail in [13]. Under these settings, the meaning of Theorem 1.11 should now be clear. From this theorem, it immediately follows that the values of total spin in the ground state coincide in Theorems 1.2 and 1.7.

In addition to the NT stability class discussed here, several other stability classes are known. Recently, it has become clear that the stability theory can describe the flat-band ferromagnetism [17]. On the other hand, some progress has been made in grounding this theory with the Tomita–Takesaki theory in operator algebras [16].