Stability of good quantum numbers in ground states

TL;DR

This paper investigates the stability of good quantum numbers, which are eigenvalues of a bounded self-adjoint operator commuting with the Hamiltonian, under perturbations, using order theory in quantum mechanics.

Contribution

It introduces a new perspective on the stability of quantum numbers by applying order theory to analyze perturbations of Hamiltonians with discrete spectra.

Findings

Good quantum numbers remain stable under certain perturbations.

Order theory provides a framework for analyzing eigenvalue stability.

Results have implications for understanding quantum state properties under perturbations.

Abstract

Let $H$ be a self-adjoint operator, bounded from below and let $O$ be a bounded self-adjoint operator with purely discrete spectrum. Suppose that (i) $E(H)=\inf \mathrm{spec}(H)$ is a simple eigenvalue, and (ii) $H$ strongly commutes with $O$. Let $\psi_H$ be the eigenvector associated with $E(H)$. By the assumptions (i) and (ii), $\psi_H$ is an eigenvector of $O$: $O\psi_H=\mu(H)\psi_H$. In the context of quantum mechanics, $\mu(H)$ is called a good quantum number. In this note, we examine the stability of $\mu(H)$ under perturbations of $H$ from a viewpoint of the order theory.

Figures

Equations

$$O{\psi }_H=\mu (H){\psi }_H.$$

$$\displaystyle e^{-\beta(\overline{sH+tH^{\prime}})}=\mbox{$\mathrm{s}$-$\displaystyle\lim_{n\to\infty}$}\Big{(}e^{-\beta sH/n}e^{-\beta tH^{\prime}/n}\Big{)}^{n},$$

$${\mathcal{E}}_{1,2}(A)={\pi }_{1,2}A{\pi }_{1,2}+{\pi }_{1,2}^{\perp }A{\pi }_{1,2}^{\perp },A\in \mathcal{B}({\mathfrak{H}}_2).$$

$${\pi }_{1,2}A{\pi }_{1,2}{\mathfrak{P}}_1={\pi }_{1,2}A{\pi }_{1,2}{\mathfrak{P}}_2\subseteq {\pi }_{1,2}A{\mathfrak{P}}_2\subseteq {\pi }_{1,2}{\mathfrak{P}}_2={\mathfrak{P}}_1.$$

$$(H_1,{\mathfrak{P}}_1)$$

$$⇢\cdots$$

$$(H_1,{\mathfrak{P}}_1)$$

$$\to \cdots$$

$${\mathcal{P}}_{{\mathfrak{H}}_{\ast },0}=\bigcup _{\mathfrak{H}\in {\mathbb{H}}_{{\mathfrak{H}}_{\ast }}}\bigcup _{\mathfrak{P}\subset \mathfrak{H}}{\mathcal{A}}_{\mathfrak{P}}^{+},$$

$$\displaystyle\mathscr{P}_{\mathfrak{H}_{*},0}(O)=\{H\in\mathscr{P}_{\mathfrak{H}_{*},0}\,|\,\mbox{$e^{isO}e^{itH}=e^{itH}e^{isO}$ for all $s,t\in\mathbb{R}$}\}.$$

$$\mu (H_j)⟨{\psi }_{H_j}|{\pi }_{j,j+1}{\psi }_{H_{j+1}}⟩$$

$$=⟨{\psi }_{H_j}|{\mathcal{E}}_{j,j+1}(O){\psi }_{H_{j+1}}⟩$$

$$=⟨{\psi }_{H_j}|{\pi }_{j,j+1}O{\psi }_{H_{j+1}}⟩$$

$$=\mu (H_{j+1})⟨{\psi }_{H_j}|{\pi }_{j,j+1}{\psi }_{H_{j+1}}⟩.$$

$$\mathfrak{M}={\mathfrak{M}}_{\ast }\otimes \mathfrak{N},\mathfrak{H}={\mathfrak{H}}_{\ast }\otimes \mathfrak{X},\Omega ={\Omega }_{\ast }\otimes {\Omega }_{\mathfrak{X}}.$$

$$S({\varrho }_{\psi }^{H_{\ast }}|{\varrho }_{{\psi }_{\ast }})=\mathrm{tr}[{\varrho }_{\psi }^{{\mathfrak{H}}_{\ast }}\log {\varrho }_{\psi }^{{\mathfrak{H}}_{\ast }}]-\mathrm{tr}[{\varrho }_{\psi }^{{\mathfrak{H}}_{\ast }}\log {\varrho }_{{\psi }_{\ast }}].$$

$${\mathfrak{P}}_{\ast }\otimes {\mathbb{R}}_{+}^n=\mathrm{coni}\{{\phi }_j\otimes e_j|{\phi }_j\in {\mathfrak{P}}_{\ast },j=1,\dots ,n\},$$

$${\mathfrak{H}}_{\mu }={\mathfrak{H}}_{\ast }\otimes {\mathbb{C}}^{n_{\mu }},{\mathfrak{P}}_{\mu }={\mathfrak{P}}_{\ast }\otimes {\mathbb{R}}_{+}^{n_{\mu }},\mu =1,2,\dots ,\ell .$$

$${\mathfrak{H}}_I={\mathfrak{H}}_{\ast }\otimes {\mathbb{C}}^{n_{{\mu }_1}}\otimes \cdots \otimes {\mathbb{C}}^{n_{{\mu }_k}},{\mathfrak{P}}_I={\mathfrak{P}}_{{}_{\ast }}\otimes {\mathbb{R}}_{+}^{n_{{\mu }_1}}\otimes \cdots \otimes {\mathbb{R}}_{+}^{n_{{\mu }_k}}$$

$$\tau \phi =\phi \otimes {\omega }_{{\mu }_{k+1}}\otimes \cdots \otimes {\omega }_{{\mu }_{k+\ell }}.$$

$${\pi }_{I_1,I_2}\psi =P_{{\omega }_{I_2\backslash I_1}}\psi ,$$

$$H_{\mathrm{MLM}}={\mathbf{S}}_A\cdot {\mathbf{S}}_B,$$

$$[S_x^{(j)},S_y^{(k)}]=i{\delta }_{xy}\sum _{\ell =1}^3{\epsilon }_{jk\ell }{S}_x^{(\ell )}.$$

$${\mathbf{S}}_{\mathrm{tot}}=\sum _{x\in \Lambda }{\mathbf{S}}_x=(S_{\mathrm{tot}}^{(1)},S_{\mathrm{tot}}^{(2)},S_{\mathrm{tot}}^{(3)})$$

$$H_{\mathrm{HH}}$$

$$(A+s{)}^{-1}={\int }_0^{\infty }d\beta e^{-\beta (A+s)},$$

$$\displaystyle e^{-\beta A}=\mbox{$\mathrm{s}$-$\displaystyle\lim_{n\to\infty}$}\Big{(}1+\frac{\beta}{n}A\Big{)}^{-n}.$$

$$e^{-\beta E(A)}\Vert \xi {\Vert }^2=⟨\xi |e^{-\beta A}\xi ⟩\le ⟨|\xi ||e^{-\beta A}|\xi |⟩\le e^{-\beta E(A)}\underset{=\Vert \xi {\Vert }^2}{\underbrace{\Vert |\xi |{\Vert }^2}}.$$

$$e^{-\beta (A-B)}=\sum _{n=0}^{\infty }{\mathcal{I}}_n(\beta ),$$

$${\mathcal{I}}_n(\beta )={\int }_{0\le s_1\le \cdots \le s_n\le \beta }B(s_1)\cdots B(s_n)e^{-\beta A}$$

$$\displaystyle e^{-\beta(A-B)}\unrhd\mathcal{I}_{n}(\beta)\ \mbox{w.r.t. $\mathfrak{P}$}$$

$$⟨\xi |e^{-\beta (A-B)}\eta ⟩\ge ⟨\xi |{\mathcal{I}}_{\ell }(\beta )\eta ⟩.$$

$$⟨\xi |{\mathcal{I}}_{\ell }(\beta )\eta ⟩={\int }_{0\le s_1\le \cdots \le s_{\ell }\le \beta }F(s_1,\dots ,s_{\ell })>0.$$

Full text

Mathematics Subject Classification (2010).

Primary: 47A63, 47B60, 47A75; Secondary: 47A55

**Keywords. **

Ground states; Eigenvalues; Positivity improving resolvents; Posets; Good quantum numbers; ferromagnetism.

1 Introduction

Let $\mathfrak{H}$ be a complex Hilbert space and let $H$ be a self-adjoint operator on $\mathfrak{H}$, bounded from below. Suppose that $E(H)=\inf\mathrm{spec}(H)$ is a simple eigenvalue, where $\mathrm{spec}(H)$ is spectrum of $H$. The eigenvector associated with $E(H)$ is denoted by $\psi_{H}$. Let $O$ be a bounded self-adjoint operator with purely discrete spectrum. Assume that $H$ strongly commutes with $O$, that is, their spectral measures commute with each other. Under this setting, we readily see that $\psi_{H}$ is an eigenvector of $O$:

[TABLE]

In quantum mechanics, suppose that a particular Hamiltonian $H$ and an operator $O$ with corresponding eigenvalues and eigenvectors are given. Then the eigenvalues are said to be “good quantum numbers” if every eigenvector remains an eigenvector of the same eigenvalue as time evolves, or $H$ strongly commutes with $O$. In this sense, the eigenvalue $\mu(H)$ can be regarded as a good quantum number.

In this note, we will examine the stability of $\mu(H)$. To be precise, let $V$ be a self-adjoint operator. We will consider a perturbation of $H$ by $V$. For simplicity, we suppose that $V$ is bounded. We continue to assume that $E(H+V)$ is a simple eigenvalue of $H+V$. Our main purpose is to answer the following question.

*When does the equation $\mu(H+V)=\mu(H)$ hold ? *

In the rest of the present note, we will provide a framework which enables us to solve the above problem. Our novel idea for constructing the framework is to apply the positivity improvingness of the resolvent of $H$.

Before we proceed, we briefly explain the motivation behind the aforementioned problem. An essence of the problem originates from the study of ferromagnetism in many-electron systems; Mathematical studies of ferromagnetism were initiated by Lieb [6], Nagaoka [12] and Thouless [16]. The origin of ferromagnetism still has been mystery and are actively examined even today, see e.g., [3, 4, 15]. In [8, 9, 10, 11], Miyao examined the stability of ferromagnetism in many-electron systems. In particular, he gave a model independent framework which describes various stability results concerning ferromagnetism in the Hubbard model [11]. Remark that in concrete applications to many-electron system, $H$ corresponds to the Hamiltonian and $O$ corresponds to the total spin operator. In the present note, we focus our attention on a mathematical aspect of the theory established in [11]. We will find that its structure is well decribed by the order theory.

The rest of the present note is organized as follows. In Section 2, we introduce some basic notions to state our main result. In particular, we focus our attention on the study of positivity improving resolvents. In Section 3, we state our main result; we provide a novel framework which solve the stability problem stated in this introduction. Section 4 is devoted to give an example. This example suggests that our framework contains rich mathematical strucutures. In Appendices A and B, we prove some operator inequalities which are useful in the main sections.

Acknowledgements

The author was partially supported by KAKENHI 18K03315.

2 Preliminary

2.1 Basic definitions

Let $\mathfrak{H}$ be a complex Hilbert space. By a convex cone, we understand a closed convex set $\mathfrak{P}\subset\mathfrak{H}$ such that $t\mathfrak{P}\subseteq\mathfrak{P}$ for all $t\geq 0$ and $\mathfrak{P}\cap(-\mathfrak{P})=\{0\}$. The dual cone of $\mathfrak{P}$ is defined by $\mathfrak{P}^{\dagger}=\{\eta\in\mathfrak{H}\,|\,\langle\eta|\xi\rangle\geq 0\ \forall\xi\in\mathfrak{P}\}.$ We say that $\mathfrak{P}$ is self-dual if $\mathfrak{P}=\mathfrak{P}^{\dagger}.$ In what follows, we always assume that $\mathfrak{P}$ is self-dual.

A vector $\xi$ is said to be positive w.r.t. $\mathfrak{P}$ if $\xi\in\mathfrak{P}$. We write this as $\xi\geq 0$ w.r.t. $\mathfrak{P}$. A vector $\eta\in\mathfrak{P}$ is called strictly positive w.r.t. $\mathfrak{P}$ whenever $\langle\xi|\eta\rangle>0$ for all $\xi\in\mathfrak{P}\backslash\{0\}$. We write this as $\eta>0$ w.r.t. $\mathfrak{P}$.

We denote by $\mathscr{B}(\mathfrak{H})$ the set of all bounded linear operators on $\mathfrak{H}$.

Definition 2.1

Let $A\in\mathscr{B}(\mathfrak{H})$.

• •

If $A\mathfrak{P}\subseteq\mathfrak{P},$111 For each subset $\mathfrak{C}\subseteq\mathfrak{H}$, $A\mathfrak{C}$ is defined by $A\mathfrak{C}=\{Ax\,|\,x\in\mathfrak{C}\}$. we then write this as $A\unrhd 0$ w.r.t. $\mathfrak{P}$. In this case, we say that $A$* preserves the positivity w.r.t. $\mathfrak{P}$.*

• •

The set of all positivity preserving operators w.r.t. $\mathfrak{P}$ is denoted by $\mathscr{P}(\mathfrak{P})$. $\mathscr{P}(\mathfrak{P})$ is a weakly closed convex cone in $\mathscr{B}(\mathfrak{H})$.

• •

We write $A\rhd 0$ w.r.t. $\mathfrak{P}$, if $A\xi>0$ w.r.t. $\mathfrak{P}$ for all $\xi\in\mathfrak{P}\backslash\{0\}$. In this case, we say that $A$* improves the positivity w.r.t. $\mathfrak{P}$.*

Let $\mathscr{L}^{2}(\mathfrak{H})$ be the set of all Hilbert-Schmidt class operators on $\mathfrak{H}$. It is well-known that $\mathscr{L}^{2}(\mathfrak{H})$ becomes a Hilbert space by introducing the norm $\langle A|B\rangle_{\mathscr{L}^{2}}=\mathrm{tr}[A^{*}B],\ \ A,B\in\mathscr{L}^{2}(\mathfrak{H})$.

Let $\mathfrak{S}(\mathfrak{H})$ be the set of all density matries on $\mathfrak{H}$.222 We say that a linear operator $\varrho$ on $\mathfrak{H}$ is a density matrix, if it is positive and trace class operator such that $\mathrm{tr}[\varrho]=1$. For given normalized vector $\psi$, we set $\varrho_{\psi}=|\psi\rangle\langle\psi|$. Trivially, we have $\varrho_{\psi}\in\mathfrak{S}(\mathfrak{H})$. If $\psi\geq 0$ w.r.t. $\mathfrak{P}$, then $\varrho_{\psi}\unrhd 0$ w.r.t. $\mathfrak{P}$, while, if $\psi>0$ w.r.t. $\mathfrak{P}$, then $\varrho_{\psi}\rhd 0$ w.r.t. $\mathfrak{P}$.

In the present note, we will examine self-adjoint operators satisfying the following conditions:

$H$ is self-adjoint and bounded from below;

• 2.

$H$ has purely discrete spectrum;

• 3.

$(H+s)^{-1}\unrhd 0$ w.r.t. $\mathfrak{P}$ for all $s>-E(H)$, where $E(H)=\inf\mathrm{spec}(H)$.

We denote by $\mathscr{A}_{\mathfrak{P}}$ the set of all operators satisfying the conditions 1.-3. above.

Proposition 2.2

Let $H,H^{\prime}\in\mathscr{A}_{\mathfrak{P}}$. If $sH+tH^{\prime}$ is essentially self-adjoint for some $s>0$ and $t>0$, then $\overline{sH+tH^{\prime}}\in\mathscr{A}_{\mathfrak{P}}$. In particular, $\mathscr{A}_{\mathfrak{P}}$ is a convex cone.

Proof. By Proposition A.1, $e^{-\beta sH}\unrhd 0$ and $e^{-\beta tH^{\prime}}\unrhd 0$ w.r.t. $\mathfrak{P}$ for all $\beta\geq 0$. By the Trotter product formula [13, Theorem S.20], we have

[TABLE]

where $\mathrm{s}$-$\displaystyle\lim_{n\to\infty}$ indicates the strong limit. Because $e^{-\beta sH/n}\unrhd 0$ and $e^{-\beta tH^{\prime}/n}\unrhd 0$ w.r.t. $\mathfrak{P}$, we see that \big{(}e^{-\beta sH/n}e^{-\beta tH^{\prime}/n}\big{)}^{n}\unrhd 0 w.r.t. $\mathfrak{P}$ for all $\beta\geq 0$ and $n\in\mathbb{N}$. Thus, the right hand side of (2.2) preserves the positivity w.r.t. $\mathfrak{P}$ for all $\beta\geq 0$. By applying Proposition A.1 again, we obtain that $\overline{sH+tH^{\prime}}\in\mathscr{A}_{\mathfrak{P}}$. $\Box$

Let $\mathscr{A}^{+}_{\mathfrak{P}}$ is the set of all self-adjoint operators satisfying 1., 2. and 3’. below:

• 3’.

$(H+s)^{-1}\rhd 0$ w.r.t. $\mathfrak{P}$ for all $s>-E(H)$.

Remark 2.3

If $H\in\mathscr{A}^{+}_{\mathfrak{P}}$, then $E(H)$ is a simple eigenvalue with strictly positive eigenvector by Theorem A.5. **

2.2 Propagation of positivity

Let $\mathfrak{H}_{1}$ and $\mathfrak{H}_{2}$ be complex Hilbert spaces, and let $\mathfrak{P}_{1}$ and $\mathfrak{P}_{2}$ be self-dual cones in $\mathfrak{H}_{1}$ and $\mathfrak{H}_{2}$, respectively. Suppose that $\mathfrak{H}_{1}$ is a closed subspace of $\mathfrak{H}_{2}$. The orthogonal projection from $\mathfrak{H}_{2}$ to $\mathfrak{H}_{1}$ is denoted by $\pi_{1,2}$. We say that the *positivity is inherited from $\mathfrak{P}_{1}$ to $\mathfrak{P}_{2}$, * if the following are satisfied:

$\mathfrak{P}_{1}=\pi_{1,2}\mathfrak{P}_{2}$;

• 2.

$\pi_{1,2}\unrhd 0$ w.r.t. $\mathfrak{P}_{2}$.

In this case, we write $\mathfrak{P}_{1}\dashrightarrow\mathfrak{P}_{2}$. As we will see, this binary relation defines a partial order333Readers are referred to [14] for partial orders.. The *conditional expectation * $\mathscr{E}_{1,2}:\mathscr{B}(\mathfrak{H}_{2})\to\mathscr{B}(\mathfrak{H}_{2})$ is defined by

[TABLE]

The following proposition is often useful:

Proposition 2.4

\mathscr{E}_{1,2}\big{(}\pi_{1,2}\mathscr{P}(\mathfrak{P}_{2})\pi_{1,2}\big{)}=\pi_{1,2}\mathscr{E}_{1,2}\big{(}\mathscr{P}(\mathfrak{P}_{2})\big{)}\pi_{1,2}=\mathscr{P}(\mathfrak{P}_{1}).**

Proof. Let $A\in\mathscr{P}(\mathfrak{P}_{2})$. We have

[TABLE]

Therefore, we have $\mathscr{E}_{1,2}(\pi_{1,2}\mathscr{P}(\mathfrak{P}_{2})\pi_{1,2})\subseteq\mathscr{P}(\mathfrak{P}_{1})$. Conversely, suppose that $A\in\mathscr{P}(\mathfrak{P}_{1})$. Then, corresponding to the decomposition $\mathfrak{H}_{2}=\mathfrak{H}_{1}\oplus\mathfrak{H}_{1}^{\perp}$, we have $A\oplus 0\in\mathscr{P}(\mathfrak{P}_{2})$. Indeed, for all $x,y\in\mathfrak{P}_{2}$, we have $\langle x|A\oplus 0y\rangle=\langle\pi_{1,2}x|A\pi_{1,2}y\rangle\geq 0$. Because $\mathscr{E}_{1,2}(A\oplus 0)=A\oplus 0$, we obtain that $\pi_{1,2}\mathscr{E}_{1,2}(\mathscr{P}(\mathfrak{P}_{2}))\pi_{1,2}\supseteq\mathscr{P}(\mathfrak{P}_{1})$. $\Box$

Definition 2.5

Let $H_{1}\in\mathscr{A}_{\mathfrak{P}_{1}}$ and $H_{2}\in\mathscr{A}_{\mathfrak{P}_{2}}$ be self-adjoint operators bounded from below. If $\mathfrak{P}_{1}\dashrightarrow\mathfrak{P}_{2}$ is satisfied, then we say that *the $\mathfrak{P}_{2}$-positivity of $H_{2}$ is inherited from the $\mathfrak{P}_{1}$-positivity of $H_{1}$ * and write this as $(H_{1},\mathfrak{P}_{1})\dashrightarrow(H_{2},\mathfrak{P}_{2})$. **

Proposition 2.6

Suppose that $(H_{1},\mathfrak{P}_{1})\dashrightarrow(H_{2},\mathfrak{P}_{2})$. Suppose that $E(H_{1})$ and $E(H_{2})$ are simple eigenvalues. Then the corresponding eigenvectors, say $\psi_{H_{1}}$ and $\psi_{H_{2}}$, are positive, namely, $\psi_{H_{1}}\geq 0$ w.r.t. $\mathfrak{P}_{1}$, and $\psi_{H_{2}}\geq 0$ w.r.t. $\mathfrak{P}_{2}$. Equivalently, we have $\varrho_{\psi_{H_{1}}}\unrhd 0$ w.r.t. $\mathfrak{P}_{1}$ and $\varrho_{\psi_{H_{2}}}\unrhd 0$ w.r.t. $\mathfrak{P}_{2}$. Moreover, $\pi_{1,2}\mathscr{E}_{1,2}(\varrho_{\psi_{H_{2}}})\pi_{1,2}\unrhd 0$ w.r.t. $\mathfrak{P}_{1}$ and $\langle\psi_{H_{1}}|\pi_{1,2}\psi_{H_{2}}\rangle\geq 0$ hold.

Proof. By Proposition A.3, we have $\psi_{H_{1}}\geq 0$ w.r.t. $\mathfrak{P}_{1}$ and $\psi_{H_{2}}\geq 0$ w.r.t. $\mathfrak{P}_{2}$, respectively. From the property $\mathfrak{P}_{1}=\pi_{1,2}\mathfrak{P}_{2}$, it holds that $\pi_{1,2}\psi_{H_{2}}\geq 0$ w.r.t. $\mathfrak{P}_{1}$. Thus, we obtain that $\langle\psi_{H_{1}}|\pi_{1,2}\psi_{H_{2}}\rangle\geq 0$. Because $\varrho_{\psi_{H_{2}}}\unrhd 0$ w.r.t. $\mathfrak{P}_{2}$, we obtain $\pi_{1,2}\mathscr{E}_{1,2}(\varrho_{\psi_{H_{2}}})\pi_{1,2}\unrhd 0$ by Proposition 2.4. $\Box$

Let $\{\mathfrak{H}_{n}\}_{n=1}^{\infty}$ be a sequence of Hilbert spaces, and let $\{\mathfrak{P}_{n}\}_{n=1}^{\infty}$ and $\{\mathfrak{P}_{n}^{\prime}\}_{n=1}^{\infty}$ be sequences of self-dual cones such that

• (i)

$\mathfrak{H}_{n}$ is a closed subspace of $\mathfrak{H}_{n+1}$;

• (ii)

$\mathfrak{P}_{n}$ and $\mathfrak{P}_{n}^{\prime}$ are self-dual cones in $\mathfrak{H}_{n}$.

Assume that $\mathscr{A}_{\mathfrak{P}_{n}}\cap\mathscr{A}_{\mathfrak{P}_{n}^{\prime}}\neq\{0\}$ for all $n\in\mathbb{N}$. Suppose that a sequence of semibounded self-adjoint operators $\{H_{n}\}_{n=1}^{\infty}$ satisfies the following:

[TABLE]

By definition, we have $\langle\psi_{H_{1}}|\pi_{1,2}\psi_{H_{2}}\rangle\langle\psi_{H_{2}}|\pi_{2,3}\psi_{H_{3}}\rangle\cdots\langle\psi_{H_{n}}|\pi_{n,n+1}\psi_{H_{n+1}}\rangle\geq 0$ and $\pi_{j,j+1}\mathscr{E}_{j,j+1}(\varrho_{\psi_{H_{j+1}}})\pi_{j,j+1}\unrhd 0$ w.r.t. $\mathfrak{P}_{j}$, where $\pi_{j,j+1}$ is the orthogonal projection from $\mathfrak{H}_{j+1}$ onto $\mathfrak{H}_{j}$. In this sense, the positivity of $\psi_{H_{1}}$ is propageted to $\psi_{H_{n}}$ for every $n\in\mathbb{N}$.

2.3 Propagation of strict positivity

Definition 2.7

Let $H_{1}\in\mathscr{A}_{\mathfrak{P}_{1}}^{+}$ and $H_{2}\in\mathscr{A}_{\mathfrak{P}_{2}}^{+}$. If $\mathfrak{P}_{1}\dashrightarrow\mathfrak{P}_{2}$ is satisfied, then we say that *the strict $\mathfrak{P}_{2}$-positivity of $H_{2}$ is inherited from the stirct $\mathfrak{P}_{1}$-positivity of $H_{1}$ * and write this as $(H_{1},\mathfrak{P}_{1})\to(H_{2},\mathfrak{P}_{2})$. By definition, we readily confirm that if $(H_{1},\mathfrak{P}_{1})\to(H_{2},\mathfrak{P}_{2})$, then we have $(H_{1},\mathfrak{P}_{1})\dashrightarrow(H_{2},\mathfrak{P}_{2})$. **

Let $H_{1}\in\mathscr{A}_{\mathfrak{P}_{1}}^{+}$ and $H_{2}\in\mathscr{A}_{\mathfrak{P}}^{+}$. As before, the ground state of $H_{j}$ is denoted by $\psi_{H_{j}},\ j=1,2$.

Theorem 2.8

If $(H_{1},\mathfrak{P}_{1})\to(H_{2},\mathfrak{P}_{2})$, then $\langle\psi_{H_{1}}|\pi_{1,2}\psi_{H_{2}}\rangle>0$. Equivalently, we have \big{\langle}\varrho_{\psi_{H_{1}}}\big{|}\mathscr{E}_{1,2}(\varrho_{\psi_{H_{2}}})\big{\rangle}_{\mathscr{L}^{2}}>0. In addition, $\pi_{1,2}\mathscr{E}_{1,2}(\varrho_{\psi_{H_{2}}})\pi_{2,1}\rhd 0$ w.r.t. $\mathfrak{P}_{1}$.

To prove Theorem 2.8, we begin with the following lemma:

Lemma 2.9

Let $A\in\mathscr{B}(\mathfrak{H})$ with $A\neq 0$. Assume that $u>0$ w.r.t. $\mathfrak{P}$. If $A\unrhd 0$ w.r.t. $\mathfrak{P}$, then $Au\neq 0$.

Proof. First, we prove the following claim: Let $A\in\mathscr{B}(\mathfrak{H})$. If $Au=0$ for all $u\in\mathfrak{P}$, then $A=0$. Indeed, by Proposition A.2, each $u\in\mathfrak{H}$ can be written as $u=v_{1}-v_{2}+i(w_{1}-w_{2})$, where $v_{1},v_{2},w_{1},w_{2}\in\mathfrak{P}$ such that $\langle v_{1}|v_{2}\rangle=0$ and $\langle w_{1}|w_{2}\rangle=0$. Thus, the assumption implies that $Au=0$ for all $u\in\mathfrak{H}$.

Assume that $Au=0$. Then, $\langle v|Au\rangle=0$ for all $v\in\mathfrak{P}$, implying that $\langle A^{*}v|u\rangle=0$. Since $u>0$ and $A^{*}v\geq 0$ w.r.t. $\mathfrak{P}$, we conclude that $A^{*}v$ must be zero. Because $v$ is arbitrary, $A^{*}=0$ by the above claim. This contradicts with the assumption $A\neq 0$. $\Box$

Proof of Theorem 2.8

Note that $\psi_{H_{1}}>0$ w.r.t. $\mathfrak{P}_{1}$ and $\psi_{H_{2}}>0$ w.r.t. $\mathfrak{P}_{2}$, respectively. Because $\mathfrak{P}_{1}=\pi_{1,2}\mathfrak{P}_{2}$ and $\pi_{1,2}\unrhd 0$ w.r.t. $\mathfrak{P}_{2}$, we obtain that $\pi_{1,2}\psi_{H_{2}}\geq 0$ w.r.t. $\mathfrak{P}_{1}$ and $\pi_{1,2}\psi_{H_{2}}\neq 0$ by Lemma 2.9. Since $\psi_{H_{1}}>0$ w.r.t. $\mathfrak{P}_{1}$, we conclude that $\langle\psi_{H_{1}}|\pi_{1,2}\psi_{H_{2}}\rangle>0$.

For each $x,y\in\mathfrak{P}\backslash\{0\}$, we have $\langle x|\mathscr{E}_{1,2}(\varrho_{\psi_{H_{2}}})y\rangle=\langle x\oplus 0|\psi_{H_{2}}\rangle\langle\psi_{H_{2}}|y\oplus 0\rangle>0$, which implies that $\pi_{1,2}\mathscr{E}(\varrho_{\psi_{H_{2}}})\pi_{1,2}\rhd 0$ w.r.t. $\mathfrak{P}_{1}$. $\Box$

As before, assume that $\mathscr{A}_{\mathfrak{P}_{n}}\cap\mathscr{A}_{\mathfrak{P}_{n}^{\prime}}\neq\{0\}$ for all $n=1,\dots,N$. Suppose that a sequence of semibounded self-adjoint operators $\{H_{n}\}_{n=1}^{N}$ satisfies the following:

[TABLE]

Then, we have $\langle\psi_{H_{1}}|\pi_{1,2}\psi_{H_{2}}\rangle\langle\psi_{H_{2}}|\pi_{2,3}\psi_{H_{3}}\rangle\cdots\langle\psi_{H_{N-1}}|\pi_{{N-1},N}\psi_{H_{N}}\rangle>0$ for each $N\in\mathbb{N}$ (equivalently, \big{\langle}\varrho_{\psi_{H_{1}}}\big{|}\mathscr{E}_{1,2}(\varrho_{\psi_{H_{2}}})\big{\rangle}_{\mathscr{L}^{2}}\cdots\big{\langle}\varrho_{\psi_{H_{N-1}}}\big{|}\mathscr{E}_{N-1,N}(\varrho_{\psi_{H_{N}}})\big{\rangle}_{\mathscr{L}^{2}}>0) by Theorem 2.8. Furthermore, $\pi_{j,j+1}\mathscr{E}_{j,j+1}(\varrho_{H_{j+1}})\pi_{j,j+1}\rhd 0$ w.r.t. $\mathfrak{P}_{j}$ for all $j\in\mathbb{N}$. In this sense, a strict positivity of $\psi_{H_{1}}$ is propageted to $\psi_{H_{N}}$. As we will see in the following sections, this property is important to examine the stability of the good quantum numbers.

Definition 2.10

We say that $(H_{1},\mathfrak{P}_{1})$ and $(H_{N},\mathfrak{P}_{N}^{\prime})$ are connected by the sequences $\{(H_{j},\mathfrak{P}_{j})\}_{j=1}^{N-1}$ and $\{(H_{j},\mathfrak{P}_{j}^{\prime})\}_{j=2}^{N}$ if (2.6) holds. We simply express this as $H_{1}\to H_{N}$. **

For a given Hilbert space $\mathfrak{H}_{*}$, let $\mathbb{H}_{\mathfrak{H}_{*}}$ be the set of all Hilbert spaces containing $\mathfrak{H}_{*}$ as a closed subspace. Let $\mathscr{P}_{\mathfrak{H}_{*},0}$ be the set of self-adjoint operators defined by

[TABLE]

where the union $\bigcup_{\mathfrak{P}\subset\mathfrak{H}}$ runs over all self-dual cones in $\mathfrak{H}$.

Proposition 2.11

The binary relation “ $\to$” is a preoder on $\mathscr{P}_{\mathfrak{H}_{*},0}$. Namely, we have the following:

• (i)

$H\to H$;

• (ii)

$H\to H^{\prime},\ \ H^{\prime}\to H^{\prime\prime}\Longrightarrow H\to H^{\prime\prime}$.

Proof. (i) Because $H\in\mathscr{P}_{\mathfrak{H}_{*},0}$, there is a self-dual cone $\mathfrak{P}$ such that $H\in\mathscr{A}_{\mathfrak{P}}^{+}$. Then we can readily check that $(H,\mathfrak{P})\to(H,\mathfrak{P})$.

(ii) By definition, $H$ and $H^{\prime}$ are connected by seqences $P=\{(H_{j},\mathfrak{P}_{j})\}_{j=1}^{N-1}$ and $P^{\prime}=\{H_{j},\mathfrak{P}_{j}^{\prime}\}_{j=2}^{N}$ with $H_{1}=H$ and $H_{N}=H^{\prime}$. Also $H^{\prime}$ and $H^{\prime\prime}$ are connected by sequences $Q=\{(K_{j},\mathfrak{Q}_{j})\}_{j=1}^{M-1}$ and $Q^{\prime}=\{(K_{j},\mathfrak{Q}_{j}^{\prime})\}_{j=2}^{M}$ with $K_{1}=H^{\prime}$ and $K_{M}=H^{\prime\prime}$. Now we define new sequences $R$ and $R^{\prime}$ by $R=P\cup Q$ and $R^{\prime}=P\cup Q^{\prime}$, then $H$ and $H^{\prime\prime}$ are connected by $R$ and $R^{\prime}$. $\Box$

Definition 2.12

Let $H_{1},H_{2}\in\mathscr{P}_{\mathfrak{H}_{*},0}$. If $H_{1}\to H_{2}$ and $H_{2}\to H_{1}$, then we write this as $H_{1}\equiv H_{2}$. The binary relation “ $\equiv$ ” is an equivalence relation on $\mathscr{P}_{\mathfrak{H}_{*},0}$. Let $\mathscr{P}_{\mathfrak{H}_{*},0}$ be the set of equivalence classes: $\mathscr{P}_{\mathfrak{H}_{*}}=\mathscr{P}_{\mathfrak{H}_{*},0}/\equiv$. The equivalence class containing $H$ is denoted by $[H]$. The binary relation “$\to$” on $\mathscr{P}_{\mathfrak{H}_{*}}$ is naturally defined by $[H_{1}]\to[H_{2}]\ \ \ \mbox{if}\ \ H_{1}\to H_{2}.$ This is a partial order on $\mathscr{P}_{\mathfrak{H}_{*}}$; namely, we have, by Proposition 2.11,

• (i)

$[H]\to[H]$;

• (ii)

$[H_{1}]\to[H_{2}],\ [H_{2}]\to[H_{1}]\Longrightarrow[H_{1}]=[H_{2}]$;

• (iii)

$[H_{1}]\to[H_{2}],\ [H_{2}]\to[H_{3}]\Longrightarrow[H_{1}]\to[H_{3}]$.

In what follows, we abbreviate $[H]\to[H^{\prime}]$ to $H\to H^{\prime}$ if no confusion arises. **

3 Stability of good quantum numbers in ground states

3.1 Main result

Let $O\in\mathscr{B}(\mathfrak{H}_{*})$ be self-adjoint. In what follows, we always assume that $O$ has purely discrete spectrum. In this section, we will explore the following class of self-adjoint operators:

[TABLE]

For each $\mathfrak{H}\in\mathbb{H}_{\mathfrak{H}_{*}}$, $O$ can be naturally extended to a self-adjoint operator on $\mathfrak{H}$.444 Indeed, corresponding to the decomposition $\mathfrak{H}=\mathfrak{H}_{*}\oplus\mathfrak{H}_{*}^{\perp}$, the natural extension of $O$ to $\mathfrak{H}$ is defined by $O\oplus 0$. We denote by $O$ this natural extension if no confusion arises. The natural extension is also denoted by the same symbol $O$.

Note that the preorder “$\to$” can be defined on $\mathscr{P}_{\mathfrak{H}_{*},0}(O)$ as well. As before, we set $\mathscr{P}_{\mathfrak{H}_{*}}(O)=\mathscr{P}_{\mathfrak{H}_{*},0}(O)/\equiv$. Then the preoder “$\to$” can be also lifted up to a partial order. We identify the equivalence class $[H]\in\mathscr{P}_{\mathfrak{H}_{*}}(O)$ with $H$ if no confusion occurs.

Proposition 3.1

Let $H,K\in\mathscr{P}_{\mathfrak{H}_{*}}(O)$. If $H\to K$, then $\mu(H)=\mu(K)$.

Proof. Suppose that $H\in\mathscr{A}_{\mathfrak{P}}^{+}$ and $K\in\mathscr{A}_{\mathfrak{Q}}^{+}$. There exist sequences $\{(H_{j},\mathfrak{P}_{j})\}_{j=1}^{N-1}$ and $\{(H_{j},\mathfrak{P}_{j}^{\prime})\}_{j=2}^{N}$ satisfying (2.6) with $H_{1}=H,\ \mathfrak{P}_{1}=\mathfrak{P},\ H_{N}=K$ and $\mathfrak{P}_{N}^{\prime}=\mathfrak{Q}$. Let $\pi_{j,j+1}$ be the orthogonal projection from $\mathfrak{H}_{j+1}$ onto $\mathfrak{H}_{j}$. Let $\mathscr{E}_{j,j+1}$ be the corresponding conditional expectation. Because $O\in\mathscr{B}(\mathfrak{H}_{*})$, we see that $\mathscr{E}_{j,j+1}(O)=O$, which implies that

[TABLE]

By applying Theorem 2.8, we obtain that $\mu(H_{j})=\mu(H_{j+1})$. Repeating this argument several times, we arrive at $\mu(H)=\mu(H_{1})=\mu(H_{2})=\cdots=\mu(H_{N})=\mu(K)$. $\Box$

Definition 3.2

Let $H_{*}\in\mathscr{P}_{\mathfrak{H}_{*}}(O)$. The $H_{*}$-stability class $\mathscr{U}_{O}(H_{*})$ is defined by $\mathscr{U}_{O}(H_{*})=\{H\in\mathscr{P}_{\mathfrak{H}_{*}}(O)\,|\,H_{*}\to H\}$. **

Theorem 3.3

For every Hamiltonian $H\in\mathscr{P}_{\mathfrak{H}_{*}}(O)$ in the $H_{*}$-stability class, we have $\mu(H)=\mu(H_{*})$.

Proof. The theorem immediately follows from Proposition 3.1. $\Box$

3.2 Basic properties of $\mathscr{U}_{O}(H_{*})$

In this subsection, we will prove two basic properties of $\mathscr{U}_{O}(H)$.

Theorem 3.4

For each $H\in\mathscr{P}_{\mathfrak{H}_{*}}(O)$, the cardinality of $\mathscr{U}_{O}(H)$ is greater than $\aleph_{0}$, the cardinality of the natural numbers. In this sense, $\mathscr{U}_{O}(H)$ is rich.

Proof. Suppose that $H$ acts in the Hilbert space $\mathfrak{H}$. Note that because $H\in\mathscr{P}_{\mathfrak{H}_{*}}(O)$, there is a self-dual cone $\mathfrak{P}$ in $\mathfrak{H}$ such that $H\in\mathscr{A}_{\mathfrak{P}}^{+}$. We consider an extended Hilbert space $\mathfrak{H}\otimes\mathbb{C}^{2}$. We define a Hamiltonian $H_{1}$ acting in $\mathfrak{H}\otimes\mathbb{C}^{2}$ by $H_{1}=H\otimes 1-1\otimes\sigma_{1},$ where $\sigma_{1}$ is the standard Pauli matrix: $\sigma_{1}=\begin{pmatrix}0&1\\ 1&0\end{pmatrix}$. Remark the following facts: \mathbb{R}^{2}_{+}=\bigg{\{}\left(\begin{array}[]{c}x\\ y\end{array}\right)\in\mathbb{C}^{2}\,\bigg{|}\,x,y\geq 0\bigg{\}} is a self-dual cone in $\mathbb{C}^{2}$, and the lowest eigenvalue of $-\sigma_{1}$ is simple with strictly positive eigenvector. Indeed, the eigenvector is given by \psi_{-\sigma_{1}}=\left(\begin{array}[]{c}1/\sqrt{2}\\ 1/\sqrt{2}\end{array}\right), which is obviously strictly positive w.r.t. $\mathbb{R}_{+}^{2}$. Now we define a self-dual cone in $\mathfrak{H}\otimes\mathbb{C}^{2}$ by \mathfrak{P}_{1}=\bigg{\{}\Psi_{1}\otimes\left(\begin{array}[]{c}1\\ 0\end{array}\right)+\Psi_{2}\otimes\left(\begin{array}[]{c}0\\ 1\end{array}\right)\,\bigg{|}\,\Psi_{1},\Psi_{2}\in\mathfrak{P}\bigg{\}}. Note that the ground state of $H_{1}$ is unique and concretely given by $\psi_{H_{1}}=\psi_{H}\otimes\psi_{-\sigma_{1}}$. Since $H\in\mathscr{A}_{\mathfrak{P}}^{+}$, it holds that $\psi_{H}>0$ w.r.t. $\mathfrak{P}$. Thus, we readily confirm that $\langle\Phi|\psi_{H_{1}}\rangle>0$ for all $\Phi\in\mathfrak{P}_{1}\backslash\{0\}$, which implies that $\psi_{H_{1}}>0$ w.r.t. $\mathfrak{P}_{1}$. By applying Theorem A.5, we conlude that $(H_{1}+s)^{-1}\rhd 0$ w.r.t. $\mathfrak{P}_{1}$ for all $s>-E(H_{1})$.

We introduce an orthogonal projection $P$ by P\Psi\otimes r=\Psi\otimes\left(\begin{array}[]{c}0\\ r_{2}\end{array}\right) for each $\Psi\in\mathfrak{H}$ and r=\left(\begin{array}[]{c}r_{1}\\ r_{2}\end{array}\right)\in\mathbb{C}^{2}. We can identify $\mathrm{ran}(P)$ with $\mathfrak{H}$ by the isometry \tau:\mathrm{ran}(P)\ni\Psi\otimes\left(\begin{array}[]{c}0\\ r_{2}\end{array}\right)\mapsto r_{2}\Psi\in\mathfrak{H}. By definition, we have $P\unrhd 0$ w.r.t. $\mathfrak{P}_{1}$ and $P\mathfrak{P}_{1}=\mathfrak{P}$ by the aforementioned identification. Hence, we readily check that $H\to H_{1}$.

Next, let us consider a further extended Hilbert space $(\mathfrak{H}\otimes\mathbb{C}^{2})\otimes\mathbb{C}^{2}$. Define a Hamiltonian $H_{2}$ by $H_{2}=H_{1}\otimes 1-1\otimes\sigma_{1}$, and define a self-dual cone $\mathfrak{P}_{2}$ by \mathfrak{P}_{2}=\bigg{\{}\Phi_{1}\otimes\left(\begin{array}[]{c}1\\ 0\end{array}\right)+\Phi_{2}\otimes\left(\begin{array}[]{c}0\\ 1\end{array}\right)\,\bigg{|}\,\Phi_{1},\Phi_{2}\in\mathfrak{P}_{1}\bigg{\}}. Using arguments similar to those in the last paragraph, we can confirm that $H_{1}\to H_{2}$, which implies that $H\to H_{2}$. Repeating this procedure, we can construct a sequence of Hamiltonians $\{H_{\ell}\}_{\ell=1}^{\infty}$ such that $H\to H_{\ell}$. Therefore, $\mathscr{U}_{O}(H)$ contains at least countably infinite number of Hamiltonians. $\Box$

Proposition 3.5

Let ${\bm{U}}_{O}$ be the set of all stability classes: ${\bm{U}}_{O}=\{\mathscr{U}_{O}(H)\,|\,H\in\mathscr{P}_{\mathfrak{H}_{*}}(O)\}$. Then ${\bm{U}}_{O}$ is a partially ordered set under set inclusion. In addition, the map $\mathscr{U}_{O}:\,\mathscr{P}_{\mathfrak{H}_{*}}(O)\to{\bm{U}}_{O}$ is monotonically decreasing, that is, if $H_{1}\to H_{2}$, then $\mathscr{U}_{O}(H_{1})\supseteq\mathscr{U}_{O}(H_{2})$.

Proof. Suppose that $H\in\mathscr{U}_{O}(H_{2})$. Then we have $H_{2}\to H$. Because $H_{1}\to H_{2}$, we conclude that $H_{1}\to H$ by Definition 2.12. Thus, $H\in\mathscr{U}_{O}(H_{1})$, i.e., $\mathscr{U}_{O}(H_{1})\supseteq\mathscr{U}_{O}(H_{2})$. $\Box$

3.3 Equivalent Hamiltonians

Let $\mathfrak{M}_{*}$ and $\mathfrak{N}$ be von Neumann algebras on a separable Hilbert spaces $\mathfrak{H}_{*}$ and $\mathfrak{X}$, respectively. Let $\Omega_{*}$ and $\Omega_{\mathfrak{X}}$ be cyclic and separating vectors for $\mathfrak{M}_{*}$ and $\mathfrak{N}$, respectively. We denote by $\mathfrak{P}_{*}$ and $\mathfrak{P}_{\mathfrak{X}}$ the natural cones associated with the pairs $\{\mathfrak{M}_{*},\Omega_{*}\}$ and $\{\mathfrak{N},\Omega_{\mathfrak{X}}\}$, respectively.555 We use $\Delta$ and $J$ to denote the modular operator and the modular conjugation associated with the pair $\{\mathfrak{M},\Omega\}$. Let $\mathcal{P}_{0}(\mathfrak{M})=\{AJAJ\,|\,A\in\mathfrak{M}\}.$ The natural cone $\mathfrak{P}$ associated with the pair $\{\mathfrak{M},\Omega\}$ is defined by $\mathfrak{P}=\overline{\mathcal{P}_{0}(\mathfrak{M})\Omega},$ where the bar denotes the strong closure. It is well-known that $\mathfrak{P}$ is a self-dual cone in $\mathfrak{H}$ [1].

We set

[TABLE]

Because $\Omega$ is a cyclic and separating vector for $\mathfrak{M}$, we can also define the natural cone $\mathfrak{P}$ associated with $\mathfrak{M}$. Physically, $\mathfrak{X}$ represents effects from environments surrounding the system described by $\mathfrak{M}_{*}$.

Let $\pi=1\otimes|\Omega_{\mathfrak{X}}\rangle\langle\Omega_{\mathfrak{X}}|$. Then $\pi$ is the orthogonal projection from $\mathfrak{H}$ to $\mathfrak{H}_{*}\cong\mathfrak{H}_{*}\otimes\Omega_{\mathfrak{X}}$, where $\mathfrak{H}_{*}\otimes\Omega_{\mathfrak{X}}=\{\varphi\otimes\Omega_{\mathfrak{X}}\,|\,\varphi\in\mathfrak{H}_{*}\}$. Note that all results in the previous sections hold true for $\mathfrak{H}_{1}=\mathfrak{H}_{*}\cong\mathfrak{H}_{*}\otimes\Omega_{\mathfrak{X}}$ and $\mathfrak{H}_{2}=\mathfrak{H}$.

For given Hamiltonian $H_{*}$ acting in $\mathfrak{H}_{*}$, let us consider $H_{*}$-stability class $\mathscr{U}_{O}(H_{*})$.

Definition 3.6

We say that the Hamiltonian $H$ is equivalent to $H_{*}$, if there is a semibounded self-adjoint operator $L$ on $\mathfrak{X}$ such that $L\in\mathscr{A}_{\mathfrak{P}_{\mathfrak{X}}}^{+}$ and $H=H_{*}\otimes 1+1\otimes L$. **

Let $H\in\mathscr{U}_{O}(H_{*})$. Let $\psi_{*}$ and $\psi$ be the normalized ground states of $H_{*}$ and $H$, respectively. The following proposition is readily confirmed.

Proposition 3.7

Suppose that $H$ is equivalent to $H_{*}$. Then there exists a normailzed vector $\omega\in\mathfrak{X}$ such that

$\psi=\psi_{*}\otimes\omega$;

• 2.

$\omega>0$* w.r.t. $\mathfrak{P}_{\mathfrak{X}}$.*

Remark 3.8

Let $\{H_{\ell}\}_{\ell=1}^{\infty}$ be the sequence of Hamiltonians constructed in the proof of Theorem 3.4. By the construction, $H_{\ell+1}$ is equivalent to $H_{\ell}$ for all $\ell\in\mathbb{N}$; thus, every $H_{\ell}$ is equivalent to $H$. In this sense, this example is rivial. As we will see in Section 4, we can construct sequences of Hamiltonians in $\mathscr{U}_{O}(H)$ which are inequivalent to each other. **

As before, we set $\varrho_{\psi_{*}}=|\psi_{*}\rangle\langle\psi_{*}|$ and $\varrho_{\psi}=|\psi\rangle\langle\psi|$. Let $\mathrm{tr}_{\mathfrak{X}}$ be the partial trace operations $\mathrm{tr}_{\mathfrak{X}}:\mathscr{B}(\mathfrak{H})\to\mathscr{B}(\mathfrak{H}_{*})$. We define a density matrix by $\varrho_{\psi}^{\mathfrak{H}_{*}}=\mathrm{tr}_{\mathfrak{X}}[\varrho_{\psi}]$. The quantum relative entropy of $\varrho_{\psi}^{\mathfrak{H}_{*}}$ to $\varrho_{\psi_{*}}$ is given by

[TABLE]

Definition 3.9

We say that $H\in\mathscr{U}_{O}(H_{*})$ is weakly equivalent to $H_{*}$, if $S(\varrho_{\psi}^{H_{*}}|\varrho_{\psi_{*}})=0$. **

Theorem 3.10

Let $H\in\mathscr{U}_{O}(H_{*})$. The following (i) and (ii) are equivalent:

• (i)

$H$* is weakly equivalent;*

• (ii)

There exists a normailzed vector $\omega\in\mathfrak{X}$ such that

$\psi=\psi_{*}\otimes\omega$;

• 2.

$\omega>0$* w.r.t. $\mathfrak{P}_{\mathfrak{X}}$.*

Remark 3.11

By Proposition 3.7, the triviality of $H$ implies the condition (ii). However, the converse is false in general. **

Proof of Theorem 3.10. (ii) $\Rightarrow$ (i): Easy.

(i) $\Rightarrow$ (ii): First, note that $S(\varrho_{\psi}^{H_{*}}|\varrho_{\psi_{*}})=0$ if and only if $\varrho_{\psi}^{H_{*}}=\varrho_{\psi_{*}}$. Thus, there exists an $x\in\mathfrak{X}$ such that $\psi=\psi_{*}\otimes x$. Because $\psi$ and $\psi_{*}$ are strictly positive, $x$ must be strictly positive. $\Box$

4 Example: construction of a lattice

In this subsection, we will show that $\mathscr{U}_{O}(H)$ is truly rich in the sense that $\mathscr{U}_{O}(H)$ contains infinitely many inequivalent Hamiltonians, and illustrate the interesting structure of $\mathscr{U}_{O}(H)$ by constructing a specific example.

Let $H_{0}$ be a self-adjoint operator on $\mathfrak{H}_{*}$, bounded from below. In this section, we assume the following condition:

(H) $e^{-\beta H_{0}}\rhd 0$ w.r.t. $\mathfrak{P}_{*}$ for all $\beta>0$.

By Proposition A.4, we have $H_{0}\in\mathscr{A}_{\mathfrak{P}_{*}}^{+}$. Suppose that $H_{0}$ commutes with $O$. Our purpose in this subsection is to examine the stability of $\mu(H_{0})$.

For each $n\in\mathbb{N}$ with $n\geq 2$, we consider a Hilbert space $\mathfrak{H}_{*}\otimes\mathbb{C}^{n}$. Then $\mathfrak{H}_{*}$ can be regarded as a closed subspace of $\mathfrak{H}_{*}\otimes\mathbb{C}^{n}$ in the following manner: $\mathfrak{H}_{*}\cong\mathfrak{H}_{*}\otimes(1/\sqrt{n},\dots,1/\sqrt{n})^{T}\subset\mathfrak{H}_{*}\otimes\mathbb{C}^{n}$, where $a^{T}$ indicates the transpose of $a$ and $\mathfrak{H}\otimes a=\{\psi\otimes a|\psi\in\mathfrak{H}\}$. Thus, $\mathfrak{H}_{*}\otimes\mathbb{C}^{n}\in\mathbb{H}_{\mathfrak{H}_{*}}$. A natural self-dual cone in $\mathfrak{H}_{*}\otimes\mathbb{C}^{n}$ is given by

[TABLE]

where $\mathbb{R}_{+}^{n}$ is a natural self-dual cone in $\mathbb{C}^{n}$: $\mathbb{R}_{+}^{n}=\{r=(r_{1},\dots,r_{n})^{T}\in\mathbb{R}^{n}\,|\,r_{j}\geq 0,j=1,\dots,n\}$, $\{e_{j}\}_{j=1}^{n}$ is a standard orthonormal system in $\mathbb{R}^{n}$ given by $e_{j}=(0,\dots,\underbrace{1}_{j},\dots 0)^{T}$, and $\mathrm{coni}(S)$ is the conical hull of $S$.

Before we proceed, we introduce a useful class of operators.

Definition 4.1

Let $\mathfrak{H}$ be a Hilbert space and let $\mathfrak{P}$ be a self-dual cone in $\mathfrak{H}$. We say that $A\in\mathscr{B}(\mathfrak{H})$ is ergodic w.r.t. $\mathfrak{P}$ if the following are satisfied:

• •

$A\unrhd 0$ w.r.t. $\mathfrak{P}$;

• •

For each $\xi,\eta\in\mathfrak{P}\backslash\{0\}$, there exists a $k\in\mathbb{N}\cup\{0\}$ such that $\langle\xi|A^{k}\eta\rangle>0$.

Let $\{n_{\mu}\}_{\mu=1}^{\ell}$ be a set of natural numbers with $\ell\geq 2$ such that $n_{1}+\cdots+n_{\ell}=N$. We set

[TABLE]

Let $X\in\mathscr{B}(\mathfrak{H}_{*})$ be self-adjoint. Let $\{Y_{\mu}\}_{\mu=1}^{\ell}$ be a family of self-adjoint operators such that $Y_{\mu}$ acts in $\mathbb{C}^{n_{\mu}}$. In what follows, we assume the following:

• (i)

$X\unrhd 0$ w.r.t. $\mathfrak{P}_{*}$;

• (ii)

$X$ has purely discrete spectrum and commutes with $O$;

• (iii)

$Y_{\mu}$ is ergodic w.r.t. $\mathbb{R}_{+}^{n_{\mu}}$ for all $\mu=1,\dots,\ell$.

Lemma 4.2

We define a self-adjoint operator $V_{\mu}$ acting in $\mathfrak{H}_{\mu}$ by $V_{\mu}=X\otimes Y_{\mu}$. Then $H_{0}-V_{\mu}\in\mathscr{A}_{\mathfrak{P}_{\mu}}^{+}$. In particular, $H_{0}-V_{\mu}\in\mathscr{U}_{O}(H_{0})$ for all $\mu=1,\dots,\ell$. If $X_{\mu}\neq 1$, then $H_{0}-V_{\mu}$ is inequivalent to $H_{0}$.

Note that by the assumptions, $H_{0}-V_{\mu}$ has purely discrete spectrum. We will prove Lemma 4.2 in Appendix B.

Next, let $\mathcal{I}$ be the set of all subsets of $\{1,\dots,{\ell}\}$. Trivially, $\mathcal{I}$ is a lattice under set inclusion. Let $\mathcal{I}^{\partial}$ be the dual poset of $\mathcal{I}$, that is, the poset with the same underlying set but whose order relation is the opposite of set inclusion. For a given $I=\{{\mu_{1}},\dots,{\mu_{k}}\}$, we set

[TABLE]

and $V_{I}=V_{\mu_{1}}+\cdots+V_{\mu_{\ell}}$. Needless to say, $\mathfrak{P}_{I}$ is defined by \mathfrak{P}_{I}=\Big{(}\cdots\big{(}(\mathfrak{P}_{*}\otimes\mathbb{R}_{+}^{n_{\mu_{1}}})\otimes\mathbb{R}_{+}^{n_{\mu_{k+1}}}\big{)}\otimes\cdots\otimes\mathbb{R}_{+}^{n_{\mu_{k-1}}}\Big{)}\otimes\mathbb{R}_{+}^{n_{\mu_{k}}}. If $I=\varnothing$, we simply set $\mathfrak{H}_{I}=\mathfrak{H}_{*},\ \mathfrak{P}_{I}=\mathfrak{P}_{*}$ and $V_{I}=0$. Note that each $V_{\mu_{j}}$ acts in $\mathfrak{H}_{I}$ in the following manner: $V_{\mu_{j}}=X\otimes(1\otimes\cdots\otimes\overbrace{Y_{\mu_{j}}}^{j-{\mathrm{th}}}\otimes\cdots\otimes 1).$ As before, $\mathfrak{H}_{*}$ can be regarded as a closed subspace of $\mathfrak{H}_{I}$: $\mathfrak{H}_{*}\cong\mathfrak{H}_{*}\otimes\omega_{I}\subset\mathfrak{H}_{I}$, where $\omega_{I}=\omega_{\mu_{1}}\otimes\cdots\otimes\omega_{\mu_{k}}\in\mathbb{C}^{n_{\mu_{1}}}\otimes\cdots\otimes\mathbb{C}^{n_{\mu_{k}}}$ with $\omega_{\mu_{i}}=(1/\sqrt{n_{\mu_{i}}},\dots,1/\sqrt{n_{\mu_{i}}})^{T}\in\mathbb{C}^{n_{\mu_{i}}}$.

Lemma 4.3

For each $I\in\mathcal{I}$, we set $H_{I}=H_{0}-V_{I}$. Then $H_{I}\in\mathscr{A}_{\mathfrak{P}_{I}}^{+}$. In particular, $H_{I}\in\mathscr{U}_{O}(H_{0})$. If $X\neq 1$, then $H_{I}$ is inequivalent to $H_{0}$.

We will provide a proof of Lemma 4.3 in Appendix B.

Let $I_{1},I_{2}\in\mathcal{I}$. If $I_{1}\subseteq I_{2}$, then $\mathfrak{H}_{I_{1}}$ can be regarded as a subspace of $\mathfrak{H}_{I_{2}}$ in the following manner: For simplicity, we consider the case where $I_{1}=\{\mu_{1},\dots,\mu_{k}\}$ and $I_{2}=I_{1}\cup\{\mu_{k+1},\dots,\mu_{k+\ell}\}$. Let $\tau$ be a linear operator from $\mathfrak{H}_{I_{1}}$ to $\mathfrak{H}_{I_{2}}$ defined by

[TABLE]

It is readily checked that $\tau$ is an isometry. By identifying $\mathfrak{H}_{I_{1}}$ with $\tau\mathfrak{H}_{I_{1}}$, $\mathfrak{H}_{I_{1}}$ can be regarded as a subspace of $\mathfrak{H}_{I_{2}}$. Note that we can extend this argument to general $I_{1}$ and $I_{2}$ with $I_{1}\subseteq I_{2}$.

Theorem 4.4

The map $H_{\bullet}:\mathcal{I}^{\partial}\ni I\mapsto H_{I}\in\mathscr{U}_{O}(H_{0})$ is order-preserving, that is, if $I_{1}\subseteq I_{2}$, then $H_{I_{1}}\to H_{I_{2}}$. In particular, $P=\{H_{I}\}_{I\in\mathcal{I}}$ is a lattice. The greatest element in $P$ is $H_{0}$, and the smallest element in $P$ is $H_{\{1,\dots,\ell\}}$.

Proof. For simplicity, we consider the case where $I_{1}=\{\mu_{1},\dots,\mu_{k}\}$ and $I_{2}=I_{1}\cup\{\mu_{k+1},\dots,\mu_{k+\ell}\}$. As explained before, $\mathfrak{H}_{I_{1}}$ can be regarded as a closed subspace of $\mathfrak{H}_{I_{2}}$ by the isometry $\tau$ defined by (4.15). Using this identification, we can identify $\mathfrak{P}_{I_{1}}$ with $\tau\mathfrak{P}_{I_{1}}=\{\psi\otimes\omega_{\mu_{k+1}}\otimes\cdots\otimes\omega_{\mu_{k+\ell}}\,|\,\psi\in\mathfrak{P}_{I_{1}}\}$. Let $\pi_{I_{1},I_{2}}$ be the orthogonal projection from $\mathfrak{H}_{I_{2}}$ to $\mathfrak{H}_{I_{1}}$:

[TABLE]

where $P_{\omega_{I_{2}\backslash I_{1}}}=1\otimes|\omega_{I_{2}\backslash I_{1}}\rangle\langle\omega_{I_{2}\backslash I_{1}}|$ with $\omega_{I_{2}\backslash I_{1}}=\omega_{\mu_{k+1}}\otimes\cdots\otimes\omega_{\mu_{k+\ell}}$. We readily confirm that $\mathfrak{P}_{I_{1}}=\pi_{I_{1},I_{2}}\mathfrak{P}_{I_{2}}$. Combining this with Lemma 4.3, we conclude the assertion in the theorem. $\Box$

Example 1

For $\ell=3$, we get the following Hasse daigram:

In the above graph, the vertices are labeld with the elements of the partilly ordered set $P$, and the edges indicate the covering relation666As for the definition of the covering relation, see [14]. . **

5 Applications to many-electron systems

In this section, we will briefly explain how the theory presented in this paper can be applied to the study of ferromagnetism. Note that the detailed proofs can be found in [11].

Let us consider a finite lattice $\Lambda$. Suppose that $\Lambda$ is bipartite: $\Lambda=A\cup B$ with $A\cap B=\varnothing$. The Marshall-Lieb-Mattis Hamiltonian is given by

[TABLE]

where ${\bm{S}}_{A}=\sum_{x\in A}{\bm{S}}_{x}$ and ${\bm{S}}_{B}=\sum_{x\in B}{\bm{S}}_{x}$; ${\bm{S}}_{x}=(S_{x}^{(1)},S_{x}^{(2)},S_{x}^{(3)})$ are the spin operators at site $x$ satisfying the standard commutation relations:

[TABLE]

The total spin operators are

[TABLE]

and ${\bm{S}}_{\mathrm{tot}}^{2}=\sum_{j=1}^{3}(S_{\mathrm{tot}}^{(j)})^{2}$ with eigenvalues $S(S+1)$. We say that a vector $\varphi$ has total spin $S$ if it satisfies ${\bm{S}}_{\mathrm{tot}}^{2}\varphi=S(S+1)\varphi$.

We set $\mathfrak{H}_{M}=\ker(S_{\mathrm{tot}}^{(3)}-M)$, the $M$-subspace. We wish to examine properties of $\mathscr{U}_{O}(H_{\mathrm{MLM}}\restriction\mathfrak{H}_{M=0})$ with $O={\bm{S}}_{\mathrm{tot}}^{2}$. The Marshall-Lieb-Mattis theorem [5, 7] claims that $\mu(H_{\mathrm{MLM}}\restriction\mathfrak{H}_{0})=S_{*}(S_{*}+1)$ with S_{*}=\big{|}|A|-|B|\big{|}/2. Hence, we have the following:

Theorem 5.1

Every Hamiltonian $H$ in $\mathscr{U}_{O}(H_{\mathrm{MLM}}\restriction\mathfrak{H}_{0})$ satisfies $\mu(H)=S_{*}(S_{*}+1)$.

Remark 5.2

Assume that the ground state of $H$ has total spin $S$. We say that the ground state of $H$ exhibits ferromagnetism, if $S$ satisfies $S=c|\Lambda|+o(|\Lambda|)$ with $c>0$ as $|\Lambda|\to\infty$. Therefore, if the ground state of $H_{\mathrm{MLM}}\restriction\mathfrak{H}_{0}$ exhibits ferromagnetism, then every Hamiltonian in $\mathscr{U}_{O}(H_{\mathrm{MLM}}\restriction\mathfrak{H}_{0})$ exhibits ferromagnetism as well. **

Does $\mathscr{U}_{O}(H_{\mathrm{MLM}}\restriction\mathfrak{H}_{0})$ contain physically interesting Hamiltonians? In [11], we provide the following answer for this question:

Theorem 5.3

Let us consider the half-filled many-electron systems on $\Lambda$. The following Hamiltonians belong to $\mathscr{U}_{O}(H_{\mathrm{MLM}}\restriction\mathfrak{H}_{0})$:

• •

The Heisenberg Hamiltonian $H_{\mathrm{Heis}}$ restricted to the $M=0$-subspace;

• •

The Hubbard Hamiltonian $H_{\mathrm{H}}$ restricted to the $M=0$-subspace;

• •

The Holstein-Hubbard Hamiltonian $H_{\mathrm{HH}}$ restricted to the $M=0$-subspace;

• •

A many-electron model coupled to the quantized radiation field $H_{\mathrm{rad}}$ restricted to the $M=0$-subspace.

In addition, these Hamiltonians satisfy the following diagram:

[TABLE]

(In the above, we abbreviate the restriction of operator $X$ to the $M=0$-subspace, i.e., $X\restriction\ker(S_{\mathrm{tot}}^{(3)})$ to $X$. ) These Hamiltonians are inequivalent to $H_{\mathrm{MLM}}\restriction\mathfrak{H}_{0}$.

Remark 5.4

• •

Theorem 5.3 indicates the stability of Lieb’s theorem under the influences from environment, e.g., the lattice vibrations and the quantum radiation field.

• •

The Marshall-Lieb-Mattis stability class $\mathscr{U}_{O}(H_{\mathrm{MLM}}\restriction\mathfrak{H}_{0})$ is one of the most important examples; except for this, the Nagaoka-Thouless stability class is examined in details [11].

Appendix A Basic properties of positivity preserving operators

A.1 Positivity preserving operators

Proposition A.1

Let $A$ be a positive self-adjoint operator. The following statements are equivalent:

• (i)

$e^{-\beta A}\unrhd 0$* w.r.t. $\mathfrak{P}$ for all $\beta\geq 0$.*

• (ii)

$(A+s)^{-1}\unrhd 0$* w.r.t. $\mathfrak{P}$ for all $s>-E(A)$, where $E(A)=\inf\mathrm{spec}(A)$.*

Proof. The proposition follows from the following elementary formulas:

[TABLE]

$\Box$

Proposition A.2

Let $\mathfrak{P}$ be a self-dual cone. Then $\mathfrak{P}$ has the following properties:

• (i)

$\mathfrak{P}\cap(-\mathfrak{P})=\{0\}$.

• (ii)

There exists a unique antilinear involution $J$ in $\mathfrak{H}$ such that $J\xi=\xi$ for all $\xi\in\mathfrak{P}$.

• (iii)

Each element $\xi\in\mathfrak{H}$ with $J\xi=\xi$ has a unique decomposition $\xi=\xi_{+}-\xi_{-}$ where $\xi_{+},\xi_{-}\in\mathfrak{P}$ and $\langle\xi_{+}|\xi_{-}\rangle=0$.

• (iv)

$\mathfrak{H}$* is linearly spanned by $\mathfrak{P}$.*

Proof. See, e.g., [1, Proof of Proposition 2.5.28 (2), (3) and (4)]. $\Box$

Proposition A.3

Let $A$ be a positive self-adjoint operator. Assume that $e^{-\beta A}\unrhd 0$ w.r.t. $\mathfrak{P}$ for all $\beta\geq 0$. Assume that $E(A)=\inf\mathrm{spec}(A)$ is an eigenvalue of $A$. Then there exists a nonzero vector $\xi\in\ker(A-E(A))$ such that $\xi\geq 0$ w.r.t. $\mathfrak{P}$.

Proof. STEP 1. Let $J$ be an antilinear involution given by Proposition A.2. Set $\mathfrak{H}_{J}=\{\xi\in\mathfrak{H}\,|\,J\xi=\xi\}$. We will show that $\ker(A-E(A))\cap\mathfrak{H}_{J}\neq\{0\}$.

To see this, let $\xi\in\ker(A-E(A))$. Then we have the decomposition $\xi=\Re\xi+i\Im\xi$ with $\Re\xi=\frac{1}{2}(1+J)\xi$ and $\Im\xi\ =\frac{1}{2i}(1-J)\xi$. Clearly , $\Re\xi,\Im\xi\in\mathfrak{H}_{J}$. Since $\xi\neq 0$, it holds that $\Re\xi\neq 0$ or $\Im\xi\neq 0$. Since $e^{-\beta A}\unrhd 0$ w.r.t. $\mathfrak{P}$ for all $\beta\geq 0$, $A$ commutes with $J$. Thus, $\Re\xi,\Im\xi\in\ker(A-E(A))\cap\mathfrak{H}_{J}$.

STEP 2. Take $\xi\in\ker(A-E(A))\cap\mathfrak{H}_{J}$. By Proposition A.2 (iii), we have a unique decomposition $\xi=\xi_{+}-\xi_{-}$, where $\xi_{\pm}\in\mathfrak{P}$ and $\langle\xi_{+}|\xi_{-}\rangle=0$. Let $|\xi|=\xi_{+}+\xi_{-}$. Then we have

[TABLE]

Thus, $|\xi|\in\ker(A-E(A))$. Clearly, $|\xi|\geq 0$ w.r.t. $\mathfrak{P}$. $\Box$

A.2 Positivity improvingness and ergodicity

Proposition A.4

Let $A$ be a positive self-adjoint operator. If $e^{-\beta A}\rhd 0$ w.r.t. $\mathfrak{P}$ for all $\beta>0$, then $(A+s)^{-1}\rhd 0$ w.r.t. $\mathfrak{P}$ for all $s>-E(A)$.

Proof. This proposition immediately follows from the formula (A.20). $\Box$

Theorem A.5

Let $A$ be a positive self-adjoint operator. Suppose that $E(A)=\inf\mathrm{spec}(A)$ is an eigenvalue. Then the following statements are equivalent:

• (i)

$(A+s)^{-1}\rhd 0$* w.r.t. $\mathfrak{P}$ for all $s>-E(A)$.*

• (ii)

$E(A)$* is a simple eigenvalue with a strictly positive eigenvector w.r.t. $\mathfrak{P}$.*

Proof. This theorem is proved in [2]. Note that the original theorem in [2] is constructed within real Hilbert spaces, however, we can readily extend it to a theorem within complex Hilbert spaces. $\Box$

Definition A.6

Let $J$ be the involution given in Proposition A.2. We set $\mathfrak{H}_{J}=\{\xi\in\mathfrak{H}\,|\,J\xi=\xi\}$. Let $A,B\in\mathscr{B}(\mathfrak{H})$. Suppose that $A\mathfrak{H}_{J}\subseteq\mathfrak{H}_{J}$ and $B\mathfrak{H}_{J}\subseteq\mathfrak{H}_{J}$. If $(A-B)\mathfrak{P}\subseteq\mathfrak{P}$, then we write this as $A\unrhd B$ w.r.t. $\mathfrak{P}$. **

Proposition A.7

Let $A$ be a positive self-adjoint operator and $B$ be a bounded self-adjoint operator. Suppose that the following conditions are satisfied:

• (i)

$e^{-\beta A}\unrhd 0$* w.r.t. $\mathfrak{P}$ for all $\beta\geq 0$;*

• (ii)

$B$* is ergodic w.r.t. $\mathfrak{P}$.*

Then $e^{-\beta(A-B)}\rhd 0$ w.r.t. $\mathfrak{P}$ for all $\beta>0$. In particular, $(A-B+s)^{-1}\rhd 0$ w.r.t. $\mathfrak{P}$ for all $s>\inf\mathrm{spec}(A-B)$.

Proof. By Propositions 2.2 and A.1, we find that $e^{-\beta(A-B)}\unrhd 0$ w.r.t. $\mathfrak{P}$ for all $\beta\geq 0$. By the Duhamel formula, we get

[TABLE]

where $\mathcal{I}_{0}(\beta)=e^{-\beta A}$ and

[TABLE]

with $B(s)=e^{-sA}Be^{sA}$. Note that the right hand side of (A.23) converges in the operator norm topology. Because $B(s_{1})\cdots B(s_{n})e^{-\beta A}\unrhd 0$ w.r.t. $\mathfrak{P}$, provided that $0\leq s_{1}\leq\cdots\leq s_{n}\leq\beta$, we see that $\mathcal{I}_{n}(\beta)\unrhd 0$ w.r.t. $\mathfrak{P}$ for all $\beta\geq 0$. Thus, we get, by Definition A.6,

[TABLE]

for all $n\in\mathbb{N}\cup\{0\}$ and $\beta\geq 0$.

For each $\xi,\eta\in\mathfrak{P}\backslash\{0\}$, there exists an $\ell\in\mathbb{N}\cup\{0\}$ such that $\langle\xi|V^{\ell}e^{-\beta A}\eta\rangle>0$ due to the ergodicity of $V$. On the other hand, by (A.25), we obtain that

[TABLE]

It suffices to show that the right hand side of (A.26) is strictly positive. To this end, let $F(s_{1},\dots,s_{\ell})=\langle\xi|B(s_{1})\cdots B(s_{\ell})e^{-\beta A}\eta\rangle$. Note that $F(0,\dots,0)=\langle\xi|V^{\ell}e^{-\beta A}\eta\rangle>0$ and $F(s_{1},\dots,s_{\ell})\geq 0$, provided that $0\leq s_{1}\leq\cdots\leq s_{\ell}\leq\beta$. Because $F$ is continuous in $s_{1},\dots,s_{\ell}$, we have

[TABLE]

Thus, we are done. $\Box$

A.3 Composition of ergodic operators

Proposition A.8

Let $\mathfrak{H}$ be a Hilbert space and let $\mathfrak{P}$ be a self-dual cone in $\mathfrak{H}$. Let $A\in\mathscr{B}(\mathfrak{H})$ and let $B\in\mathscr{B}(\mathbb{C}^{n})$. Suppose that $A$ and $B$ satisfy the following conditions:

• (i)

$A$* is ergodic w.r.t. $\mathfrak{P}$.*

• (ii)

$B$* is ergodic w.r.t. $\mathbb{R}_{+}^{n}$.*

Then $A\otimes 1+1\otimes B$ is ergodic w.r.t. $\mathfrak{P}\otimes\mathbb{R}_{+}^{n}$.

Proof. Set $C=A\otimes 1+1\otimes B$. Take $\varphi,\psi\in(\mathfrak{P}\otimes\mathbb{R}_{+}^{n})\backslash\{0\}$, arbitrarily. We can express $\varphi$ and $\psi$ as $\varphi=\sum_{j=1}^{n}\xi_{j}\otimes e_{j}$ and $\psi=\sum_{j=1}^{n}\eta_{j}\otimes e_{j}$, where $\xi_{j},\eta_{j}\in\mathfrak{P}$, and $\{e_{j}\}$ is the standard orthonormal system in $\mathbb{R}^{n}$. Because $\varphi\neq 0$ and $\psi\neq 0$, there exist $p,q\in\mathbb{N}\cup\{0\}$ such that $\xi_{p}\neq 0$ and $\eta_{q}\neq 0$. Thus, we have

[TABLE]

By the assumptions, there exist $M,N\in\mathbb{N}\cup\{0\}$ such that $\langle\xi_{p}|A^{M}\eta_{q}\rangle>0$ and $\langle e_{p}|B^{N}e_{q}\rangle>0$. By the binomial theorem and Definition A.6, we have

[TABLE]

Combining (A.28) and (A.29), we get

[TABLE]

Thus, we are done. $\Box$

Appendix B Proof of Lemmas 4.2 and 4.3

B.1 A general proposition

We begin with a general proposition.

Proposition B.1

Let $H_{0},O$ and $X$ be self-adjoint operators acting in $\mathfrak{H}_{*}$ satisfying the all assumptions in Section 4. Let $Y\in\mathscr{B}(\mathbb{C}^{n})$ be a self-adjoint operator satisfying the following condtion:

(A)* $Y$ is ergodic w.r.t. $\mathbb{R}_{+}^{n}$.*

Then $H=H_{0}\otimes 1-X\otimes Y\in\mathscr{A}_{\mathfrak{P}_{*}\otimes\mathbb{R}_{+}^{n}}^{+}$.

Proof. By the Duhamel formula, we have the norm convergent expansion:

[TABLE]

where $\mathcal{J}_{0}(\beta)=e^{-\beta H_{0}}\otimes 1$ and

[TABLE]

with $X(s)=e^{-sH_{0}}Xe^{sH_{0}}$. Because $X(s_{1})\cdots X(s_{j})e^{-\beta H_{0}}\unrhd 0$ w.r.t. $\mathfrak{P}_{*}$, provided that $0\leq s_{1}\leq\cdots\leq s_{j}\leq\beta$, we obtain that $\mathcal{J}_{j}(\beta)\unrhd 0$ w.r.t. $\mathfrak{P}_{*}\otimes\mathbb{R}_{+}^{n}$. Thus, we get

[TABLE]

Choose $\varphi,\psi\in(\mathfrak{P}_{*}\otimes\mathbb{R}_{+}^{n})\backslash\{0\}$, arbitrarily. Using an argument similar to that in the proof of Proposition A.8, we can find $p,q\in\mathbb{N}$ such that $\varphi\geq\xi_{p}\otimes e_{p}$ and $\psi\geq\eta_{q}\otimes e_{q}$ w.r.t. $\mathfrak{P}_{*}\otimes\mathbb{R}_{+}^{n}$ with $\xi_{p},\eta_{q}\in\mathfrak{P}_{*}\backslash\{0\}$. Because $Y$ is ergodic w.r.t. $\mathbb{R}_{+}^{n}$, there exists an $\ell\in\mathbb{N}\cup\{0\}$ such that $\langle e_{p}|Y^{\ell}e_{q}\rangle>0$. For this $\ell$, we claim that

[TABLE]

provided that $0<s_{1}<s_{2}<\cdots<s_{\ell}<\beta$. To this end, observe that $Xe^{-(\beta-s_{\ell})H_{0}}\eta_{q}\geq 0$ and $Xe^{-(\beta-s_{\ell})H_{0}}\eta_{q}\neq 0$ by Lemma 2.9. Hence, $X(s_{\ell})e^{-\beta H_{0}}\eta_{q}=e^{-s_{\ell}H_{0}}(Xe^{-(\beta-s_{\ell})H_{0}}\eta_{q})>0$ w.r.t. $\mathfrak{P}_{*}$ if $0<s_{\ell}<\beta$. Repeating this argument, we see that $X(s_{1})\cdots X(s_{\ell})e^{-\beta H_{0}}\eta_{q}>0$ w.r.t. $\mathfrak{P}_{*}$, provided that $0<s_{1}<s_{2}<\cdots<s_{\ell}<\beta$. Therefore, we conclude (B.34). To sum, we obtain that

[TABLE]

Thus, we are done. $\Box$

B.2 Proof of Lemma 4.2

By Proposition B.1, we readily confirm that $H_{0}-V_{\mu}\in\mathscr{A}_{\mathfrak{P}_{\mu}}^{+}$. Recall the identification $\mathfrak{H}_{*}\cong\mathfrak{H}_{*}\otimes\omega_{\mu}\subset\mathfrak{H}_{\mu}$, where $\omega_{\mu}=(1/\sqrt{n_{\mu}},\dots,1/\sqrt{n_{\mu}})^{T}\in\mathbb{C}^{n_{\mu}}$. Let $\pi$ be the orthogonal projection from $\mathfrak{H}_{\mu}$ to $\mathfrak{H}_{*}$ defined by $\pi\psi\otimes a=\langle\omega_{\mu}|a\rangle\psi$ for each $\psi\in\mathfrak{H}_{*}$ and $a\in\mathbb{C}^{n_{\mu}}$. We readily check that $\pi\mathfrak{P}_{\mu}=\mathfrak{P}_{*}$, which implies that $H_{0}\to H_{0}-V_{\mu}$. Hence, $H_{0}-V_{\mu}\in\mathscr{U}_{O}(H_{0})$. $\Box$

B.3 Proof of Lemma 4.3

Lemma B.2

For each $I=\{\mu_{1},\dots,\mu_{k}\}\in\mathcal{I}$, let

[TABLE]

Then, for each $I\in\mathcal{I}$, the following holds true:

• (E)

$Y_{I}$* is ergodic w.r.t. $\mathbb{R}_{+}^{n_{\mu_{1}}}\otimes\cdots\otimes\mathbb{R}_{+}^{n_{\mu_{k}}}$.*

Proof. We will prove Lemma B.2 by induction.

Suppose that (E) holds true for every $I\in\mathcal{I}$ with $|I|=k$. Our goal is to prove (E) for every $I\in\mathcal{I}$ with $|I|=k+1$. For a given $I=\{\mu_{1},\dots,\mu_{k+1}\}\in\mathcal{I}$, we set $\tilde{I}=\{\mu_{1},\dots,\mu_{k}\}$. Thus, $I=\tilde{I}\cup\{\mu_{k+1}\}$ holds. Corresponding to this, $Y_{I}$ can be expressed as $Y_{I}=Y_{\tilde{I}}\otimes 1+1\otimes Y_{\mu_{k+1}}$. Because $Y_{\tilde{I}}$ is ergodic w.r.t. $\mathbb{R}_{+}^{n_{\mu_{1}}}\otimes\cdots\otimes\mathbb{R}_{+}^{n_{\mu_{k}}}$, we can apply Proposition A.8 and conclude that $Y_{I}$ is ergodic w.r.t. $\mathbb{R}_{+}^{n_{\mu_{1}}}\otimes\cdots\otimes\mathbb{R}_{+}^{n_{\mu_{k+1}}}$. $\Box$

Proof of Lemma 4.3

Write $I=\{\mu_{1},\dots,\mu_{k}\}$. Recall the identification $\mathfrak{H}_{*}\cong\mathfrak{H}_{*}\otimes\omega_{I}\subset\mathfrak{H}_{I}$. By Proposition B.1 and Lemma B.2, we see that $H_{I}\in\mathscr{A}_{\mathfrak{P}_{I}}^{+}$. Let $\pi_{I}$ be the orthogonal projection from $\mathfrak{H}_{I}$ to $\mathfrak{H}_{*}$ defined by $\pi_{I}\Psi\otimes b=\langle\omega_{I}|b\rangle\Psi$ for every $\Psi\in\mathfrak{H}_{*}$ and $b\in\mathbb{C}^{n_{\mu_{1}}}\otimes\cdots\otimes\mathbb{C}^{n_{\mu_{k}}}$. We confirm that $\pi_{I}\mathfrak{P}_{I}=\mathfrak{P}_{*}$ holds, which implies that $H_{0}\to H_{I}$. Thus, $H_{I}\in\mathscr{U}_{O}(H_{0})$. $\Box$