# Classification of symmetry protected topological phases in quantum spin
chains

Yoshiko Ogata  Graduate School of Mathematical Sciences The University of
Tokyo, Komaba, Tokyo, 153-8914, Japan Supported in part by the Grants-in-Aid
for Scientific Research, JSPS.

###### Abstract

We consider the classification problem of symmetry protected topological (SPT)
phases on quantum spin systems. SPT phases are gapped short-range-entangled
quantum phases with a symmetry $G$. We explain that in one and two-dimensional
quantum spin systems, there are $H^{2}(G,U(1))$/ $H^{3}(G,U(1))$-valued
invariant, confirming a physicists conjecture.

###### Contents

  1. 1 Symmetry protected topological phases
    1. 1.1 Finite dimensional quantum mechanics
    2. 1.2 Basic facts about $C^{*}$-dynamical systems
    3. 1.3 Quantum spin systems
    4. 1.4 SPT-phases
    5. 1.5 Split property
    6. 1.6 Automorphic equivalence
  2. 2 SPT-phases in $1$-dimensional systems
    1. 2.1 $H^{2}(G,{\rm U}(1))$ and projective representations
    2. 2.2 Split property of unique gapped ground state in $1$-d
    3. 2.3 Matrix product state
    4. 2.4 $H^{2}(G,{\rm U}(1))$-index for MPS with on-site symmetry
    5. 2.5 $H^{2}(G,{\rm U}(1))$-index for SPT with on-site symmetry
    6. 2.6 Lieb-Schultz-Mattis (LSM) type theorems
    7. 2.7 A classification of pure states on quantum spin chains satisfying the split property with on-site finite group symmetries
    8. 2.8 An easy derivation of $H^{2}(G,{\rm U}(1))$
    9. 2.9 SPT phases with the reflection symmetry
  3. 3 SPT-phases in $2$-dimensional systems
    1. 3.1 Derivation of $H^{3}(G,{\rm U}(1))$ out of cocycle action
    2. 3.2 Action of $\beta_{g}^{U}$ on automorphisms
    3. 3.3 Action of $\beta^{U}$ on $\omega_{\Phi}$
    4. 3.4 Definition of $H^{3}(G,{\rm U}(1))$-valued index
    5. 3.5 What $h(\Phi)$ represents
    6. 3.6 Two-dimensional Dijkgraaf-Witten model

##  1 Symmetry protected topological phases

Phase transitions in macroscopic systems have been among central topics of
study in theoretical physics. Traditional examples of phase transitions were
those between a disordered phase and an ordered phase. It is well known that
such a phase transition can be characterized by identifying a suitable order
parameter, which is nothing but an extensive sum of some local operators. A
notable example is the ferromagnetic phase transition in the Ising model,
where the order parameter is the bulk magnetization. This is the topic of the
lecture note by Michael Aizenman in the present volume.

In recent years it has become clear that certain quantum many-body systems
exhibit phase transitions between two phases which have no orders. Such phase
transitions cannot be characterized by standard order parameters, and are
called “topological phase transitions” [No]. We note that here the term
“topological” should not be interpreted in its strict mathematical sense.

In the present lecture we focus on mathematical aspects of the problem of
symmetry protected topological phases in quantum spin systems. The problem has
its root in Haldane’s discovery (which brought him the 2016 Nobel prize in
physics [No]) that the antiferromagnetic Heisenberg chain with spin 1 has
“exotic” low energy properties, essentially different from those of its spin
$1/2$ counterpart. The problem has been extensively studied from experimental,
numerical, theoretical, and mathematical points of view. After the
introduction by Gu and Wen [GW] of the notion of symmetry protected
topological (SPT) phases, it has become clear that the model studied by
Haldane belongs to a nontrivial SPT phase characterized by an index that takes
value in the second group cohomology. The problem of SPT phases has been
further developed by physicists, and it has been conjectured that SPT phases
in $\nu$ dimensional quantum spin systems are classified by an index that
takes value in the $(\nu+1)$-th group cohomology of the relevant symmetry
group.

Let us be slightly more precise. A quantum spin system is characterized by a
self-adjoint operator called Hamiltonian, which governs the time evolution of
the system. In short SPT phases are equivalence classes of Hamiltonians
satisfying certain conditions on the symmetry as well as low energy
properties. Let us consider the set of all Hamiltonians with some fixed
symmetry, which have a unique gapped ground state in the bulk. In two or
higher dimensions, in order to exclude models with so called “topological
order”, we further assume that all Hamiltonians in the set can be smoothly
deformed into a common trivial gapped Hamiltonian without closing the gap. We
say two such Hamiltonians are equivalent, if they can be smoothly deformed
into each other, without breaking the symmetry. We call an equivalence class
of this classification, a symmetry protected topological (SPT) phase. In this
lecture, we consider this problem in operator algebraic framework. In this
framework, what we deal with is not Hamiltonians, but a time evolution
(strongly continuous one parameter group of automorphisms on a
$C^{*}$-algebra). The reason for considering the problem in this framework is
because it enables us to treat infinite system directly. It is essential to
treat the infinite system directly for analysis of SPT phases for us, because
the invariant of our classification can be most naturally defined on the
infinite system. The situation is quite analogous to that of Fredholm index.
Recall that the Fredholm index of an operator $T$ (with $\dim\ker T,\dim\ker
T^{*}<\infty$ and $T{\mathcal{H}}$ closed) on a Hilbert space ${\mathcal{H}}$
is defined by $\mathop{\mathrm{ind}}\nolimits T:=\dim\ker T-\dim\ker T^{*}$.
By linear algebra, this is always $0$ if ${\mathcal{H}}$ is of finite
dimensional. However, it is not the case if the dimension of
$\dim{\mathcal{H}}$ is infinite. For example, on
${\mathcal{H}}=l^{2}({\mathbb{N}})$, the unilateral shift

$\displaystyle
S\left(\xi_{1},\xi_{2},\xi_{3},\ldots\right):=\left(0,\xi_{1},\xi_{2},\xi_{3},\ldots\right),\quad\left(\xi_{1},\xi_{2},\xi_{3},\ldots\right)\in
l^{2}({\mathbb{N}})$

has $\ker S=\\{0\\}$, $\ker S^{*}={\mathbb{C}}(1,0,0,0,\ldots)$ and
$\mathop{\mathrm{ind}}\nolimits S=0-1=-1$. From this, we see that in order to
get non-zero Fredholm index, one need to consider infinite dimensional Hilbert
spaces. Quite analogous phenomena occurs in SPT-theory.

In this section, we introduce the SPT-phases in operator algebraic framework.

###  1.1 Finite dimensional quantum mechanics

In order to motivate us for the operator algebraic framework of quantum
statistical mechanics, we first recall finite dimensional quantum mechanics in
this subsection.

In finite-dimensional quantum mechanics, physical observables are represented
by elements of $\mathop{\mathrm{M}}\nolimits_{n}$, the algebra of $n\times
n$-matrices. Each positive matrix $\rho$ with
$\mathop{\mathrm{Tr}}\nolimits\rho=1$ (called a density matrix) defines a
physical state by

$\omega_{\rho}:\mathop{\mathrm{M}}\nolimits_{n}\ni
A\mapsto\mathop{\mathrm{Tr}}\nolimits\left(\rho A\right)\in{\mathbb{C}}.$

We call this map $\omega_{\rho}$ a sate. This corresponds to the expectation
values of each physical observables $A\in\mathop{\mathrm{M}}\nolimits_{n}$, in
the physical state $\omega_{\rho}$. When a state ${\omega_{\rho}}$ can not be
written as a convex combination of two different states, it is said to be
$pure$. A state $\omega_{\rho}$ is pure if and only if $\rho$ is a rank one
projection. Time evolution is given by a self-adjoint matrix $H$, called
Hamiltonian, via a formula

$\mathop{\mathrm{M}}\nolimits_{n}\ni
A\mapsto\tau_{t}(A):=e^{it{H}}Ae^{-it{H}},\quad t\in{\mathbb{R}}.$

Let $p$ be the spectral projection of $H$ corresponding to the lowest
eigenvalue. A state ${\omega_{\rho}}(A):=\mathop{\mathrm{Tr}}\nolimits{\rho}A$
on $\mathop{\mathrm{M}}\nolimits_{n}$ is said to be a ground state of $H$ if
the support of $\rho$ is under $p$. Ground state is unique if and only if $p$
is a rank one projection, i.e., the lowest eigenvalue of $H$ is non-
degenerated. In this case, the unique ground state is
$\omega_{p}(A):=\mathop{\mathrm{Tr}}\nolimits{p}A$, and it is pure because $p$
is rank one. Let $G$ be a finite group and suppose that there is a group
action
$\beta:G\to\mathop{\mathrm{Aut}}\nolimits(\mathop{\mathrm{M}}\nolimits_{n})$
given by unitaries $V_{g}$, $g\in G$

${\beta_{g}}(A):=\mathop{\mathrm{Ad}}\nolimits\left({V_{g}}\right)\left(A\right),\quad
A\in\mathop{\mathrm{M}}\nolimits_{d},\quad g\in G.$

If a Hamiltonian $H$ satisfies ${\beta_{g}}({H})={H}$ for all $g\in G$, we say
$H$ is $\beta$-invariant. If a $\beta$-invariant Hamiltonian $H$ has a unique
ground state $\omega_{p}(A):=\mathop{\mathrm{Tr}}\nolimits{p}A$, then this
unique ground state $\omega_{p}$ is $\beta$-invariant
$\omega_{p}\left(\beta_{g}(A)\right)=\omega_{p}(A)$,
$A\in\mathop{\mathrm{M}}\nolimits_{n}$, because the spectral projection $p$ is
$\beta$-invariant, i.e., $\beta_{g}(p)=p$.

###  1.2 Basic facts about $C^{*}$-dynamical systems

In order to extend the finite dimensional quantum mechanics to infinite ones,
we move to general $C^{*}$-dynamical systems. In general $C^{*}$-dynamical
systems, physical observables are represented by elements of a
$C^{*}$-algebra. A $C^{*}$-algebra ${\mathfrak{A}}$ is a Banach $*$-algebra
with norm satisfying

$\left\|A^{*}A\right\|=\left\|A\right\|^{2},\quad A\in{\mathfrak{A}}.$

In this lecture, we always consider $C^{*}$-algebras with a unit $\mathbb{I}$,
i.e., a unital $C^{*}$-algebra. The algebra of $n\times n$ matrices
$\mathop{\mathrm{M}}\nolimits_{n}$ is a $C^{*}$-algebra with uniform norm

$\left\|A\right\|:=\sup\left\\{\left\|A\xi\right\|\mid\xi\in{\mathbb{C}}^{n},\;\left\|\xi\right\|=1\right\\}.$

A linear functional $\omega$ over a $C^{*}$-algebra ${\mathfrak{A}}$ is
defined to be positive if

$\omega(A^{*}A)\geq 0,$

for all $A\in{\mathfrak{A}}$. In physical settings, we regard a
$C^{*}$-algebra ${\mathfrak{A}}$ an algebra of physical local observables and
$\omega(A)$ denotes the expectation value of an observable
$A\in{\mathfrak{A}}$ in a state $\omega$. A positive linear functional
$\omega$ on ${\mathfrak{A}}$ with $\left\|\omega\right\|=1$ is called a state.
A state $\omega$ over a $C^{*}$-algebra ${\mathfrak{A}}$ is defined to be pure
if it can not be written as a convex combination of two different states.
Recall that physical state in finite quantum mechanics is a map
$\omega_{\rho}:\mathop{\mathrm{M}}\nolimits_{n}\to{\mathbb{C}}$ given by a
density matrix. In fact, this $\omega_{\rho}$ is a positive normalized linear
functional on $\mathop{\mathrm{M}}\nolimits_{n}$, namely it is a state in the
above sense. It is pure if and only if $\rho$ is a one-rank projection.

A representation of a $C^{*}$-algebra ${\mathfrak{A}}$ is defined to be a pair
$({\mathcal{H}},\pi)$, where ${\mathcal{H}}$ is a Hilbert space and $\pi$ is a
$*$-homomorphism of ${\mathfrak{A}}$ into ${\mathcal{B}}({\mathcal{H}})$ (the
set of all bounded operators on ${\mathcal{H}}$.) Two representations of a
$C^{*}$-algebra ${\mathfrak{A}}$, $({\mathcal{H}}_{1},\pi_{1})$,
$({\mathcal{H}}_{2},\pi_{2})$ are unitary equivalent if there is a unitary
$u:{\mathcal{H}}_{1}\to{\mathcal{H}}_{2}$ such that

$\displaystyle\mathop{\mathrm{Ad}}\nolimits(u)\circ\pi_{1}(A)=\pi_{2}(A),\quad
A\in{\mathfrak{A}}.$

Given a state, we may associate a corresponding representation.

######  Theorem 1.1.

For each state $\omega$ on ${\mathfrak{A}}$, there exist a representation
$\pi_{\omega}$ of ${\mathfrak{A}}$ on a Hilbert space ${\mathcal{H}}_{\omega}$
and a unit vector $\Omega_{\omega}\in{\mathcal{H}}_{\omega}$ such that

$\displaystyle\omega(A)=\left\langle\Omega_{\omega},\pi_{\omega}(A)\Omega_{\omega}\right\rangle,\quad
A\in{\mathfrak{A}},\quad\text{and}\quad{\mathcal{H}}_{\omega}=\overline{\pi_{\omega}({\mathfrak{A}})\Omega_{\omega}}.$
(1.1)

Here, $\overline{\cdot}$ denotes the norm closure. It is unique up to unitary
equivalence.

The latter condition in (1.1) means that the vector $\Omega_{\omega}$ is
cyclic for $\pi_{\omega}({\mathfrak{A}})$ in ${\mathcal{H}}_{\omega}$. We use
the convention of mathematical physics for inner product : namely, it is
linear for the right variable. The representation
$({\mathcal{H}}_{\omega},\pi_{\omega},\Omega_{\omega})$ is called a Gelfand-
Naimark-Segal (GNS)-representation/GNS-triple.

An injective $*$-homomorphism of a $C^{*}$-algebra ${\mathfrak{A}}$ onto
itself is called a $*$-automorphism of ${\mathfrak{A}}$. We denote by
$\mathop{\mathrm{Aut}}\nolimits({\mathfrak{A}})$, the set of all
$*$-automorphisms on ${\mathfrak{A}}$. For example, for a unitary
$V\in{\mathfrak{A}}$, $\mathop{\mathrm{Ad}}\nolimits(V)(A):=VAV^{*}$,
$A\in{\mathfrak{A}}$ defines a $*$-automorphism on ${\mathfrak{A}}$. Such a
$*$-automorphism is called an inner automorphism. In this lecture, we
occasionally write $\alpha=({\rm{inner}})$ when
$\alpha=\mathop{\mathrm{Ad}}\nolimits(V)$ with some unitary
$V\in{\mathfrak{A}}$. A $*$-automorphism which is not inner is said to be
outer.

For a given representation $({\mathcal{H}},\pi)$ of a $C^{*}$-algebra
${\mathfrak{A}}$, we say an automorphism $\alpha$ of ${\mathfrak{A}}$ is
implementable by a unitary in $({\mathcal{H}},\pi)$ if there is a unitary $u$
on ${\mathcal{H}}$ satisfying

$\mathop{\mathrm{Ad}}\nolimits(u)\left(\pi(A)\right)=\pi\circ\alpha(A),\quad
A\in{\mathfrak{A}}.$

If a state $\omega$ is invariant under some
$\alpha\in\mathop{\mathrm{Aut}}\nolimits({\mathfrak{A}})$, then it can be
implemented in the GNS-representation by a unitary. Namely the following
theorem holds.

######  Theorem 1.2.

If $\omega\circ\alpha=\omega$, then there is a unitary $V_{\alpha}$ on
${\mathcal{H}}_{\omega}$ such that

$\displaystyle
V_{\alpha}\pi_{\omega}(A)V_{\alpha}^{*}=\pi_{\omega}\left(\alpha(A)\right),\quad
V_{\alpha}\Omega_{\omega}=\Omega_{\omega},\quad A\in{\mathfrak{A}}.$

Note from the condition that

$\displaystyle
V_{\alpha}\pi_{\omega}(A)\Omega_{\omega}=\pi_{\omega}\left(\alpha(A)\right)\Omega_{\omega},\quad
A\in{\mathfrak{A}}.$

In fact, that is how $V_{\alpha}$ is defined.

In particular, let $\gamma:G\to\mathop{\mathrm{Aut}}\nolimits({\mathfrak{A}})$
be a group action of $G$ on a $C^{*}$-algebra ${\mathfrak{A}}$, and suppose
that a state $\omega$ is invariant under $\gamma$. Then from the Theorem, we
get unitaries $V_{g}:=V_{\gamma_{g}}$. It is a genuine unitary representation
because

$\displaystyle
V_{g}V_{h}\pi_{\omega}(A)\Omega_{\omega}=\pi_{\omega}\circ\gamma_{g}\gamma_{h}(A)\Omega_{\omega}=\pi_{\omega}\circ\gamma_{gh}(A)\Omega_{\omega}=V_{gh}\pi_{\omega}(A)\Omega_{\omega}.$

Using the GNS representation and the Hahn-Banach theorem, we can show that any
$C^{*}$-algebra is isomorphic to a norm-closed self-adjoint algebra of bounded
operators on a Hilbert space.

For a subset ${\mathcal{M}}$ of operators on a Hilbert space ${\mathcal{H}}$,
let ${\mathcal{M}}^{\prime}$ denote its commutant, i.e., the set of all
bounded operators on ${\mathcal{H}}$ commuting with every operator in
${\mathcal{M}}$. A von Neumann algebra on a Hilbert space ${\mathcal{H}}$ is a
$*$-subalgebra ${\mathcal{M}}$ of ${\mathcal{B}}({\mathcal{H}})$ (the set of
all bounded operators on ${\mathcal{H}}$) such that
${\mathcal{M}}={\mathcal{M}}^{\prime\prime}$. A center of a von Neumann
algebra is defined as
${\mathcal{Z}}({\mathcal{M}}):={\mathcal{M}}\cap{\mathcal{M}}^{\prime}$. A von
Neumann algebra ${\mathcal{M}}$ is called a factor if it has a trivial center.
Given a state $\omega$ on a $C^{*}$-algebra, we obtain a von Neumann algebra
$\pi_{\omega}({\mathfrak{A}})^{\prime\prime}$ out of the GNS representation of
$\omega$. We call this von Neumann algebra a von Neumann algebra associated to
$\omega$.

A von Neumann algebra is said to be a type $I$-factor if it is isomorphic to
${\mathcal{B}}({\mathcal{H}})$ for some Hilbert space ${\mathcal{H}}$. Type
$I$ factors will play an important role in the analysis of SPT phases. From
this point of view, the following theorems are useful.

######  Theorem 1.3.

Let ${\mathcal{H}}_{1}$, ${\mathcal{H}}_{2}$ be Hilbert spaces and
$\tau:{\mathcal{B}}({\mathcal{H}}_{1})\to{\mathcal{B}}({\mathcal{H}}_{2})$ a
$*$-isomorphism. Then there is a unitary
$u:{\mathcal{H}}_{1}\to{\mathcal{H}}_{2}$ such that $\tau(x)=uxu^{*}$,
$x\in{\mathcal{B}}({\mathcal{H}}_{1})$.

######  Theorem 1.4.

The following conditions are equivalent for a state $\omega$:

(1)

The representation $({\mathcal{H}}_{\omega},\pi_{\omega})$ is irreducible,
i.e., the only closed subspaces of ${\mathcal{H}}_{\omega}$ which are
invariant under the action of $\pi_{\omega}({\mathfrak{A}})$ are the trivial
subspaces $\\{0\\}$ and ${\mathcal{H}}_{\omega}$.

(2)

$\omega$ is pure,

(3)

$\pi_{\omega}({\mathfrak{A}})^{\prime\prime}={\mathcal{B}}({\mathcal{H}}_{\omega})$,

(4)

$\pi_{\omega}({\mathfrak{A}})^{\prime}={\mathbb{C}}\mathbb{I}$.

Another notion which plays an important role in the analysis of SPT phases is
quasi-equivalence. In probability theory, a probability measure $\lambda$ is
absolutely continuous with respect to another probability measure $\mu$ if
there is a positive $h\in L^{1}(\mu)$ satisfying

$\int f\;d\lambda=\int f\;h\;d\mu,$

for all bounded measurable functions $f$. Furthermore, two measures are
equivalent if they are absolutely continuous to each other. Quasi-equivalence
is a non-commutative version of it.

If $({\mathcal{H}},\pi)$ is a representation of a $C^{*}$-algebra
${\mathfrak{A}}$, then a state $\varphi$ of ${\mathfrak{A}}$ is said to be
$\pi$-normal if there exists a positive trace-class operator $\rho$ (called a
density operator) on ${\mathcal{H}}$ with
$\mathop{\mathrm{Tr}}\nolimits(\rho)=1$ such that

$\displaystyle\varphi(A)=\mathop{\mathrm{Tr}}\nolimits\left(\rho\pi(A)\right),\quad\text{for
all}\quad A\in{\mathfrak{A}}.$ (1.2)

Two representations $\pi_{1}$ and $\pi_{2}$ of a $C^{*}$-algebra
${\mathfrak{A}}$ are said to be quasi-equivalent, written
$\pi_{1}\sim_{q.e.}\pi_{2}$ if each $\pi_{1}$-normal state is $\pi_{2}$-normal
and conversely.

######  Theorem 1.5.

The following conditions are equivalent for representations $\pi_{1}$ and
$\pi_{2}$ of ${\mathfrak{A}}$:

(1)

$\pi_{1}\sim_{q.e.}\pi_{2}$

(2)

there exists an isomorphism
$\tau:\pi_{1}({\mathfrak{A}})^{\prime\prime}\to\pi_{2}({\mathfrak{A}})^{\prime\prime}$
such that $\tau\left(\pi_{1}(A)\right)=\pi_{2}(A)$ for all
$A\in{\mathfrak{A}}$.

Two states $\omega_{1}$ and $\omega_{2}$ of ${\mathfrak{A}}$ are said to be
quasi-equivalent if $\pi_{\omega_{1}}$ and $\pi_{\omega_{2}}$ are quasi-
equivalent. Although any states on a matrix algebra are quasi-equivalent, that
is not the case for infinite systems. Indeed, the following physical
interpretation can be made. By the cyclicity of the GNS vector
$\Omega_{\omega}$ in the GNS-representation (1.1), if $\omega_{1}$ and
$\omega_{2}$ are quasi-equivalent, then $\omega_{2}$ can be approximated
arbitrarily well in norm topology of ${\mathfrak{A}}^{*}$ by states of the
form

$\displaystyle\sum_{n=1}^{\infty}\omega_{1}\left(A_{n}^{*}\cdot A_{n}\right),$

with some $A_{n}\in{\mathfrak{A}}$. Regarding elements in ${\mathfrak{A}}$ as
local observables, it may be interpreted that $\omega_{2}$ can be obtained
from $\omega_{1}$ via some local perturbation. Because the converse also
holds, heuristically we may say that two states $\omega_{1}$ and $\omega_{2}$
being quasi-equivalent means that they are macroscopically the same.

Two representations $\pi_{1}$ and $\pi_{2}$ of a $C^{*}$-algebra
${\mathfrak{A}}$ are mutually singular if none of $\pi_{2}$-normal state is
$\pi_{1}$-normal and converse. One can regard this as an analog of mutually
singular measures (like $\delta$-function and Lebesgue measure) in probability
theory.

Recall that a von Neumann algebra is a factor if it has a trivial center.

######  Theorem 1.6.

Let $({\mathcal{H}},\pi)$ be a representation of a $C^{*}$-algebra
${\mathfrak{A}}$. If $\pi({\mathfrak{A}})^{\prime\prime}$ is a factor, then
any subrepresentation of $\pi$ is quasi-equivalent to $\pi$.

We say that a state $\omega$ is a factor state if associated von Neumann
algebra is a factor. Physically, it can be regarded as a homogeneous state, in
the sense that it cannot be written as a mixture of macroscopically distinct
states (like a convex combination of all spin up state and all spin down
states). A pure state is of course homogeneous, which even forbids mixture of
macroscopically same states.

If an automorphism $\alpha$ on a $C^{*}$-algebra ${\mathfrak{A}}$ is
implementable (by a unitary operator $u$) in a GNS-representation
$({\mathcal{H}}_{\omega},\pi_{\omega})$ of a state $\omega$ on
${\mathfrak{A}}$, then $\alpha$ preserves the set of $\pi_{\omega}$-normal
states. Indeed, let $\varphi$ be a $\pi_{\omega}$-normal state. Then there is
a density operator $\rho$ on ${\mathcal{H}}_{\omega}$ satisfying

$\varphi(A):=\mathop{\mathrm{Tr}}\nolimits\left(\rho\pi_{\omega}(A)\right),\quad
A\in{{\mathfrak{A}}}.$

We then have

$\displaystyle\varphi\circ\alpha(A)=\mathop{\mathrm{Tr}}\nolimits\left(\rho\pi_{\omega}\circ\alpha(A)\right)=\mathop{\mathrm{Tr}}\nolimits\left(\rho
u\pi_{\omega}(A)u^{*}\right)=\mathop{\mathrm{Tr}}\nolimits\left(\left(u^{*}\rho
u\right)\pi_{\omega}(A)\right),\quad A\in{{\mathfrak{A}}}.$

Hence the state $\varphi\circ\alpha$ is given by a density operator $u^{*}\rho
u$. This means that $\varphi\circ\alpha$ is $\omega$-normal. From the above
interpretation of macroscopic equivalence, $\alpha$ being implementable on the
GNS representation of $\omega$ means that it does not move $\omega$
macroscopically.

We also prepare tensor product of $C^{*}$-algebras. Let ${\mathfrak{A}}$,
${\mathfrak{B}}$ be $C^{*}$-algebras with faithful representations
$({\mathcal{H}}_{{\mathfrak{A}}},\pi_{{\mathfrak{A}}})$,
$({\mathcal{H}}_{{\mathfrak{B}}},\pi_{{\mathfrak{B}}})$. The the spatial
$C^{*}$-norm $\left\|\cdot\right\|$ on algebraic tensor product
${\mathfrak{A}}\odot{\mathfrak{B}}$ is defined by

$\displaystyle\left\|\sum_{i}a_{i}\otimes
b_{i}\right\|:=\left\|\sum_{i}\pi_{{\mathfrak{A}}}(a_{i})\otimes\pi_{{\mathfrak{B}}}(b_{i})\right\|_{{\mathcal{B}}({\mathcal{H}}_{{\mathfrak{A}}}\otimes{\mathcal{H}}_{{\mathfrak{B}}})}.$

The right-hand side means the norm as an element of
${\mathcal{B}}({\mathcal{H}}_{{\mathfrak{A}}}\otimes{\mathcal{H}}_{{\mathfrak{B}}})$.
The completion of ${\mathfrak{A}}\odot{\mathfrak{B}}$ with respect to this
norm is denoted by ${\mathfrak{A}}\otimes{\mathfrak{B}}$. This is the spacial
tensor product of ${\mathfrak{A}}$ and ${\mathfrak{B}}$. For states
$\omega_{{\mathfrak{A}}}$, $\omega_{{\mathfrak{B}}}$ on ${\mathfrak{A}}$,
${\mathfrak{B}}$, there exists a unique state
$\omega_{{\mathfrak{A}}}\otimes\omega_{{\mathfrak{B}}}$ on
${\mathfrak{A}}\otimes{\mathfrak{B}}$ such that

$\displaystyle\left(\omega_{{\mathfrak{A}}}\otimes\omega_{{\mathfrak{B}}}\right)(A\otimes
B)=\omega_{{\mathfrak{A}}}(A)\omega_{{\mathfrak{B}}}(B),\quad
A\in{\mathfrak{A}},\;B\in{\mathfrak{B}}.$ (1.3)

Furthermore, for representations
$({\mathcal{H}}_{\mathfrak{A}},\pi_{\mathfrak{A}})$,
$({\mathcal{H}}_{\mathfrak{B}},\pi_{\mathfrak{B}})$ of
${\mathfrak{A}},{\mathfrak{B}}$, there is a unique representation
$\pi_{{\mathfrak{A}}}\otimes\pi_{{\mathfrak{B}}}$ of
${\mathfrak{A}}\otimes{\mathfrak{B}}$ on
${\mathcal{H}}_{{\mathfrak{A}}}\otimes{\mathcal{H}}_{{\mathfrak{B}}}$ such
that

$\left(\pi_{{\mathfrak{A}}}\otimes\pi_{{\mathfrak{B}}}\right)(A\otimes
B)=\pi_{{\mathfrak{A}}}(A)\otimes\pi_{{\mathfrak{B}}}(B),\quad
A\in{\mathfrak{A}},\quad B\in{\mathfrak{B}}.$

In the analysis of SPT-phases we encounter the following situation.

######  Lemma 1.7.

Let $\omega_{{\mathfrak{A}}}$, $\omega_{{\mathfrak{B}}}$ be pure states on
$C^{*}$-algebras ${\mathfrak{A}}$, ${\mathfrak{B}}$, with GNS triples
$({\mathcal{H}}_{{\mathfrak{A}}},\pi_{{\mathfrak{A}}},\Omega_{{\mathfrak{A}}})$
and
$({\mathcal{H}}_{{\mathfrak{B}}},\pi_{{\mathfrak{B}}},\Omega_{{\mathfrak{B}}})$.
Let $\alpha_{{\mathfrak{A}}}$, $\alpha_{{\mathfrak{B}}}$ be automorphisms on
${\mathfrak{A}}$, ${\mathfrak{B}}$ such that

$\displaystyle\omega_{{\mathfrak{A}}}\alpha_{{\mathfrak{A}}}\otimes\omega_{{\mathfrak{B}}}\alpha_{{\mathfrak{B}}}=\left(\omega_{{\mathfrak{A}}}\otimes\omega_{{\mathfrak{B}}}\right)\circ\mathop{\mathrm{Ad}}\nolimits
V,$

for some unitary $V\in{\mathfrak{A}}\otimes{\mathfrak{B}}$. Then there is a
unitary $u_{{\mathfrak{B}}}$ on ${\mathcal{H}}_{\mathfrak{B}}$ such that

$\displaystyle\mathop{\mathrm{Ad}}\nolimits(u_{\mathfrak{B}})\circ\pi_{{\mathfrak{B}}}(A)=\pi_{{\mathfrak{B}}}\circ\alpha_{{\mathfrak{B}}}(A),\quad
A\in{\mathfrak{B}}.$

  * Proof.

First, note that
$({\mathcal{H}}_{{\mathfrak{A}}}\otimes{\mathcal{H}}_{{\mathfrak{B}}},\pi_{{\mathfrak{A}}}\otimes\pi_{{\mathfrak{B}}},\Omega_{{\mathfrak{A}}}\otimes\Omega_{{\mathfrak{B}}})$
is a GNS representation of
$\omega_{{\mathfrak{A}}}\otimes\omega_{{\mathfrak{B}}}$. By the assumption,
$\omega_{{\mathfrak{A}}}\otimes\omega_{{\mathfrak{B}}}$ is invariant under

$\left(\alpha_{{\mathfrak{A}}}\otimes\alpha_{{\mathfrak{B}}}\right)\circ\mathop{\mathrm{Ad}}\nolimits(V^{*}).$

Then, from Theorem 1.2, there is a unitary $w$ on
${\mathcal{H}}_{{\mathfrak{A}}}\otimes{\mathcal{H}}_{{\mathfrak{B}}}$ such
that

$\displaystyle\mathop{\mathrm{Ad}}\nolimits(w)\left(\pi_{{\mathfrak{A}}}\otimes\pi_{{\mathfrak{B}}}\right)=\left(\pi_{{\mathfrak{A}}}\otimes\pi_{{\mathfrak{B}}}\right)\left(\alpha_{{\mathfrak{A}}}\otimes\alpha_{{\mathfrak{B}}}\right).$

(We absorbed $\mathop{\mathrm{Ad}}\nolimits V^{*}$ part to
$\mathop{\mathrm{Ad}}\nolimits(w)$.) From this, we get

$\mathop{\mathrm{Ad}}\nolimits(w)\left(\mathbb{I}_{{\mathcal{H}}_{{\mathfrak{A}}}}\otimes\pi_{{\mathfrak{B}}}({\mathfrak{B}})^{\prime\prime}\right)\subset\mathbb{I}_{{\mathcal{H}}_{{\mathfrak{A}}}}\otimes\pi_{{\mathfrak{B}}}({\mathfrak{B}})^{\prime\prime},\quad\mathop{\mathrm{Ad}}\nolimits(w^{*})\left(\mathbb{I}_{{\mathcal{H}}_{{\mathfrak{A}}}}\otimes\pi_{{\mathfrak{B}}}({\mathfrak{B}})^{\prime\prime}\right)\subset\mathbb{I}_{{\mathcal{H}}_{{\mathfrak{A}}}}\otimes\pi_{{\mathfrak{B}}}({\mathfrak{B}})^{\prime\prime}.$

Hence we obtain

$\mathop{\mathrm{Ad}}\nolimits(w)\left(\mathbb{I}_{{\mathcal{H}}_{{\mathfrak{A}}}}\otimes\pi_{{\mathfrak{B}}}({\mathfrak{B}})^{\prime\prime}\right)=\mathbb{I}_{{\mathcal{H}}_{{\mathfrak{A}}}}\otimes\pi_{{\mathfrak{B}}}({\mathfrak{B}})^{\prime\prime}.$

Now, because $\omega_{{\mathfrak{B}}}$ is a pure state, we have
$\pi_{{\mathfrak{B}}}({\mathfrak{B}})^{\prime\prime}={\mathcal{B}}({\mathcal{H}}_{{\mathfrak{B}}})$
by Theorem 1.4. This means there is a $*$-automorphism $\tau$ on
${\mathcal{B}}({\mathcal{H}}_{{\mathfrak{B}}})$ such that

$\mathop{\mathrm{Ad}}\nolimits(w)\left(\mathbb{I}_{{\mathcal{H}}_{{\mathfrak{A}}}}\otimes
x\right)=\mathbb{I}_{{\mathcal{H}}_{{\mathfrak{A}}}}\otimes\tau(x),\quad
x\in{\mathcal{B}}({\mathcal{H}}_{{\mathfrak{B}}}).$

By Theorem 1.3, a $*$-isomorphism
$\tau:{\mathcal{B}}({\mathcal{H}}_{{\mathfrak{B}}})\to{\mathcal{B}}({\mathcal{H}}_{{\mathfrak{B}}})$
is implemented by a unitary
$u_{\mathfrak{B}}\in{\mathcal{H}}_{{\mathfrak{B}}}$, i.e.

$\tau(x)=\mathop{\mathrm{Ad}}\nolimits(u_{\mathfrak{B}})(x),\quad
x\in{\mathcal{B}}({\mathcal{H}}_{{\mathfrak{B}}}).$

Hence from all the equations above, we obtain

$\displaystyle\begin{split}&\left(\mathbb{I}_{{\mathcal{H}}_{{\mathfrak{A}}}}\otimes\pi_{{\mathfrak{B}}}\alpha_{{\mathfrak{B}}}(A)\right)=\mathop{\mathrm{Ad}}\nolimits(w)\left(\mathbb{I}_{{\mathcal{H}}_{{\mathfrak{A}}}}\otimes\pi_{{\mathfrak{B}}}(A)\right)\\\
&=\mathbb{I}_{{\mathcal{H}}_{{\mathfrak{A}}}}\otimes\tau\pi_{{\mathfrak{B}}}(A)=\mathbb{I}_{{\mathcal{H}}_{{\mathfrak{A}}}}\otimes\mathop{\mathrm{Ad}}\nolimits(u_{\mathfrak{B}})\pi_{{\mathfrak{B}}}(A),\quad
A\in{\mathfrak{B}}.\end{split}$

This proves the Lemma.  $\square$

The following Lemma is sometimes useful.

######  Lemma 1.8.

Let ${\mathfrak{A}}$, ${\mathfrak{B}}$ be $C^{*}$-algebras. Let $\omega$ be a
pure state on ${\mathfrak{A}}\otimes{\mathfrak{B}}$. Then the GNS
representation $\pi_{\omega|_{{\mathfrak{A}}}}$ of $\omega|_{{\mathfrak{A}}}$
(the restriction of $\omega$ to
${\mathfrak{A}}\simeq{\mathfrak{A}}\otimes{\mathbb{C}}\mathbb{I}_{{\mathfrak{B}}}$
), and the restriction of the GNS representation $\pi_{\omega}$ of $\omega$ to
${\mathfrak{A}}$, $\pi_{\omega}|_{{\mathfrak{A}}}$ are quasi-equivalent.

  * Proof.

Let $({\mathcal{H}}_{\omega},\pi_{\omega},\Omega_{\omega})$ be a GNS
representation of $\omega$. First we show that
$\pi_{\omega}\left({\mathfrak{A}}\right)^{\prime\prime}$ is a factor. To see
this, note that

$\displaystyle\pi_{\omega}\left({\mathfrak{A}}\right)^{\prime\prime}\cap\pi_{\omega}\left({\mathfrak{A}}\right)^{\prime}\subset\pi_{\omega}\left({\mathfrak{B}}\right)^{\prime}\cap\pi_{\omega}\left({\mathfrak{A}}\right)^{\prime}=\pi_{\omega}\left({\mathfrak{A}}\otimes{\mathfrak{B}}\right)^{\prime}={\mathbb{C}}\mathbb{I},$
(1.4)

because $\omega$ is pure. (Recall Theorem 1.4.) Hence
$\pi_{\omega}\left({\mathfrak{A}}\right)^{\prime\prime}$ is a factor. Let

$\displaystyle{\mathcal{K}}:=\overline{\pi_{\omega}({\mathfrak{A}})\Omega_{\omega}},\quad
p:\;\;\text{projection onto
}{\mathcal{K}},\quad\pi:=\pi_{\omega}(\cdot)p|_{{\mathfrak{A}}}$

The projection $p$ belongs to
$\pi_{\omega}\left({\mathfrak{A}}\right)^{\prime}$, and one can check that
$({\mathcal{K}},\pi,\Omega_{\omega})$ is a GNS representation of
$\omega|_{{\mathfrak{A}}}$. Note that $\pi$ is a sub-representation of
$\pi_{\omega}|_{{\mathfrak{A}}}$. Because
$\pi_{\omega}\left({\mathfrak{A}}\right)^{\prime\prime}$ is a factor, Theorem
1.6 guarantees that $\pi$ and $\pi_{\omega}|_{{\mathfrak{A}}}$ are quasi-
equvalent.  $\square$

###  1.3 Quantum spin systems

Let $d\in{\mathbb{N}}$ and $\nu\in{\mathbb{N}}$ be fixed. Physically,
$\frac{d-1}{2}$ denotes the size of on-site spin (spin quantum number) and
$\nu$ denotes the spacial dimension. We denote by
${\mathfrak{S}}_{{\mathbb{Z}}^{\nu}}$, the set of all finite subsets of
${\mathbb{Z}}^{\nu}$. For each finite subset
$\Lambda\in{\mathfrak{S}}_{{\mathbb{Z}}^{\nu}}$, we associate a finite
dimensional $C^{*}$-algebra

$\displaystyle{\mathcal{A}}_{\Lambda}:=\bigotimes_{\Lambda}\mathop{\mathrm{M}}\nolimits_{d}.$

Here, $\mathop{\mathrm{M}}\nolimits_{d}$ is the algebra of $d\times d$
-matrices. If $\Lambda_{1},\Lambda_{2}\in{\mathfrak{S}}_{{\mathbb{Z}}^{\nu}}$
satisfy an inclusion relation $\Lambda_{1}\subset\Lambda_{2}$, then we may
embed ${\mathcal{A}}_{\Lambda_{1}}$ into ${\mathcal{A}}_{\Lambda_{2}}$ as

$\displaystyle\iota_{\Lambda_{1},\Lambda_{2}}:{\mathcal{A}}_{\Lambda_{1}}\to{\mathcal{A}}_{\Lambda_{2}},\quad\iota_{\Lambda_{1},\Lambda_{2}}(A):=A\otimes\mathbb{I}_{\Lambda_{2}\setminus\Lambda_{1}},\quad
A\in{\mathcal{A}}_{\Lambda_{1}}.$

Clearly, we have
$\iota_{\Lambda_{2},\Lambda_{3}}\iota_{\Lambda_{1},\Lambda_{2}}=\iota_{\Lambda_{1},\Lambda_{3}}$
if $\Lambda_{1}\subset\Lambda_{2}\subset\Lambda_{3}$. Then the
$\nu$-dimensional quantum spin system ${\mathcal{A}}_{{\mathbb{Z}}^{\nu}}$ is
the $C^{*}$-inductive limit of this inductive net
$(\iota_{\Lambda_{1},\Lambda_{2}},{\mathcal{A}}_{\Lambda})$. Namely,
${\mathcal{A}}_{{\mathbb{Z}}^{\nu}}$ is the unique $C^{*}$-algebra satisfying
the following conditions.

(i)

There is a net of $*$-homomorphisms
$\mu_{\Lambda}:{\mathcal{A}}_{\Lambda}\to{\mathcal{A}}_{{\mathbb{Z}}^{\nu}}$,
with $\Lambda\in{\mathfrak{S}}_{{\mathbb{Z}}^{\nu}}$, such that
$\mu_{\Lambda_{1}}=\mu_{\Lambda_{2}}\circ\iota_{\Lambda_{1},\Lambda_{2}}$ if
$\Lambda_{1}\subset\Lambda_{2}$, and

(ii)

If
$({\mathcal{B}},\\{\lambda_{\Lambda}\\}_{\Lambda\in{\mathfrak{S}}_{{\mathbb{Z}}^{\nu}}})$
is a system, where ${\mathcal{B}}$ is a $C^{*}$-algebra,
$\lambda_{\Lambda}:{\mathcal{A}}_{\Lambda}\to{\mathcal{B}}$ is a
$*$-homomorphism for each $\Lambda\in{\mathfrak{S}}_{{\mathbb{Z}}^{\nu}}$, and
where
$\lambda_{\Lambda_{1}}=\lambda_{\Lambda_{2}}\circ\iota_{\Lambda_{1},\Lambda_{2}}$
for any $\Lambda_{1},\Lambda_{2}\in{\mathfrak{S}}_{{\mathbb{Z}}^{\nu}}$ with
$\Lambda_{1}\subset\Lambda_{2}$, then there is one and only one
$*$-homomorphism $\lambda:{\mathcal{A}}_{{\mathbb{Z}}^{\nu}}\to{\mathcal{B}}$
such that $\lambda_{\Lambda}=\lambda\circ\mu_{\Lambda}$.

The subset
$\cup_{\Lambda\in{\mathfrak{S}}_{{\mathbb{Z}}^{\nu}}}\mu_{\Lambda}\left({\mathcal{A}}_{\Lambda}\right)$,
which is dense in ${\mathcal{A}}_{{\mathbb{Z}}^{\nu}}$, is called a local
algebra, and we denote it by ${\mathcal{A}}_{\rm loc}$. Because
$\mu_{\Lambda}$ is injective, we identify ${\mathcal{A}}_{\Lambda}$ and
$\mu_{\Lambda}\left({\mathcal{A}}_{\Lambda}\right)$ and omit the symbol
$\mu_{\Lambda}$ For each infinite subset $\Gamma$, we may define
${\mathcal{A}}_{\Gamma}$ in exactly the same manner. The $C^{*}$-algebra
${\mathcal{A}}_{\Gamma}$ can be regarded as a $C^{*}$-subalgebra of
${\mathcal{A}}_{{\mathbb{Z}}^{\nu}}$ that we will regard it so. We say an
element $A$ has a support in $\Gamma$ if it belongs to
${\mathcal{A}}_{\Gamma}$. If an automorphism $\alpha$ acts trivially on
${\mathcal{A}}_{\Gamma^{c}}$ for some $\Gamma\subset{\mathbb{Z}}^{\nu}$, we
say that $\alpha$ has a support in $\Gamma$. In the analysis of SPT-phases,
this support becomes important.

Corresponding to the space translation of ${\mathbb{Z}}^{\nu}$, we may define
a ${\mathbb{Z}}^{\nu}$-action
$\tau:{\mathbb{Z}}^{\nu}\to\mathop{\mathrm{Aut}}\nolimits\left({\mathcal{A}}_{{\mathbb{Z}}^{\nu}}\right)$
on ${\mathcal{A}}_{{\mathbb{Z}}^{\nu}}$ by

$\tau_{\bf y}\left(A^{(\bf x)}\right)=\left(A^{(\bf x+y)}\right)$

for $A\in\mathop{\mathrm{M}}\nolimits_{d}$ and ${\bf
x,y}\in{\mathbb{Z}}^{\nu}$. Here $A^{(\bf x)}$ indicates an element in
${\mathcal{A}}_{{\mathbb{Z}}^{\nu}}$ with $A$ in the $\bf x$-site of the
tensor product of ${\mathcal{A}}_{{\mathbb{Z}}^{\nu}}$ and the unit in any
other site. This $\tau$ is called the space translation on
${\mathcal{A}}_{{\mathbb{Z}}^{\nu}}$.

In operator algebraic framework, physical models are specified with a map
called interactions. An interaction $\Phi$ is a map

$\Phi:{\mathfrak{S}}_{{\mathbb{Z}}^{\nu}}\to{\mathcal{A}}_{\rm loc}$

satisfying

$\displaystyle\Phi(X)=\Phi(X)^{*}\in{\mathcal{A}}_{X}$

for all $X\in{\mathfrak{S}}_{{\mathbb{Z}}^{\nu}}$. Physically, this $\Phi(X)$
indicates an interaction term between spins inside of $X$. An interaction
$\Phi$ is said to be translationally invariant, if it satisfies the following
condition

$\tau_{\bf x}\left(\Phi(X)\right)=\Phi(X+{\bf x}),\quad
A\in{\mathfrak{S}}_{{\mathbb{Z}}^{\nu}},\quad{\bf x}\in{\mathbb{Z}}^{\nu}.$

The easiest type of interaction is an on-site interaction, satisfying

$\displaystyle\Phi(X)=0,\quad\text{if}\quad|X|\neq 1.$ (1.5)

It means, the only possibly non-zero interaction terms are of the form
$\Phi(\\{\bf x\\})$, with ${\bf x}\in{\mathbb{Z}}^{\nu}$. In this case, all
interaction terms commute with each other.

Physically, we are more interested in interactions which do have non-zero
interaction terms between different sites of ${\mathbb{Z}}^{\nu}$. For
example, let $\\{S_{j}\\}_{j=1,2,3}$ be generators of the irreducible
representation of ${\mathfrak{su}(2)}$ on ${\mathbb{C}}^{d}$. Then an
interaction of ${\mathcal{A}}_{{\mathbb{Z}}^{\nu}}$ given by

$\displaystyle\Phi(\\{x,x+1\\})=\sum_{j=1}^{3}\,S^{(x)}_{j}S^{(x+1)}_{j},$
(1.6)

is called antiferromagnetic Heisenberg chain and has been extensively studied.

Now, given an interaction, we would like to define a dynamics on
${\mathcal{A}}_{{\mathbb{Z}}^{\nu}}$ out of it. In order for that, we need to
assume that $\Phi$ is “suitably local”. The simplest one among such good
conditions is uniformly bounded and finite range. An interaction is of finite
range if there exists an $m\in{\mathbb{N}}$ such that $\Phi(X)=0$ for $X$ with
a diameter larger than $m$. It is uniformly bounded if it satisfies

$\displaystyle\sup_{X\in{\mathfrak{S}}_{{\mathbb{Z}}^{\nu}}}\left\|\Phi(X)\right\|<\infty.$

We can extensively relax these conditions. See [NSY].

Given a good interaction, we may define a $C^{*}$-dynamics, i.e., strongly
continuous one parameter group of automorphisms on
${\mathcal{A}}_{{\mathbb{Z}}^{\nu}}$. For an interaction $\Phi$ and a finite
set $\Lambda\subset{{\mathbb{Z}}^{\nu}}$, we define the local Hamiltonian by

$\left(H_{\Phi}\right)_{\Lambda}:=\sum_{X\subset{\Lambda}}\Phi(X).$ (1.7)

For a uniformly bounded finite range interaction $\Phi$, the limit

$\alpha_{t}^{\Phi}(A)=\lim_{\Lambda\to{\mathbb{Z}}^{\nu}}e^{it(H_{\Phi})_{\Lambda}}Ae^{-it(H_{\Phi})_{\Lambda}},\quad
t\in{\mathbb{R}},\quad A\in{\mathcal{A}}_{{\mathbb{Z}}^{\nu}}$ (1.8)

exists and defines a dynamics $\alpha^{\Phi}$ on
${\mathcal{A}}_{{\mathbb{Z}}^{\nu}}$. The reason for us to consider the
dynamics $\alpha^{\Phi}$ instead of Hamiltonians, is because there is no
mathematically meaningful limit of local Hamiltonians
$\left(H_{\Phi}\right)_{\Lambda}$ as $\Lambda\to{\mathbb{Z}}^{\nu}$ in (1.7).
But the limit (1.8) makes sense. For the same reason, a ground state is
defined in terms of the dynamics $\alpha_{\Phi}$.

######  Definition 1.9.

Let $\delta_{\Phi}$ be the generator of $\alpha^{\Phi}$. A state $\omega$ on
${\mathcal{A}}_{{\mathbb{Z}}^{\nu}}$ is called an $\alpha^{\Phi}$-ground state
if the inequality

$-i\omega\left(A^{*}{\delta_{\Phi}}\left(A\right)\right)\geq 0$

holds for any element $A$ in the domain ${\mathcal{D}}({\delta_{\Phi}})$ of
${\delta_{\Phi}}$.

We occasionally say ground state of $\Phi$ instead of $\alpha^{\Phi}$-ground
state.

Let $({\mathcal{H}}_{\varphi},\pi_{\varphi},\Omega_{\varphi})$ be the GNS
triple of a $\alpha^{\Phi}$-ground state $\varphi$. Then there exists a unique
positive operator $H_{\varphi,\Phi}$ on ${\mathcal{H}}_{\varphi}$ such that
$e^{itH_{\varphi,\Phi}}\pi_{\varphi}(A)\Omega_{\varphi}=\pi_{\varphi}(\tau_{t}^{\Phi}(A))\Omega_{\varphi}$,
for all $A\in{\mathcal{A}}$ and $t\in\mathbb{R}$. We call this
$H_{\varphi,\Phi}$ the bulk Hamiltonian associated with $\varphi$. Note that
$\Omega_{\varphi}$ is an eigenvector of $H_{\varphi,\Phi}$ with eigenvalue
$0$. See [BR2] for the general theory.

Let us consider the corresponding condition for a finite quantum system
$\mathop{\mathrm{M}}\nolimits_{n}$ with dynamics given by a Hamiltonian $H$

$\displaystyle\alpha_{t}(A)=e^{itH}Ae^{-itH},\quad t\in{\mathbb{R}},\quad
A\in{\mathop{\mathrm{M}}\nolimits_{n}}.$ (1.9)

Let $P$ be the spectral projection of $H$ corresponding to the lowest
eigenvalue $E_{0}$. Recall that a state $\omega$ on
$\mathop{\mathrm{M}}\nolimits_{n}$ is given by a density matrix $\rho$ with
the formula $\omega(A)=\mathop{\mathrm{Tr}}\nolimits\rho A$. Let $s(\omega)$
be the range projection of this $\rho$. Then one can check that $\omega$ is an
$\alpha$-ground state if and only if the support $s(\omega)$ of $\omega$
satisfies $s(\omega)\leq P$. Recall that this is the very definition of ground
state, in finite quantum mechanics. In fact note that the generator $\delta$
of $\alpha$ in (1.9) is $\delta(A)=i[H,A]$. If $s(\omega)\leq P$, then we have

$\displaystyle-i\omega\left(A^{*}{\delta}\left(A\right)\right)=\omega\left(A^{*}(H-E_{0})A\right)\geq
0,\quad A\in\mathop{\mathrm{M}}\nolimits_{n}.$

Conversely, suppose that $-i\omega\left(A^{*}{\delta}\left(A\right)\right)\geq
0$ for all $A\in\mathop{\mathrm{M}}\nolimits_{n}$. For unit eigenvectors
$\xi,\eta$ of $H$ with $H\xi=E_{0}\xi$, $H\eta=E\eta$, for $E>E_{0}$, set
$A\in\mathop{\mathrm{M}}\nolimits_{n}$ to be a matrix satisfying
$A\zeta=\left\langle\eta,\zeta\right\rangle\xi$ for any
$\zeta\in{\mathbb{C}}^{n}$. Substituting this $A$, we get

$\displaystyle
0\leq-i\omega\left(A^{*}{\delta}\left(A\right)\right)=\left(E_{0}-E\right)\left\langle\eta,\rho\eta\right\rangle$

Because $E_{0}-E<0$, this means that
$\left\langle\eta,\rho\eta\right\rangle=0$ for any such $\eta$. Hence we
conclude that $\rho P=\rho$, namely, $s(\omega)\leq P$.

Therefore, our definition in operator algebraic framework can be regarded as a
generalization of this physical definition of a ground state.

Note, in general, there can be many states satisfying the condition in
Definition 1.9. Namely, the ground state does not need to be unique. However,
what we are interested in this lecture is the situation that it is unique.
Furthermore, we require the existence of gap.

######  Definition 1.10.

Suppose that there is a unique $\alpha^{\Phi}$-ground state $\omega_{\Phi}$.
Then we say $\Phi$ has a unique gapped ground state in the bulk if there
exists a $\gamma>0$ such that

$\displaystyle-i\omega_{\Phi}\left(A^{*}{\delta_{\Phi}}\left(A\right)\right)\geq\gamma\omega_{\Phi}(A^{*}A),\quad\text{for
all }A\in{\mathcal{D}}({\delta_{\Phi}})\;\text{with}\;\omega_{\Phi}(A)=0.$
(1.10)

Let us consider the corresponding condition for a finite system
$\mathop{\mathrm{M}}\nolimits_{n}$ with dynamics (1.9). Then the above
condition corresponds to the situation that “the lowest eigenvalue of $H$ is
non-degenerated and the difference between the lowest eigenvalue and the
second lowest eigenvalue is at least $\gamma$”.

Let $\Lambda_{N}:=[-N,N]^{\nu}\cap{\mathbb{Z}}^{\nu}$. Let $\omega_{N,\Phi}$
be a ground state of the local Hamiltonian
$\left(H_{\Phi}\right)_{\Lambda_{N}}$ on ${\mathcal{A}}_{\Lambda_{N}}$. Then,
any wk$*$-accumulation point of $\omega_{N,\Phi}$ for $N\to\infty$ is an
$\alpha^{\Phi}$-ground state. For this reason, there is always an
$\alpha^{\Phi}$-ground state. Let
$\sigma\left(\left(H_{\Phi}\right)_{\Lambda_{N}}\right)$ be the spectrum of
$\left(H_{\Phi}\right)_{\Lambda_{N}}$, and $E_{N}$ its infimum. Suppose that
there is a $\gamma>0$ such that

$\sigma\left(\left(H_{\Phi}\right)_{\Lambda_{N}}\right)\setminus\\{E_{N}\\}\subset[E_{N}+\gamma,\infty)$

holds for any $N\in{\mathbb{N}}$. If furthermore the $\alpha^{\Phi}$-ground
state $\omega_{\Phi}$ is unique, then we have

$\omega_{\Phi}={\mathrm{wk}}*-\lim_{N\to\infty}\omega_{N,\Phi}$

and it is a unique gapped ground state with gap at least $\gamma$.

One remarkable property of unique gapped ground state is the exponential decay
of correlation functions.

######  Theorem 1.11 ([HK] [NS06] [NS09]).

Let $\Phi$ be a uniformly bounded finite range interaction with a unique
gapped ground state $\omega_{\Phi}$. Then the correlation functions of
$\omega_{\Phi}$ decay exponentially fast : there exists $\mu>0$ and a constant
$C>0$ such that for all $A\in{\mathcal{A}}_{X}$, $B\in{\mathcal{A}}_{Y}$, with
finite $X,Y\subset{\mathbb{Z}}^{\nu}$

$\left|\omega_{\Phi}\left(AB\right)-\omega_{\Phi}(A)\omega_{\Phi}(B)\right|\leq
C\left\|A\right\|\left\|B\right\||X|{e^{-\mu d(X,Y)}}.$

This means $\omega_{\Phi}$ is “almost like a product state”.

In the theory of SPT-phases, we are interested in interactions satisfying some
given symmetry. In this lecture, we consider on-site finite group symmetry in
one and two-dimensional quantum spin systems and reflection symmetry in one
dimensional quantum systems.

On-site finite group symmetry is given as follows. We fix a finite group $G$
and a (projective) unitary representation $U$ of $G$ on ${\mathbb{C}}^{d}$.
Then there is a unique automorphism $\beta_{g}$ satisfying

$\displaystyle\beta_{g}(A)=\left(\bigotimes_{x\in\Lambda}U(g)\right)A\left(\bigotimes_{x\in\Lambda}U(g)^{*}\right),\;g\in
G,\;\Lambda\in{\mathfrak{S}}_{{\mathbb{Z}}^{\nu}},\;A\in{\mathcal{A}}_{\Lambda}.$

Clearly, this gives an action of $G$ on ${\mathcal{A}}_{{\mathbb{Z}}^{\nu}}$,
i.e., $\beta_{g}\beta_{h}=\beta_{gh}$ for $g,h\in G$. We call this action of
$G$, an on-site symmetry given by $G$ and $U$. Likewise, for each subset
$\Gamma\subset{\mathbb{Z}}^{\nu}$, we may define an action
$\beta_{g}^{\Gamma}$. We say an interaction $\Phi$ is $\beta$-invariant if
$\beta_{g}(\Phi(X))=\Phi(X)$ for all $X\in{\mathfrak{S}}_{{\mathbb{Z}}^{\nu}}$
and $g\in G$. For a ground state $\varphi$ of a $\beta$-invariant interaction
$\Phi$, one can check that $\varphi\circ\beta_{g}$ is also a ground state of
$\Phi$. Therefore, if a $\beta$-invariant interaction $\Phi$ has a unique
ground state $\omega_{\Phi}$, the ground state is $\beta$-invariant,
$\omega_{\Phi}\circ\beta_{g}=\omega_{\Phi}$.

For example, let $U$ be a projective representation of
$G={\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}$ on ${\mathbb{C}}^{d}$, given by

$\displaystyle U\left([1],[0]\right)=e^{i\pi S_{1}},\quad
U\left([0],[1]\right)=e^{i\pi S_{2}},\quad U\left([1],[1]\right)=e^{i\pi
S_{3}}$

Here, $\\{S_{j}\\}_{j=1,2,3}$ are generators of the irreducible representation
of ${\mathfrak{su}(2)}$ on ${\mathbb{C}}^{d}$. If $d$ is odd, this gives a
genuine representation while if $d$ is even, it is a projective
representation. One can check that antiferromagnetic Heisenberg chain (1.6) is
invariant under this symmetry.

Reflection symmetry in one dimensional quantum systems is given as follows.
Let $R:{\mathbb{Z}}\to{\mathbb{Z}}$ be the reflection : $R(i):=-i-1$,
$i\in{\mathbb{Z}}$, and
$\Theta_{R}\in\mathop{\mathrm{Aut}}\nolimits({\mathcal{A}}_{{\mathbb{Z}}})$ an
automorphism corresponding to $R$. More precisely, $\Theta_{R}$ is defined as
follows. By $Q^{(j)}$, $j\in{\mathbb{Z}}$, we denote the element of
${\mathcal{A}}_{{\mathbb{Z}}}$ with $Q\in\mathop{\mathrm{M}}\nolimits_{d}$ in
the $j$-th component of the tensor product of ${\mathcal{A}}_{{\mathbb{Z}}}$
and the unit in any other component. The reflection $\Theta_{R}$ is the unique
$*$-automorphism on ${\mathcal{A}}_{{\mathbb{Z}}}$ which satisfies

$\displaystyle\Theta_{R}\left(Q^{(j)}\right)=Q^{(-j-1)},\quad\text{for
all}\;Q\in\mathop{\mathrm{M}}\nolimits_{d}\;\text{and}\;j\in{\mathbb{Z}}.$

An interaction $\Phi$ is reflection invariant if
$\Theta_{R}(\Phi(X))=\Phi(R(X))$ for all $X\in{\mathfrak{S}}_{\mathbb{Z}}$.
One can check that antiferromagnetic Heisenberg chain (1.6) is invariant under
this symmetry.

###  1.4 SPT-phases

We denote the set of all uniformly bounded finite range interactions with
unique gapped ground state by ${\mathcal{P}}_{U.G.}$. First, we introduce an
equivalence relation $\sim$ in this set ${\mathcal{P}}_{U.G.}$ as follows.:

######  Definition 1.12 (Definition of $\sim$).

Two interactions $\Phi_{0},\Phi_{1}\in{\mathcal{P}}_{U.G.}$ are equivalent
(denoted as $\Phi_{0}\sim\Phi_{1}$) if there is a path ${\mathcal{P}}_{U.G.}$
in $\Phi:[0,1]\to{\mathcal{P}}_{U.G.}$ satisfying the following conditions.

(i)

The path connects given interactions, i.e., $\Phi(0)=\Phi_{0}$ and
$\Phi(1)=\Phi_{1}$.

(ii)

For each $X\in{\mathfrak{S}}_{{\mathbb{Z}}^{\nu}}$, the matrix-valued function
$[0,1]\ni s\mapsto\Phi_{s}(X)\in{\mathcal{A}}_{X}$ is smooth,

(iii)

The gap is uniformly bounded from below by some $\gamma>0$ along the path.
Namely, for each $s\in[0,1]$, there is a $\gamma_{s}>0$ satisfying (1.10) for
$\Phi=\Phi(s)$, and $\inf_{s\in[0,1]}\gamma_{s}>0$.

(iv)

The path of expectation values $[0,1]\ni s\mapsto\omega_{s}(A)\in{\mathbb{C}}$
of sub-exponentially localized elements
$A\in{\mathcal{A}}_{{\mathbb{Z}}^{\nu}}$ with respect to the ground state
$\omega_{s}$ is regular (in the sense that it is differentiable and the
derivative is not too large compared to some norm of $A$) with respect to
$s\in[0,1]$.

For more precise description of the conditions on the path, see subsection 1.6
Assumption 1.21 Assumption 1.25. A part of these assumptions are for technical
reason, namely we would like to apply Theorem 1.26. Physically, this condition
means that there is no phase transition along the path.

Among many interactions in ${\mathcal{P}}_{U.G.}$, what corresponds to SPT is
the ones with short-range entanglement. Recall the definition of on-site
interaction (1.5). Fix some on-site interaction $\Phi_{0}$ which belongs to
${\mathcal{P}}_{U.G.}$. Such an interaction can be constructed as follows. For
each $x\in{\mathbb{Z}}^{\nu}$, fix any unit vector
$\xi_{x}\in{\mathbb{C}}^{d}$. Let $p_{x}$ be an orthogonal projection on
${\mathbb{C}}^{d}$ onto the subspace ${\mathbb{C}}\xi_{x}$. Set $\Phi_{0}$ as

$\displaystyle\Phi_{0}\left(\\{x\\}\right)=\mathbb{I}-p_{x},\quad
x\in{\mathbb{Z}}^{\nu},$

and $\Phi_{0}(X)=0$ for any other $X\in{\mathfrak{S}}_{{\mathbb{Z}}^{\nu}}$.
Then $\Phi_{0}\in{\mathcal{P}}_{U.G.}$ and its unique ground state is of the
form

$\displaystyle\omega_{\Phi_{0}}=\bigotimes_{x\in{\mathbb{Z}}^{\nu}}\rho_{\xi_{x}},$

where $\rho_{\xi_{x}}$ is a vector state on $\mathop{\mathrm{M}}\nolimits_{d}$
given by $\rho_{\xi_{x}}(A)=\left\langle\xi_{x},A\xi_{x}\right\rangle$. Note
that any on-site interaction can be smoothly deformed into an on-site
interaction of this form inside of ${\mathcal{P}}_{U.G.}$, just by smoothly
flattering the spectrum of each $\Phi(\\{x\\})$. Furthermore, any on-site
interactions of this form can be deformed into each other by rotating
$\xi_{x}$ smoothly. Therefore, the following set does not depend on the choice
of $\Phi_{0}$:

$\displaystyle{\mathcal{P}}_{U.G.}^{0}:=\left\\{\Phi\in{\mathcal{P}}_{U.G.}\mid\Phi\sim\Phi_{0}.\right\\}.$

As we see in the next subsection, this set physically represents interactions
whose unique gapped ground state is short-range entangled. This excludes
exotic states with “topological order” (or long-range entanglement) such as
the ground states of the Kitaev toric code model [Ki]. In general, a pure
state with a non-trivial super selection sector has long-range entanglement
[NO].

Now we introduce the on-site symmetry $\beta$ to the game. We are interested
in the set

$\displaystyle{\mathcal{P}}_{U.G.{{\beta}}}^{0}:=\left\\{\Phi\in{\mathcal{P}}_{U.G.}^{0}\mid\beta\text{-invariant}\right\\}.$
(1.11)

We introduce an equivalence relation $\sim_{\beta}$ on this set
${\mathcal{P}}_{U.G.{{\beta}}}^{0}$.

######  Definition 1.13 (Definition of $\sim_{\beta}$).

Two interactions $\Phi_{0},\Phi_{1}\in{\mathcal{P}}_{U.G.{{\beta}}}^{0}$ are
equivalent (denoted as $\Phi_{0}\sim_{\beta}\Phi_{1}$) if there is a path in
$\Phi:[0,1]\to{\mathcal{P}}_{U.G.{{\beta}}}^{0}$ satisfying the conditions
(i)-(iv) of Definition 1.12.

The equivalence class of ${\mathcal{P}}_{U.G.{{\beta}}}^{0}$ with this
classification are the SPT-phases.

######  Definition 1.14 (Symmetry Protected Topological (SPT) phases).

Each equivalence class of ${{\mathcal{P}}}_{U.G.{\beta}}^{0}$ with respect to
$\sim_{\beta}$ is called Symmetry Protected Topological (SPT) phases.

This notion of SPT-phases was introduced by Gu and Wen in [GW]. This is the
object we would like to classify. Analogously, we can define SPT phases
${\mathcal{P}}_{U.G.{{\Theta_{R}}}}^{0}$ for the reflection symmetry
$\Theta_{R}$.

###  1.5 Split property

The split property, studied extensively in operator algebra has turned out to
be quite important in analysis of SPT phases. In this section we introduce the
notion.

Let us consider an inclusion of two von Neumann algebras
${\mathcal{N}}\subset{\mathcal{M}}$. We say the inclusion is split, if there
is a type $I$ factor ${\mathcal{R}}$ such that
${\mathcal{N}}\subset{\mathcal{R}}\subset{\mathcal{M}}$. The inclusion which
shows up in this lecture is the simplest case, that is, ${\mathcal{N}}$ itself
is a type $I$ factor. In particular, we will use the following terminology in
this lecture.

######  Definition 1.15.

Let $\Gamma_{1},\Gamma_{2}\subset{\mathbb{Z}}^{\nu}$ be non-empty disjoint
subsets of ${\mathbb{Z}}^{\nu}$. A pure state $\omega$ on
${\mathcal{A}}_{\Gamma_{1}\cup\Gamma_{2}}$ satisfies the split property with
respect to $\Gamma_{1}$-$\Gamma_{2}$, if the von Neumann algebra
$\pi_{\omega}({\mathcal{A}}_{\Gamma_{1}})^{\prime\prime}$ is a type $I$
factor.

It is beneficial to heed in mind the following Theorems about type $I$
factors.

######  Theorem 1.16 (Theorem 1.31 V[T]).

Let ${\mathcal{M}}$ be a type $I$ factor acting on a Hilbert space
${\mathcal{H}}$. Then there are Hilbert spaces ${\mathcal{H}}_{1}$,
${\mathcal{H}}_{2}$ and a unitary
$U:{\mathcal{H}}\to{\mathcal{H}}_{1}\otimes{\mathcal{H}}_{2}$ such that

$\displaystyle\mathop{\mathrm{Ad}}\nolimits(U)({\mathcal{M}})={\mathcal{B}}({\mathcal{H}}_{1})\otimes{\mathbb{C}}\mathbb{I}_{{\mathcal{H}}_{2}}.$

######  Theorem 1.17 (Theorem 2.30 V[T]).

Let ${\mathcal{M}}_{1}$, ${\mathcal{M}}_{2}$ be von Neumann algebras acting on
Hilbert spaces ${\mathcal{H}}_{1}$, ${\mathcal{H}}_{2}$. Let ${\mathcal{M}}$
be a von Neumann algebra acting on ${\mathcal{H}}_{1}\otimes{\mathcal{H}}_{2}$
generated by the algebraic tensor product
${\mathcal{M}}_{1}{\odot}{\mathcal{M}}_{2}$. Then ${\mathcal{M}}$ is of type
$I$ if and only if ${\mathcal{M}}_{1}$ and ${\mathcal{M}}_{2}$ are both of
type $I$.

Using such properties, we can show the following.

######  Proposition 1.18.

For a pure state $\omega$ on ${\mathcal{A}}_{\Gamma_{1}\cup\Gamma_{2}}$,
following conditions are equivalent.

(i)

The state $\omega$ satisfies the split property with respect to
$\Gamma_{1}$-$\Gamma_{2}$

(ii)

There are states $\omega_{1}$, $\omega_{2}$ on ${\mathcal{A}}_{\Gamma_{1}}$,
${\mathcal{A}}_{\Gamma_{2}}$ such that $\omega$ and
$\omega_{1}\otimes\omega_{2}$ are quasi-equivalent.

(iii)

There are irreducible representations $({\mathcal{H}}_{1},\pi_{1})$,
$({\mathcal{H}}_{2},\pi_{2})$ of ${\mathcal{A}}_{\Gamma_{1}}$,
${\mathcal{A}}_{\Gamma_{2}}$ and a unit vector
$\Omega\in{\mathcal{H}}_{1}\otimes{\mathcal{H}}_{2}$ such that
$({\mathcal{H}}_{1}\otimes{\mathcal{H}}_{2},\pi_{1}\otimes\pi_{2},\Omega)$ is
a GNS-triple of $\omega$.

Recalling the “physical” interpretation that “two states are quasi-equivalent
means they are macroscopically the same”, the above proposition tells us that
$\omega$ satisfies the split property with respect to
$\Gamma_{1}$-$\Gamma_{2}$ if it is macroscopically product state with respect
to the $\Gamma_{1}$-$\Gamma_{2}$ cut.

For a pure state with split property, the following holds.

######  Lemma 1.19.

[O7] Let $\Gamma_{1},\Gamma_{2}\subset{\mathbb{Z}}^{\nu}$ be non-empty
disjoint subsets of ${\mathbb{Z}}^{\nu}$. Let $\omega$ be a pure state on
${{\mathcal{A}}_{\Gamma_{1}}}\otimes{{\mathcal{A}}_{\Gamma_{2}}}$ and
$\varphi_{{{\mathcal{A}}_{\Gamma_{1}}}}$,
$\varphi_{{{\mathcal{A}}_{\Gamma_{2}}}}$ states on
${{\mathcal{A}}_{\Gamma_{1}}}$, ${{\mathcal{A}}_{\Gamma_{2}}}$ respectively.
Assume that $\omega$ is quasi-equivalent to
$\varphi_{{{\mathcal{A}}_{\Gamma_{1}}}}\otimes\varphi_{{{\mathcal{A}}_{\Gamma_{2}}}}$.
Then for any pure states $\psi_{{{\mathcal{A}}_{\Gamma_{1}}}}$,
$\psi_{{{\mathcal{A}}_{\Gamma_{2}}}}$ on ${{\mathcal{A}}_{\Gamma_{1}}}$,
${{\mathcal{A}}_{\Gamma_{2}}}$, there are automorphisms
$\gamma_{{{\mathcal{A}}_{\Gamma_{1}}}}\in\mathop{\mathrm{Aut}}\nolimits\left({{\mathcal{A}}_{\Gamma_{1}}}\right)$,
$\gamma_{{{\mathcal{A}}_{\Gamma_{2}}}}\in\mathop{\mathrm{Aut}}\nolimits\left({{\mathcal{A}}_{\Gamma_{2}}}\right)$
and a unitary
$u\in{\mathcal{U}}\left({{\mathcal{A}}_{\Gamma_{1}}}\otimes{{\mathcal{A}}_{\Gamma_{2}}}\right)$
such that

$\displaystyle\omega=\left(\left(\psi_{{{\mathcal{A}}_{\Gamma_{1}}}}\circ\gamma_{{{\mathcal{A}}_{\Gamma_{1}}}}\right)\otimes\left(\psi_{{{\mathcal{A}}_{\Gamma_{2}}}}\circ\gamma_{{{\mathcal{A}}_{\Gamma_{2}}}}\right)\right)\circ\mathop{\mathrm{Ad}}\nolimits(u).$
(1.12)

If $\psi_{{{\mathcal{A}}_{\Gamma_{1}}}}$ and
$\varphi_{{{\mathcal{A}}_{\Gamma_{1}}}}$ are quasi-equivalent, then we may set
$\gamma_{{{\mathcal{A}}_{\Gamma_{1}}}}=\mathop{\mathrm{id}}\nolimits_{{{\mathcal{A}}_{\Gamma_{1}}}}$.

###  1.6 Automorphic equivalence

For the investigation of the equivalence relation $\sim$, automorphic
equivalence on smooth path of interactions plays an important role.
Automorphic equivalence started as Hasting’s adiabatic Lemma [H2][HW]. After
seminal mathematical development in [BMNS] [NSY], it is now a strong
mathematical machinery to analyze gapped state phases. It basically says the
following : given a smooth path of gapped local Hamiltonians, there is a
concrete smooth path of automorphisms interpolating ground state spaces.
Original Hastings’s adiabatic Lemma was about finite systems. In the framework
given by [BMNS] [NSY], the existence of gap for finite systems is assumed as
well, and the thermodynamic limit taken afterwords. An infinite version,
assuming the gap in the infinite system was derived in [MO].

In this subsection, we review Hasting’s original adiabatic Lemma, and
introduce its infinite dimensional version. Let

$H:[0,1]\ni s\to H(s)\in\mathop{\mathrm{M}}\nolimits_{n}$

be a smooth path of self-adjoint matrices of size $n$. In physical settings
this $H(s)$ is a Hamiltonian. We consider the situation that there is a gap
between the lowest eigenvalue of $H(s)$ and the rest of the spectrum, which is
uniform in $s$. What we are interested in is the lowest eigenvalue eigenspace
of the Hamiltonian i.e, the ground state space. For each parameter
$s\in[0,1]$, let $E(s)$ be the lowest eigenvalue of $H(s)$ and $P(s)$ be the
spectral projection of $H(s)$ corresponding to the eigenvalue $E(s)$. As we
saw in subsection 1.3, a ground state of $H(s)$ is a positive linear
functional on $\mathop{\mathrm{M}}\nolimits_{n}$ with support under $P(s)$. We
assume that there is a constant $\gamma>0$ such that
$\sigma(H(s))\setminus\\{E(s)\\}\subset[E(s)+\gamma,\infty)$ for any
$s\in[0,1]$. Here $\sigma(H(s))$ indicates the spectrum of $H(s)$. Namely, we
require that there is no cross-over of the ground state energy along the path.
The Hastings adiabatic Lemma is that there is a path of unitaries following an
explicit differential equation, which maps $P(0)$, to $P(s)$.

######  Theorem 1.20 (Hastings[H2]).

In the setting given above, there is a smooth path of unitaries
$U:[0,1]\to\mathop{\mathrm{M}}\nolimits_{n}$ such that

$\displaystyle\begin{split}&P(s)=U(s)P(0)U(s)^{*},\quad s\in[0,1],\\\
&\frac{d}{ds}U(s)=iD(s)U(s),\quad U(0)=\mathbb{I},\\\
&D(s):=\int_{-\infty}^{\infty}dtW_{\gamma}(t)e^{itH(s)}H^{\prime}(s)e^{-itH(s)}.\end{split}$

Here, $W_{\gamma}$ is a $L^{1}$-function on ${\mathbb{R}}$ with Fourier
transformation satisfying

$\displaystyle\hat{W}_{\gamma}(k)=-\frac{i}{\sqrt{2\pi}k},\quad
k\notin(-\gamma,\gamma).$ (1.13)

This theorem says that there is a smooth path of unitaries which maps ground
state space $P(0)$ to $P(s)$. An important point for application is that the
path of unitaries is given by quite a concrete differential equation. For the
choice of $W_{\gamma}$, there is some freedom. The proof goes as follows. Let
$s_{0}\in[0,1]$ and $\Gamma$ be a contour on ${\mathbb{C}}$ having $E(s)$
inside and and $\sigma(H(s))\setminus\\{E(s)\\}$ outside for $s$ near $s_{0}$.
Then our $P(s)$ can be represented as

$\displaystyle P(s):=\frac{1}{2\pi i}\int_{\Gamma}dz(z-H(s))^{-1}$

near $s_{0}$. Now we derive a differential equation satisfied by $P(s)$. In
order to do so, we first simply differentiate the above representation of
$P(s)$. We then obtain

$\displaystyle\frac{d}{ds}P(s)=\frac{1}{2\pi
i}\int_{\Gamma}dz(z-H(s))^{-1}H^{\prime}(s)(z-H(s))^{-1}.$ (1.14)

Now we make a very simple but key observation about a path of orthogonal
projetions. That is

$\displaystyle P(s)P^{\prime}(s)P(s)=(1-P(s))P^{\prime}(s)(1-P(s))=0.$ (1.15)

This can be obtained by differentiating $P(s)^{2}=P(s)$,

$\displaystyle P(S)P^{\prime}(s)+P^{\prime}(s)P(s)=P^{\prime}(s).$

Multiplying $P(s)$ (resp. $1-P(s)$) from left and right, we obtain (1.15).
Therefore, we can rewrite (1.14), picking up off-diagonal terms:

$\displaystyle\begin{split}&\frac{d}{ds}P(s)\\\ &=\frac{1}{2\pi
i}\int_{\Gamma}dz\left(\begin{gathered}P(s)(z-H(s))^{-1}H^{\prime}(s)(z-H(s))^{-1}\left(1-P(s)\right)\\\
+(1-P(s))(z-H(s))^{-1}H^{\prime}(s)(z-H(s))^{-1}P(s)\end{gathered}\right)\\\
&=P(s)H^{\prime}(s)\frac{1}{2\pi
i}\left(\int_{\Gamma}dz(z-E(s))^{-1}(z-H(s))^{-1}\left(1-P(s)\right)\right)\\\
&\quad+\frac{1}{2\pi
i}\left(\int_{\Gamma}dz(1-P(s))(z-E(s))^{-1}(z-H(s))^{-1}\right)H^{\prime}(s)P(s)\\\
&=P(s)H^{\prime}(s)(E(s)-H(s))^{-1}\left(1-P(s)\right)+(1-P(s))(E(s)-H(s))^{-1}H^{\prime}(s)P(s).\end{split}$

In the last line we used the fact that $\Gamma$ has $E(s)$ inside and and
$\sigma(H(s))\setminus\\{E(s)\\}$ outside. On the other hand, using the
property of the Fourier transform (1.13), the last term can be written in
terms of $D(s)$ as

$\displaystyle
P(s)H^{\prime}(s)(E(s)-H(s))^{-1}\left(1-P(s)\right)+(1-P(s))(E(s)-H(s))^{-1}H^{\prime}(s)P(s)$
$\displaystyle=i\left[D(s),P(s)\right].$

Hence $P(s)$ satisfies the differential equation

$\displaystyle\frac{d}{ds}P(s)=i\left[D(s),P(s)\right].$

This proves the Theorem.

Now we would like to apply this Theorem to the Heisenberg dynamics given by
time-dependent interactions. First, let us introduce paths of interactions.

######  Assumption 1.21.

Let $\Phi(\cdot~{};s):\mathfrak{S}_{\mathbb{Z}^{\nu}}\to\mathcal{A}_{\rm loc}$
be a family of uniformly bounded, finite range interactions parameterized by
$s\in[0,1]$. We assume the following:

(i)

For each $X\in{\mathfrak{S}}_{{\mathbb{Z}}^{\nu}}$, the map $[0,1]\ni
s\to\Phi(X;s)\in{\mathcal{A}}_{X}$ is continuous and piecewise $C^{1}$. We
denote by $\dot{\Phi}(X;s)$ the corresponding derivatives. The interaction
obtained by differentiation is denoted by $\dot{\Phi}(s)$, for each
$s\in[0,1]$.

(ii)

There is a number $R\in\mathbb{N}$ such that
$X\in{\mathfrak{S}}_{{\mathbb{Z}}^{\nu}}$ and
$\mathop{\mathrm{diam}}\nolimits{X}\geq R$ imply $\Phi(X;s)=0$, for all
$s\in[0,1]$.

(iii)

Interactions are bounded as follows

$\displaystyle\sup_{s\in[0,1]}\sup_{X\in{\mathfrak{S}}_{{\mathbb{Z}}^{\nu}}}\left(\left\|\Phi\left(X;s\right)\right\|+|X|\left\|\dot{\Phi}\left(X;s\right)\right\|\right)<\infty.$

(iv)

Setting

$\displaystyle
b(\varepsilon):=\sup_{Z\in{\mathfrak{S}}_{{\mathbb{Z}}^{\nu}}}\sup_{s,s_{0}\in[0,1],0<|s-s_{0}|<\varepsilon}\left\|\frac{\Phi(Z;s)-\Phi(Z;s_{0})}{s-s_{0}}-\dot{\Phi}(Z;s_{0})\right\|$

for each $\varepsilon>0$, we have $\lim_{\varepsilon\to 0}b(\varepsilon)=0$.

We would like to apply Theorem 1.20 to path of local Hamiltonians satisfying
Assumption 1.21. For each $s\in[0,1]$ we have local Hamiltonians

$\displaystyle H_{\Lambda,\Phi}(s):=\sum_{Z\subset\Lambda}\Phi(Z,s),\quad
s\in[0,1],\quad\Lambda\in{\mathfrak{S}}_{{\mathbb{Z}}^{2}}.$

In addition to Assumption 1.21, we need to require a uniform gap in the
spectrum of these local Hamiltonians.

######  Assumption 1.22.

(i)

There is $\gamma>0$ such that

$\displaystyle\sigma\left(H_{\Lambda,\Phi}(s)\right)\setminus\left\\{\inf\sigma\left(H_{\Lambda,\Phi}(s)\right)\right\\}\subset\left[\inf\sigma\left(H_{\Lambda,\Phi}(s)\right)+\gamma,\infty\right),\quad\Lambda\in{\mathfrak{S}},\quad
s\in[0,1].$

(ii)

For each $s\in[0,1]$, there exists a unique $\tau_{\Phi(s)}$-ground state
$\omega_{\Phi(s)}$.

Applying Theorem 1.20 to the path of local Hamiltonians $H_{\Lambda,\Phi}(t)$,
we obtain a path of unitaries $V_{\Lambda}(s)$ in ${\mathcal{A}}_{\Lambda}$
given by a differential equation

$\displaystyle\frac{d}{ds}V_{\Lambda}(s)=i\left(\int_{-\infty}^{\infty}dtW_{\gamma}(t)e^{itH_{\Lambda,\Phi}(s)}H_{\Lambda,\dot{\Phi}}(s)e^{-itH_{\Lambda,\Phi}(s)}\right)V_{\Lambda}(s),$
(1.16)

with $V(0)=\mathbb{I}$, interpolating ground state projections. Because

$e^{itH_{\Lambda,\Phi}(s)}H_{\Lambda,\dot{\Phi}}(s)e^{-itH_{\Lambda,\Phi}(s)}=\sum_{Z\subset\Lambda}\left(e^{itH_{\Lambda,\Phi}(s)}\dot{\Phi}(Z,t)e^{-itH_{\Lambda,\Phi}(s)}\right),\quad
t\in[0,1],$

is the summation of “almost local” terms thanks to the Lieb-Robinson bound
[NSY], it can be written in terms of interaction, not finite range, but
satisfying some locality. As a result, the following theorem is obtained.

######  Theorem 1.23 ([BMNS][NSY]).

Let $\Phi$ be a path of interactions satisfying Assumption 1.21 and Assumption
1.22. Then the thermodynamic limit

$\alpha_{s}(A):=\lim_{\Lambda\to{\mathbb{Z}}^{\nu}}\mathop{\mathrm{Ad}}\nolimits\left(V_{\Lambda}(s)\right)(A),\quad
A\in{\mathcal{A}}_{{\mathbb{Z}}^{\nu}}$

exists and defines a strongly continuous family of automorphisms, satisfying

$\omega_{\Phi(s)}=\omega_{\Phi(0)}\circ\alpha_{s},\quad s\in[0,1].$

Note that in Assumption 1.22, we assume the existence of the uniform gap in
finite systems. This is against our policy that we should start from infinite
systems directly. Motivated by this, we introduced an infinite-version of
Theorem 1.23. There, we assume the gap in the bulk. We now explain about this
infinite version.

For $\Gamma\subset{\mathbb{Z}}^{\nu}$, we denote by
$\Pi_{\Gamma}:{\mathcal{A}}\to{\mathcal{A}}_{\Gamma}$ the conditional
expectation with respect to the trace state $\tau$, i.e., a linear map
satisfying

$\displaystyle\Pi_{\Gamma}(a\otimes b)=a\tau(b),\quad
a\in{\mathcal{A}}_{\Gamma},\quad b\in{\mathcal{A}}_{\Gamma^{c}}.$

For each $N\in{\mathbb{N}}$, recall that
$\Lambda_{N}:=[-N,N]^{\nu}\cap{\mathbb{Z}}^{\nu}$. Let us consider the
following subset of ${\mathcal{A}}$.

######  Definition 1.24.

Let $f:(0,\infty)\to(0,\infty)$ be a continuous decreasing function with
$\lim_{t\to\infty}f(t)=0$. For each $A\in{\mathcal{A}}$, let

$\displaystyle\left\|A\right\|_{f}:=\left\|A\right\|+\sup_{N\in\mathbb{N}}\left(\frac{\left\|A-{\mathbb{E}}_{\Lambda_{N}}(A)\right\|}{f(N)}\right).$
(1.17)

We denote by ${\mathcal{D}}_{f}$ the set of all $A\in{\mathcal{A}}$ such that
$\left\|A\right\|_{f}<\infty$.

The set ${\mathcal{D}}_{f}$ is a $*$-algebra which is a Banach space with
respect to the norm $\left\|\cdot\right\|_{f}$. Instead of Assumption 1.22, we
assume the following.

######  Assumption 1.25.

(i)

For each $s\in[0,1]$, there exists a unique $\tau_{\Phi(s)}$-ground state
$\omega_{\Phi(s)}$.

(ii)

There exists a $\gamma>0$ such that

$\displaystyle-i\omega_{\Phi(s)}\left(A^{*}{\delta_{\Phi(s)}}\left(A\right)\right)\geq\gamma\omega_{\Phi(s)}(A^{*}A),\quad\text{for
all
}A\in{\mathcal{D}}({\delta_{\Phi(s)}})\;\text{with}\;\omega_{\Phi(s)}(A)=0,$
(1.18)

for all $s\in[0,1]$.

(iii)

There exists $0<\beta<1$ satisfying the following: Set
$\zeta(t):=e^{-t^{\beta}}$. Then for each $A\in D_{\zeta}$,

$[0,1]\ni s\mapsto\omega_{\Phi(s)}(A)$

is differentiable with respect to $s$, and there is a constant $C_{\zeta}$
such that:

$\displaystyle\left|\frac{d{\omega_{\Phi(s)}}(A)}{ds}\right|\leq
C_{\zeta}\left\|A\right\|_{\zeta},$ (1.19)

for any $A\in D_{\zeta}$.

One can show that Assumption 1.22 implies Assumption 1.25. Namely, Assumption
1.25 is a weaker condition. Nevertheless, we obtained the following.

######  Theorem 1.26 ([MO]).

Let $\Phi$ be a path of interactions satisfying Assumption 1.21 and Assumption
1.25. Then the conclusion of Theorem 1.23 holds.

Although the result of Theorem 1.26 is analogous to that of Theorem 1.23,
there is a crucial difference about the proof. The proof of Hasting’s theorem
fully uses the fact that they are working on operators on the same Hilbert
space. In particular, the property $P(s)^{2}=P(s)$ of the orthogonal
projection was a key input for the analysis. On the other hand, for infinite
version, the GNS representation of $\omega_{\Phi(s)}$ for different value of
$s$ s are mutually singular in general. This requires a new idea.

In order to explain the point of the proof of Theorem 1.26, we now prove
Hastings’s Lemma following the strategy of Theorem 1.26. Assume that $P(s)$ in
the setting of Theorem 1.20 to be of one rank (namely, each $H(s)$ has a
unique ground state), and set $\omega_{s}:=\mathop{\mathrm{Tr}}\nolimits
P(s)(\cdot)$, and $\alpha_{s}:=\mathop{\mathrm{Ad}}\nolimits(U(s)^{*})$. It
suffices to show that

$\frac{d}{ds}\omega_{s}\alpha_{s}^{-1}(A)=0,$

for any $A$. From the definition of $\alpha_{s}$, this means

$\displaystyle
0=\dot{\omega}_{s}\circ\alpha_{s}^{-1}(A)+\omega_{s}\left(i\left[D(s),\alpha_{s}^{-1}(A)\right]\right).$
(1.20)

The question is what is $\dot{\omega}_{s}$. To consider it, we recall the fact
that $\omega_{s}$ is a ground state of $H(s)$. This means it is invariant
under $i[H(s),\cdot]$. Therefore, we have

$\omega_{s}\circ\left(i[H(s),\cdot]\right)=0.$

Differentiating this, we obtain

$\displaystyle\dot{\omega}_{s}\left(i[H(s),A]\right)+\omega_{s}\left(i[\dot{H}(s),A]\right)=0.$
(1.21)

If $i[H(s),\cdot]$ was invertible, this would give us a representation of
$\dot{\omega}_{s}$ in terms of $\omega_{s}$. Unfortunately, $i[H(s),\cdot]$ is
not invertible. In order to solve the issue, an operation called Hasting’s
operator $I_{s}$ plays an important role. Let $w_{\gamma}$ be a non-negative
$L^{1}$-function such that

$\mathop{\mathrm{supp}}\nolimits\hat{w}_{\gamma}\subset(-\gamma,\gamma),\quad\text{and}\quad\int
dtw_{\gamma}(t)=1.$

In fact we may take this $\omega_{\gamma}$ and $W_{\gamma}$ (in Theorem 1.20),
as even and odd functions respectively so that vanishing at infinity and

$\frac{d}{dt}W_{\gamma}(t)=-\omega_{\gamma}(t),\quad t\in[0,\infty).$

(See [NSY] section 6.) Set

$I_{s}\left(A\right):=\int
dtw_{\gamma}(t)\mathop{\mathrm{Ad}}\nolimits\left(e^{iH(s)t}\right)(A).$

Because of the support condition of $\hat{w}_{\gamma}$, we have

$\displaystyle\omega_{s}\left(B^{*}I_{s}(A)\right)=\omega_{s}(B^{*})\omega_{s}(A),$
(1.22)

for any $A,B$. Indeed, we have

$\displaystyle\omega_{s}\left(B^{*}I_{s}(A)\right)=\int
dtw_{\gamma}(t)\mathop{\mathrm{Tr}}\nolimits\left(P(s)B^{*}e^{i\left(H(s)-E(s\right)t}A\right)$
$\displaystyle=\mathop{\mathrm{Tr}}\nolimits\left(P(s)B^{*}\left(\hat{w}_{\gamma}(-H(s)+E(s))\right)A\right).$

Now, note that because of the spectral gap
$\sigma(H(s))\setminus\\{E(s)\\}\subset[E(s)+\gamma,\infty)$, and the support
property of $\hat{w}_{\gamma}$,
$\mathop{\mathrm{supp}}\nolimits\hat{w}_{\gamma}\subset(-\gamma,\gamma)$,
$\hat{w}_{\gamma}(H(s)-E(s))$ is proportional to
$P(s)$,$\hat{w}_{\gamma}(H(s)-E(s))=cP(s)$. Setting $A=B=\mathbb{I}$, we see
that $c=1$. Hence we obtain (1.22). The same argument allows us to prove
(1.22) (for $\omega_{s}=\omega_{\Phi(s)}$) in infinite system, on GNS Hilbert
space of $\omega_{\Phi(s)}$, using the bulk Hamiltonian
$H_{\omega_{\Phi(s)},\Phi(s)}$ instead of $H(s)$ and an orthogonal projection
onto ${\mathbb{C}}\Omega_{\Omega_{\Phi(s)}}$ instead of $P(s)$. The point is
everything in the argument is well-defined even in infinite setting.

In particular, from (1.22), with $A$ replaced by $A^{*}$ and $B$ by
$\left(I_{s}(A)-\omega_{s}(A)\right)^{*}$, we have

$\displaystyle\omega_{s}\left(\left(I_{s}(A)-\omega_{s}(A)\right)\left(I_{s}(A)-\omega_{s}(A)\right)^{*}\right)=0.$

This means

$\displaystyle\left(\mathbb{I}-P(s)\right)\left(I_{s}(A)-\omega_{s}(A)\right)=\left(I_{s}(A)-\omega_{s}(A)\right).$
(1.23)

Now differentiating (1.22), we have

$\displaystyle\dot{\omega}_{s}\left(B^{*}I_{s}(A)\right)+\omega_{s}\left(B^{*}\dot{I}_{s}(A)\right)=\dot{\omega}_{s}(B^{*})\omega_{s}(A)+\omega_{s}(B^{*})\dot{\omega}_{s}(A).$

Substituting $B=\mathbb{I}-P(s)$ to this equation, and recalling (1.23), we
obtain

$\displaystyle\dot{\omega}_{s}(I_{s}(A))=\dot{\omega}_{s}\left(I_{s}(A)-\omega_{s}(A)\right)=\dot{\omega}_{s}\left(\mathbb{I}-P(s)\left(I_{s}(A)-\omega_{s}(A)\right)\right)=0.$

Here we used the clear fact that $\dot{\omega}_{s}(\mathbb{I})=0$. The same
argument allows us to prove $\dot{\omega}_{\Phi(s)}(I_{s}(A))=0$, for infinite
system if we replace the role of $\mathbb{I}-P(s)$ by approximate identity of
the left kernel associated to $\omega_{\Phi(s)}$.

Hence we have obtained $\dot{\omega}_{s}(I_{s}(A))=0$. A good thing about it
is that the difference between $I_{s}(A)$ and $A$ is given by

$\displaystyle A-I_{s}(A)=-\int
dtw_{\gamma}(t)\int_{0}^{t}du\;i\left[H(s),\mathop{\mathrm{Ad}}\nolimits\left(e^{iH(s)u}\right)(A)\right],$

and $i[H(s),\cdot]$ shows up. Substituting this to $\dot{\omega}_{s}$, and
recalling $\dot{\omega}_{s}(I_{s}(A))=0$, we obtain

$\displaystyle\begin{split}&\dot{\omega}_{s}\left(A\right)=\dot{\omega}_{s}\left(A-I_{s}(A)\right)\\\
&=-\int
dtw_{\gamma}(t)\int_{0}^{t}du\;\dot{\omega}_{s}\left(i\left[H(s),\mathop{\mathrm{Ad}}\nolimits\left(e^{iH(s)u}\right)(A)\right]\right).\end{split}$

Now we apply (1.21) to the last term. Then we obtain

$\displaystyle\dot{\omega}_{s}\left(A\right)=\int
dtw_{\gamma}(t)\int_{0}^{t}du\;\omega_{s}\left(i\left[\dot{H}(s),\mathop{\mathrm{Ad}}\nolimits\left(e^{iH(s)u}\right)(A)\right]\right)$
$\displaystyle=\int
dtW_{\gamma}(t)\omega_{s}\left(i\left[\dot{H}(s),\mathop{\mathrm{Ad}}\nolimits\left(e^{iH(s)t}\right)(A)\right]\right)$
$\displaystyle=-\int
dtW_{\gamma}(t)\omega_{s}\left(i\left[\mathop{\mathrm{Ad}}\nolimits\left(e^{iH(s)t}\left(\dot{H}(s)\right)\right)(A)\right]\right)$
$\displaystyle=-\omega_{s}\left(i\left[D(s),A\right]\right).$

Now we use this formula to prove the desired equation (1.20). Indeed we have

$\displaystyle\dot{\omega}_{s}\circ\alpha_{s}^{-1}(A)+\omega_{s}\left(i\left[D(s),\alpha_{s}^{-1}(A)\right]\right)$
$\displaystyle=-\omega_{s}\left(i\left[D(s),\alpha_{s}^{-1}(A)\right]\right)+\omega_{s}\left(i\left[D(s),\alpha_{s}^{-1}(A)\right]\right)=0.$

Hence we have proven the Theorem. The important part here is that we use only
objects, which has its infinite version. For example the commutator
$[H(s),\cdot]$, still makes sense although $H(s)$ itself does not have any
meaning in infinite system.

As a dynamics given by (time-dependent) local interactions, the automorphism
in Theorem 1.26 satisfies some quasi local property. For example, it satisfies
the Lieb-Robinson bound. Among such properties, what is important in analysis
of SPT-phases is what we call factorization property.

Factorization property in $1$-dimensional system is the following.

######  Proposition 1.27.

[O4] Let $\nu=1$. The automorphism $\alpha_{s}$ in Theorem 1.26 satisfies a
factorization property such that

$\alpha_{s}=({\rm{inner}})\circ\left(\alpha_{L}\otimes\alpha_{R}\right),$

where $\alpha_{L}$ is an automorphism on ${\mathcal{A}}_{(-\infty,-1]}$,
$\alpha_{R}$ an automorphism on ${\mathcal{A}}_{[0,\infty)}$.

A two-dimensional version we use is as follows. For $0<\theta<\frac{\pi}{2}$,
a cone $C_{\theta}$ is defined by

$\displaystyle
C_{\theta}:=\left\\{(x,y)\in{\mathbb{Z}}^{2}\mid|y|\leq\tan\theta\cdot|x|\right\\}.$
(1.24)

Furthermore, $H_{L}$, $H_{R}$,$H_{U}$, $H_{D}$ denotes half left/right and
upper/lower planes, and $C_{\theta,L}:=C_{\theta}\cap H_{L}$,
$C_{\theta,R}:=C_{\theta}\cap H_{R}$, $C_{\theta,U}:=C_{\theta}\cap H_{U}$,
$C_{\theta,D}:=C_{\theta}\cap H_{D}$.

######  Proposition 1.28.

Let $\nu=2$. For any $0<\theta<\frac{\pi}{2}$, there is
$\alpha_{L}\in\mathop{\mathrm{Aut}}\nolimits{\mathcal{A}}_{H_{L}}$,
$\alpha_{R}\in\mathop{\mathrm{Aut}}\nolimits{\mathcal{A}}_{H_{R}}$, and
$\Theta\in\mathop{\mathrm{Aut}}\nolimits{\mathcal{A}}_{\left(C_{\theta}\right)^{c}}$
such that

$\displaystyle\alpha_{s}=({\rm{inner}})\circ\left(\alpha_{L}\otimes\alpha_{R}\right)\circ\Theta.$
(1.25)

Actually, $\alpha_{s}$ can be cut in many directions simultaneously. Using
such properties, we can also show the following.

######  Lemma 1.29.

For any $0<\theta^{\prime}<\theta<\frac{\pi}{2}$, and
$\eta_{L}^{\prime}\in\mathop{\mathrm{Aut}}\nolimits{\mathcal{A}}_{C_{\theta^{\prime},L}}$,
$\eta_{R}^{\prime}\in\mathop{\mathrm{Aut}}\nolimits{\mathcal{A}}_{C_{\theta^{\prime},R}}$,
there are automorphisms
$\eta_{L}\in\mathop{\mathrm{Aut}}\nolimits{\mathcal{A}}_{C_{\theta,L}}$,
$\eta_{R}\in\mathop{\mathrm{Aut}}\nolimits{\mathcal{A}}_{C_{\theta,R}}$ such
that

$\alpha_{s}\circ\left(\eta_{L}^{\prime}\otimes\eta_{R}^{\prime}\right)\circ\alpha_{s}^{-1}=({\rm{inner}})\left(\eta_{L}\otimes\eta_{R}\right).$

Factorization property is quite simple but strong analytical property. It also
enables us to show that the existence of a non-trivial super selection sector
results in long-range entanglement [NO].

##  2 SPT-phases in $1$-dimensional systems

In this section, we study SPT-phases in $1$-dimensional systems. We will see
that there is a $H^{2}(G,{\rm U}(1))$-valued index for SPT with on-site
symmetry, and ${\mathbb{Z}}_{2}$-valued index for SPT with reflection
symmetry. They generalize the indices considered for Matrix product states.

###  2.1 $H^{2}(G,{\rm U}(1))$ and projective representations

For SPT-phases in one dimension with on-site symmetry, physicists conjectured
that there is a $H^{2}(G,{\rm U}(1))$-valued invariant of the classification.
In this subsection, we recall the definition of $H^{2}(G,{\rm U}(1))$ and its
relation to projective representations.

A $2$-cochain $C^{2}(G,{\rm U}(1))$ is a map from $G^{\times 2}$ to ${\rm
U}(1)$. It is said to be a $2$-cocycle if the relation

$\displaystyle\sigma(gh,k)\sigma(g,h)=\sigma(g,hk)\sigma(h,k)$

holds. The set of all $2$-cocycles are denoted by $Z^{2}(G,{\rm U}(1))$. We
introduce an equivalence relation to $Z^{2}(G,{\rm U}(1))$ by
$\sigma^{\prime}\simeq\sigma$ if there exists $c:G\mapsto{\mathbb{T}}$ such
that

$\displaystyle\sigma^{\prime}(g,h)=\sigma(g,h)\frac{c(gh)}{c(g)c(h)}.$

The equivalence classes forms a group, which is the second group cohomology
$H^{2}(G,{U(1)})$.

The second group cohomology $H^{2}(G,{U(1)})$ shows up naturally, when we
consider projective representations. A projective representation of $G$ on a
Hilbert space ${\mathcal{H}}$ is a map from $G$ to a unitary operators
${\mathcal{U}}({\mathcal{H}})$ on ${\mathcal{H}}$ satisfying

$\displaystyle u(g)u(h)=\sigma(g,h)u(gh),\quad g,h\in G.$

Here, $\sigma(g,h)$ is a ${\rm U}(1)$-phase. Using the associativity of $u$,
we can show that $\sigma$ is a $2$-cocycle. Indeed, we have

$\displaystyle
u(g)u(h)u(k)=\sigma(g,h)u(gh)u(k)=\sigma(g,h)\sigma(gh,k)u(ghk),$

and

$\displaystyle
u(g)u(h)u(k)=\sigma(h,k)u(g)u(hk)=\sigma(h,k)\sigma(g,hk)u(ghk).$

Comparing these equations, we obtain $2$-cocycle relation for $\sigma$.

Note that given a group action $G$ on ${\mathcal{B}}({\mathcal{H}})$, for a
Hilbert space ${\mathcal{H}}$, we get some projective representation. Let
$\gamma:G\to\mathop{\mathrm{Aut}}\nolimits\left({\mathcal{B}}({\mathcal{H}})\right)$
be a group homomorphism. Then by Theorem 1.3, there is a unitary $u_{g}$ on
${\mathcal{H}}$ such that

$\displaystyle\mathop{\mathrm{Ad}}\nolimits(u_{g})(x)=\gamma_{g}(x),\quad
x\in{\mathcal{B}}({\mathcal{H}}).$

Note that there is some ${\rm U}(1)$-ambiguity about the choice of $u_{g}$:
for any map $c:G\to{\rm U}(1)$, we also have

$\displaystyle\mathop{\mathrm{Ad}}\nolimits(c_{g}u_{g})(x)=\gamma_{g}(x),\quad
x\in{\mathcal{B}}({\mathcal{H}}).$ (2.1)

This $u:G\to{\mathcal{U}}\left({\mathcal{H}}\right)$ is a projective group
representation. Indeed, we have

$\displaystyle\mathop{\mathrm{Ad}}\nolimits(u_{g}u_{h})(x)=\gamma_{g}\gamma_{h}(x)=\gamma_{gh}(x)=\mathop{\mathrm{Ad}}\nolimits(u_{gh})(x),\quad
x\in{\mathcal{B}}({\mathcal{H}}).$

This means there is some $\sigma(g,h)\in{\rm U}(1)(1)$ such that

$\displaystyle u_{g}u_{h}=\sigma(g,h)u_{gh}.$

Namely, $u$ is a projective representation, and hence $\sigma$ is a
$2$-cocycle. The ${\rm U}(1)$-ambiguity of the choice $c$ (2.1) results in
ambiguity of the $2$-cocycle $\sigma$. If we choose $c_{g}u_{g}$ instead of
$u_{g}$, then the $2$-cocycle obtained is

$\displaystyle\tilde{\sigma}(g,h)=\sigma(g,h)c_{g}c_{h}\bar{c_{gh}}.$

However, from this equation, we see that the resulting second cohomology class
does not depend on such a choice, namely we have

$\left[\tilde{\sigma}\right]_{H^{2}(G,{\rm
U}(1))}=\left[\sigma\right]_{H^{2}(G,{\rm U}(1))}.$

Hence, given a group action $\gamma$ on ${\mathcal{B}}({\mathcal{H}})$, we
obtain an associated second cohomology class without ambiguity.

###  2.2 Split property of unique gapped ground state in $1$-d

An important property of the unique gapped ground state in one dimension is
the split property. Using this property, one can easily define indices, which
turns out to be invariants of the classification. More precisely, the
following Theorem is crucial in order to define the SPT-index in one-
dimensional systems.

######  Theorem 2.1 (Matsui ’13).

Let $\omega_{\Phi}$ be a unique gapped $\alpha^{\Phi}$-ground state for
$\Phi\in{\mathcal{P}}_{U.G.}$. Then $\omega_{\Phi}$ satisfies the split
property with respect to ${\mathcal{A}}_{L}$ and ${\mathcal{A}}_{R}$, i.e.,
$\pi_{\omega_{\Phi,R}}({\mathcal{A}}_{R})^{\prime\prime}$ is a type I factor.

Thanks to this theorem, known for any unique gapped ground state, we can
classify

$\displaystyle{\mathcal{P}}_{U.G.{{\beta}}}:=\left\\{\Phi\in{\mathcal{P}}_{U.G.}\mid\beta\text{-invariant}\right\\}.$

not only the SPT-phases ${\mathcal{P}}_{U.G.{{\beta}}}^{0}$ (1.11). Although
it is expected that ${\mathcal{P}}_{U.G.}^{0}={\mathcal{P}}_{U.G.}$ for one
dimensional systems, there is no result proving it. Therefore, at least at
this stage of research, classification of ${\mathcal{P}}_{U.G.}$ is stronger
result than classification of ${\mathcal{P}}_{U.G.}^{0}$.

###  2.3 Matrix product state

Analysis of SPT-phases was initially carried out for matrix product states. In
this section, we introduce matrix product states.

In general, even constructing an example of gapped Hamiltonians is a highly
non-trivial problem. The non-commutativity of quantum systems makes proving
the existence of spectral gap a very hard problem . This is the case already
for local Hamiltonians. If the interaction terms $\Phi(X)$ in local
Hamiltonians

$\displaystyle\left(H_{\Phi}\right)_{\Lambda}:=\sum_{X\subset{\Lambda}}\Phi(X).$

mutually commute, we can consider the joint distribution. But typically, they
don’t commute.

Fortunately, for one-dimensional systems, there is a recipe and technics by
Fannes-Nachtergaele-Werner [FNW] and [N] to construct a class of Hamiltonians
with unique gapped ground state. This is sometimes called Matrix product state
(MPS) Hamiltonians. In this section, we explain this recipe.

Recently there was great progress in higher dimensions. The technics from
[FNW] and [N] has turned out to be useful even in higher dimensional systems.
See [ALLNY] [LN] and [LSW].

Throughout this section, we fix some orthonormal basis of $\mathbb{C}^{d}$,
$\\{\psi_{\mu}\\}_{\mu=1}^{d}$, and denote by $e_{\mu\nu}$, $\mu,\nu=1,\ldots
d$ the system of matrix unit corresponding to it. The interactions given by
[FNW] is a type of interactions so called parent Hamiltonians. A parent
Hamiltonian is made out of a sequence of subspaces of
$\bigotimes_{i=0}^{l-1}{\mathbb{C}}^{d}$, $l\in\mathbb{N}$, which satisfies
the condition called the intersection property.

######  Definition 2.2.

We say that a sequence of subspaces
$\\{{\mathcal{D}}_{N}\\}_{N\in\mathbb{N}}$,
${\mathcal{D}}_{N}\subset\bigotimes_{i=0}^{N-1}\mathbb{C}^{d}$,
$N\in\mathbb{N}$, satisfies the intersection property, if there exists an
$m\in\mathbb{N}$, such that the relation

${\mathcal{D}}_{N}=\bigcap_{x=0}^{N-m}({\mathbb{C}}^{d})^{\otimes
x}\otimes{\mathcal{D}}_{m}\otimes({\mathbb{C}}^{d})^{\otimes N-m-x},$

holds for all $N\geq m$.

Let ${\mathcal{D}}_{N}\subset\bigotimes_{i=0}^{N-1}\mathbb{C}^{d}$,
$N\in\mathbb{N}$ be subspaces satisfying the intersection property as above.
Let $h_{m}$ be the orthogonal projection onto the orthogonal complement of
${\mathcal{D}}_{m}$ in $\otimes_{i=0}^{m-1}\mathbb{C}^{d}$. We set

$\displaystyle\Phi(X):=\left\\{\begin{gathered}\tau_{x}\left(h_{m}\right),\quad\text{if}\;X=[x,x+m-1]\\\
0,\quad\text{otherwise}\end{gathered}\right..$

Here $\tau_{x}$ denotes the space translation on ${\mathbb{Z}}$. Then, by the
intersection property, we see that

$\ker\left(\left(H_{\Phi}\right)_{[0,N-1]}\right)={\mathcal{D}}_{N}$

for all $N\geq m$. Hamiltonians given in this way are called parent
Hamiltonians. Parent Hamiltonians are nice in the sense that 1.we know that
the lowest eigenvalues of local Hamiltonians are zero, and 2.we know how the
ground state spaces of local Hamiltonians, i.e.,
$\ker\left(\left(H_{\Phi}\right)_{[0,N-1]}\right)={\mathcal{D}}_{N}$ look
like. Now we would like to construct a sequence of spaces 1. satisfying the
intersection property, and 2. the corresponding parent local Hamiltonian
$\ker\left(\left(H_{\Phi}\right)_{[0,N-1]}\right)$ is gapped with respect to
the open boundary conditions. This is carried out by a linear map determined
by a $d$-tuple of matrices. First, we prepare an $d$-tuple of $k\times k$
matrices ${\boldsymbol{v}}:=(v_{1},\ldots,v_{d})$ and $m\in\mathbb{N}$. Define
a subspace ${\mathcal{G}}_{m,{\boldsymbol{v}}}$ of
$\bigotimes_{i=0}^{m-1}{\mathbb{C}}^{d}$ by the range of the following map
$\Gamma_{m,{\boldsymbol{v}}}:\mathop{\mathrm{M}}\nolimits_{k}\to\bigotimes_{i=0}^{m-1}{\mathbb{C}}^{d}$,

$\displaystyle\Gamma_{m,{\boldsymbol{v}}}\left(X\right)=\sum_{\mu_{0},\ldots,\mu_{m-1}\in\\{1,\cdots,d\\}}\left(\mathop{\mathrm{Tr}}\nolimits
X\left(v_{\mu_{0}}v_{\mu_{1}}\cdots
v_{\mu_{m-1}}\right)^{*}\right)\bigotimes_{i=0}^{m-1}\psi_{\mu_{i}},\quad
X\in\mathop{\mathrm{M}}\nolimits_{k}.$

Let $h_{m,{\boldsymbol{v}}}$ be the orthogonal projection onto
${\mathcal{G}}_{m,{\boldsymbol{v}}}^{\perp}$ in
$\otimes_{i=0}^{m-1}\mathbb{C}^{d}$. Then we set

$\displaystyle\Phi_{m,{\boldsymbol{v}}}(X):=\left\\{\begin{gathered}\tau_{x}\left(h_{m,{\boldsymbol{v}}}\right),\quad\text{if}\;X=[x,x+m-1]\\\
0,\quad\text{otherwise}\end{gathered}\right..$

Of course, for a random choice of ${\boldsymbol{v}}$, the spaces
$\\{{\mathcal{G}}_{N,{\boldsymbol{v}}}\\}_{N}$ does not satisfy the
intersection property, and the local Hamiltonian given by
$h_{m,{\boldsymbol{v}}}$ does not have a gap. We are interested in some
sufficient condition for ${\boldsymbol{v}}$ to satisfy these properties. The
sufficient condition which was introduced in [FNW] is the following equivalent
properties.

######  Theorem 2.3.

For
${\boldsymbol{v}}:=(v_{1},\ldots,v_{d})\in\mathop{\mathrm{M}}\nolimits_{k}^{\times
d}$, let
$T_{\boldsymbol{v}}:\mathop{\mathrm{M}}\nolimits_{k}\to\mathop{\mathrm{M}}\nolimits_{k}$
be the completely positive map given by

$\displaystyle T_{\boldsymbol{v}}(A)=\sum_{i=1}^{d}v_{i}Av_{i}^{*},\quad
A\in\mathop{\mathrm{M}}\nolimits_{k}.$

Assume that the spectral radius $r_{\boldsymbol{v}}$ of $T_{\boldsymbol{v}}$,
is $1$. The following properties are equivalent.

  1. 1.

There exist a unique faithful state $\varphi_{\boldsymbol{v}}$ and a strictly
positive element $e_{\boldsymbol{v}}\in\mathop{\mathrm{M}}\nolimits_{k}$
satisfying

$\lim_{N\to\infty}T_{\boldsymbol{v}}^{N}(A)=\varphi_{\boldsymbol{v}}(A)e_{\boldsymbol{v}},\quad
A\in\mathop{\mathrm{M}}\nolimits_{k}.$

  2. 2.

The spectrum $\sigma\left(T_{\boldsymbol{v}}\right)$ of $T_{\boldsymbol{v}}$
satisfies $\sigma\left(T_{\boldsymbol{v}}\right)\cap{\mathbb{T}}=\\{1\\}$. $1$
is a non degenerate eigenvalue of $T_{\boldsymbol{v}}$. There exist a faithful
$T_{\boldsymbol{v}}$-invariant state $\varphi_{\boldsymbol{v}}$ and a strictly
positive $T_{\boldsymbol{v}}$-invariant element
$e_{\boldsymbol{v}}\in\mathop{\mathrm{M}}\nolimits_{k}$.

  3. 3.

There exists an $m\in\mathbb{N}$ such that
${\mathcal{K}}_{m}({\boldsymbol{v}})=\mathop{\mathrm{M}}\nolimits_{k}$, where

${\mathcal{K}}_{m}({\boldsymbol{v}}):=\mathop{\mathrm{span}}\nolimits\left\\{v_{\mu_{1}}v_{\mu_{2}}\cdots
v_{\mu_{m}}\mid(\mu_{1},\mu_{2},\ldots,\mu_{l})\subset\\{1,\ldots,d\\}^{\times
m}\right\\}.$

When $d$-tuple of matrices ${\boldsymbol{v}}$ satisfies these (equivalent)
conditions, we say that it is primitive.

Given a primitive $d$-tuple ${\boldsymbol{v}}$, we can construct a matrix
product state (MPS), which was initially called a finitely correlated state.
Let $\varphi_{\boldsymbol{v}}$ be the state, and $e_{\boldsymbol{v}}$ the
positive element given in (1) of the above theorem. Then the formula

$\displaystyle\omega_{{\boldsymbol{v}}}\left(\bigotimes_{i=1}^{l}e_{\mu_{i},\nu_{i}}\right)=\varphi_{\boldsymbol{v}}\left(v_{\mu_{1}}\cdots
v_{\mu_{l}}e_{{\boldsymbol{v}}}v_{\nu_{l}}^{*}\cdots
v_{\nu_{1}}^{*}\right),\quad\mu_{i},\nu_{i}\in\mathcal{S},\quad
i=1,\ldots,l,\quad l\in\mathbb{N},$ (2.2)

defines a translationally invariant state on ${\mathcal{A}}_{{\mathbb{Z}}}$.
This is the matrix product state generated by ${\boldsymbol{v}}$.

Here is the Theorem of Fannes-Nachtergaele-Werner :

######  Theorem 2.4 ([FNW]).

Assume that $\Gamma_{m-1,{\boldsymbol{v}}}$ is injective. Then
${\Phi_{m,{\boldsymbol{v}}}}$ has a unique gapped ground state
$\omega_{{\boldsymbol{v}}}$.

######  Remark 2.5.

In fact, the uniqueness of the ground state in our sense (Definition 1.9 )
requires some additional analysis. See [M1], [O3].

Hence, we got a class of gapped Hamiltonians by the FNW-recipe. It is nice to
have a systematic way of constructing gapped Hamiltonians. However, the power
of MPS is not only that. If an interaction satisfies some (a bit strong)
conditions, we can show that the ground state is MPS.

######  Theorem 2.6 ([M1](See also [O3] for primitivity)).

Let $\Phi$ be a translation invariant finite range interaction with a unique
ground state $\omega_{\Phi}$ on ${\mathcal{A}}_{{\mathbb{Z}}}$. Suppose that
there exists a constant $c\in\mathbb{N}$ such that

$\displaystyle 1\leq\dim\ker\left(\left(H_{\Phi}\right)_{[1,N]}\right)\leq c,$
(2.3)

for all $N\in\mathbb{N}$. Then $\omega_{\Phi}$ is a MPS, namely there is a
primitive ${\boldsymbol{v}}$ such that
$\omega_{\Phi}=\omega_{{\boldsymbol{v}}}$.

This theorem follows from the following Theorem from operator algebra theory.

######  Theorem 2.7 ([A]).

Let ${\mathcal{H}}$ be a separable infinite dimensional Hilbert space, and
$d\in\mathbb{N}$. Let $\Xi:B({\mathcal{H}})\to B({\mathcal{H}})$ be a unital
endomorphism of $B({\mathcal{H}})$ such that
$\left(\Xi\left(B({\mathcal{H}})\right)\right)^{\prime}$ is isomorphic to
$\mathop{\mathrm{M}}\nolimits_{d}$. Then there exist $S_{i}\in
B({\mathcal{H}})$, $i=1,\ldots,d$ such that

$\displaystyle
S_{i}^{*}S_{j}=\delta_{ij},\quad\sum_{j=1}^{d}S_{j}xS_{j}^{*}=\Xi(x),\quad
x\in B({\mathcal{H}}).$ (2.4)

Note that $S_{j}$s in the theorem form a representation of Cuntz algebra
[BR2].

Let $\omega_{\Phi}$ be a unique gapped ground state for an interaction $\Phi$
which satisfies the conditions in Theorem 2.6. By Theorem 2.1, there is an
irreducible representation $({\mathcal{K}},\pi)$ of ${\mathcal{A}}_{R}$,
quasi-equivalent to the GNS representation of
$\omega_{\Phi}|_{{\mathcal{A}}_{R}}$. Because $\omega_{\Phi}$ is
translationally invariant, the space translation extends uniquely to an
endomorphism on
$\pi({\mathcal{A}}_{R})^{\prime\prime}={\mathcal{B}}({\mathcal{K}})$. Applying
Theorem 2.7 to it, we obtain operators $S_{\mu}\in B({\mathcal{H}}_{\omega})$
with $\mu=1,\ldots d$ satisfying the following:

$\displaystyle\begin{split}&S_{\mu}^{*}S_{\nu}=\delta_{\mu\nu}\mathbb{I},\\\
&\sum_{\mu=1,\ldots,d}S_{\mu}\pi(A)S_{\mu}^{*}=\pi\circ\tau_{1}(A),\quad
A\in{\mathcal{A}}_{R}.\\\
&\pi\left(e_{\mu\nu}\otimes\mathbb{I}_{[1,\infty)}\right)=S_{\mu}S_{\nu}^{*}\quad\text{for
all}\quad\mu,\nu=1,\ldots,d.\end{split}$ (2.5)

Here $e_{\mu\nu}\otimes\mathbb{I}_{[1,\infty)}$ indicates an element
$e_{\mu\nu}$ in ${\mathcal{A}}_{\\{0\\}}=\mathop{\mathrm{M}}\nolimits_{d}$
embedded into ${\mathcal{A}}_{R}$. From this, we have

$\displaystyle\pi\left(\bigotimes_{k=0}^{l-1}e_{\mu_{k},\nu_{k}}\right)=S_{\mu_{0}}\cdots
S_{\mu_{l-1}}S_{\nu_{l-1}}^{*}\cdots S_{\nu_{0}}^{*},$ (2.6)

for all $l\in\mathbb{N}$, $\mu_{k},\nu_{k}=1,\ldots,d$. Because $\pi$ and the
GNS representation of $\omega_{\Phi}|_{{\mathcal{A}}_{R}}$ are quasi-
equivalent, there is a density operator on ${\mathcal{K}}$ such that

$\omega_{\Phi}(A)=\mathop{\mathrm{Tr}}\nolimits\rho\pi(A),\quad
A\in{\mathcal{A}}_{R}.$

Because of the first inequality of (2.3) and the uniqueness of the ground
state, $\omega_{\Phi}$ has to be frustration-free, namely, it should satisfy
$\omega_{\Phi}(\Phi(X))=0$ for any $X\in\mathfrak{S}_{{\mathbb{Z}}}$. From
this and the second inequality of (2.3), one can show that the support of
$\rho$ is of finite rank. (See [M1].) The $n$-tuple ${\boldsymbol{v}}$
generating the finitely correlated state $\omega$, in Theorem 2.6 is given as
a restriction of $S_{i}$s to this range of $\rho$.

Recall that it is believed that any interactions $\Phi_{1},\Phi_{2}$ with
unique gapped ground state in one-dimension are equivalent
$\Phi_{1}\sim\Phi_{2}$. Although in general, it is still an open problem, for
the class of interactions satisfying the conditions in Theorem 2.6, we can say
it is true.

######  Theorem 2.8 ([O3]).

Let $\Phi_{1},\Phi_{2}$ be uniformly bounded finite range interactions
satisfying the assumptions in Theorem 2.6. Then we have
$\Phi_{1}\sim\Phi_{2}$. Furthermore, they can be connected without breaking
the space translation invariance.

In the physicist’s approach on SPT-phases, the uniqueness theorem of MPS gets
important. In quantum information theory, it is called the fundamental theorem
of MPS. It is due to Fannes-Nachtergaele-Werner again.

######  Theorem 2.9 ([FNW2]).

Suppose that primitive
${\boldsymbol{v}}\in\mathop{\mathrm{M}}\nolimits_{k}^{\times d}$ and
$\tilde{\boldsymbol{v}}\in\mathop{\mathrm{M}}\nolimits_{\tilde{k}}^{\times d}$
give the same state
$\omega_{{\boldsymbol{v}}}=\omega_{\tilde{\boldsymbol{v}}}$. Then
$k=\tilde{k}$ and there is a $c\in{\mathbb{T}}$, a unitary $k\times k$ matrix
$U$, such that

$\tilde{v}_{\mu}=cUv_{\mu}U^{*},\quad\mu=1,\ldots,d.$

###  2.4 $H^{2}(G,{\rm U}(1))$-index for MPS with on-site symmetry

Now let us consider MPS which are invariant under the on-site symmetry $\beta$
given by a unitary representation $U$ of a finite group $G$. In [PTBO1],
[PTBO2], [PWSVC] a $H^{2}(G,{\rm U}(1))$-valued index was obtained by the
following argument from $\beta$-invariant MPS. In this section we explain
about this.

Let $\omega_{{\boldsymbol{v}}}$ be a $\beta$-invariant MPS given by a
primitive ${\boldsymbol{v}}\in\mathop{\mathrm{M}}\nolimits_{k}^{\times d}$.
Because $\omega_{{\boldsymbol{v}}}$ is $\beta_{g}$-invariant, we can check
that $d$-tuple $\tilde{\boldsymbol{v}}(g)$ given by

$\tilde{v}_{\mu}(g):=\sum_{\nu=1}^{d}U(g)_{\nu\mu}v_{\nu},\quad g\in
G,\quad\mu=1,\ldots,d$

also gives the same state. Namely, we have

$\omega_{\tilde{\boldsymbol{v}}(g)}=\omega_{{\boldsymbol{v}}}.$

Then, by the fundamental theorem of MPS, there is $c(g)\in{\mathbb{T}}$, a
$k\times k$ unitary matrix $u(g)$, such that

$\displaystyle
c(g)u(g)v_{\mu}u(g)^{*}=\tilde{v}_{\mu}(g)=\sum_{\nu=1}^{d}U(g)_{\nu\mu}v_{\nu}.$

Now, let us consider $\tilde{v}_{\mu}(gh)$. From above, we have

$\displaystyle\tilde{v}_{\mu}(gh)=c(gh)u(gh)v_{\mu}u(gh)^{*}.$

On the other hand, by the definition of $\tilde{v}_{\mu}(gh)$, we have

$\displaystyle\tilde{v}_{\mu}(gh)=\sum_{\nu=1}^{d}U(gh)_{\nu\mu}v_{\nu}=\sum_{\nu,\lambda=1}^{d}U(g)_{\nu\lambda}U(h)_{\lambda\mu}v_{\nu}$
$\displaystyle=\sum_{\lambda=1}^{d}U(h)_{\lambda\mu}\tilde{v}_{\lambda}(g)=c(g)u(g)\sum_{\lambda=1}^{d}U(h)_{\lambda\mu}v_{\lambda}u(g)^{*}$
$\displaystyle=c(g)c(h)u(g)u(h)v_{\mu}u(h)^{*}u(g)^{*}.$

Hence we obtain

$\displaystyle
c(gh)u(gh)v_{\mu}u(gh)^{*}=c(g)c(h)u(g)u(h)v_{\mu}u(h)^{*}u(g)^{*},$

for all $\mu=1,\ldots,d$. Recalling (3) of Theorem 2.3, we then see that
$c(gh)=c(g)c(h)$ and

$\displaystyle u(gh)xu(gh)^{*}=u(g)u(h)xu(h)^{*}u(g)^{*},$

for all $x\in\mathop{\mathrm{M}}\nolimits_{k}$. This means there is some
$\sigma(g,h)\in{\rm U}(1)$ such that

$\displaystyle u(g)u(h)=\sigma(g,h)u(gh),\quad g,h\in G.$

Hence we obtain a projective representation, and a second cohomology
$[\sigma]_{H^{2}(G,{\rm U}(1))}\in H^{2}(G,{\rm U}(1))$. Here we considered
linear representation of $G$ but anti-linear ones can be treated analogously.
This is the index obtained in [PTBO1], [PTBO2], [PWSVC].

###  2.5 $H^{2}(G,{\rm U}(1))$-index for SPT with on-site symmetry

The derivation of $H^{2}(G,{\rm U}(1))$-valued index given in the previous
subsection is elegant and nice, but there is a crucial problem for our use.
That is, it is defined only on a subset of ${\mathcal{P}}_{U.G.\beta}^{0}$,
i.e., MPS. In order to think about classification problem, we definitely need
some index which are defined all the ${\mathcal{P}}_{U.G.\beta}^{0}$.

Such an index can be derived from split property. In fact, the existence of a
projective representation associated to state with split property is automatic
as we saw in subsection 2.1, and it has been noticed for operator algebraist
from long time ago [M2]. Thanks to Matsui’s theorem, we know that our unique
ground state $\omega_{\Phi}$ for $\beta$-invariant
$\Phi\in{\mathcal{P}}_{U.G.}$ satisfies the split property. From Theorem 2.1,
and Proposition 1.18, the GNS triple of $\omega_{\Phi}$ is of the form
$({\mathcal{K}}_{L}\otimes{\mathcal{K}}_{R},\rho_{L}\otimes\rho_{R},\Omega)$
with irreducible representations $({\mathcal{K}}_{L},\rho_{L})$,
$({\mathcal{K}}_{R},\rho_{R})$ of ${\mathcal{A}}_{L}$, ${\mathcal{A}}_{R}$ and
a unit vector $\Omega\in{\mathcal{K}}_{L}\otimes{\mathcal{K}}_{R}$. In this
representation, we have

$\displaystyle\left(\left(\rho_{L}\otimes\rho_{R}\right)\left({\mathcal{A}}_{R}\right)\right)^{\prime\prime}={\mathbb{C}}\mathbb{I}_{{\mathcal{K}}_{L}}\otimes{\mathcal{B}}({\mathcal{K}}_{R}),\quad\left(\left(\rho_{L}\otimes\rho_{R}\right)\left({\mathcal{A}}_{L}\right)\right)^{\prime\prime}={\mathcal{B}}({\mathcal{K}}_{L})\otimes{\mathbb{C}}\mathbb{I}_{{\mathcal{K}}_{R}}.$

From now on, we use this structure of GNS triple repeatedly. Furthermore,
because $\omega_{\Phi}$ is a unique gapped ground state of $\beta$-invariant
interaction, it is $\beta$-invariant. Therefore, by Theorem 1.2, there is a
unitary representation $V$ of $G$ on
${\mathcal{K}}_{L}\otimes{\mathcal{K}}_{R}$ such that

$\displaystyle\mathop{\mathrm{Ad}}\nolimits\left(V_{g}\right)\circ\left(\rho_{L}\otimes\rho_{R}\right)=\rho_{L}\beta_{g}^{L}\otimes\rho_{R}\beta_{g}^{R},\quad
V_{g}\Omega=\Omega.$

If we restrict this to ${\mathcal{A}}_{R}$, then we get

$\displaystyle\mathop{\mathrm{Ad}}\nolimits\left(V_{g}\right)\left(\mathbb{I}_{{\mathcal{K}}_{L}}\otimes\rho_{R}(A)\right)=\mathbb{I}_{{\mathcal{K}}_{L}}\otimes\rho_{R}\beta_{g}^{R}(A),\quad
A\in{\mathcal{A}}_{R}.$

From this and irreducibility of $\rho_{R}$, we obtain a $*$-automorphism
$\gamma_{g}$ of ${\mathcal{B}}({\mathcal{K}}_{R})$ such that

$\displaystyle\mathop{\mathrm{Ad}}\nolimits\left(V_{g}\right)\left(\mathbb{I}_{{\mathcal{K}}_{L}}\otimes
x\right)=\mathbb{I}_{{\mathcal{K}}_{L}}\otimes\gamma_{g}(x),\quad
x\in{\mathcal{B}}({\mathcal{K}}_{R}),$
$\displaystyle\gamma_{g}\rho_{R}(A)=\rho_{R}\beta_{g}^{R}(A),\quad
A\in{\mathcal{A}}_{R}.$

Note that
$\gamma:G\to\mathop{\mathrm{Aut}}\nolimits\left({\mathcal{B}}({\mathcal{K}}_{R})\right)$
is a group action. By Theorem 1.3, there is a unitary $u_{R}(g)$ on
${\mathcal{K}}_{R}$ such that

$\displaystyle\mathop{\mathrm{Ad}}\nolimits(u_{R}(g))(x)=\gamma_{g}(x),\quad
x\in{\mathcal{B}}({\mathcal{K}}_{R}).$

Because $\gamma$ is a group action
$u_{R}:G\to{\mathcal{U}}\left({\mathcal{K}}_{R}\right)$ is a projective
representation. Hence we got a projective representation $u_{R}$ of $G$ on
${\mathcal{K}}_{R}$ such that

$\displaystyle\mathop{\mathrm{Ad}}\nolimits(u_{R}(g))\rho_{R}(A)=\rho_{R}\beta_{g}^{R}(A),\quad
A\in{\mathcal{A}}_{R}.$

Let $h_{R}$ be the second group cohomology class associated to the projective
representation. Using Theorem 1.3, one can show that this value $h_{R}$ is
independent of the choice of
${\mathcal{K}}_{R},\rho_{R},{\mathcal{K}}_{L},\rho_{L}$ (see [O4]). This value
turned out to be the desired invariant.

######  Theorem 2.10 ([O4]).

This second cohomology class $h_{\Phi}:=h_{R}\in H^{2}(G,{\rm U}(1))$ is an
invariant of $\sim_{\beta}$.

Now, let us investigate what this $H^{2}(G,{\rm U}(1))$-valued index
represents. By the same argument as the derivation of $u_{R}$, there is a
projective representation $u_{L}$ of $G$ on ${\mathcal{K}}_{L}$ such that

$\displaystyle\mathop{\mathrm{Ad}}\nolimits(u_{L}(g))\rho_{L}(A)=\rho_{L}\beta_{g}^{L}(A),\quad
A\in{\mathcal{A}}_{L}.$

Let $\sigma_{L}$, $\sigma_{R}$ be $2$-cocycles associated to projective
representations $u_{L}$, $u_{R}$. Now both of $V_{g}$ and $u_{L}(g)\otimes
u_{R}(g)$ implements $\beta_{g}$

$\displaystyle\mathop{\mathrm{Ad}}\nolimits\left(V_{g}\right)\circ\left(\rho_{L}\otimes\rho_{R}\right)=\rho_{L}\beta_{g}^{L}\otimes\rho_{R}\beta_{g}^{R}=\mathop{\mathrm{Ad}}\nolimits(u_{L}(g)\otimes
u_{R}(g))\circ\left(\rho_{L}\otimes\rho_{R}\right).$

Because $\rho_{L}\otimes\rho_{R}$ is irreducible, this means that

$V_{g}=c_{g}(u_{L}(g)\otimes u_{R}(g)),\quad g\in G,$

with some ${\rm U}(1)$-phase $c_{g}$. Absorbing this $c_{g}$ to $u_{L}(g)$, we
may assume that $c_{g}=1$. Note that it does not change the 2nd cohomology
class associated to $u_{L}$. Hence we obtain

$\displaystyle V_{g}=u_{L}(g)\otimes u_{R}(g),\quad g\in G.$ (2.7)

From the fact that $V$ is a genuine representation, while $u_{L}$, $u_{R}$ are
projective representations with $2$-cocycles $\sigma_{L}$, $\sigma_{R}$, we
get

$\displaystyle u_{L}(gh)\otimes u_{R}(gh)=V_{gh}=V_{g}V_{h}$
$\displaystyle=u_{L}(g)u_{L}(h)\otimes
u_{R}(g)u_{R}(h)=\sigma_{L}(g,h)\sigma_{R}(g,h)u_{L}(gh)\otimes u_{R}(gh).$

Hence we obtain the relation

$\displaystyle\sigma_{L}(g,h)\sigma_{R}(g,h)=\mathbb{I}.$

Let us think about what this means. First, $V$ is a genuine representation
corresponding to the $\beta$-action of $G$ on the whole chain. It splits into
left and right, to projective representations $u_{L}(g)$ and $u_{R}(g)$. Let
us consider the special case that $\omega_{\Phi}$ is of product form with
respect to left-right cut,
$\omega_{\Phi}=\omega_{\Phi,L}\otimes\omega_{\Phi,R}$. Note that
$\omega_{\Phi,L}$/ $\omega_{\Phi,R}$ is a pure
$\beta^{L}$/$\beta^{R}$-invariant state on
${\mathcal{A}}_{L}$/${\mathcal{A}}_{R}$. Let
$({\mathcal{H}}_{\omega_{\Phi,L}},\pi_{\omega_{\Phi,L}},\Omega_{\omega_{\Phi,L}})$,
$({\mathcal{H}}_{\omega_{\Phi,R}},\pi_{\omega_{\Phi,R}},\Omega_{\omega_{\Phi,R}})$
be GNS representations of $\omega_{\Phi,L}$, $\omega_{\Phi,R}$. Because
$\omega_{\Phi,L}$, $\omega_{\Phi,R}$ are pure, the representations
$\pi_{\omega_{\Phi,L}}$, $\pi_{\omega_{\Phi,R}}$ are irreducible. Because of
the $\beta^{L}$/$\beta^{R}$-invariance, there are genuine unitary
representations $u_{L}$, $u_{R}$ of $G$ on ${\mathcal{H}}_{\omega_{\Phi,L}}$,
${\mathcal{H}}_{\omega_{\Phi,R}}$, implementing $\beta^{L}$/$\beta^{R}$, and
fixing $\Omega_{\omega_{\Phi,L}}$, $\Omega_{\omega_{\Phi,R}}$. The triple

$\left({\mathcal{H}}_{\omega_{\Phi,L}}\otimes{\mathcal{H}}_{\omega_{\Phi,R}},\pi_{\omega_{\Phi,L}}\otimes\pi_{\omega_{\Phi,R}},\Omega_{\omega_{\Phi,L}}\otimes\Omega_{\omega_{\Phi,R}}\right)$

is a GNS triple of $\omega_{\Phi}=\omega_{\Phi,L}\otimes\omega_{\Phi,R}$, and

$V_{g}:=u_{L}(g)\otimes u_{R}(g)$

is the genuine representation implementing $\beta$ on
${\mathcal{A}}_{{\mathbb{Z}}}$ and
$V_{g}\left(\Omega_{\omega_{\Phi,L}}\otimes\Omega_{\omega_{\Phi,R}}\right)=\Omega_{\omega_{\Phi,L}}\otimes\Omega_{\omega_{\Phi,R}}$.
Hence in this product form case, the genuine representation $V$ split into
left and right genuine representations $u_{L},u_{R}$. In terms of cohomology,
$1\in H^{2}(G,{\rm U}(1))$ split into $1\cdot 1$. Namely, when there is no
correlation at all between left and right half infinite chains, the cohomology
split is trivial. From this, we can say that $h_{R}$ indicates how left-half
and right-half are correlated, in terms of the group action.

Lastly, we explain how to prove the stability of the index, Theorem 2.10.
Namely, we would like to show that our index is invariant with respect to
$\sim_{\beta}$. Let $\Phi_{1},\Phi_{2}\in{\mathcal{P}}_{U.G.}$ be
$\beta$-invariant interactions such that $\Phi_{1}\sim_{\beta}\Phi_{2}$. A key
observation is the following.

######  Proposition 2.11.

[O4] If $\Phi_{1}\sim_{\beta}\Phi_{2}$, then there are automorphisms
$\alpha_{L}\in\mathop{\mathrm{Aut}}\nolimits({\mathcal{A}}_{L})$ and
$\alpha_{R}\in\mathop{\mathrm{Aut}}\nolimits({\mathcal{A}}_{R})$ satisfying

(i)

$\omega_{\Phi_{2}}$ and
$\omega_{\Phi_{1}}\circ\left(\alpha_{L}\otimes\alpha_{R}\right)$ are quasi-
equivalent,

(ii)

$\alpha_{L}\circ\beta_{g,L}=\beta_{g,L}\circ\alpha_{L}$, and
$\alpha_{R}\circ\beta_{g,R}=\beta_{g,R}\circ\alpha_{R}$ for all $g\in G$.

(i) says that we can move from $\omega_{\Phi_{1}}$ to $\omega_{\Phi_{2}}$
without changing the correlation between left-right chains, by
$\alpha_{L}\otimes\alpha_{R}$. (ii) says that each $\alpha_{L}$ and
$\alpha_{R}$ do not change the representations of each sides. Hence left-half
and right-half correlation in terms of the group action, can not be changed by
$\alpha_{L}\otimes\alpha_{R}$. This proposition is nothing but the
factorization property of the automorphic equivalence explained in subsection
1.6.

To proceed the proof, let $u_{R,1}$, $u_{R,2}$,
$({\mathcal{K}}_{R,1},\rho_{R,1})$, $({\mathcal{K}}_{R,2},\rho_{R,2})$ be the
projective representations and irreducible representations given above,
corresponding to $\omega_{\Phi_{1}}$ and $\omega_{\Phi_{2}}$. They satisfy

$\displaystyle\mathop{\mathrm{Ad}}\nolimits\left({u_{R,1}(g)}\right)\circ\rho_{R,1}(A)=\rho_{R,1}\circ\beta_{g,R}(A),\quad\mathop{\mathrm{Ad}}\nolimits\left({u_{R,2}(g)}\right)\circ\rho_{R,2}(A)=\rho_{R,2}\circ\beta_{g,R}(A),$

for $A\in{\mathcal{A}}_{R}$. Let $\pi_{1}$, $\pi_{2}$ be GNS representations
of $\omega_{\Phi_{1}}$ and $\omega_{\Phi_{2}}$. Then we have

$\displaystyle\pi_{2}\sim_{\rm
q.e.}\pi_{1}\circ\left(\alpha_{L}\otimes\alpha_{R}\right),$

by (i) of Proposition 2.11. From this, we have

$\displaystyle\rho_{R,2}\sim_{\rm
q.e.}\left.\pi_{2}\right|_{{\mathcal{A}}_{R}}\sim_{\rm
q.e.}\left.\pi_{1}\right|_{{\mathcal{A}}_{R}}\circ\alpha_{R}\sim_{\rm
q.e.}\rho_{R,1}\circ\alpha_{R}.$

Hence there exists a $*$-isomorphism
$\tau:{\mathcal{B}}({\mathcal{K}}_{1})\to{\mathcal{B}}({\mathcal{K}}_{2})$
such that

$\displaystyle\rho_{R,2}=\tau\circ\rho_{R,1}\circ\alpha_{R}.$

By Theorem 1.3 and the irreducibility of $\rho_{R,1},\rho_{R,2}$, there exists
a unitary $W:{\mathcal{K}}_{1}\to{\mathcal{K}}_{2}$ such that
$\tau=\mathop{\mathrm{Ad}}\nolimits(W)$. Using this, we get

$\displaystyle\mathop{\mathrm{Ad}}\nolimits(u_{R,2}(g))\circ\rho_{2,R}=\rho_{2,R}\circ\beta_{g,R}=\tau\circ\rho_{R,1}\circ\alpha_{R}\circ\beta_{g,R}$

Now recall that $\alpha_{R}$ and $\beta_{g,R}$ commutes. Therefore,

$\displaystyle\tau\circ\rho_{R,1}\circ\alpha_{R}\circ\beta_{g,R}=\tau\circ\rho_{R,1}\circ\beta_{g,R}\circ\alpha_{R}$
$\displaystyle=\tau\circ\mathop{\mathrm{Ad}}\nolimits\left({u_{R,1}(g)}\right)\circ\rho_{R,1}\circ\alpha_{R}=\tau\circ\mathop{\mathrm{Ad}}\nolimits\left({u_{R,1}(g)}\right)\circ\tau^{-1}\circ\rho_{R,2}$
$\displaystyle=\mathop{\mathrm{Ad}}\nolimits\left(\mathop{\mathrm{Ad}}\nolimits_{W}(u_{R,1}(g))\right)\circ\rho_{R,2}$

Hence there is $c_{g}\in U(1)$ such that

$\displaystyle
u_{R,2}(g)=c_{g}\mathop{\mathrm{Ad}}\nolimits_{W}(u_{R,1}(g))\quad g\in G.$

From this, $u_{R,1}$ and $u_{R,2}$ have the same second cohomology class. The
same proof goes through for anti-linear actions.

###  2.6 Lieb-Schultz-Mattis (LSM) type theorems

The cohomology valued index $h_{\Phi,\beta}$ is important as an invariant of
our classification of SPT phases. However, there is still another use of it,
that is, Lieb-Schultz-Mattis (LSM) type theorem which we explain in this
subsection.

It is a hard problem to decide if a given interaction has a unique gapped
ground state or not. For example, Haldane conjectured in 1983 that
Antiferromagnetic Heisenberg chain

$\displaystyle\Phi_{Heisenberg}(\\{x,x+1\\})=\sum_{j=1}^{3}\,S^{(x)}_{j}S^{(x+1)}_{j},$

does not have a unique gapped ground state if $d$ is even, while it does if
$d$ is odd. The latter part of the conjecture is still an open problem. Lieb-
Schultz-Mattis [LSM] and Affleck-Lieb [AL] proved affirmatively the first part
of the conjecture. They proved the following : if interaction has ${\rm U}(1)$
on-site symmetry and space translation symmetry and the spin-size $d$ is even,
then it cannot have a unique gapped ground state. They used the continuity of
${\rm U}(1)$ in an ingenious way, to prove the existence of low-energy
excitation. There has been a lot of progress after that [H3][Os][NS07][M2],
but traditionally, the symmetry has been some continuous ones. Recently, it
was claimed by physicists that Lieb-Schultz-Mattis type theorem holds for some
discrete symmetries. It is proved for matrix product states parent
Hamiltonians [CGW], [WPVZ] [Fu].

Using our $H^{2}(G,{\rm U}(1))$-valued index, we can prove their claim. Let
$G$ be a finite group and let
$\beta_{g}:=\bigotimes_{{\mathbb{Z}}}\mathop{\mathrm{Ad}}\nolimits(U_{g})$
with $U$ a projective representation of $G$ on ${\mathbb{C}}^{d}$.

######  Theorem 2.12 ([OT], [OTT]).

If the second cohomology class $h$ of $U$ is non-trivial, then there is no
translation invariant interaction in ${\mathcal{P}}_{U.G.\beta}$.

  * Proof.

Suppose that there exists an interaction $\Phi\in{\mathcal{P}}_{U.G.\beta}$
which is translation invariant. Let $h_{\Phi,\beta}^{x}\in H^{2}(G,U(1))$ be
an index associated to
$\left.\omega_{\Phi}\right|_{{\mathcal{A}}_{[x,\infty)\cap{\mathbb{Z}}}}$ for
each $x\in{\mathbb{Z}}$. (Note that we can cut the chain anywhere.) Then we
have

$h_{\Phi,\beta}^{x}=h+h_{\Phi,\beta}^{x+1},\quad\text{and}\quad
h_{\Phi,\beta}^{x}=h_{\Phi,\beta}^{x+1}.$

Hence $h=0$.  $\square$

Recall that antiferromagnetic Heisenberg chain $\Phi_{Heisenberg}$ has the
${\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}$-symmetry. There, cohomology class of
the projective representation $U$ of ${\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}$
is trivial if $d$ is odd, nontrivial if $d$ is even. Hence we have
$\Phi_{Heisenberg}\notin{\mathcal{P}}_{U.G.\beta}$, if $d$ is even. This gives
an alternative proof for the first half of Haldane conjecture. Analogously, we
can prove the following, using the $H^{2}(G,{\rm U}(1))$-index.

######  Theorem 2.13 ([OTT]).

If there is $\Phi\in{\mathcal{P}}_{U.G.\beta}$, origin-reflection invariant,
then $h=2h_{1}$ with some $h_{1}\in H^{2}(G,{\rm U}(1))$.

###  2.7 A classification of pure states on quantum spin chains satisfying the
split property with on-site finite group symmetries

We obtained an invariant of the classification of SPT phases for on-site
finite group symmetry, It is an open problem if they are complete invariant.

However, for on-site symmetry, we still can say that it is a complete
invariant of some (philosophically) analogous classification. That is, a
classification of pure states on quantum spin chains satisfying the split
property with on-site finite group symmetries,

The classification we consider here is the following.

######  Definition 2.14 ([O5]).

Let $\omega_{0}$, $\omega_{1}$ be pure $\beta$-invariant states on
${\mathcal{A}}_{{\mathbb{Z}}}$ satisfying the split property with respect to
${\mathcal{A}}_{L}$, ${\mathcal{A}}_{R}$.
We write $\omega_{0}\sim_{{\rm split},\beta}\omega_{1}$ if there exist
automorphisms $\alpha_{L}$ and $\alpha_{R}$ on ${\mathcal{A}}_{L}$,
${\mathcal{A}}_{R}$ such that

(i)

$\omega_{1}$ and $\omega_{0}\circ\left(\alpha_{L}\otimes\alpha_{R}\right)$ are
quasi-equivalent,

(ii)

$\alpha_{L}\circ\beta_{g,L}=\beta_{g,L}\circ\alpha_{L}$, and
$\alpha_{R}\circ\beta_{g,R}=\beta_{g,R}\circ\alpha_{R}$ for all $g\in G$.

How can this be (philosophically) analogous to our original classification?
The interpretation comes from rough identification that

gapped

$\approx$ ground state is almost product with respect to left-right cut

$\approx$ ground state has the split property.

These $\approx$ are not $=$. However, philosophically, they share the concept.
From this analogy, the existence of smooth path of interactions with unique
gapped ground state is rephrased as the existence of a map which does not
change the correlation between left and right chains. In (i) of the
classification, we are considering an automorphism
$\alpha_{L}\otimes\alpha_{R}$ of product form. This is a kind of map which
does not change the correlation between left and right chains at all. Hence,
we regard this (i) as an analog of $\sim$. With this interpretation, adding
the condition (ii) corresponds to $\sim_{\beta}$, that is, the requirement the
symmetry should be preserved along the path.

Note that the $H^{2}(G,{\rm U}(1))$-valued index is actually defined not only
for $\omega_{\Phi}$ with $\beta$-invariant $\Phi\in{\mathcal{P}}_{U.G.}$, but
it is also defined for a $\beta$-invariant pure split state $\omega$. Let us
denote that $H^{2}(G,{\rm U}(1))$-valued index by $h(\omega)$, for
$\beta$-invariant pure split state $\omega$.

Then we obtain the following.

######  Theorem 2.15.

[O5] The $H^{2}(G,{\rm U}(1))$-valued index is a complete invariant of
$\sim_{{\rm split},\beta}$.

Homogeneity of pure state space of separable simple $C^{*}$-algebras has been
studied extensively from old time in the operator algebra community [P], [B],
[FKK], [KOS]. For the proof of Theorem 2.15, we used the machinery developed
in these papers. The point of our analysis, compared to theirs is how to
encode the $H^{2}(G,{\rm U}(1))$ to the problem. We encode the information of
$[\sigma]_{H^{2}(G,U(1))}$ to the twisted dynamical system
$\Sigma^{(\sigma)}=(G,{\mathcal{A}}_{R},\beta_{R},\sigma)$, and consider the
twisted crossed product $C^{*}(\Sigma^{(\sigma)})$ of $\Sigma^{(\sigma)}$.
Then each $\omega$ with $h(\omega)=[\sigma]_{H^{2}(G,U(1))}$ gives a covariant
representation of $\Sigma^{(\sigma)}$. We then consider the homogeneity
problem for $C^{*}\left(\Sigma^{(\sigma)}\right)$.

###  2.8 An easy derivation of $H^{2}(G,{\rm U}(1))$

In subsection 2.1, we derived $H^{2}(G,{\rm U}(1))$-valued index for SPT
phases with on-site symmetry, using the split propety, proven for all the
unique gapped ground states. It relied on Matsui’s Theorem, which is a
corollary of Hasting’s area law [H1]. Hasting’s area law is quite a big
theorem. However, if you are interested in only on SPT phases, then there is a
handy way to derive the $H^{2}(G,{\rm U}(1))$-valued index. The point is, if
$\Phi$ is in a SPT phase, by definition, we have $\Phi\sim\Phi_{0}$, with the
fixed reference on-site interaction $\Phi_{0}$. Using the automorphic
equivalence, this means there is an automorphism $\alpha$ given by local
interactions such that $\omega_{\Phi}=\omega_{\Phi_{0}}\circ\alpha$. Recall
that $\omega_{\Phi_{0}}$ is a pure product state. Therefore, there are pure
states $\omega_{L}$, $\omega_{R}$ on ${\mathcal{A}}_{L}$, ${\mathcal{A}}_{R}$
such that

$\omega_{\Phi_{0}}=\omega_{L}\otimes\omega_{R}.$

On the other hand, as we saw in subsection 1.6, $\alpha$ satisfies a
factorization property

$\alpha=({\rm{inner}})\circ\left(\alpha_{L}\otimes\alpha_{R}\right).$

Now, because $\omega_{\Phi}$ is a unique ground state of $\beta$-invariant
interaction $\Phi$, it is $\beta$-invariant. From this and above
decompositions of $\omega_{\Phi_{0}}$, $\alpha$, we get

$\displaystyle\left(\omega_{L}\alpha_{L}\otimes\omega_{R}\alpha_{R}\right)\circ({\rm{inner}})=\omega_{\Phi}=\omega_{\Phi}\circ\beta_{g}=\left(\omega_{L}\alpha_{L}\beta_{g}^{L}\otimes\omega_{R}\alpha_{R}\beta_{g}^{R}\right)\circ({\rm{inner}}).$

Because $\omega_{L}$, $\omega_{R}$ are pure states, we may apply Lemma 1.7.
Let $({\mathcal{H}}_{R},\pi_{R},\Omega_{R})$ be a GNS representation of
$\omega_{R}$. Because $\omega_{R}$ is pure, $\pi_{R}$ is irreducible. Applying
this Lemma, we obtain a unitary $u_{g}$ on ${\mathcal{H}}_{R}$ such that

$\displaystyle\mathop{\mathrm{Ad}}\nolimits(u_{g})\circ\pi_{R}\circ\alpha_{R}(A)=\pi_{R}\circ\alpha_{R}\circ\beta_{g}^{R}(A),\quad
A\in{{\mathcal{A}}_{R}}.$

Note that

$\displaystyle\mathop{\mathrm{Ad}}\nolimits(u_{g}u_{h})\circ\pi_{R}\circ\alpha_{R}(A)=\pi_{R}\circ\alpha_{R}\circ\beta_{g}^{R}\beta_{h}^{R}(A)$
$\displaystyle=\pi_{R}\circ\alpha_{R}\circ\beta_{gh}^{R}(A)=\mathop{\mathrm{Ad}}\nolimits(u_{gh})\circ\pi_{R}\circ\alpha_{R}(A),$

for each $A\in{{\mathcal{A}}_{R}}$ and $g,h\in G$. Because $\pi_{R}$ is
irreducible, this implies

$\mathop{\mathrm{Ad}}\nolimits(u_{g}u_{h})(x)=\mathop{\mathrm{Ad}}\nolimits(u_{gh})(x)$

for all $x\in{\mathcal{B}}({\mathcal{H}})$. Hence there is some
$\sigma(g,h)\in{\rm U}(1)$ such that

$u_{g}u_{h}=\sigma(g,h)u_{gh},\quad g,h\in G.$

Namely, $u_{g}$ is a projective representation. In fact this is the very
projective representation $u_{R}(g)$ we found in subsection 2.1.

###  2.9 SPT phases with the reflection symmetry

At the end of this section, we consider SPT-phases with the reflection
symmetry. Let us first explain the ${\mathbb{Z}}_{2}$-index introduced by
Pollmann.et.al for the reflection symmetry [PTBO1][PTBO2]. Let
$\omega_{{\boldsymbol{v}}}$ be a $\Theta_{R}$-invariant MPS given by a
primitive ${\boldsymbol{v}}\in\mathop{\mathrm{M}}\nolimits_{k}^{\times d}$.
Let $\rho_{\boldsymbol{v}}$ be the density matrix corresponding to
$\varphi_{\boldsymbol{v}}$. (Recall Theorem 2.3.) Replacing $v_{\mu}$ with
$e_{{\boldsymbol{v}}}^{\frac{1}{2}}v_{\mu}e_{{\boldsymbol{v}}}^{-\frac{1}{2}}$,
we may assume that $e_{\boldsymbol{v}}=\mathbb{I}$. Because
$\omega_{{\boldsymbol{v}}}$ is $\Theta_{R}$-invariant, we can check that

$\tilde{v}_{\mu}:=\rho_{{\boldsymbol{v}}}^{-\frac{1}{2}}\bar{v_{\mu}}^{*}\rho_{{\boldsymbol{v}}}^{\frac{1}{2}},\quad\mu=1,\ldots,d$

also give the same state. Here, $\bar{\cdot}$ indicates the complex
conjugation with respect to a CONS of $\mathop{\mathrm{M}}\nolimits_{k}$
diagonalizing $\rho_{{\boldsymbol{v}}}$. Namely, we have

$\omega_{\tilde{\boldsymbol{v}}}=\omega_{{\boldsymbol{v}}}.$

Then, using the fundamental theorem of MPS (Theorem 2.4), one can show that
there is a $c\in{\mathbb{T}}$, a $k\times k$ anti-unitary matrix $\theta$,
such that

$\displaystyle v_{\mu}=c\rho_{{\boldsymbol{v}}}^{-\frac{1}{2}}\theta
v_{\mu}^{*}\theta\rho_{{\boldsymbol{v}}}^{\frac{1}{2}},\quad\mu=1,\ldots,d.$
(2.8)

(See [O6] Lemma 5.1 for the detailed proof.) As in section 2.4, using the
primitivity of ${\boldsymbol{v}}$, we can conclude that

$\theta^{2}=\mathbb{I}\quad\text{or}\quad\theta^{2}=-\mathbb{I}.$

This signature is Pollmann et.al.’s ${\mathbb{Z}}_{2}$-index.

Now, as before, we would like to extend it to general $\omega_{\Phi}$, for
$\Theta_{R}$-invariant $\Phi\in{\mathcal{P}}_{U.G.}$. Let
$\Phi\in{\mathcal{P}}_{U.G.}$ be a $\Theta_{R}$-invariant interaction. By the
split property and the reflection symmetry of $\omega_{\Phi}$, the GNS of
$\omega_{\Phi}$ is of the form

$\displaystyle({\mathcal{H}}\otimes{\mathcal{H}},\hat{\pi}:=\left(\pi\circ\Theta_{R}|_{{\mathcal{A}}_{L}}\right)\otimes\pi,\Omega).$

Let $\Gamma$ be the unitary implementing $\Theta_{R}$, i.e.,

$\displaystyle\Gamma\hat{\pi}(A)\Omega=\hat{\pi}\circ\Theta_{R}(A)\Omega,\quad
A\in{\mathcal{A}}.$

Then there is a $\sigma=\pm 1$ such that

$\displaystyle\Gamma\left(\xi\otimes\eta\right)={\sigma}\left(\eta\otimes\xi\right),\quad\text{for
all}\quad\xi,\eta\in{\mathcal{H}}.$

This $\sigma$ is our ${\mathbb{Z}}_{2}$-valued index.

######  Definition 2.16 (O ’19).

Define the index of $\Phi\in{\mathcal{P}}_{U.G.,\Theta_{R}}$ by
$\sigma_{\Phi,\Theta_{R}}:=\sigma$.

From the left-right symmetric form,

$\displaystyle({\mathcal{H}}\otimes{\mathcal{H}},\hat{\pi}:=\left(\pi\circ\Theta_{R}|_{{\mathcal{A}}_{L}}\right)\otimes\pi,\Omega),$

the index $\sigma$ allows a characterization with the Tomita-Takesaki theory.
We briefly explain it here. For simplicity, we assume that the GNS-vector
$\Omega$ is cyclic and separating for
${\mathbb{C}}\mathbb{I}\otimes{\mathcal{B}}({\mathcal{H}})$ in
${\mathcal{H}}\otimes{\mathcal{H}}$. (Otherwise, we restrict
${\mathbb{C}}\mathbb{I}\otimes{\mathcal{B}}({\mathcal{H}})$ to the support of
$\Omega$.) By this simplification, there is an anti-liner operator on
${\mathcal{H}}\otimes{\mathcal{H}}$ with domain
$\left({\mathbb{C}}\mathbb{I}\otimes{\mathcal{B}}({\mathcal{H}})\right)\Omega$
such that

$\displaystyle S\left(\mathbb{I}\otimes
x\right)\Omega:=\left(\mathbb{I}\otimes x^{*}\right)\Omega,\quad
x\in{\mathcal{B}}({\mathcal{H}}).$ (2.9)

This $S$ is closable. Let

$\bar{S}=J\Delta^{\frac{1}{2}}$

be the polar decomposition of $\bar{S}$. The anti-linear operator $J$ is
called the modular conjugation and the positive operator $\Delta$ is called
the modular operator. By a general theory of Tomita-Takesaki theory (see
[BR1]), we have

$\displaystyle\Delta\Omega=J\Omega=\Omega.$ (2.10)

Then the following holds.

######  Proposition 2.17 ([O6]).

There is an anti-unitary $\theta$ on ${\mathcal{H}}$ such that

$\displaystyle J\left(\mathbb{I}\otimes x\right)J^{*}=\theta
x\theta^{*}\otimes\mathbb{I},\quad\theta^{2}=\sigma_{\Phi,\Theta_{R}}\mathbb{I}.$

This characterization can be used to show that our index is the generalization
of Pollmann et.al.’s index on MPS. Roughly speaking, it can be understood as
follows. Let us consider a $\Theta_{R}$-invariant MPS
$\omega_{\boldsymbol{v}}$ given by a primitive ${\boldsymbol{v}}$. For
simplicity, we still pretend that the GNS-vector $\Omega$ of
$\omega_{{\boldsymbol{v}}}$ is cyclic and separating for
${\mathbb{C}}\mathbb{I}\otimes{\mathcal{B}}({\mathcal{H}})$ in
${\mathcal{H}}\otimes{\mathcal{H}}$, although it is not true for MPS.(See the
original paper [O6] for proper formulation.) In this case, we have

$\displaystyle\Delta^{\frac{1}{2}}\left(\mathbb{I}\otimes
x\right)\Omega=\left(\mathbb{I}\otimes\rho_{\boldsymbol{v}}^{\frac{1}{2}}x\rho_{\boldsymbol{v}}^{-\frac{1}{2}}\right)\Omega,\quad
x\in B({\mathcal{H}}).$ (2.11)

Recall that we may take $v_{\mu}$ as a restriction of $S_{\mu}$ in (2.5) to
some finite dimensional subspace. For simplicity, let us omit the difference
between $S_{\mu}$ and $v_{\mu}$. The point is, in our setting, there are two
ways to send operators in ${\mathbb{C}}\mathbb{I}\otimes B({\mathcal{H}})$ to
operators in ${\mathcal{B}}({\mathcal{H}})\otimes{\mathbb{C}}\mathbb{I}$. One
is by modular theory, that is,

$\displaystyle\begin{split}&\left(\mathbb{I}\otimes\rho_{\boldsymbol{v}}^{\frac{1}{2}}\theta
S_{\mu}\theta^{*}\rho_{\boldsymbol{v}}^{-\frac{1}{2}}\right)\Omega=J\Delta^{\frac{1}{2}}\left(\mathbb{I}\otimes\rho_{\boldsymbol{v}}^{-\frac{1}{2}}\theta
S_{\mu}^{*}\theta^{*}\rho_{\boldsymbol{v}}^{\frac{1}{2}}\right)\Omega\\\
&=J\left(\mathbb{I}\otimes\theta
S_{\mu}^{*}\theta^{*}\right)\Omega=J\left(\mathbb{I}\otimes\theta
S_{\mu}^{*}\theta^{*}\right)J^{*}\Omega\\\
&=\left(S_{\mu}^{*}\otimes\mathbb{I}\right)\Omega.\end{split}$

Here, we used (2.9), (2.11), (2.10), and Proposition 2.17. On the other hand,
using the fact that $\omega_{\boldsymbol{v}}$ is translationally invariant, we
can show that there is some $a\in{\mathbb{T}}$ such that

$\displaystyle a\left(\mathbb{I}\otimes
S_{\mu}^{*}\right)\Omega=\left(S_{\mu}^{*}\otimes\mathbb{I}\right)\Omega,\quad\mu=1,\ldots,d.$

To understand it heuristically, recall that the left side of
${\mathcal{H}}\otimes{\mathcal{H}}$ corresponds to left infinite chain, while
the right-hand side corresponds to the right infinite chain. Comparing these
equation, and the assumption that $\Omega$ is separating for
${\mathbb{C}}\mathbb{I}\otimes{\mathcal{B}}({\mathcal{H}})$, we obtain

$\displaystyle\rho_{\boldsymbol{v}}^{\frac{1}{2}}\theta
S_{\mu}\theta^{*}\rho_{\boldsymbol{v}}^{-\frac{1}{2}}=aS_{\mu}^{*},\quad\mu=1,\ldots,d.$

This corresponds to (2.8). Therefore, the signature of $\theta^{2}$ is the
Pollmann et.al.’s index. However, from Proposition 2.17, we have
$\theta^{2}=\sigma_{\Phi,\Theta_{R}}\mathbb{I}$. Hence our index coincides
with Pollman’s index in MPS.

##  3 SPT-phases in $2$-dimensional systems

In this section we explain the derivation of $H^{3}(G,{\rm U}(1))$-valued
index for $2$-dimensional SPT obtained in [O7]. Before starting the analysis,
we fix some notation. We denote by $H_{L}$, $H_{R}$, $H_{U}$, $H_{D}$, the
left, right, upper, lower half-planes i.e.,

$\displaystyle H_{L}:=\left\\{(x,y)\in{\mathbb{Z}}^{2}\mid
x\leq-1\right\\},\quad H_{R}:=\left\\{(x,y)\in{\mathbb{Z}}^{2}\mid 0\leq
x\right\\},$ $\displaystyle H_{U}:=\left\\{(x,y)\in{\mathbb{Z}}^{2}\mid 0\leq
y\right\\},\quad H_{D}:=\left\\{(x,y)\in{\mathbb{Z}}^{2}\mid
y\leq-1\right\\}.$

For each $\Gamma\subset{\mathbb{Z}}^{\nu}$, let $\beta_{g}^{\Gamma}$ be the
restriction of $\beta_{g}$ to ${\mathcal{A}}_{\Gamma}$. With a bit abuse of
notation, we also use the same symbol $\beta_{g}^{\Gamma}$ to denote
$\beta_{g}^{\Gamma}\otimes\mathop{\mathrm{id}}\nolimits_{{\mathcal{A}}_{\Gamma^{c}}}$.
We use notation $\beta_{g}:=\beta_{g}^{{\mathbb{Z}}^{2}}$,
$\beta_{g}^{U}:=\beta_{g}^{H_{U}}$, $\beta_{g}^{RU}:=\beta_{g}^{H_{R}\cap
H_{U}}$, $\beta_{g}^{LU}:=\beta_{g}^{H_{L}\cap H_{U}}$. Throughout this
section, we fix a trivial $\Phi_{0}\in{\mathcal{P}}_{U.G.}^{0}$. Recall that
its unique gapped ground state $\omega_{\Phi_{0}}$ is of product form

$\displaystyle\omega_{\Phi_{0}}=\bigotimes_{x\in{\mathbb{Z}}^{\nu}}\rho_{\xi_{x}}.$

We denote by $\omega_{R}$, $\omega_{L}$, $\omega_{U}$, $\omega_{D}$, the
restriction of $\omega_{\Phi_{0}}$ to ${\mathcal{A}}_{H_{R}}$,
${\mathcal{A}}_{H_{L}}$, ${\mathcal{A}}_{H_{U}}$, ${\mathcal{A}}_{H_{D}}$
respectively. Because $\omega_{\Phi_{0}}$ is a pure state of a product form,
both of $\omega_{R}$ and $\omega_{L}$ are pure. Let
$({\mathcal{H}}_{L},\pi_{L},\Omega_{L})$,
$({\mathcal{H}}_{R},\pi_{R},\Omega_{R})$ be GNS-triples of $\omega_{L}$,
$\omega_{R}$, respectively. Then $({\mathcal{H}}_{0},\pi_{0},\Omega_{0})$,
with ${\mathcal{H}}_{0}={\mathcal{H}}_{L}\otimes{\mathcal{H}}_{R}$,
$\pi_{0}:=\pi_{L}\otimes\pi_{R}$, $\Omega_{0}:=\Omega_{L}\otimes\Omega_{R}$ is
a GNS triple for $\omega_{\Phi_{0}}$. Note that all of $\pi_{L}$, $\pi_{R}$,
$\pi_{0}$ are irreducible because $\omega_{L}$, $\omega_{R}$, $\omega_{0}$ are
pure.

###  3.1 Derivation of $H^{3}(G,{\rm U}(1))$ out of cocycle action

In one dimensional system, we obtained a $H^{2}(G,{\rm U}(1))$-valued index
out of projective representations. According to physicists, we are supposed to
find a $H^{3}(G,{\rm U}(1))$-valued index. How can we find such an index?

We can find a hint in literatures of operator algebra on the classification of
cocycle actions. [C], [J].

Let $\\{\gamma_{g}\\}_{g\in G}$ be automorphisms on a factor ${\mathcal{M}}$.
Assume that there are unitaries $\\{u(g,h)\\}_{g,h\in
G}\subset{\mathcal{U}}({\mathcal{M}})$ such that

$\displaystyle\gamma_{g}\gamma_{h}=\mathop{\mathrm{Ad}}\nolimits\left(u(g,h)\right)\gamma_{gh}.$
(3.1)

This $\gamma$ is called a cocycle action. Using this relation twice, we obtain

$\displaystyle\gamma_{g}\gamma_{h}\gamma_{k}=\mathop{\mathrm{Ad}}\nolimits\left(u(g,h)\right)\gamma_{gh}\gamma_{k}=\mathop{\mathrm{Ad}}\nolimits\left(u(g,h)u(gh,k)\right)\gamma_{ghk}.$

Here we used the relation (3.1) to $\gamma_{g}\gamma_{h}$ and then to
$\gamma_{gh}\gamma_{k}$. On the other hand, we have

$\displaystyle\gamma_{g}\gamma_{h}\gamma_{k}=\gamma_{g}\mathop{\mathrm{Ad}}\nolimits\left(u(h,k)\right)\gamma_{hk}=\mathop{\mathrm{Ad}}\nolimits\left(\gamma_{g}\left(u(h,k)\right)\right)\gamma_{g}\gamma_{hk}$
$\displaystyle=\mathop{\mathrm{Ad}}\nolimits\left(\gamma_{g}\left(u(h,k)\right)u(g,hk)\right)\gamma_{ghk}.$

Here we used the relation (3.1) to $\gamma_{h}\gamma_{k}$ and then
$\gamma_{g}\gamma_{hk}$. By the associativity, of course above two equations
should give the same value. Hence we get

$\displaystyle\mathop{\mathrm{Ad}}\nolimits\left(u(g,h)u(gh,k)\right)(a)=\mathop{\mathrm{Ad}}\nolimits\left(\gamma_{g}\left(u(h,k)\right)u(g,hk)\right)(a),\quad
a\in{\mathcal{M}}.$

This means

$\displaystyle\left(u(g,h)u(gh,k)\right)^{*}\left(\gamma_{g}\left(u(h,k)\right)u(g,hk)\right)\in{\mathcal{M}}\cap{\mathcal{M}}^{\prime}={\mathbb{C}}\mathbb{I}.$

Namely, there exists some phase $c(g,h,k)$ such that

$\displaystyle
u(g,h)u(gh,k)=c(g,h,k)\gamma_{g}\left(u(h,k)\right)u(g,hk),\quad g,h,k\in G.$
(3.2)

Using this relation repeatedly, we can show that $c:G^{\times 3}\to{\rm U}(1)$
satisfies the $3$-cocycle relation. Indeed, for $g,h,k,f\in G$, by repeated
use of (3.2), we get

$\displaystyle u(g,h)u(gh,k)u(ghk,f)=\left[u(g,h)u(gh,k)\right]u(ghk,f)$
$\displaystyle=c(g,h,k)\gamma_{g}\left(u(h,k)\right)u(g,hk)u(ghk,f)$
$\displaystyle=c(g,h,k)\gamma_{g}\left(u(h,k)\right)\cdot\left[u(g,hk)u(ghk,f)\right]$