Interacting Fermi systems in the tracial state

Heide Narnhofer ∗

Fakultät für Physik

Universität Wien

We argue that for Fermi systems on lattices or the continuum with interaction
invariant under a kind of Galilei transformation the time evolution is either
weakly asymptotically abelian or at least $\eta-$abelian in the tracial state
but not norm asymptotically abelian.

Keywords: time invariant states, interacting systems

∗ E–mail address: heide.narnhofer@ univie.ac.at

##  1 Introduction

In order to understand the convergence to equilibrium in the Eigenstate-
Thermalization-Hypothesis-approach (ETH) [1], [2] local operators are
expressed by eigenstates of the hamiltonian and the resulting spectrum given
as difference of the eigenvalues of the hamiltonian determines the time
dependence of these operators. However these eigenstates exist only for finite
systems. We have to turn to the thermodynamic limit of infinite systems. This
limit is best described by the $C^{*}-$algebra of quasilocal operators and we
will concentrate on fermions with creation and annihilation operators on the
lattice or on the continuum. Under appropriate assumptions the time evolution
is given as a continuous automorphism group [3] or [4]. This again allows to
define the Arveson spectrum [5] of a quasilocal operator in correspondence to
the spectrum for a finite system. But now this spectrum is only defined as a
set, that in general is infinite and cannot be used to control convergence
properties or relations between different operators. This changes if we
consider time invariant states. In these states the time evolution is given by
unitary operators and their spectral decomposition is inherited by the
Arveson-spectrum. However the spectral decomposition depends on the invariant
state. But it allows to determine some behaviour on the level of the
$C^{*}$-algebra that may be relevant with respect to the existence of other
invariant states.

As first result we will exclude for a large class of interactions the
existence of operators periodic under time evolution and also the possibility
that local operators remain local for all times. This can be compared to the
Lieb-Robinson bound [6] that shows that in general we can expect spreading
with some finite velocity but cannot guarantee that spreading really happens
for all operators. With interaction between different lattice-points operators
really get delocalized. It remains to characterize this delocalisation in more
detail. For quasifree evolution it is well known that the result is much
stronger: creation and annihilation operators anticommute asymptotically and
therefore time translated even products of creation and annihilation operators
commute asymptotically in time with local operators, therefore escape to
infinity. But already for the XY-model (though not a Fermi-system but closely
related to) it was shown [7] that the time evolution is not norm-
asymptotically abelian for the total algebra, though this holds for the even
subalgebra. Nevertheless some local operators neither commute nor anticommute
asymptotically, therefore these time-translated operators consist of a local
part combined with a delocalized part. Nevertheless the expectation-values of
their products factorize in the limit for time-invariant states with GNS-
representation that is cyclic and separating. The system is weakly
asymptotically abelian.

If we examine the existing models for which it is known that only a single
time invariant state exists [8], [9], [10], [11], [12] in all these models
neither norm asymptotic abelianess nor strong asymptotic abelianess is
satisfied. Operators are delocalized but they still do not commute with some
local operators. All these models do not have a physical interpretation. But
they serve as a hint, that a kind of algebraic entanglement created in the
course of time between distances far apart might be responsible for the
approach to equilibrium as a stronger fact than approach to just some
invariant state and might reduce the existence of invariant states other than
equilibrium-states.

In order to study whether interacting systems are not strongly asymptotically
abelian but keep non trivial commutations with local operators we will study
the behaviour in the tracial state. Here the quasilocal algebra can be
extended to a doubled quasilocal algebra. For quasifree evolutions we have a
stronger symmetry relation that is responsible for norm asymptotic abelianess,
but its violation by interaction enables to keep algebraic entanglement with
local operators in the course of time.

In this note we first define in section 1 the systems we want to consider:
Fermi-lattice systems and Galilei invariant continuous systems, i. e. systems
for which time evolution is defined as approximately local automorphisms. In
section 2 we discuss the possibilities how time evolution can be localized or
delocalized. In section 3 we work in the representation given by the tracial
state and transfer the result on continuous systems [13] that the systems gets
delocalized in the course of time to Fermi-lattice systems. In section 4 we
show that in the course of time operators do not get completely delocalized if
the system is interacting.

Finally we discuss why the failing of strong asymptotic abelianess might be of
relevance for thermodynamical behaviour. For odd operators that anticommute it
is well know that their expectation value has to vanish in invariant states.
Now we increased the class of operators that do not asymptotically commute on
the algebraic level. We give hints how these algebraic correlations can give
additional restrictions on the invariant states based on the arguments used
for anticommutativity in comparison to the consequences of randomness in [10].

##  2 The Fermi-algebra

The Fermi algebra is built by creation and annihilation operators
$a^{*}(f),a(g)$ satisfying

$a(f)a(g)+a(g)a(f)=0,\quad a^{*}(f)a(g)+a(g)a^{*}(f)=\langle g|f\rangle$ (1)

where $f$ is a square-integrable function either on the lattice or on the
continuum. The even algebra ${\cal A}_{e}$ is built by even polynomials in
creation and annihilation operators. We pick some $f_{0}$ which defines the
unitary

$U_{0}=a(f_{0})+a^{*}(f_{0})\quad U_{0}^{2}=1$ (2)

This unitary implements the automorphism $\alpha_{0}(A)=U_{0}AU_{0}$ on the
algebra and also a non-inner automorphism on the even algebra. It evolves in
time. We are interested in its limit for large times. Anticommutativity on the
odd algebra implements commutativity on the even algebra and holds, if the
automorphism implemented by $\tau_{t}U_{0}$ becomes trivial.

We will mainly concentrate in our proofs on Fermi algebras on the lattice. We
consider time evolutions that are implemented by

$H_{\Lambda}=\sum_{x,y}a^{*}_{x}a_{y}f(x-y)+\sum_{xyz}a^{*}_{x}a^{*}_{y}a_{z}a_{-x-y-z}v(x,y,z)\quad
x,y,z\in\Lambda$ (3)

with taking the limit $\Lambda$ becoming the total lattice, which defines a
time-automorphism-group that is invariant under space translations. [3]. We
are inspired on results for continuous Fermi-systems with Galilei-invariance.
Here the time evolution is given by the hamiltonian in dimension $\nu$

$H=\frac{1}{2m}\int dx^{\nu}\nabla a^{*}(x)\nabla a(x)+\int
d^{\nu}pd^{\nu}p^{\prime}d^{\nu}qd^{\nu}q^{\prime}a^{*}_{pq}a^{*}_{p^{\prime}q^{\prime}}w(p-p^{\prime})v(q-q^{\prime})a_{p^{\prime}q^{\prime}}a_{pq}=K+V.$
(4)

Here $a_{pq}=a(W(p,q)f)$ is an annihilation operator smeared with an $f$ that
is translated by the one-particle Weyl operators $W(p,q)=e^{i(qP+pX)}.$ As
concrete example we take as in [4] $f$ as a Gauss function so that with
appropriate normalization in $\nu$ dimensions with the notation

$[a^{*}(x),a(y)]_{+}=\delta^{\nu}(x-y)$ $a(f(x))=a(f)=\int d^{\nu}xf(x)a(x)$
$a_{pq}=\pi^{-\frac{\nu}{4}}\int
d^{\nu}xe^{-\frac{(q-x)^{2}}{2}+ipx}a(x)=a(|p,q\rangle)$

where $|p,q\rangle$ are coherent states. The cutoff in the interaction is
introduced to guarantee that the time-evolution exists as continuous
automorphism-group. This is shown as for the lattice-system on the basis of
perturbation-theory.

##  3 Localization and delocalization

Classical dynamical systems are given by a probability space equipped with a
finite probability measure $\mu$ and a continuous measurable map $T(t)$ for
which the measure is invariant $\mu\circ T_{t}=\mu.$ The system is ergodic if
the average over $f(T_{t}x)$ is $\mu$-almost constant. This can be improved by
demanding that the average is replaced by the limit $t\rightarrow\pm\infty.$
The quantum analog is a quasilocal $C^{*}$-algebra, here the Fermi-algebra
built by creation and annihilation operators $a(f),a^{*}(g)$ satisfying

$a(f)a(g)=-a(g)a(f),\quad a(f)a^{*}(g)+a^{*}(g)a(f)=\langle f|g\rangle$ (5)

where $f(x)$ is a function either over $R^{\nu}$ or over the lattice. Space-
translation is given by $\sigma_{x}a(f(y))=a(f(x+y)).$ Local subalgebras
${\cal A}_{\Lambda}$ are built by creation and annihilation operators with
support of $f$ in $\Lambda.$ Time evolution is given for lattices by

$\frac{d}{dt}\tau_{t}A=i[\sum\sigma_{x}H_{\Lambda},A]$ (6)

with $H_{\Lambda}$ localized in $\Lambda$. The existence of the time evolution
as continuous automorphism-group on the algebra is guaranteed [3] and this can
be generalized for $H$ sufficiently approximated by local operators and also
for continuous systems with interactions where the point interaction is
slightly smeared out [4].

###### Definition 1

The time evolution is strictly localized if for any local operator $A\in{\cal
A}_{\Lambda}$ there exists a $\bar{\Lambda}$ such that $\tau_{t}A\in{\cal
A}_{\bar{\Lambda}}\forall t$. This can be characterized by

$[\tau_{t}A,B]=0\forall t\in R,\forall B\in{\cal A}_{\bar{\Lambda}^{C}},B\quad
even$ (7)

.

This holds evidently if there is no interaction between different lattice
points.

###### Definition 2

The time evolution is delocalized, if it is not strictly localized.

This can be specified:

###### Definition 3

The time evolution is completely delocalized, if for every localized A and
every $t_{0}\in R$ there exists a $\Lambda$ such that

$[\tau(t)A,B]=0\quad\forall t>t_{0},\forall B\in{\cal A}_{\Lambda}$ (8)

A slightly weaker form is

###### Definition 4

The evolution is norm asymptotically abelian on the even subalgebra , if

$norm-\lim[\tau_{t}A,B]=0\quad A\quad even.$ (9)

This is shown for quasifree time evolutions $\tau_{t}a(f)=a(e^{iht}f)$ where
$h$ has absolutely continuous spectrum that is nowhere constant [18].
Similarily we can define strong and weak asymptotical abelianess by replacing
the norm limit by the strong limit in a representation of the algebra or by
the weak limit. As it will turn out, which kind of limit holds will depend on
the representation of the algebra, whereas norm limit is independent of the
representation.

The fact that the group $R$ is ameanable guarantees that invariant states
exist. For the considered time evolution also equilibrium states satisfying
the KMS condition exist obtained by limiting considerations [3]. Asymptotic
abelianess also in its weaker form can be used to obtain extremal invariant
states [15]. It also guarantees that local perturbations of time invariant
states return to the unperturbed state. Further it can be used under the
additional assumption of dynamical stability [16] or passivity [17] to prove
that the extremal invariant state is an equilibrium state. However the fact
that quasifree states that are space-translation invariant are also invariant
under all quasifree time translations that commute with space translations
shows that norm asymptotic abelianess cannot explain approach to equilibrium
without further restrictions on the initial state.

##  4 Delocalization for interacting systems

For continuous Fermi systems with Galilei invariant interaction it is shown
[13] that they are weakly asymptotically abelian in the tracial state. The
proof is based on the fact that acting on the system with the Galilei
automorphism $\gamma(g)a(f(x))=a(e^{igx}f(x))$ combined with the Galilei
invariant time evolution space translation is added to the free evolution
$\gamma(g)\tau_{0}(t)\gamma(-g)a((f)=\sigma(gt)\tau_{0}(t)a(f)$. The
interaction was adjusted to keep Galilei invariance, which connects space,
boost, gauge and time transformations

$\sigma_{x}a(f(y))=a(f(x+y))\quad\gamma_{b}a(f(y))=a(e^{iby}f(y))\quad\nu_{\alpha}a(f)=e^{i\alpha}a(f)\quad\tau_{t},$
(10)

such that

$\sigma_{x}\circ\nu_{\alpha}=\nu_{\alpha}\circ\sigma_{x},\quad\gamma_{b}\circ\nu_{\alpha}=\nu_{\alpha}\circ\gamma_{b},\quad\gamma_{b}\circ\sigma_{x}=\sigma_{x}\circ\gamma_{b}\circ\nu_{-bx}$
$\tau_{t}\circ\nu_{\alpha}=\nu_{\alpha}\circ\tau_{t},\quad\tau_{t}\circ\sigma_{x}=\sigma_{x}\circ\tau_{t}\quad\tau_{t}\circ\gamma_{b}=\gamma_{b}\circ\tau_{t}\circ\sigma_{bt}\circ\nu_{-b^{2}t/2}.$

The advantage of the above model is the fact that the time evolution is
related to the space translation by Galilei invariance so that it can inherit
weak asymptotical abelianess [13],[9]: small perturbation by the Galilei
automorphism adds a contribution of space translations and the possible
separation of space-translation and time-translation combined with a smearing
of the Galilei transformation implies that the decay of the spacial
correlation functions is inherited by the time correlation functions. This
tells us especially, that no local operator remains strictly local in the
course of time. Notice that only one-time correlations could be controlled,
multi-time correlations remained as an open problem.

With an appropriate replacement of the Galilei-automorphism we can prove a
slightly weaker result for lattice systems:

###### Theorem 1

Let the time evolution be implemented by terms of the form $\sum
a^{*}_{x_{1}}..a^{*}_{x_{k}}a_{y_{1}}..a_{y_{k}}\delta(x_{1}+..x_{k}-y_{1}-..y_{k}).$
Consider the automorphism $\gamma(g)a_{x}=e^{igx}a_{x}$. Then the time
evolution in the tracial state has apart from the GNS vector $|\Omega\rangle$
a continuous spectrum.

Proof: By assumption on the time evolution

$\sigma(x)\gamma(g)\tau(t)\gamma(-g)\sigma(-x)=\gamma(g)\tau(t)\gamma(-g)=\tau_{g}(t)$
(11)

The tracial state is invariant under all automorphisms that can be locally
approximated. Therefore these automorphisms are implemented by unitaries that
let the GNS vector $|\Omega\rangle$ invariant. Assume
$\tau(t)A|\Omega\rangle=e^{iEt}A|\Omega\rangle$ for some operator $A$ that can
be approximated arbitrary well by a local operator. Choose $g$ such that
$||\gamma(g)A-A|\Omega\rangle||<\epsilon.$ Then also

$||\tau(t)(\gamma(g)A-A)||=||\gamma(-g)\tau(t)(\gamma(g)A-A)|\Omega\rangle||=||\tau_{g}(t)A-e^{iEt}\gamma(-g)A|\Omega\rangle||<2\epsilon.$

By assumption on the time evolution also $\tau_{g}(t)$ commutes with space
translation. Therefore with $\sigma_{n}\gamma(g)\sigma_{-n}=\gamma_{n}(g)$
also $\gamma_{n}(g)\tau(t)\gamma_{n}(-g)=\tau_{g}(t)$ and thus

$||(e^{iHt}-e^{iEt})e^{igG_{n}}A|\Omega\rangle||\leq\epsilon\quad\forall
n,t,g\leq g_{0}$

With increasing $n$ $\gamma_{n}(g)$ rotates locally with increasing velocity.
Therefore varying over $t$ and $n$ it follows that $A$ does not only
approximate an eigenvector for $H$ with eigenvalue $E$ but also an eigenvector
for $G$ with eigenvalue $0.$ Such an eigenvector corresponds to a quasilocal
operator of the commutative subalgebra built by $a^{*}_{x}a_{x}$ . But
according to our choice of the hamiltonian this excludes that they are
eigenoperators of the time evolution.

The fact, that there are no eigenvectors and therefore also no eigenoperators
for the time evolution implies

###### Theorem 2

For interacting systems as in Theorem (1) there are no local operators that
remain strictly local in the course of time.

Proof: For a unitary-group with only continuous spectrum every subspace that
is invariant under the evolution is infinite dimensional. Local operators
create a finite dimensional subspace. Therefore no vector respectively
operator can stay in a finite dimensional subspace respectively local algebra.

Whereas for the continuous system it was shown that time correlations
$\omega(A\tau(t)B)-\omega(A)\omega(B)$ vanish for $t\rightarrow\infty$ and
therefore the system is weakly asymptotically abelian at least in the tracial
state, we only obtain that the spectrum of the unitary implementing the time
automorphism is apart from the GNS vector continuous. We did not succeed to
prove that it is absolutely continuous so that we could apply Riemann-
Lebesgue. A further argument on the control of the continuous spectrum would
be necessary. It remains so far that the invariant mean of local operators
$\eta\tau(t)A=\omega(A)$ and commutes with all local operators.

##  5 The tracial state as the Vacuum state over an extended algebra

In [20] it was shown that applying the results of scattering theory in the
vacuum state the time automorphism also for interacting systems is though not
in norm so nevertheless strongly asymptotically abelian. Scattering theory
does not exist in temperature states [21], and in general we do not have a
replacement to control long time behaviour. This is different for the tracial
state, that is invariant under all approximately local automorphisms and
allows a concrete construction of the GNS-representation that will enable us
to perform estimates also for interacting systems.

Let $V\in{\cal A}$ with $V=V^{*}$ implement an automorphism on ${\cal A}.$ In
the GNS-representation of the tracial state let

$V|\Omega\rangle=V^{\prime}|\Omega\rangle\quad V^{\prime}\in{\cal A}^{\prime}$
(12)

###### Theorem 3

Under the assumption that $st-\lim_{t\rightarrow\pm\infty}\tau(t)VA\tau(t)V=A$
it follows that $st-\lim_{t\rightarrow\pm\infty}\tau(t)V\tau(t)V^{\prime}=1$

Proof:

$st-\lim_{t\rightarrow\pm\infty}\tau(t)VA\tau(t)V|\Omega\rangle=A|\Omega\rangle=$
(13)
$=st-\lim_{t\rightarrow\pm\infty}\tau(t)VA\tau(t)V^{\prime}|\Omega\rangle=st-\lim_{t\rightarrow\pm\infty}\tau(t)V\tau(t)V^{\prime}A|\Omega\rangle\quad\forall
A\in{\cal A}$

where $A|\Omega\rangle$ is dense in the Hilbertspace. $V$ itself converges
weakly to $0$. But in the tracial state it is not necessary to control its
action as automorphism. By combination with the commutant we have only to
control the convergence of a well defined unitary operator to $1.$

If we replace $V$ by an operator that is assumed to be asymptotically
anticommutative, then with

$WA_{e}|\Omega\rangle=A_{e}|\Omega\rangle\quad
WA_{o}|\omega\rangle=-A_{o}|\Omega\rangle$ (14)

for even and odd operators we have to replace
$st-\lim_{t\rightarrow\pm\infty}\tau(t)V\tau(t)V^{\prime}=W.$

We turn to our system of interest and specify $V$ and $V^{\prime}:$

###### Theorem 4

With the observable algebra ${\cal A}(a)$ built by creation and annihilation
operators $a(f),a^{*}(f)$ and the algebra ${\cal A}(b)$ built by
$b(f)=WJa(\bar{f})J$, J the modular conjugation, the GNS representation of the
tracial state coincides with the vacuum state over an extended Fermi-algebra
built by creation and annihilation operator $A(f),A^{*}(f),B(f),B^{*}(f)$
connected with the initial algebra by a Bogoliubov transformation

$a(f)=\frac{1}{\sqrt{2}}(A(f)+B^{*}(\bar{f})),$ (15)

With

$b(f)=\frac{1}{\sqrt{2}}(A(f)-B^{*}(\bar{f}))$ (16)
$[a(f),b(g)]_{+}=0,[a(f),b^{*}(g)]_{+}=0,[b(f),b(g)]_{+}=0,[b(f)^{*},b(g)]_{+}=\langle
g|f\rangle$ (17)

The even algebra ${\cal A}(b)_{e}$ commutes with ${\cal A}(a).$ Let
$|\Omega\rangle$ be the vacuum vector for ${\cal A}(A,B)$ and $H_{e}$
respectively $H_{o}$ be the subspaces built from $|\Omega\rangle$ by even
respectively odd polynomials in $A,B,A^{*},B^{*}$ and let

$W|\Psi_{e}\rangle=|\Psi_{e}\rangle\quad
W|\Psi_{o}\rangle=-|\Psi_{o}\rangle,\quad|\Psi_{e,o}\rangle\in H_{e,o}$ (18)

then $Wb(f),b^{*}(g)W$ commute with ${\cal A}(a)$ and create the commutant
${\cal A}(a)^{\prime}$.

Proof: Both the vacuum state over ${\cal A}(A,B)$ and the tracial state over
${\cal A}(a)$ are determined by the two point function

$2\omega(a(f)a^{*}(g))=\langle
g|f\rangle=\frac{1}{2}\omega(A(f)+B^{*}(\bar{f})(A^{*}(g)+B(g)))$ (19)

Further with $J$ the modular conjugation

$a(f)|\Omega\rangle=Ja(\bar{f})|\Omega\rangle=\frac{1}{\sqrt{2}}B^{*}(\bar{f})|\Omega\rangle=Wb(f)|\Omega\rangle,\quad
Ja(\bar{f})J=Wb(f).$ (20)

The time evolution is implemented by an operator satisfying
$H=-JHJ,H|\Omega\rangle=0$ . Acting on ${\cal A}(a)$ it is implemented by a
sequence of local operators $h_{\Lambda}$ that we can express term by term by
$A(f),B(f),A^{*}(g)B^{*}(g)$ First we concentrate on quadratic terms that
contribute to $h_{\Lambda}$:

$2a^{*}(f)a(g)=A^{*}(f)A(g)+B(\bar{f})A(g)+A^{*}(f)B^{*}(\bar{g})+B(\bar{f})B^{*}(\bar{g})$
(21)

These terms do not annihilate the vacuum. However the effect on ${\cal A}(a)$
remains unchanged if we add counterparts from the commutant, including
c-numbers:

$b^{*}(f)b(g)=A^{*}(f)A(g)-B(\bar{f})A(g)-A^{*}(f)B^{*}(\bar{g})+B(\bar{f})B^{*}(\bar{g})$
(22)

Therefore we choose as contributions to the hamiltonian

$a^{*}(f)a(g)+b^{*}(f)b(g)-\langle
g|f\rangle=A^{*}(f)A(g)-B^{*}(\bar{g})B(\bar{g})$ (23)

Notice that the necessary c-number renormalization tends to infinity in
agreement with the fact that the time evolution is not an inner automorphism
of ${\cal A}(a).$ We observe that a quasifree time evolution on ${\cal A}(a)$
becomes a quasifree time evolution also on ${\cal A}(A,B)$ and decouples
between ${\cal A}(A)$ and ${\cal A}(B).$ Notice that different to the vacuum
state the hamiltonian implementing the time-evolution for the extended algebra
is not bounded from below. It is by construction invariant under the modular
conjugation combined with the reflection $\gamma A(f)=B(f)$, which coincides
with time reflection individually in ${\cal A}(a)$ and ${\cal A}(b)$.
Therefore time limits in different time-directions can coincide. On the basis
of the $C^{*}$ algebra we have convergence in norm for the automorphisms
implemented by $a_{0}+a^{*}_{0}$, on the basis of the corresponding unitary
given in Theorem (3) we can write it as

$U=VV^{\prime}W=2P-1$ (24)

where expressed in the Bogoliubov-transformed creation and annihilation-
operators

$P=A_{0}A_{0}^{*}+B_{0}B_{0}^{*}$ (25)

and we have strong convergence of $P$ to unity in both time-directions and
similarily strong convergence to $0$ for $1-P.$ This holds for quasifree time
evolution and we have to examine what changes for interacting systems. In more
generality we observe:

###### Lemma 1

Let $U$ be a unitary with $U=U^{*}.$ Let
$st-\lim_{t\rightarrow\pm\infty}\tau(t)U=1$. Then with $U=2P-1$
$st-\lim_{t\rightarrow\pm\infty}\tau_{t}P=1$, whereas with $Q=1-P$
$st-\lim_{t\rightarrow\pm\infty}\tau_{t}Q=0.$

Proof: In the strong limit eigenvalues can disappear but cannot be created.
Therefore $U$ must contain an infinite dimensional projection, that in both
time directions converge to one and has therefore to be invariant under time
reverse.

###### Lemma 2

Assume that $\tau_{t}A=e^{iHt}Ae^{-iHt}$ is strongly asymptotically abelian on
the even algebra. Then it follows that
$st-\lim_{t\rightarrow\pm\infty}[H,P]=0.$

Proof: Under the assumption of strong asymptotically abelianess according to
(24) and the lemma

$st\lim_{t\rightarrow\infty}e^{iHt}(P-e^{i\epsilon H}Pe^{-i\epsilon
H})e^{-iHt}=0\forall\epsilon$

Taking the limit $\epsilon\rightarrow 0$ reduces to convergence of the time
derivative.

We imply these observations to time evolution without and with interaction.
Since we know, that quasifree automorphisms are norm asymptotically
antiabelian, in the combination $U=(a_{0}+a^{*}_{0})(b_{0}-b^{*}_{0})$ the
time-evolution of this unitary satisfies the requirements in the lemma and
converges in the indicated way. Expressed in the Bogoliubov transformed
algebras convergence happens separately for the gauge invariant parts of
${\cal A}(A)$ and ${\cal A}(B)$. However interaction mixes between these gauge
invariant parts. We concentrate on the contributions of the interaction. They
are given by polynomials of forth order in creation and annihilation
operators. Expressed by ${\cal A}(A,B)$ they contain contributions with four
creation operators, with three creation operators with two creation operators
and with one creation operator. The contribution from the commutant
necessarily has to annihilate the term with four creation operators in order
to satisfy $H|\Omega\rangle=0.$ This holds if we choose

$a^{*}(f)a^{*}(g)a(g)a(f)-b^{*}(f)b^{*}(g)b(g)b(f)$ (26)

and observe that it contains terms like

$A^{*}(f)B^{*}(f)B^{*}(g)B(g)$ (27)

where we use that for the expression following from the contributions of $b$
the terms with four, two and zero creation operators cancel and only the terms
with three and one remain, where in addition the annihliation operator has to
be commuted to the right which gives an additional quadratic contribution. We
observe two facts: On one hand the time evolution on the algebra ${\cal A}(b)$
is given by the kinetic part together with an interaction similar as for the
observable algebra but with opposite sign (Compare (20) and (21)). If we
believe that the sign of the interaction is relevant for stability properties
of the system, we might expect that the resulting instability in the commutant
can have consequences also in the algebra. On the other hand the tracial state
for the commutant is the limit of increasing negative temperature, for which
stability arguments are turned around. Relevant is the fact, that time
reflection takes place in the algebra of observables, whereas the modular
conjugation acts as time reflection combined with a reflection between algebra
and commutant. For quasifree time evolution we have the additional symmetry
between ${\cal A}(A)$ and ${\cal A}(B)$ and thus the necessary relation
between modular conjugation and time-reflection. That does not hold for
interacting systems. If we express the hamiltonian by the operators $A$ and
$B$, it is not the sum of operators belonging either to ${\cal A}(A)$ or to
${\cal A}(B)$. Further it is not invariant under the gauge transformation of
either ${\cal A}(A)$ or ${\cal A}(B).$

The interaction contains contributions

$(A_{x}-A_{x}^{*}-B_{x}^{*}+B_{x})(A_{y}+B_{y}^{*}+A_{y}^{*}+B_{y})=2(A_{x}B_{y}^{*}+B_{x}^{*}A_{y}+A_{x}A_{x}^{*})$

and corresponding terms reflecting the symmetry between ${\cal A}(A)$ and
${\cal A}(B)$ that is opposite to the symmetry for the quasifree evolution.
They contribute to the timederivative

$\frac{d}{dt}P=i[H,(A_{x}A_{x}^{*}+B_{x}B_{x}^{*})])$

Especially for strictly local interaction the time derivative remains strictly
local and can be evaluated to be an operator without eigenvalue $0$. Therefore
it can converge weakly to 0, but not strongly, as it would be necessary and we
run into a contradiction for strong asymptotic abelianess. This already can be
seen in the model of [22] with
$H=\sum_{x}\Pi_{j\in\Lambda}\sigma_{z}^{j+x}f(\Lambda)$ where the quasifree
part of the hamiltonian is missing. Here the system is even strictly
localized. We have an abelian subalgebra that is pointwise invariant under
time-evolution. The remaining operators, especially the creation and
anhiliation operators for the Fermi system respectively
$\sigma_{x},\sigma_{y}$ localized at the point $0$ for the corresponding
lattice system converge weakly to $0$ [23] though they stay localized.
Delocalization is produced by adding the quasefree part to the hamiltonian and
this delocalization is improved by the interaction creating nontrivial
commutation-relations between local regions that are far apart.

We summarize our observations in the tracial state and their consequences for
the $C^{*}-$ algebra:

###### Theorem 5

For Fermi lattice systems with interaction the time evolution is not strictly
localized but also not norm asymptotically abelian.

As a by-product we observe, that the kind of asymptotic abelianess can be
state-dependent, in the groundstate we have strong asymptotic abelianess [20],
in the tracial state at its best weak asymptotic abelianess.

##  6 Consequences

We have demonstrated that for quasifree and for interacting systems local
operators do not remain local for increasing time. Comparing with a strictly
local evolution where we can construct invariant states by choosing locally an
invariant state depending on the position and take arbitrary products of these
states, nonlocality produces constraints between far separated regions. We
observe, that there are more constraints if the time evolution is not strongly
asymptotic abelian: Choosing a subalgebra ${\cal M}_{\nu}$ with dimension
$\nu$ of our observable algebra we can write

${\cal A}(a)={\cal M}_{\nu}\otimes{\cal A}_{M}$ (28)

With $m\in{\cal M}_{\nu}$ we consider $(\tau_{t}m)_{ik}=$ as element in ${\cal
A}_{M}$. If the time automorphism is norm asymptotially abelian then
$\lim_{t}[(\tau_{t}m)_{ii}-(\tau_{t}m)_{jj}]=0,\lim_{t}(\tau_{t}m)_{ij}=\lim_{t}(\tau_{t}m)_{ji}=0.$
If they do not commute then $||(\tau_{t}m)_{ij}||>0$ for arbitrary large $t$.
An invariant state is given by functionals $\omega_{ik}$ over ${\cal A}_{M}$
with $\omega(m_{ij})=\omega_{ij}.$ $(\tau_{t}m)_{ij}$ varies with $t$ and time
invariance implies constraints for $\omega_{ik}$ If the algebraic correlations
that exist for interacting systems and are not trivial but are random which
was assumed for all operators in the case of [10] then it follows that
$\omega_{ij}=0=\omega(m_{ij})$ and we obtain a condition on the local level.
Similarily correlations between annihilation operators $a(f)$ give
restrictions, for example gauge invariance of the state. In general it seems
plausible that with increasing number of algebraic correlations also the
possibility for an invariant state is reduced.

Galilei invariance allowed that spacial clustering is inherited by time
clustering. However the proof fails for multiclustering, and in fact for norm
asymptotically abelian time evolution

$\lim_{t\rightarrow\infty}\omega(AB_{t}CD_{t})=\omega(AC)\omega(BD)\omega(AB_{t}CD_{t})=\omega(AC)\omega(BD)$
(29)

whereas for a time evolution that is weakly asymptotically abelian or
$\eta-$abelian

$\lim_{t\rightarrow\infty}|\omega(AB_{t}A^{*}B^{*}_{t})|<\omega(AA^{*})\omega(BB^{*}).$
(30)

Let us assume that states extremely invariant under space translation cluster
uniformly and this clustering is inherited by time translation

$\lim_{t\rightarrow\infty}P_{\Lambda_{2}^{C}}e^{iHt}P_{\Lambda_{1}}=|\Omega\rangle\langle\Omega|$
(31)

where $\Lambda_{2}$ containing $\Lambda_{1}$ are local regions,$\Lambda^{C}$
the complement of $\Lambda$ and $P_{\Lambda}$ the projection operator onto
${\cal A}_{\Lambda}|\Omega\rangle.$ Under this assumption together with norm
asymptotic abelianess

$\lim_{t_{j}-t_{i}\rightarrow\infty}\omega(\tau_{t_{1}}A_{1}\tau_{t_{2}}A_{2}...\tau_{t_{k}}A_{k})=\omega(A_{j})\omega_{i\neq
j}(\Pi\tau_{t_{i}}A_{i})$ (32)

independently of the ordering and the size of $t_{i}$ and we can control
multiclustering. This makes it possible to control local perturbations

$\langle\Omega|Ae^{it(H+\lambda V)}Be^{-it(H+\lambda V)}A^{*}|\omega\rangle.$
(33)

By expanding in the coupling constant and taking the limit
$t\rightarrow\infty$ we can obtain an invariant state. In KMS-states where the
time automorphism coincides with the modular automorphism we can construct
this invariant state by the time ordered expression

$|\Omega_{V}\rangle={\cal T}\int_{0}^{\beta/2}d\gamma
e^{\tau_{i\gamma}V}|\Omega\rangle$ (34)

where now the integration runs over a finite region and does not refer to
asymptotic behaviour, therefore can also be applied for systems that are just
weakly asymptotically abelian. An other possiblity to construct the state
invariant under the perturbation but staying in the follium is looking for the
vector in the natural cone satisfying

$(H+\lambda(V-JVJ))|\Omega_{V}\rangle=0=|\Omega+\sum\lambda^{n}B_{n}\rangle$
(35)

which is solved by using the modular operator $M$ that we now may assume to be
different from $H$ but by invariance commutes with $H$. Expanding in $\lambda$
gives

$B_{1}|\Omega\rangle=-\frac{1}{H}(V-JVJ)|\Omega\rangle=-\frac{1}{H}(1-e^{-M/2})V|\Omega\rangle$
(36) $B_{n}|\Omega\rangle=\frac{1}{H}B_{n-1}|\Omega\rangle.$ (37)

Considering

$\lim_{\epsilon\rightarrow 0}\frac{-i}{H+i\epsilon}=\lim_{\epsilon\rightarrow
0}\int_{0}^{\infty}dte^{it(H+i\epsilon)}$ (38)

we are dealing with an unbounded operator $\frac{1}{H}$, but from clustering
in time it follows that $(V-e^{-M/2}V)|\Omega\rangle$ belongs to its domain.
However for $B_{n}|\Omega\rangle$ we would need higher order of time
correlation functions that for weakly asymptotically abelian or $\eta-$abelian
systems are not available.

Therefore we have to be aware that differently as for KMS-states local
perturbations of the dynamics can lead to states that are not normal with
respect to the initial state, i.e. that already the local perturbations can
create infinite energy in the course of time. We observe different behaviour
with respect to the degree of abelianess, but also the special role of KMS-
states. Extending the local perturbation to global perturbation as it is
possible for KMS-states without violating the KMS-property with respect to the
new dynamics is even more problematic. All these are indications of the
sensibility of weakly asymptotically systems and increasing restrictions for
invariant states.

##  7 Conclusion

We have concentrated on Fermi-systems on the lattice. Our considerations can
with appropriate modifications be generalized to lattice systems and Galilei-
invariant lattice systems. For these systems the tracial state exists. In the
corresponding GNS-representation every approximately local automorphism can be
represented by unitaries. The relevant automorphisms are space translation,
time translation with and without interaction, Galilei-transformation
respectively its replacement for lattice-systems and gauge-transformation. The
relation between time evolution and Galilei transformation was used to prove,
that no local operator is an eigenoperator of the time evolution and
consequently becomes delocalized in the course of time. Next we used gauge-
translation and space translation to construct a local structure also on the
commutant and extended this local structure to all operators in the
representation, including the operators of the commutant. The modular
conjugation which is an antilinear automorphism of the total algebra respects
the local structure. The relation between modular conjugation and time
reflection allows strong abelianess for quasifree evolution, but for
interaction that react differently on the antilinearity algebraic correlations
between local operators and local operators shifted in time remain.

Finally we search for arguments why weak asymptotic abelianess might reduce
the possibility for time invariant states as suggested by examples of discrete
time evolution. On one hand it offers additional areas of randomness that
create restrictions on space correlations. On the other hand we refer to the
demand of dynamical stability i.e. the existence of invariant states under
local perturbations without leaving the follium. This holds for KMS-states,
but otherwise the necessary analyticity properties get out of control for
interacting systems.

We are optimistic that further analysis that concentrates on long range
effects in time on the algebraic level and their effect on long range
correlations in space for invariant states can lead to deeper understanding of
the importance and the power of interaction with respect to thermodynamics.

## References

  * [1] M. Srednicki: Phys. Rev.E 56/2 888 (1994)
  * [2] J.M. Deutsch: Rep. Prog. Phys. 81 082001 (2018)
  * [3] O. Bratteli, D.W. Robinson: Operator Algebras and Quantum Statistical Mechanics I and II, Springer (1981)
  * [4] H.Narnhofer, W. Thirring: Phys. Rev. Lett. 64/16 1863 (1991)
  * [5] W. Arveson: J. Func. Anal. 15 217 (1974)
  * [6] E.H. Lieb, D.W. Robinson: Commun. Math. Phys. 28 251 (1972)
  * [7] H. Araki: Publ. RIMS, Kyoto Univ. 20 277 (1984)
  * [8] H. Narnhofer, E. Stoermer, W. Thirring: Ergod. Th. Dynam. Sys. 15(1995) 961
  * [9] H. Narnhofer, W. Thirring: Lett. Math. Phys. 35 (1995) 145
  * [10] B. Baumgartner, F. Benatti, H. Narnhofer: J. Phys.A: Math. Theor. 43 (2010) 115301
  * [11] M. Bozijko, B. Kuemmerer, R. Speicher: Commun. Math. Phys. 185 129 (1997)
  * [12] H. Narnhofer, W. Thirring: Rep. Math. Phys. 45/2 239 (2000)
  * [13] H. Narnhofer, W. Thirring: Int. Journ.of mod. Phys. A 6, 2937 (1991)
  * [14] H. Narnhofer: Acta Phys. Austr. 31 349 (1970)
  * [15] S. Doplicher, R.V. Kadison, D. Kastler, D.W. Robinson: Commun. Math. Phys. 6 101 (1967)
  * [16] R. Haag, D. Kastler, E.B. Trych-Pohlmeyer: Commun. Math. Phys.5 215 (1967)
  * [17] W. Pusz, S.L. Woronowicz: Commun. Math. Phys. 58 273 (1978)
  * [18] H. Narnhofer: Acta Phys. Austr. bf31 349 (1970)
  * [19] H. Narnhofer: J.Phys.A: Math. Theor. 41, 315201 (208)
  * [20] Ch. Jaekel: Lett. Math. Phys. 21 343 (1991)
  * [21] M. Requardt, H. Narnhofer, W. Thirring: Commun. Math. Phys. 92 (1983) 247
  * [22] G.G. Emch : J. Math. Phys. 7 1198 (1966)
  * [23] C. Radin: J. Math. Phys. 11 2945 (1970)