# Low-temperature asymptotics for transverse autocorrelator of the
magnetically polarized Ising chain studied by ordinary and nested Dyson
equations

P. N. Bibikov

(Russian State Hydrometeorological University, Saint-Petersburg, Russia)

###### Abstract

Suggesting two versions for the Plakida-Tserkovnikov algorithm breakdown in
the low-temperature regime, we derive ordinary and nested Dyson equations for
the transverse autocorrelator of the magnetically polarized Ising chain. Using
them we get the corresponding low-temperature asymptotics for the
autocorrelator. We show that the ordinary Dyson equation results in a correct
$o({\rm e}^{-\beta E_{gap}})$ account of the magnon creation process, while
the nested Dyson equation additionally gives the correct $o({\rm e}^{-\beta
E_{gap}})$ contribution associated with transitions from magnons to bound two-
magnon states. The obtained result may be useful for the extension of the
suggested approach on the polarized $XXZ$-chain.

##  1 Introduction

Spectral densities of the form [1]

$I(\omega,T)=-\frac{1}{\pi(1-{\rm e}^{-\beta\omega})}{\rm
Im}\lim_{N\rightarrow\infty}\langle\langle
O,O^{\dagger}\rangle\rangle_{\omega},$ (1)

are important experimentally measurable dynamical characteristics of magnetic
compounds [2, 3, 4]. Here $O$ is an appropriate fluctuation operator, $N$ is
the number of sites in a magnetic lattice, and for a pair of arbitrary
operators ${\cal A}$ and ${\cal B}$ there are two equivalent definitions of
the two-real-time retarded Green function [1]

$\langle\langle{\cal A},{\cal
B}\rangle\rangle_{\omega}\equiv\frac{1}{i}\int_{0}^{\infty}dt{\rm
e}^{izt}\langle[{\cal A}(t),{\cal
B}]\rangle=\frac{1}{i}\int_{0}^{\infty}dt{\rm e}^{izt}\langle[{\cal A},{\cal
B}(-t)]\rangle,\qquad z\equiv\omega+i\epsilon.$ (2)

As usual

${\cal A}(t)\equiv{\rm e}^{i\hat{H}t}{\cal A}{\rm
e}^{-i\hat{H}t},\qquad\langle{\cal A}\rangle\equiv\frac{1}{Z(T)}{\rm
tr}\Big{(}{\rm e}^{-\beta\hat{H}}{\cal A}\Big{)},\qquad Z(T)={\rm
tr}\Big{(}{\rm e}^{-\beta\hat{H}}\Big{)},$ (3)

where $\hat{H}$ is the corresponding Hamiltonian. From (2) readily follows [1]
that

$\displaystyle z\langle\langle{\cal A},{\cal
B}\rangle\rangle_{\omega}=\langle[{\cal A},{\cal
B}]\rangle+\langle\langle{\cal A},[\hat{H},{\cal B}]\rangle\rangle_{\omega},$
(4) $\displaystyle z\langle\langle{\cal A},{\cal
B}\rangle\rangle_{\omega}=\langle[{\cal A},{\cal
B}]\rangle+\langle\langle[{\cal A},\hat{H}],{\cal B}\rangle\rangle_{\omega}.$
(5)

Besides $\langle\langle{\cal A},{\cal B}\rangle\rangle_{\omega}$ possesses the
spectral representation

$\langle\langle{\cal A},{\cal
B}\rangle\rangle_{\omega}=\frac{1}{Z(T,N)}\sum_{\mu,\nu}\frac{{\rm e}^{-\beta
E_{\nu}}-{\rm e}^{-\beta E_{\mu}}}{z+E_{\nu}-E_{\mu}}\langle\nu|{\cal
A}|\mu\rangle\langle\mu|{\cal B}|\nu\rangle,$ (6)

where $\mu$ and $\nu$ parameterize the eigenbasis of $\hat{H}$ ($E_{\mu}$ and
$E_{\nu}$ are the corresponding energies). If ${\cal A}=O$ and ${\cal
B}=O^{\dagger}$, then (6) is reduced to

$\langle\langle
O,O^{\dagger}\rangle\rangle_{\omega}=\frac{1}{Z(T,N)}\sum_{\mu,\nu}\frac{{\rm
e}^{-\beta E_{\nu}}-{\rm e}^{-\beta
E_{\mu}}}{z+E_{\nu}-E_{\mu}}|\langle\nu|O|\mu\rangle|^{2}.$ (7)

According to the well known relation ${\rm
Im}(x+i\epsilon)^{-1}=-\pi\delta(x)$, the substitution of (7) into (1) yields

$I(\omega,T)=\lim_{N\rightarrow\infty}\frac{1}{Z(T,N)}\sum_{\mu,\nu}{\rm
e}^{-\beta
E_{\nu}}|\langle\nu|O|\mu\rangle|^{2}\delta(\omega+E_{\nu}-E_{\mu}).$ (8)

The so called $XXZ$ model related to Hamiltonian

$\hat{H}^{(XXZ)}=-\sum_{n=1}^{N}\Big{[}\frac{J_{\bot}}{2}\Big{(}{\bf
S}^{+}_{n}{\bf S}^{-}_{n+1}+{\bf S}^{-}_{n}{\bf
S}^{+}_{n+1}\Big{)}+J_{z}\Big{(}{\bf S}^{z}_{n}{\bf
S}^{z}_{n+1}-\frac{1}{4}\Big{)}+h\Big{(}{\bf
S}_{n}^{z}-\frac{1}{2}\Big{)}\Big{]},$ (9)

is one of the most fundamental models in the theory of 1D magnetic compounds
[5]. Here, ${\bf S}_{n}^{z}$ and ${\bf S}_{n}^{\pm}={\bf S}_{n}^{x}\pm i{\bf
S}_{n}^{y}$ is a triple of spin-1/2 operators attached to the $n$-th site of
the chain

$[{\bf S}_{m}^{z},{\bf S}_{n}^{\pm}]=\pm\delta_{mn}{\bf
S}_{n}^{\pm},\qquad[{\bf S}_{m}^{+},{\bf S}_{n}^{-}]=2\delta_{mn}{\bf
S}_{n}^{z},$ (10)

and without a loss of generality we postulate

$h>0.$ (11)

Hamiltonian (9) acts in the tensor product of $N$ local ${\mathbb{C}}^{2}$
spaces attached to the sites of the chain ${\cal
H}={\mathbb{C}}^{2}\otimes\dots\otimes{\mathbb{C}}^{2}$. Under the condition
(11) supplemented by

$h+J_{z}-|J_{\bot}|>0,$ (12)

the ground state of (9) has a polarized form

$|\emptyset\rangle=|\uparrow\rangle\otimes\dots\otimes|\uparrow\rangle,$ (13)

where $|\uparrow\rangle$ and $|\downarrow\rangle$ are spin-up and spin-down
local polarized states

${\bf S}^{z}|\uparrow\rangle=\frac{1}{2}|\uparrow\rangle,\qquad{\bf
S}^{z}|\downarrow\rangle=-\frac{1}{2}|\downarrow\rangle.$ (14)

Moreover under (11) and (12) the spectrum of (9) is gapped and the lowest-
energy excitations form the magnon branch [6]

$|k\rangle={\bf
S}^{-}(-k)|\emptyset\rangle,\qquad\hat{H}|\emptyset\rangle=E_{magn}(k)|\emptyset\rangle.$
(15)

parameterized by crystal momentum $k$. Here

${\bf S}(q)\equiv\frac{1}{\sqrt{N}}\sum_{n=1}^{N}{\rm e}^{-iqn}{\bf S}_{n},$
(16)

and

$E_{magn}(k)=h+J_{z}-J_{\bot}\cos{k},$ (17)

so (12) takes the following form

$E_{gap}=h+J_{z}-|J_{\bot}|>0.$ (18)

.

One of the most important dynamical characteristics of the $XXZ$ chain is its
dynamical structure factor (DSF) $S(\omega,q,T)$ which corresponds to the
substitution of $O={\bf S}^{+}(q)$ and $O^{\dagger}={\bf S}^{-}(-q)$ into (1).
In the the gapped phase (12), (13), the spectral expansion (7) at $T=0$ yields
for $S(\omega,q,T)$ a singular delta-function lineshape [7]

$S(\omega,q,0)=\delta(\omega-E_{magn}(q)).$ (19)

This follows from the one-to-one correspondence (17) between the crystal
momentum and the energy of a magnon created from the ground state by the
fluctuation operator ${\bf S}^{-}(-q)$

$|\emptyset\rangle\xrightarrow{{\bf S}^{-}(-q)}|q\rangle.$ (20)

At $T>0$, due to the existence of thermally activated states, the delta-
singularity (19) broadens into a smooth line-shape. This broadening may be
well illustrated if one consider, the transition of a thermally activated
magnon with crystal momentum $k_{0}$ into a scattering (not bound) state of
two magnons with crystal momentums $k_{1}$ and $k_{2}$

$|k_{0}\rangle\xrightarrow{{\bf S}^{-}(-q)}|k_{1},k_{2},scatt\rangle.$ (21)

Due to the conservation of total crystal momentum, one should have in (21)

$k_{1}+k_{2}=k_{0}+q.$ (22)

According to (22) the delta-function in (8) will give

$\omega=E_{magn}(k_{1})+E_{magn}(k_{2})-E_{magn}(k_{1}+k_{2}-q).$ (23)

But for a given $q$, equation (23) does not fix $\omega$.

From the presented example, follows an instructive idea that the low-
temperatures broadening may be well elaborated by cluster expansion with an
account of even one- and two-magnon states. However, as it is well known (and
also discussed in [7]), the delta-peak singularity cannot be removed by direct
cluster expansion of (8). An adequate approach for an account of the thermal
broadening in a DSF line-shape based on the Dyson equation for the real two-
time Green retarded functions (2) was suggested by N. M. Plakida in [8, 9] and
then developed by Yu. A. Tserkovnikov in [10]. Within this approach, a two-
real-time retarded Green function is represented in a continued fraction form.
Since in the general case the latter is infinite, one should effectively break
it down in order to get a reduced tractable result. In [7], this program was
realized for evaluation of the low-temperature asymptotics of the transverse
magnetic susceptibility

$\chi(\omega,q,T)=\lim_{N\rightarrow\infty}\langle\langle{\bf S}^{+}(q),{\bf
S}^{-}(-q)\rangle\rangle_{\omega},$ (24)

for the special version of (9) related to $J_{z}=0$ (the so-called $XX$ model
[11]). Following [8, 9], $\chi(\omega,q,T)$ has been represented in the form

$\chi(\omega,q,T)=\frac{1}{z-E_{magn}(q)-\Sigma(\omega,q,T)},\qquad\Sigma(\omega,q,0)=0,$
(25)

associated with the Dyson equation. With the use of one- and two-magnon states
the self-energy $\Sigma(\omega,q,T)$ has been evaluated up to the order
$o({\rm e}^{-\beta E_{gap}})$. In the resonance region of $\omega-q$ plane
where ${\rm Im}\Sigma(\omega,q,T)>0$, the line-shape of the DSF corresponding
to (25) has a single broad peak.

An extension of this result on the XXZ model (9) seems very desirable.
However, in this way, additionally to (20) and (21), one has to deal with a
contribution from a new process, which is a bound state creation

$|k_{0}\rangle\xrightarrow{{\bf S}^{-}(-q)}|k,bound\rangle,$ (26)

(spectrum of the $XX$ model is free from bound states [11]). As it will be
shown in the present paper, the process (26) is responsible for a thermally
activated peak, which has been discussed over the last fifty years [12, 13,
14, 15, 16].

In order to get an experience in account of (26) before attacking the general
Hamiltonian (9), we shall treat here its reduced $(J_{\bot}=0)$ version

$\hat{H}^{(Is)}=-\sum_{n=1}^{N}\Big{[}J_{z}\Big{(}{\bf S}^{z}_{n}{\bf
S}^{z}_{n+1}-\frac{1}{4}\Big{)}+h\Big{(}{\bf
S}_{n}^{z}-\frac{1}{2}\Big{)}\Big{]},$ (27)

known as the Ising model [17, 18]. The DSF corresponding to (27) is equal to
the transverse autocorrelator obtained long ago [17]

$\langle\langle{\bf S}^{+}(q),{\bf
S}^{-}(-q)\rangle\rangle_{\omega}=\langle\langle{\bf S}_{n}^{+},{\bf
S}_{n}^{-}\rangle\rangle_{\omega}.$ (28)

Formula (28) immediately follows from the equality $\langle\langle{\bf
S}_{m}^{+},{\bf S}_{n}^{-}\rangle\rangle_{\omega}=0$ valid for $m\neq n$. The
latter, in its turn, may be readily obtained from the definition (2), (3)
under a substitution of (27). As it has been shown in [17]

$\langle\langle{\bf S}_{n}^{+},{\bf
S}_{n}^{-}\rangle\rangle_{\omega}=\frac{A_{+}(T)}{z-\omega_{+}}+\frac{A_{0}(T)}{z-\omega_{0}}+\frac{A_{-}(T)}{z-\omega_{-}},$
(29)

where

$\omega_{+}=h+J_{z},\qquad\omega_{0}=h,\qquad\omega_{-}=h-J_{z},$ (30)

and

$\displaystyle A_{\pm}(T)=\frac{\sigma_{1}(T)}{2}\pm
2\sigma_{2}(T)+2\sigma_{3}(T),\qquad A_{0}(T)=\sigma_{1}(T)-4\sigma_{3}(T),$
(31) $\displaystyle\sigma_{1}(T)\equiv\langle{\bf
S}^{z}_{n}\rangle,\qquad\sigma_{2}(T)\equiv\langle{\bf S}^{z}_{n}{\bf
S}^{z}_{n+1}\rangle,\qquad\sigma_{3}(T)\equiv\langle{\bf S}^{z}_{n-1}{\bf
S}^{z}_{n}{\bf S}^{z}_{n+1}\rangle,$ (32)

(according to translation invariance of (27), all expressions in (32) do not
depend on $n$).

Since as in [7] we are interesting in the low-temperature polarized phase it
is instructive to obtain the corresponding asymptotics for the coefficients in
(31). This may be done with the use of the low-energy spectrum of (27) and the
following representations

$A_{j}(T)=\langle F_{n}^{(j)}\rangle,\qquad j=+,0,-,$ (33)

where

$F^{(\pm)}_{n}=\frac{1}{2}{\bf S}_{n}^{z}\pm{\bf V}_{n}^{z}+2{\bf
W}_{n}^{z},\qquad F_{n}^{(0)}={\bf S}_{n}^{z}-4{\bf W}_{n}^{z},$ (34)

or equivalently

$\displaystyle F^{(+)}_{n}=2(1-Q_{n-1}){\bf S}^{z}_{n}(1-Q_{n+1}),$
$\displaystyle F^{(0)}_{n}=2{\bf S}^{z}_{n}(Q_{n-1}+Q_{n+1}-2Q_{n-1}Q_{n+1}),$
$\displaystyle F^{(-)}_{n}=2Q_{n-1}{\bf S}^{z}_{n}Q_{n+1}.$ (35)

Here

${\bf V}^{j}_{n}\equiv({\bf S}_{n-1}^{z}+{\bf S}_{n+1}^{z}){\bf
S}_{n}^{j},\qquad{\bf W}^{j}_{n}\equiv{\bf S}_{n-1}^{z}{\bf S}_{n+1}^{z}{\bf
S}_{n}^{j},\qquad j=+,-,z,$ (36)

and for

$Q_{n}\equiv\frac{1}{2}-{\bf S}^{z}_{n},$ (37)

one has

$Q_{m}|\emptyset\rangle=0,\qquad[Q_{m},{\bf S}^{-}_{n}]=\delta_{mn}{\bf
S}^{-}_{n}.$ (38)

The spectrum of the Ising model is well known [18] and the actual low-energy
states will be the following

$\displaystyle|m\rangle={\bf
S}^{-}_{m}|\emptyset\rangle,\qquad\hat{H}|m\rangle=E_{1}|m\rangle$
$\displaystyle|m_{1},m_{2};isol\rangle={\bf S}^{-}_{m_{1}}{\bf
S}^{-}_{m_{2}}|\emptyset\rangle,\qquad m_{2}-m_{1}>1,$
$\displaystyle\hat{H}|m_{1},m_{2};isol\rangle=E_{2,isol}|m_{1},m_{2};isol\rangle,$
$\displaystyle|m,m+1;string\rangle={\bf S}^{-}_{m}{\bf
S}^{-}_{m+1}|\emptyset\rangle,$
$\displaystyle\hat{H}|m,m+1;string\rangle=E_{2,string}|m,m+1;string\rangle$
$\displaystyle|m,m+1,m+2;string\rangle={\bf S}^{-}_{m}{\bf S}^{-}_{m+1}{\bf
S}^{-}_{m+2}|\emptyset\rangle,$
$\displaystyle\hat{H}|m,m+1,m+2;string\rangle=E_{3,string}|m,m+1,m+2;string\rangle.$
(39)

Comparing (39) with the corresponding excitations of the $XXZ$ chain [6], one
may conclude that $|m\rangle$ are the Ising limits of one-magnon states,
$|m_{1},m_{2};isol\rangle$ are the corresponding limits of scattering two-
magnon states. Finally $|m,\dots,m+M;string\rangle$ are related to bound
$M$-magnon states. The corresponding energies

$E_{1}=E_{gap}=h+J_{z},\qquad E_{M,isol}=M(h+J_{z})=ME_{1},\qquad
E_{M,string}=Mh+J_{z},$ (40)

may be readily obtained by elementary calculations. The equality
$E_{gap}=E_{1}$ (which in fact is the reduced form of (18) related to
$J_{\bot}=0$) follows from (40) and (12).

According to (14), (38) and (35), one has

$\displaystyle F^{(+)}_{n}|\emptyset\rangle=|\emptyset\rangle,\qquad
F^{(+)}_{n}|m\rangle=(1-\delta_{m-1,n}-\delta_{m+1,n}-2\delta_{mn})|m\rangle$
$\displaystyle F_{n}^{(0)}|\emptyset\rangle,\qquad
F_{n}^{(0)}|m\rangle=(\delta_{m-1,n}+\delta_{m+1,n})|m\rangle,$ $\displaystyle
F_{n}^{(-)}|\emptyset\rangle=F_{n}^{(-)}|m\rangle=0,\qquad
F_{n}^{(-)}|m_{1},m_{2},isol\rangle=\delta_{m_{1},n-1}\delta_{m_{2},n+1}|m_{1},m_{2},isol\rangle.$
(41)

So with the use of the notation

$\zeta\equiv{\rm e}^{-\beta E_{gap}}={\rm e}^{-\beta(h+J_{z})},$ (42)

(33) and (41) yield

$\displaystyle
A_{+}(T)=\frac{1+(N-4)\zeta+o(\zeta)}{1+N\zeta+o(\zeta)}=1-4\zeta+o(\zeta),$
$\displaystyle A_{0}(T)=\frac{2\zeta+o(\zeta)}{1+o(1)}=2\zeta+o(\zeta),\qquad
A_{-}(T)=\frac{\zeta^{2}+o(\zeta^{2})}{1+o(1)}=\zeta^{2}+o(\zeta^{2}).$ (43)

According to (43), the $\omega_{+}$-peak has a zero activation energy, while
the activation energies of $\omega_{0}$\- and $\omega_{-}$-peaks are
correspondingly $E_{gap}$ and $2E_{gap}$.

A physical interpretation of the expansions (43) becomes elementary under an
account of the following relations

$\omega_{+}=E_{1}-0=E_{2,isol}-E_{1},\qquad\omega_{0}=E_{2,string}-E_{1}\qquad\omega_{-}=E_{3,string}-E_{2,isol}.$
(44)

As it follows from (44), in the few-particle level the $\omega_{+}$-peak
originates from the processes (20) and (21). The $\omega_{0}$-peak originates
from (26). At last, the $\omega_{-}$-peak originates from the new process

$|m,m+2;isol\rangle\xrightarrow{{\bf S}^{-}(-q)}|m,m+1,m+2;string\rangle.$
(45)

In principle, formulas (29)-(32) may be reproduced by the Plakida-Tserkovnikov
approach because in this special case, the corresponding continued fraction is
finite. This result, however, is useless for generalization on the $XXZ$
chain, whose DSF should have an infinite continued fraction representation.
That is why in the present paper, working within the Plakida-Tserkovnikov
approach, we try to reproduce the formula (29) and the low-temperature
asymptotics (43) up to the order $o(\zeta)$ by breaking the continued
fraction. This is done by cluster expansion of the corresponding high-level
Green functions (which naturally appear in the Plakida-Tserkovnikov algorithm)
according to the spectral formula (7).

As a result, we show by direct calculation that the correct $o(\zeta)$
approximation for the $A_{+}(T)$ term in (29) (which agrees both with (43) and
(30)) may be reproduced under the breakdown of the Plakida-Tserkovnikov
algorithm after its first cycle. The corresponding reduction of the continued
fraction has the usual form (25) associated with the ordinary Dyson equation.
The $A_{0}(T)$ term is also reproduced within this approach; however, its
denominator does not agree with (29). The $A_{-}(T)$ term cannot be obtained
within the approach based on the ordinary Dyson equation. A more complicated
reduction is based on the breakdown of the Plakida-Tserkovnikov algorithm
after its second cycle. It yields the nested Dyson equation and gives correct
$o(\zeta)$ expansions (which agree both with (43) and (30)) both for
$A_{+}(T)$ and $A_{0}(T)$ terms in (29).

##  2 Derivation of the ordinary Dyson equation

###  2.1 Extraction of the first order irreducible parts

Let

$\theta_{0}(T)\equiv\frac{\langle[[O_{0},\hat{H}],O_{0}^{\dagger}]\rangle}{\langle[O_{0},O_{0}^{\dagger}]\rangle},\qquad
O_{0}\equiv O,\qquad O_{0}^{\dagger}\equiv O^{\dagger}.$ (46)

Since for an arbitrary operator ${\cal A}$ it will be ${\rm Tr}({\rm
e}^{-\beta\hat{H}}[{\cal A},\hat{H}])=0$, we have

$\theta_{0}(T)-\bar{\theta}_{0}(T)=\frac{\langle[[O_{0},\hat{H}],O_{0}^{\dagger}]\rangle-\langle[[O_{0}^{\dagger},\hat{H}],O_{0}]\rangle}{\langle[O_{0},O_{0}^{\dagger}]\rangle}=\frac{\langle[[O_{0},O_{0}^{\dagger}],\hat{H}]\rangle}{\langle[O_{0},O_{0}^{\dagger}]\rangle}=0.$
(47)

According to (46), a two (temperature-dependent) adjoint operators

$O_{1}(T)\equiv[O_{0},\hat{H}]-\theta_{0}(T)O_{0},\qquad
O_{1}^{\dagger}(T)=[\hat{H},O_{0}^{\dagger}]-\theta_{0}(T)O_{0}^{\dagger},$
(48)

which will be called the first order irreducible parts of $O_{0}$ and
$O_{0}^{\dagger}$, satisfy the first order irreducibility conditions

$\langle[O_{1}(T),O_{0}^{\dagger}]\rangle=0,\qquad\langle[O_{0},O_{1}^{\dagger}(T)]\rangle=0.$
(49)

Substitutions of ${\cal B}=O_{0}^{\dagger}$ into (4) and ${\cal A}=O_{0}$ into
(5) under an account of (46)-(48) yield

$\displaystyle(z-\theta_{0}(T))\langle\langle{\cal
A},O_{0}^{\dagger}\rangle\rangle_{\omega}=\langle[{\cal
A},O_{0}^{\dagger}]\rangle+\langle\langle{\cal
A},O_{1}^{\dagger}(T)\rangle\rangle_{\omega},$ (50)
$\displaystyle(z-\theta_{0}(T))\langle\langle O_{0},{\cal
B}\rangle\rangle_{\omega}=\langle[O_{0},{\cal B}]\rangle+\langle\langle
O_{1}(T),{\cal B}\rangle\rangle_{\omega}.$ (51)

Taking now ${\cal A}=O_{0}$ in (50) and ${\cal B}=O_{0}^{\dagger}$ in (51),
one readily gets

$\displaystyle(z-\theta_{0}(T))\langle\langle
O_{0},O_{0}^{\dagger}\rangle\rangle_{\omega}=\langle[O_{0},O_{0}^{\dagger}]\rangle+\langle\langle
O_{0},O_{1}^{\dagger}(T)\rangle\rangle_{\omega},$ (52)
$\displaystyle(z-\theta_{0}(T))\langle\langle
O_{0},O_{0}^{\dagger}\rangle\rangle_{\omega}=\langle[O_{0},O_{0}^{\dagger}]\rangle+\langle\langle
O_{1}(T),O_{0}^{\dagger}\rangle\rangle_{\omega}.$ (53)

At the same time, according to (49), the substitution of ${\cal A}=O_{1}(T)$
into (50) gives

$(z-\theta_{0}(T))\langle\langle
O_{1}(T),O_{0}^{\dagger}\rangle\rangle_{\omega}=\langle\langle
O_{1}(T),O_{1}^{\dagger}(T)\rangle\rangle_{\omega}.$ (54)

###  2.2 Introduction of the first level irreducible Green function

At this stage, we should express $\langle\langle
O_{0},O_{0}^{\dagger}\rangle\rangle_{\omega}$ from $\langle\langle
O_{1}(T),O^{\dagger}_{1}(T)\rangle\rangle_{\omega}^{(1)}$, where for two
arbitrary operators ${\cal A}$ and ${\cal B}$, the first level irreducible
Green function is defined as

$\langle\langle{\cal A},{\cal
B}\rangle\rangle_{\omega}^{(1)}\equiv\langle\langle{\cal A},{\cal
B}\rangle\rangle_{\omega}-\frac{\langle\langle{\cal
A},O_{0}^{\dagger}\rangle\rangle_{\omega}\langle\langle O_{0},{\cal
B}\rangle\rangle_{\omega}}{\langle\langle
O_{0},O_{0}^{\dagger}\rangle\rangle_{\omega}}.$ (55)

From definition (55) immediately follows the first level irreducibility
condition

$\langle\langle{\cal A}+\xi_{0}O_{0},{\cal
B}+\eta_{0}O_{0}^{\dagger}\rangle\rangle_{\omega}^{(1)}=\langle\langle{\cal
A},{\cal
B}\rangle\rangle_{\omega}^{(1)},\qquad\xi_{0},\,\eta_{0}\in{\mathbb{C}},$ (56)

which, in its turn, gives

$\langle\langle{\cal
A},O_{0}^{\dagger}\rangle\rangle_{\omega}^{(1)}=\langle\langle O_{0},{\cal
B}\rangle\rangle_{\omega}^{(1)}=0.$ (57)

According to (54)

$\langle\langle
O_{1}(T),O_{1}^{\dagger}(T)\rangle\rangle_{\omega}\cdot\langle\langle
O_{0},O_{0}^{\dagger}\rangle\rangle_{\omega}=\langle\langle
O_{1}(T),O_{0}^{\dagger}\rangle\rangle_{\omega}(z-\theta_{0}(T))\cdot\langle\langle
O_{0},O_{0}^{\dagger}\rangle\rangle_{\omega}.$ (58)

The substitution of the right side of (52) into the right side of (58) yields

$\langle\langle
O_{1}(T),O_{1}^{\dagger}(T)\rangle\rangle_{\omega}\langle\langle
O_{0},O_{0}^{\dagger}\rangle\rangle_{\omega}=\langle\langle
O_{1}(T),O_{0}^{\dagger}\rangle\rangle_{\omega}\Big{(}\langle[O_{0},O_{0}^{\dagger}]\rangle+\langle\langle
O_{0},O_{1}^{\dagger}(T)\rangle\rangle_{\omega}\Big{)},$ (59)

or equivalently

$\langle[O_{0},O_{0}^{\dagger}]\rangle\langle\langle
O_{1}(T),O_{0}^{\dagger}\rangle\rangle_{\omega}=\langle\langle
O_{1}(T),O_{1}^{\dagger}(T)\rangle\rangle_{\omega}^{(1)}\langle\langle
O_{0},O_{0}^{\dagger}\rangle\rangle_{\omega}.$ (60)

Extracting now $\langle\langle
O_{1}(T),O_{0}^{\dagger}\rangle\rangle_{\omega}$ in the left side of (60) from
(53), one readily gets the ordinary Dyson equation

$\langle[O_{0},O_{0}^{\dagger}]\rangle\Big{(}(z-\theta_{0}(T))\langle\langle
O_{0},O_{0}^{\dagger}\rangle\rangle_{\omega}-\langle[O_{0},O_{0}^{\dagger}]\rangle\Big{)}=\langle\langle
O_{1}(T),O_{1}^{\dagger}(T)\rangle\rangle_{\omega}^{(1)}\langle\langle
O_{0},O_{0}^{\dagger}\rangle\rangle_{\omega},$ (61)

or equivalently

$\langle\langle O,O^{\dagger}\rangle\rangle_{\omega}\equiv\langle\langle
O_{0},O_{0}^{\dagger}\rangle\rangle_{\omega}=\cfrac{\langle[O_{0},O_{0}^{\dagger}]\rangle}{z-\theta_{0}(T)-\cfrac{\langle\langle
O_{1}(T),O_{1}^{\dagger}(T)\rangle\rangle_{\omega}^{(1)}}{\langle[O_{0},O_{0}^{\dagger}]\rangle}}\,.$
(62)

A correspondence between (62) and the more usual form (25) was studied in [7].

##  3 Derivation of the nested Dyson equation

###  3.1 Equations on the first level irreducible Green functions

Let ${\cal A}$ and ${\cal B}$ be first order irreducible operators so that

$\langle[{\cal A},O_{0}^{\dagger}]\rangle=\langle[O_{0},{\cal B}]\rangle=0.$
(63)

The substitution of (63) into (4) and (5) yields

$\displaystyle z\langle\langle O_{0},{\cal
B}\rangle\rangle_{\omega}=\langle\langle O_{0},[\hat{H},{\cal
B}]\rangle\rangle_{\omega},$ (64) $\displaystyle z\langle\langle{\cal
A},O_{0}^{\dagger}\rangle\rangle_{\omega}=\langle\langle[{\cal
A},\hat{H}],O_{0}^{\dagger}\rangle\rangle_{\omega}.$ (65)

Representing now $z\langle\langle{\cal A},{\cal
B}\rangle\rangle^{(1)}_{\omega}$ in two equivalent forms

$\displaystyle z\langle\langle{\cal A},{\cal
B}\rangle\rangle^{(1)}_{\omega}=z\langle\langle{\cal A},{\cal
B}\rangle\rangle_{\omega}-\frac{z\langle\langle{\cal
A},O_{0}^{\dagger}\rangle\rangle_{\omega}\cdot\langle\langle O_{0},{\cal
B}\rangle\rangle_{\omega}}{\langle\langle
O_{0},O_{0}^{\dagger}\rangle\rangle_{\omega}},$ $\displaystyle
z\langle\langle{\cal A},{\cal
B}\rangle\rangle^{(1)}_{\omega}=z\langle\langle{\cal A},{\cal
B}\rangle\rangle_{\omega}-\frac{\langle\langle{\cal
A},O_{0}^{\dagger}\rangle\rangle_{\omega}\cdot z\langle\langle O_{0},{\cal
B}\rangle\rangle_{\omega}}{\langle\langle
O_{0},O_{0}^{\dagger}\rangle\rangle_{\omega}},$ (66)

and utilizing (4), (5), (64) and (65), one readily gets

$\displaystyle z\langle\langle{\cal A},{\cal
B}\rangle\rangle^{(1)}_{\omega}=\langle[{\cal A},{\cal
B}]\rangle+\langle\langle{\cal A},[\hat{H},{\cal
B}]\rangle\rangle_{\omega}-\frac{\langle\langle{\cal
A},O_{0}^{\dagger}\rangle\rangle_{\omega}\langle\langle O_{0},[\hat{H},{\cal
B}]\rangle\rangle_{\omega}}{\langle\langle
O_{0},O_{0}^{\dagger}\rangle\rangle_{\omega}},$ $\displaystyle
z\langle\langle{\cal A},{\cal B}\rangle\rangle^{(1)}_{\omega}=\langle[{\cal
A},{\cal B}]\rangle+\langle\langle[{\cal A},\hat{H}],{\cal
B}\rangle\rangle_{\omega}-\frac{\langle\langle[{\cal
A},\hat{H}],O_{0}^{\dagger}\rangle\rangle_{\omega}\langle\langle O_{0},{\cal
B}\rangle\rangle_{\omega}}{\langle\langle
O_{0},O_{0}^{\dagger}\rangle\rangle_{\omega}},$ (67)

or equivalently

$\displaystyle z\langle\langle{\cal A},{\cal
B}\rangle\rangle^{(1)}_{\omega}=\langle[{\cal A},{\cal
B}]\rangle+\langle\langle{\cal A},[\hat{H},{\cal
B}]\rangle\rangle_{\omega}^{(1)},$ (68) $\displaystyle z\langle\langle{\cal
A},{\cal B}\rangle\rangle^{(1)}_{\omega}=\langle[{\cal A},{\cal
B}]\rangle+\langle\langle[{\cal A},\hat{H}],{\cal
B}\rangle\rangle_{\omega}^{(1)}.$ (69)

Equations (68) and (69) are the first level analogs of (4) and (5).

###  3.2 Extraction of the second order irreducible operator

After the analogy of (48), we define the second order irreducible part of
$O_{0}$

$O_{2}(T)\equiv[O_{1}(T),\hat{H}]-\theta_{1}(T)O_{1}(T)-\rho_{1}(T)O_{0},$
(70)

where

$\theta_{1}(T)=\bar{\theta}_{1}(T)=\frac{\langle[[O_{1}(T),\hat{H}],O_{1}^{\dagger}(T)]\rangle}{\langle[O_{1}(T),O_{1}^{\dagger}(T)]\rangle},\qquad\rho_{1}(T)=\frac{\langle[[O_{1}(T),\hat{H}],O_{0}^{\dagger}]\rangle}{\langle[O_{0},O_{0}^{\dagger}]\rangle}.$
(71)

The proof of the relation $\bar{\theta}_{1}(T)=\theta_{1}(T)$ is similar to
(47). Let us also prove that

$\bar{\rho}_{1}(T)=\rho_{1}(T).$ (72)

Really, according to (48) and (46),

$\rho_{1}(T)=\frac{\langle[[O_{1}(T),\hat{H}],O_{0}^{\dagger}]\rangle}{\langle[O_{0},O_{0}^{\dagger}]\rangle}=\frac{\langle[[[O_{0},\hat{H}],\hat{H}],O_{0}^{\dagger}]\rangle}{\langle[O_{0},O_{0}^{\dagger}]\rangle}-\theta_{0}^{2}(T).$
(73)

So (72) is equivalent to

$\langle[O_{0},[\hat{H},[\hat{H},O_{0}^{\dagger}]]]\rangle=\langle[[[O_{0},\hat{H}],\hat{H}],O_{0}^{\dagger}]\rangle.$
(74)

The latter, in its turn, follows from (3) and the equality

$\displaystyle[O_{0},[\hat{H},[\hat{H},O_{0}^{\dagger}]]]-[[[O_{0},\hat{H}],\hat{H}],O_{0}^{\dagger}]$
$\displaystyle=[(O_{0}O_{0}^{\dagger}+O_{0}^{\dagger}O_{0}),\hat{H}^{2}]-2[(O_{0}^{\dagger}\hat{H}O_{0}+O_{0}\hat{H}O_{0}^{\dagger}),\hat{H}].$
(75)

Equations (70)-(72) result in the following second order irreducibility
conditions

$\langle[O_{2}(T),O_{1}^{\dagger}(T)]\rangle=\langle[O_{2}(T),O_{0}^{\dagger}]\rangle=\langle[O_{1}(T),O_{2}^{\dagger}(T)]\rangle=\langle[O_{0},O_{2}^{\dagger}(T)]\rangle=0.$
(76)

The following equations

$\displaystyle(z-\theta_{1}(T))\langle\langle
O_{1}(T),O_{1}^{\dagger}(T)\rangle\rangle_{\omega}^{(1)}=\langle[O_{1}(T),O_{1}^{\dagger}(T)]\rangle+\langle\langle
O_{1}(T),O_{2}^{\dagger}(T)\rangle\rangle_{\omega}^{(1)},\quad$ (77)
$\displaystyle(z-\theta_{1}(T))\langle\langle
O_{1}(T),O_{1}^{\dagger}(T)\rangle\rangle_{\omega}^{(1)}=\langle[O_{1}(T),O_{1}^{\dagger}(T)]\rangle+\langle\langle
O_{2}(T),O_{1}^{\dagger}(T)\rangle\rangle_{\omega}^{(1)},\quad$ (78)
$\displaystyle(z-\theta_{1}(T))\langle\langle
O_{2}(T),O_{1}^{\dagger}(T)\rangle\rangle_{\omega}^{(1)}=\langle\langle
O_{2}(T),O_{2}^{\dagger}(T)\rangle\rangle_{\omega}^{(1)},$ (79)

may be readily obtained from (68) and (69) in the same manner as (52)-(54)
were obtained from (4) and (5) (in former case, however, one has to
additionally account for (57)).

###  3.3 Introduction of the second level irreducible Green function

Let

$\langle\langle{\cal A},{\cal
B}\rangle\rangle_{\omega}^{(2)}\equiv\langle\langle{\cal A},{\cal
B}\rangle\rangle_{\omega}^{(1)}-\frac{\langle\langle{\cal
A},O_{1}^{\dagger}(T)\rangle\rangle_{\omega}^{(1)}\langle\langle
O_{1}(T),{\cal B}\rangle\rangle_{\omega}^{(1)}}{\langle\langle
O_{1}(T),O_{1}^{\dagger}(T)\rangle\rangle_{\omega}^{(1)}},$ (80)

be the second level irreducible Green function. According to (80) and (56) for
two arbitrary ${\cal A}$ and ${\cal B}$ and four arbitrary complex numbers
$\xi_{1}$, $\xi_{0}$, $\eta_{1}$ and $\eta_{0}$, one has the second level
irreducibility property

$\langle\langle{\cal A}+\xi_{1}O_{1}(T)+\xi_{0}O_{0},{\cal
B}+\eta_{1}O_{1}^{\dagger}(T)+\eta_{0}O_{0}^{\dagger}\rangle\rangle_{\omega}^{(2)}=\langle\langle{\cal
A},{\cal B}\rangle\rangle_{\omega}^{(2)}.$ (81)

In the same manner as it was done for (62), one may prove that

$\langle\langle
O_{1}(T),O_{1}^{\dagger}(T)\rangle\rangle_{\omega}^{(1)}=\cfrac{\langle[O_{1}(T),O_{1}^{\dagger}(T)]\rangle}{z-\theta_{1}(T)-\cfrac{\langle\langle
O_{2}(T),O_{2}^{\dagger}(T)\rangle\rangle_{\omega}^{(2)}}{\langle[O_{1}(T),O_{1}^{\dagger}(T)]\rangle}}\,.$
(82)

By itself, (82) is an ordinary Dyson equation for $\langle\langle
O_{1}(T),O_{1}^{\dagger}(T)\rangle\rangle_{\omega}^{(1)}$. At the same time,
the combination of (82) and (62) results in the nested Dyson equation for
$\langle\langle O,O^{\dagger}\rangle\rangle_{\omega}$

$\langle\langle
O,O^{\dagger}\rangle\rangle_{\omega}=\cfrac{\langle[O_{0},O_{0}^{\dagger}]\rangle}{z-\theta_{0}(T)-\cfrac{\langle[O_{1}(T),O_{1}^{\dagger}(T)]\rangle}{z-\theta_{1}(T)-\cfrac{\langle\langle
O_{2}(T),O_{2}^{\dagger}(T)\rangle\rangle^{(2)}_{\omega}}{\langle[O_{1}(T),O_{1}^{\dagger}(T)]\rangle}}}.$
(83)

Although the iteration process may be continued [10], one has to effectively
break it in order to obtain a tractable result. In our further treatment of
the Ising model, we shall suggest two such breakdowns.

##  4 Ordinary Dyson equation for the Ising chain

###  4.1 Evaluation of $\theta_{0}(T)$

Taking

$O\equiv O_{0}={\bf S}^{+}_{n},\qquad O\equiv O_{0}^{\dagger}={\bf
S}^{-}_{n},\qquad\hat{H}\equiv\hat{H}^{(Is)},$ (84)

and accounting for (31) and (32), one obviously has

$\langle[O_{0},O_{0}^{\dagger}]\rangle=2\langle{\bf
S}^{z}_{n}\rangle=A_{+}(T)+A_{0}(T)+A_{-}(T).$ (85)

So according to (43),

$\langle[O_{0},O_{0}^{\dagger}]\rangle=2\langle{\bf
S}^{z}_{n}\rangle=1-2\zeta+o(\zeta).$ (86)

From (84), (27) and (36), it follows that

$[O_{0},\hat{H}]=h{\bf S}^{+}_{n}+J_{z}{\bf
V}^{+}_{n},\qquad[\hat{H},O_{0}^{\dagger}]=h{\bf S}^{-}_{n}+J_{z}{\bf
V}^{-}_{n}.$ (87)

Following (10) and (36)

$[{\bf V}^{+}_{n},{\bf S}^{-}_{n}]=[{\bf S}^{+}_{n},{\bf V}^{-}_{n}]=2{\bf
V}^{z}_{n}.$ (88)

From (84), (87), and (88) one readily gets

$[[O_{0},\hat{H}],O_{0}^{\dagger}]=[O_{0},[\hat{H},O^{\dagger}_{0}]]=2(h{\bf
S}^{z}_{n}+J_{z}{\bf V}^{z}_{n}),$ (89)

and the substitution of (86) and (89) into (46) with account for (30) yields

$\theta_{0}(T)=h+J_{z}\frac{\langle{\bf V}^{z}_{n}\rangle}{\langle{\bf
S}^{z}_{n}\rangle}=\omega_{+}-J_{z}\gamma(T),$ (90)

where

$\gamma(T)\equiv\frac{\sigma_{1}(T)-2\sigma_{2}(T)}{\sigma_{1}(T)}=\frac{A_{0}(T)+2A_{-}(T)}{A_{+}(T)+A_{0}(T)+A_{-}(T)}.$
(91)

According to (43),

$\gamma(T)=\frac{2\zeta+o(\zeta)}{1+o(1)}=2\zeta+o(\zeta).$ (92)

Hence, the substitution of (92) into (90) gives

$\theta_{0}(T)=\omega_{+}-2J_{z}\zeta+o(\zeta).$ (93)

Substitutions of (87) and (90) into (48) result in

$O_{1}(T)=J_{z}[{\bf V}^{+}_{n}+(\gamma(T)-1){\bf S}^{+}_{n}],\qquad
O_{1}^{\dagger}(T)=J_{z}[{\bf V}^{-}_{n}+(\gamma(T)-1){\bf S}^{-}_{n}].$ (94)

###  4.2 Breaking down the algorithm

According to (62), $\langle\langle
O_{0},O_{0}^{\dagger}\rangle\rangle_{\omega}$ may be expressed from
$\langle\langle O_{1}(T),O_{1}^{\dagger}(T)\rangle\rangle_{\omega}^{(1)}$. At
the same time, the definition (55) for $\langle\langle
O_{1}(T),O_{1}^{\dagger}(T)\rangle\rangle_{\omega}^{(1)}$ implies the
knowledge of $\langle\langle O_{0},O_{0}^{\dagger}\rangle\rangle_{\omega}$.
Due to the gap this vicious circle may be partially broken with the use of
special effective operators

$O_{1}^{\rm eff}=J_{z}({\bf V}^{+}_{n}-{\bf S}^{+}_{n}),\qquad{O_{1}^{\rm
eff}}^{\dagger}=J_{z}({\bf V}^{-}_{n}-{\bf S}^{-}_{n}),$ (95)

or equivalently

$O_{1}^{\rm eff}=-J_{z}(Q_{n-1}+Q_{n+1}){\bf S}^{+}_{n},\qquad{O_{1}^{\rm
eff}}^{\dagger}=-J_{z}(Q_{n-1}+Q_{n+1}){\bf S}^{-}_{n}.$ (96)

Comparing (94) and (95), one readily gets

$O_{1}(T)=O_{1}^{\rm eff}+J_{z}\gamma(T)O_{0},\qquad
O_{1}^{\dagger}(T)={O_{1}^{\rm eff}}^{\dagger}+J_{z}\gamma(T)O_{0},$ (97)

so, according to (56),

$\langle\langle
O_{1}(T),O_{1}^{\dagger}(T)\rangle\rangle_{\omega}^{(1)}=\langle\langle
O_{1}^{eff},{O_{1}^{\rm eff}}^{\dagger}\rangle\rangle_{\omega}^{(1)}.$ (98)

From (96) and (38), immediately follows the first order effectiveness relation

${O_{1}^{\rm
eff}}^{\dagger}|\emptyset\rangle=0\Longleftrightarrow\langle\emptyset|O_{1}^{\rm
eff}=0,$ (99)

under which, the spectral expansion (7) yields

$\langle\langle O_{0},{O_{1}^{\rm
eff}}^{\dagger}\rangle\rangle_{\omega}=O(\zeta),\qquad\langle\langle
O_{1}^{\rm
eff},O_{0}^{\dagger}\rangle\rangle_{\omega}=O(\zeta),\qquad\langle\langle
O_{1}^{\rm eff},{O_{1}^{\rm eff}}^{\dagger}\rangle\rangle_{\omega}=O(\zeta).$
(100)

At the same time (also according to (7)),

$\langle\langle O_{0},O_{0}^{\dagger}\rangle\rangle_{\omega}=O(1).$ (101)

Under (100) and (101), from (98) and the definition (55), it follows that

$\langle\langle
O_{1}(T),O_{1}^{\dagger}(T)\rangle\rangle_{\omega}^{(1)}=\langle\langle
O_{1}^{\rm eff},{O_{1}^{\rm
eff}}^{\dagger}\rangle\rangle_{\omega}+O(\zeta^{2}).$ (102)

Now, the substitution of (102) into (62) gives the ordinary Dyson equation

$\langle\langle
O,O^{\dagger}\rangle\rangle_{\omega}=\cfrac{\langle[O_{0},O_{0}^{\dagger}]\rangle}{z-\theta_{0}(T)-\cfrac{\langle\langle
O_{1}^{\rm eff},{O_{1}^{\rm
eff}}^{\dagger}\rangle\rangle_{\omega}}{\langle[O_{0},O_{0}^{\dagger}]\rangle}+o(\zeta)}\,.$
(103)

The right side of (103) may be readily evaluated. Really, according to (96)
and (38),

$\displaystyle O_{1}^{\rm eff}|m\rangle=O_{1}^{\rm
eff}|m_{1},m_{2};isol\rangle=0,$ $\displaystyle O_{1}^{\rm
eff}|m,m+1;string\rangle=-J_{z}(\delta_{m+1,n}|m\rangle+\delta_{m,n}|m+1\rangle).$
(104)

Substituting now (99) and (104) into (7), one gets

$\langle\langle O_{1}^{\rm eff},{O_{1}^{\rm
eff}}^{\dagger}\rangle\rangle_{\omega}=\sum_{m,\tilde{m}}\frac{\zeta|\langle\tilde{m}|O_{1}^{\rm
eff}|m,m+1;string\rangle|^{2}}{z+E_{1}-E_{2,string}}+o(\zeta).$ (105)

From (104), (105), (30) and (40), it follows that

$\langle\langle O_{1}^{\rm eff},{O_{1}^{\rm
eff}}^{\dagger}\rangle\rangle_{\omega}=\frac{2J_{z}^{2}\zeta}{z-\omega_{0}}+o(\zeta).$
(106)

So, the substitution of (106) into (103) yields

$\langle\langle{\bf S}_{n}^{+},{\bf
S}_{n}^{-}\rangle\rangle_{\omega}=\cfrac{2\langle{\bf
S}^{z}_{n}\rangle}{z-\omega_{+}+2J_{z}\zeta-\cfrac{2J_{z}^{2}\zeta}{z-\omega_{0}}+o(\zeta)}=\frac{(1-2\zeta+o(\zeta))(z-\omega_{0})}{(z-\omega_{+})(z-\omega_{0}+2J_{z}\zeta)+o(\zeta)},$
(107)

or in the polar sum form

$\langle\langle{\bf S}_{n}^{+},{\bf
S}_{n}^{-}\rangle\rangle_{\omega}=\frac{1-4\zeta+o(\zeta)}{z-\omega_{+}+o(\zeta)}+\frac{2\zeta+o(\zeta)}{z-\omega_{0}+2J_{z}\zeta+o(\zeta)}.$
(108)

We see that the first term in (108), which is responsible for the processes
(20) and (21), agrees with the exact formulas (29), (30) and (43). At the same
time, denominator in the second term of (108) (responsible for the process
(26)) is different from the corresponding one in (29). Based on this result,
we conclude that the ordinary Dyson equation (103) adequately describes in the
$o(\zeta)$ order only the processes related to the creation of a single
magnon.

##  5 Nested Dyson equation for the Ising chain

###  5.1 Evaluation of $\theta_{1}(T)$

According to (71), the evaluation of $\theta_{1}(T)$ requires the knowledge of
$\langle[O_{1}(T),O_{1}^{\dagger}(T)]\rangle$ and
$\langle[[O_{1}(T),\hat{H}],O_{1}^{\dagger}(T)]\rangle$. In this subsection,
we shall successively calculate them up to the order $o(\zeta)$.

  * •

According to (94), (49) and (84),

$\langle[O_{1}(T),O_{1}^{\dagger}(T)]\rangle=J_{z}\langle[{\bf
V}^{+}_{n},O_{1}^{\dagger}(T)]\rangle=J_{z}^{2}(\langle[{\bf V}^{+}_{n},{\bf
V}^{-}_{n}]\rangle+(\gamma(T)-1)\langle[{\bf
V}^{+}_{n},O_{0}^{\dagger}]\rangle).$ (109)

At the same time,

$\langle[{\bf
V}^{+}_{n},O_{0}^{\dagger}]\rangle=\langle[O_{1}(T)-(\gamma(T)-1)O_{0},O_{0}^{\dagger}]\rangle=(1-\gamma(T))\langle[O_{0},O_{0}^{\dagger}]\rangle.$
(110)

Hence, (109), (110), (85) and the commutation relation

$[{\bf V}_{n}^{+},{\bf V}_{n}^{-}]={\bf S}_{n}^{z}+4{\bf W}_{n}^{z},$ (111)

yield

$\langle[O_{1}(T),O_{1}^{\dagger}(T)]\rangle=J_{z}^{2}[(1-2(1-\gamma(T))^{2})\sigma_{1}(T)+4\sigma_{3}(T)],$
(112)

or, according to (91) and (31),

$\displaystyle\langle[O_{1}(T),O_{1}^{\dagger}(T)]\rangle=J_{z}^{2}[\sigma_{1}(T)-4(1-\gamma(T))\sigma_{2}(T)+4\sigma_{3}(T)]$
$\displaystyle=2J_{z}^{2}[2\gamma(T)\sigma_{2}(T)+A_{-}(T)].$ (113)

Taking now into account that according to (32) $\sigma_{2}(0)=1/4$ and using
expansions (43) and (92), one readily gets from (113)

$\langle[O_{1}(T),O_{1}^{\dagger}(T)]\rangle=2J_{z}^{2}\zeta+o(\zeta).$ (114)

  * •

According to (84), (87) and (94),

$[{\bf S}_{n}^{+},\hat{H}]=O_{1}(T)+[h+(1-\gamma(T))J_{z}]O_{0}.$ (115)

In the same manner, the formula

$[{\bf V}^{+}_{n},\hat{H}]=h{\bf V}^{+}_{n}+J_{z}\Big{(}\frac{1}{2}{\bf
S}^{+}_{n}+2{\bf W}^{+}_{n}\Big{)},$ (116)

(which directly follows from (10), (27) and (36)) may be represented in the
following form

$[{\bf V}^{+}_{n},\hat{H}]=2J_{z}{\bf
W}^{+}_{n}+\frac{h}{J_{z}}O_{1}(T)+\Big{(}(1-\gamma(T))h+\frac{J_{z}}{2}\Big{)}O_{0}.$
(117)

So, according to (94) and (49),

$\langle[[O_{1}(T),\hat{H}],O_{1}^{\dagger}(T)]\rangle=(h+(\gamma(T)-1)J_{z})\langle[O_{1}(T),O_{1}^{\dagger}(T)]\rangle+2J_{z}^{2}\langle[{\bf
W}_{n}^{+},O_{1}^{\dagger}(T)]\rangle,$ (118)

and the substitution of (118) into (71) results in

$\theta_{1}(T)=h+(\gamma(T)-1)J_{z}+2J_{z}^{2}\frac{\langle[{\bf
W}_{n}^{+},O_{1}^{\dagger}(T)]\rangle}{\langle[O_{1}(T),O_{1}^{\dagger}(T)]\rangle}.$
(119)

  * •

Using definitions (94) and (36), one may readily check that

$2[{\bf W}_{n}^{+},O_{1}^{\dagger}(T)]=J_{z}({\bf
V}^{z}_{n}-4(1-\gamma(T)){\bf W}^{z}_{n}).$ (120)

Hence

$2\langle[{\bf
W}_{n}^{+},O_{1}^{\dagger}(T)]\rangle=2J_{z}[\sigma_{2}(T)-2(1-\gamma(T))\sigma_{3}(T)].$
(121)

Taking now into account that according to (31) and (91)

$\displaystyle
4(1-\gamma(T))\sigma_{3}(T)=(1-\gamma(T))[2A_{-}(T)-\sigma_{1}(T)+4\sigma_{2}(T)]$
$\displaystyle=2[(1-\gamma(T))A_{-}(T)+(1-2\gamma(T))\sigma_{2}(T)],$ (122)

one readily gets from (121)

$2\langle[{\bf
W}_{n}^{+},O_{1}^{\dagger}(T)]\rangle=2J_{z}[2\gamma(T)\sigma_{2}(T)+(\gamma(T)-1)A_{-}(T)].$
(123)

Now, the comparison between (113) and (123) yields

$2J_{z}\langle[{\bf
W}_{n}^{+},O_{1}^{\dagger}(T)]\rangle=\langle[O_{1}(T),O_{1}^{\dagger}(T)]\rangle+2J_{z}^{2}(\gamma(T)-2)A_{-}(T).$
(124)

Finally, the substitution of (124) into (119) results in

$\theta_{1}(T)=\omega_{0}+J_{z}\gamma(T)+2J_{z}^{3}\frac{(\gamma(T)-2)A_{-}(T)}{\langle[O_{1}(T),O_{1}^{\dagger}(T)]\rangle},$
(125)

and with the use of expansions (92), (43) and (114), one has

$\theta_{1}(T)=\omega_{0}+o(\zeta).$ (126)

###  5.2 Breaking down the algorithm

According to (84) and (94), operators ${\bf S}_{n}^{+}$ and ${\bf V}_{n}^{+}$
are linear combinations of $O_{0}$ and $O_{1}(T)$. So, using (94), (115), (117
) and (81), one may conclude that for two arbitrary complex numbers $\xi_{1}$
and $\xi_{0}$, one has

$\langle\langle
O_{2}(T),O_{2}^{\dagger}(T)\rangle\rangle^{(2)}_{\omega}=\langle\langle\tilde{O}_{2}(T),\tilde{O}_{2}^{\dagger}(T)\rangle\rangle^{(2)}_{\omega}.$
(127)

where

$\tilde{O}_{2}(T)=2J_{z}^{2}{\bf W}_{n}^{+}+\xi_{1}{\bf V}_{n}^{+}+\xi_{0}{\bf
S}_{n}^{+}.$ (128)

As a result,

$\langle\langle
O_{2}(T),O_{2}^{\dagger}(T)\rangle\rangle^{(2)}_{\omega}=\langle\langle
O_{2}^{\rm eff},{O_{2}^{\rm eff}}^{\dagger}\rangle\rangle^{(2)}_{\omega},$
(129)

where

$O_{2}^{\rm eff}=J_{z}^{2}\Big{(}2{\bf W}_{n}^{+}+\frac{1}{2}{\bf
S}_{n}^{+}-{\bf V}_{n}^{+}\Big{)},$ (130)

or, according to (37),

$O_{2}^{\rm eff}=2J_{z}^{2}Q_{n-1}{\bf S}_{n}^{+}Q_{n+1}.$ (131)

From (131) and (38), immediately follows the second order effectiveness
relation

${O_{2}^{\rm eff}}^{\dagger}|\emptyset\rangle={O_{2}^{\rm
eff}}^{\dagger}|m\rangle=0,$ (132)

and, moreover,

${O_{2}^{\rm eff}}^{\dagger}|m,m+1\rangle={O_{2}^{\rm
eff}}^{\dagger}|m_{1},m_{2};isol\rangle=0,\qquad m_{2}-m_{1}>2.$ (133)

Under (132) spectral expansion (7) yields

$\langle\langle O_{j}^{\rm eff},{O_{2}^{\rm
eff}}^{\dagger}\rangle\rangle_{\omega}=O(\zeta^{2}),\qquad\langle\langle
O_{2}^{\rm eff},{O_{j}^{\rm
eff}}^{\dagger}\rangle\rangle_{\omega}=O(\zeta^{2}),\qquad j=0,1,2,$ (134)

(where $O_{0}^{\rm eff}\equiv O_{0}$). In its turn, at $j=0$ the substitution
of (134) into (55) with account for (101) gives

$\langle\langle O_{2}^{\rm eff},{O_{2}^{\rm
eff}}^{\dagger}\rangle\rangle^{(1)}_{\omega}=\langle\langle O_{2}^{\rm
eff},{O_{2}^{\rm
eff}}^{\dagger}\rangle\rangle_{\omega}+O(\zeta^{4})=O(\zeta^{2}).$ (135)

In similar manner, from (100), (101) and (134), it follows that

$\displaystyle\langle\langle O_{2}^{\rm eff},{O_{1}^{\rm
eff}}^{\dagger}\rangle\rangle^{(1)}_{\omega}=\langle\langle O_{2}^{\rm
eff},{O_{1}^{\rm
eff}}^{\dagger}\rangle\rangle_{\omega}+O(\zeta^{3})=O(\zeta^{2}),$
$\displaystyle\langle\langle O_{1}^{\rm eff},{O_{2}^{\rm
eff}}^{\dagger}\rangle\rangle^{(1)}_{\omega}=\langle\langle O_{1}^{\rm
eff},{O_{2}^{\rm
eff}}^{\dagger}\rangle\rangle_{\omega}+O(\zeta^{3})=O(\zeta^{2}).$ (136)

According to (106) and (136),

$\frac{\langle\langle O_{2}^{\rm eff},{O_{1}^{\rm
eff}}^{\dagger}\rangle\rangle^{(1)}_{\omega}\langle\langle
O_{1}^{eff},{O_{2}^{\rm
eff}}^{\dagger}\rangle\rangle^{(1)}_{\omega}}{\langle\langle O_{1}^{\rm
eff},{O_{1}^{\rm eff}}^{\dagger}\rangle\rangle^{(1)}_{\omega}}=O(\zeta^{3}).$
(137)

Finally, the substitution of (135) and (137) into (80) with account for (129)
yields

$\langle\langle
O_{2}(T),O_{2}^{\dagger}(T)\rangle\rangle^{(2)}_{\omega}=\langle\langle
O_{2}^{\rm eff},{O_{2}^{\rm
eff}}^{\dagger}\rangle\rangle_{\omega}+o(\zeta^{2}).$ (138)

According to (133), the evaluation of $\langle\langle O_{2}^{\rm
eff},{O_{2}^{\rm eff}}^{\dagger}\rangle\rangle_{\omega}$ in the $o(\zeta^{2})$
order needs only to take into account the process (45). From (39) and (131)
one has

${O_{2}^{\rm
eff}}^{\dagger}|m_{1},m_{2},isol\rangle=2J_{z}^{2}\delta_{m_{1},n-1}\delta_{m_{2},n+1}|n-1,n,n+1\rangle.$
(139)

A substitution of (139) into (7) gives

$\langle\langle O_{2}^{\rm eff},{O_{2}^{\rm
eff}}^{\dagger}\rangle\rangle_{\omega}=\frac{4J_{z}^{4}\zeta^{2}}{z+E_{2,isol}-E_{3,string}}+o(\zeta^{2}).$
(140)

So, according to (40) and (138),

$\langle\langle
O_{2}(T),O_{2}^{\dagger}(T)\rangle\rangle_{\omega}^{(2)}=\frac{4J_{z}^{4}\zeta^{2}}{z-\omega_{-}}+o(\zeta^{2}).$
(141)

Finally, the substitution of (86), (93), (126) and (141) into (83) results in

$\langle\langle{\bf S}_{n}^{+},{\bf
S}_{n}^{-}\rangle\rangle_{\omega}=\cfrac{1-2\zeta+o(\zeta)}{z-\omega_{+}+2J_{z}\zeta+o(\zeta)-\cfrac{2J_{z}^{2}\zeta+o(\zeta)}{z-\omega_{0}-\cfrac{2J_{z}^{2}\zeta+o(\zeta)}{z-\omega_{-}+o(\zeta)}}}.$
(142)

After a rather cumbersome calculation, (142) may be transformed into a polar
sum form

$\langle\langle{\bf S}_{n}^{+},{\bf
S}_{n}^{-}\rangle\rangle_{\omega}=\frac{1-4\zeta+o(\zeta)}{z-\omega_{+}+o(\zeta)}+\frac{2\zeta+o(\zeta)}{z-\omega_{0}+o(\zeta)}-\frac{4\zeta^{3}+o(\zeta^{3})}{z-\omega_{-}-2J\zeta+o(\zeta)},$
(143)

which, according to (43), reproduces the first two terms of (29) up to the
order $o(\zeta)$.

##  6 Summary and discussion

In the present paper, continuing the line of research which has been suggested
in [7], we studied the transverse autocorrelator of the magnetically polarized
Ising chain within the Plakida-Tserkovnikov algorithm [8, 9, 10] reduced by
two versions of its breakdown. The first one yields the ordinary Dyson
equation (62) [7], while the second version - the nested one (83). The
obtained results were compared with the exact formula (29) [17]. It has been
shown that in the $o({\rm e}^{-\beta E_{gap}})$ order, the obtained ordinary
Dyson equation completely reproduces only one term of the exact
autocorrelator. The latter is responsible for the creation of a single magnon.
At the same time, the nested Dyson equation additionally reproduces in the
$o({\rm e}^{-\beta E_{gap}})$ order the term responsible for the
transformation of a single magnon into a pair of neighboring magnons (an
analog of a bound two-magnon state). The third term in the exact formula (29)
[17], which is related to the transition from a pair of isolated magnons
(analog of a scattering two-magnon state) into a triple of neighboring magnons
(an analog of a bound three-magnon state) and has the activation energy ${\rm
e}^{-2\beta E_{gap}}$, cannot be correctly reproduced within the approaches
based on (62) or (83).

The main aim of the paper was to suggest the convenient breakdowns and to
compare the obtained results with those of the exact formula (29) [17]. That
is why some details of the obtained results are quite surprising and not
totally clear for the author.

  * •

Despite the fact that all the calculations in Sect. 4 were performed in the
$o({\rm e}^{-\beta E_{gap}})$ order, the obtained second term in (108) does
not agree (within this order) with the corresponding term in the exact formula
(29).

  * •

Formula (108), which correctly describes in the $o({\rm e}^{-\beta E_{gap}})$
order only the (20) and (21) processes, was obtained from (103). The latter
however contains in the denominator the term $\frac{\langle\langle O_{1}^{\rm
eff},{O_{1}^{\rm
eff}}^{\dagger}\rangle\rangle_{\omega}}{\langle[O_{0},O_{0}^{\dagger}]\rangle}$,
which accounts for the process (26). But if we remove this term from (103)
then the resulting formula analogous to (108) will be incorrect even for the
processes (20) and (21).

  * •

For the evaluation of the second term in (143) originated from transitions
(26) of a single magnon into a string of two neighboring magnons, one has to
account for the process (45) related to the transition of two isolated
(scattering) magnons into a string of three neighboring magnons (three-magnon
bound state). But contrary to the former one, which has the activation energy
$E_{gap}$, the latter one has the energy $2E_{gap}$.

We suppose that the calculations developed in this paper will be a guideline
for the future investigation of the $XXZ$-chain.

The author is grateful to S. B. Rutkevich for a helpful discussion.

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