#  General Formula for the Green’s Function Approach to the Spin-1/2
Antiferromagnetic Heisenberg Model

Daiki Sasamoto<EMAIL_ADDRESS>Takao Morinari
<EMAIL_ADDRESS>Course of Studies on Materials Science,
Graduate School of Human and Environmental Studies, Kyoto University, Kyoto
606-8501, Japan

###### Abstract

A wide range of analytical and numerical methods are available to study
quantum spin systems. However, the complexity of spin correlations and
interactions limits their applicability to specific temperature ranges. The
analytical approach utilizing Green’s function has proved advantageous, as it
allows for formulation without restrictions on the presence of long-range
order and facilitates estimation of the spin excitation spectrum and
thermodynamic quantities across the entire temperature range. In this work, we
present a generalized formulation of the Green’s function method that can be
applied to diverse spin systems. As specific applications, we consider the
hypercubic lattice and the $J_{1}$-$J_{2}$ model. For the cubic lattice case,
the Green’s function approach provides a good estimation for the transition
temperature. Regarding the $J_{1}$-$J_{2}$ model, we include nematic
correlations in the analysis and find no signature of such correlations,
though accurate numerical calculations are required in the presence of strong
frustration. Although our focus is on the spin one-half antiferromagnetic
Heisenberg model on an arbitrary lattice, the Green’s function approach can be
generalized to incorporate other interactions and higher spin values.

##  I Introduction

Quantum antiferromagnetic Heisenberg models are of utmost importance in
exploring intriguing phenomena. Of particular significance is the case of
spin-1/2, where quantum correlation effects are most pronounced. A well-known
example is cuprate superconductorsKeimer  _et al._ (2015), where high-
temperature superconductivity emerges upon hole doping in the quasi-two-
dimensional spin-1/2 Heisenberg antiferromagnet. The rich physics of cuprate
superconductors is likely linked to the phenomenon of quantum spin liquid.
Originating from Anderson’s proposal of the resonating valence bond
stateAnderson (1973), extensive theoretical and experimental research has been
devoted to spin liquid statesBalents (2010); Savary and Balents (2017); Zhou
_et al._ (2017), which may host exotic quantum states.

The exploration of novel quantum ground states poses a challenge due to their
absence of long-range order. In cases where magnetic long-range order exists,
mean-field theory can be applied, and quantum fluctuations around it can be
considered. However, our particular interest lies in states lacking long-range
order and exhibiting high entanglement. Analytically investigating such
systems is a formidable task, necessitating the adoption of sophisticated
numerical algorithms, such as the density-matrix renormalization group, tensor
network algorithms, and others.

Our interest in quantum spin liquid states extends beyond their ground states.
We must also explore their finite-temperature properties, especially when
considering high-temperature superconductivity and practical quantum devices.
At finite temperatures, quantum Monte Carlo serves as a powerful numerical
algorithm. However, it may encounter challenges, particularly in cases
involving frustration, where the negative sign problem can arise.

In this paper, we employ a double-time Green’s function methodTyablikov and
Bonch-Bruevich (1962); Zubarev (1960). This approach does not rely on long-
range order and can be applied to both the ground state and finite-temperature
properties. By employing a decoupling scheme in the equations of motion that
preserves the spin rotational symmetryKondo and Yamaji (1972), we derive a
general formula independent of spatial dimensionality and the range of the
exchange interaction. As specific applications, we investigate the three-
dimensional antiferromagnetic Heisenberg model and the $J_{1}$-$J_{2}$ model.
Our results demonstrate that for the former case, the magnetic transition
temperature is in good agreement with the quantum Monte Carlo resultSandvik
(1998). For the latter case, we develop a self-contained formulation of the
theory that does not require inputs from other methods. This highlights the
versatility and accuracy of our Green’s function approach in studying diverse
spin systems and finite-temperature properties.

The rest of the paper is organized as follows. In Sec. II, we derive a general
theoretical formula. In Sec. III, we apply this formula to the $d$-dimensional
hypercubic lattice. Specifically, we estimate the transition temperature for
the case of $d=3$ and demonstrate its good agreement with the quantum Monte
Carlo result. In Sec. IV, we investigate the $J_{1}$-$J_{2}$ model. We compute
the correlation functions at zero temperature. Despite using only a single
parameter for the decoupling approximation, our results are in good agreement
with previous works employing several decoupling parameters. We also present
the spin-wave dispersion.

##  II Formulation

We consider a general spin-1/2 quantum Heisenberg antiferromagnet on an
arbitrary lattice, with the Hamiltonian expressed as:

$H=\sum_{\langle i,j\rangle}J_{ij}\bm{S}_{i}\cdot\bm{S}_{j},$ (1)

where $\bm{S}_{i}=(S_{i}^{x},S_{i}^{y},S_{i}^{z})$ represents the spin-1/2
operator at site $i$. The parameter $J_{ij}=J_{ji}$ denotes the interaction
strength between the spins at sites $i$ and $j$, and $\langle i,j\rangle$
indicates the sum over a general pair of spins on the lattice.

In the double-time Green’s function method, we study the equation of motion
for the spin Green’s function. To facilitate this analysis, we find it
convenient to work with the Matsubara Green’s function rather than the real-
time Green’s function. Denoting the imaginary time by $\tau$ and introducing
the imaginary time ordering operator $T_{\tau}$, the Matsubara Green’s
function is defined as follows:

$\mathcal{G}_{j}(\tau)=-\langle T_{\tau}S_{0}^{+}(\tau)S_{j}^{-}(0)\rangle,$
(2)

where $S_{j}^{\pm}=S_{j}^{x}\pm iS_{j}^{y}$ and $A\left(\tau\right)$ is
defined as $A\left(\tau\right)={e^{\tau H}}A{e^{-\tau H}}$ for any operator
$A$. $0$ represents a site in the bulk of the system.

In order to systematically compute the equation of motion, we first present a
general formula. Suppose $A$ and $B$ are bosonic operators. We define the
following Green’s function:

${{\cal
G}_{AB}}(\tau)=-\langle{T_{\tau}}A(\tau)B(0)\rangle\equiv{\left\langle{A}|{B}\right\rangle_{\tau}}.$
(3)

The equation of motion is given by

$\frac{\partial}{{\partial\tau}}{{\cal
G}_{AB}}(\tau)=-\langle{T_{\tau}}\left[{H,A(\tau)}\right]B(0)\rangle-\delta\left(\tau\right)\left\langle{\left[{A,B}\right]}\right\rangle.$
(4)

The Fourier transform of ${{\cal G}_{AB}}\left(\tau\right)$ is defined by:

${{\cal
G}_{AB}}\left({i{\omega_{n}}}\right)=\int_{0}^{\beta}{d\tau}{e^{i{\omega_{n}}\tau}}{\left\langle{A}|{B}\right\rangle_{\tau}}\equiv{\left\langle{A}|{B}\right\rangle_{i{\omega_{n}}}}.$
(5)

Here the bosonic Matsubara frequency $\omega_{n}$ takes the values
$\omega_{n}=\frac{2\pi n}{\beta}$, where $n$ is an integer, and
$\beta=\frac{1}{k_{\rm B}T}$ is the inverse temperature with $k_{\rm B}$ being
the Boltzmann constant and $T$ being the temperature. After the Fourier
transformation of Eq. (4), we obtain the following equation:

$i{\omega_{n}}{\left\langle{A}|{B}\right\rangle_{i{\omega_{n}}}}={\left\langle{{\left[{A,H}\right]}}|{B}\right\rangle_{i{\omega_{n}}}}+\left\langle{\left[{A,B}\right]}\right\rangle.$
(6)

This is the general equation of motion. We can use Eq. (6) with $A$ replaced
by ${\left[{A,H}\right]}$ to derive the equation of motion for the first term
in the right-hand side. By repeating similar steps, we can derive higher-order
equations of motion. Furthermore, by rewriting Eq. (3) as:

${{\cal
G}_{AB}}(\tau)=-\left\langle{{T_{\tau}}A\left(0\right)B\left({-\tau}\right)}\right\rangle,$
(7)

we can derive the following equation of motionTserkovnikov (1981):

$i{\omega_{n}}{\left\langle{A}|{B}\right\rangle_{i{\omega_{n}}}}=-{\left\langle{A}\mathrel{\left|{\vphantom{A{\left[{B,H}\right]}}}\right.\kern-1.2pt}{{\left[{B,H}\right]}}\right\rangle_{i{\omega_{n}}}}+\left\langle{\left[{A,B}\right]}\right\rangle.$
(8)

Applying Eq. (6) to Eq. (2) and computing $\left[{S_{0}^{+},H}\right]$ we
obtain the following equation of motion:

$i\omega_{n}\langle
S_{0}^{+}|S_{j}^{-}\rangle_{i\omega_{n}}=J_{\bm{\delta}_{1}}\langle
S_{0}^{z}S_{1}^{+}|S_{j}^{-}\rangle_{i\omega_{n}}-J_{\bm{\delta}_{1}}\langle
S_{0}^{+}S_{1}^{z}|S_{j}^{-}\rangle_{i\omega_{n}}.$ (9)

Here, we assume that there is no long-range order, so that

$\left\langle{\left[{S_{0}^{+},S_{j}^{-}}\right]}\right\rangle=2{\delta_{j,0}}\left\langle{S_{0}^{z}}\right\rangle=0.$
(10)

In Eq. (9), we denote 1 as the site interacting with site 0, and
$\bm{\delta}_{1}$ represents the displacement vector connecting site 0 and
site 1. The summation with respect to site 1 is implicit to simplify the
notation.

The equation of motions for the two terms in the right hand side of Eq. (9)
are

$\displaystyle
i{\omega_{n}}{\left\langle{{S_{0}^{z}S_{1}^{+}}}|{{S_{j}^{-}}}\right\rangle_{i{\omega_{n}}}}$
$\displaystyle=$
$\displaystyle{J_{{\bm{\delta}_{2}}}}{\left\langle{{S_{0}^{z}S_{1}^{z}S_{1+2}^{+}}}|{{S_{j}^{-}}}\right\rangle_{i{\omega_{n}}}}-{J_{{\bm{\delta}_{2}}}}{\left\langle{{S_{0}^{z}S_{1}^{+}S_{1+2}^{z}}}|{{S_{j}^{-}}}\right\rangle_{i{\omega_{n}}}}+\frac{1}{2}{J_{{\bm{\delta}_{2}}}}{\left\langle{{S_{0}^{+}S_{2}^{-}S_{1}^{+}}}|{{S_{j}^{-}}}\right\rangle_{i{\omega_{n}}}}$
(11)
$\displaystyle-\frac{1}{2}{J_{{\bm{\delta}_{2}}}}{\left\langle{{S_{0}^{-}S_{2}^{+}S_{1}^{+}}}|{{S_{j}^{-}}}\right\rangle_{i{\omega_{n}}}}+2{{\delta}_{1,j}}\left\langle{S_{0}^{z}S_{1}^{z}}\right\rangle-{{\delta}_{0,j}}\left\langle{S_{0}^{-}S_{1}^{+}}\right\rangle,$

and

$\displaystyle
i{\omega_{n}}{\left\langle{{S_{0}^{+}S_{1}^{z}}}|{{S_{j}^{-}}}\right\rangle_{i{\omega_{n}}}}$
$\displaystyle=$
$\displaystyle\frac{1}{2}{J_{{\bm{\delta}_{2}}}}{\left\langle{{S_{0}^{+}S_{1}^{+}S_{1+2}^{-}}}|{{S_{j}^{-}}}\right\rangle_{i{\omega_{n}}}}-\frac{1}{2}{J_{{\bm{\delta}_{2}}}}{\left\langle{{S_{0}^{+}S_{1}^{-}S_{1+2}^{+}}}|{{S_{j}^{-}}}\right\rangle_{i{\omega_{n}}}}+{J_{{\bm{\delta}_{2}}}}{\left\langle{{S_{0}^{z}S_{2}^{+}S_{1}^{z}}}|{{S_{j}^{-}}}\right\rangle_{i{\omega_{n}}}}$
(12)
$\displaystyle-{J_{{\bm{\delta}_{2}}}}{\left\langle{{S_{0}^{+}S_{2}^{z}S_{1}^{z}}}|{{S_{j}^{-}}}\right\rangle_{i{\omega_{n}}}}-{{\delta}_{1,j}}\left\langle{S_{0}^{+}S_{1}^{-}}\right\rangle+2{{\delta}_{0,j}}\left\langle{S_{0}^{z}S_{1}^{z}}\right\rangle.$

Here, $1+2$ represents the site ${\bm{\delta}}_{1}+{\bm{\delta}}_{2}$. It is
important to note that sites 1 and 2 can be the same site in this expression.

Now we apply the decoupling approximation introduced by Kondo and YamajiKondo
and Yamaji (1972). That is, for example,

$\langle
S_{0}^{z}S_{1}^{z}S_{2}^{+}|S_{j}^{-}\rangle_{i\omega_{n}}\simeq\alpha\langle
S_{0}^{z}S_{1}^{z}\rangle\langle S_{2}^{+}|S_{j}^{-}\rangle_{i\omega_{n}},$
(13)

where $\alpha$ is a parameter to be determined. This is understood as
followsKondo and Yamaji (1972):

$\displaystyle{\left\langle{{S_{0}^{z}S_{1}^{z}S_{2}^{+}}}|{{S_{j}^{-}}}\right\rangle_{i{\omega_{n}}}}$
$\displaystyle=$
$\displaystyle{\left\langle{{S_{0}^{z}S_{1}^{z}\left[{\alpha+2\left({1-\alpha}\right)S_{2}^{z}}\right]S_{2}^{+}}}|{{S_{j}^{-}}}\right\rangle_{i{\omega_{n}}}}$
(14) $\displaystyle\simeq$
$\displaystyle\left\langle{S_{0}^{z}S_{1}^{z}\left[{\alpha+2\left({1-\alpha}\right)S_{2}^{z}}\right]}\right\rangle{\left\langle{{S_{2}^{+}}}|{{S_{j}^{-}}}\right\rangle_{i{\omega_{n}}}}$
$\displaystyle=$
$\displaystyle\alpha\left\langle{S_{0}^{z}S_{1}^{z}}\right\rangle{\left\langle{{S_{2}^{+}}}|{{S_{j}^{-}}}\right\rangle_{i{\omega_{n}}}}.$

After applying this decoupling scheme, we obtain

$\displaystyle i{\omega_{n}}{{\cal F}_{j}}\left({i{\omega_{n}}}\right)$ (15)
$\displaystyle\simeq$
$\displaystyle\frac{1}{2}\left[{\left({1-{{\delta}_{1,2}}}\right){J_{{\bm{\delta}_{2}}}}\alpha{c_{1-2}}+{{\delta}_{1,2}}{J_{{\bm{\delta}_{1}}}}}\right]{{\cal
G}_{j}}\left({i{\omega_{n}}}\right)-\frac{1}{2}\left[{\left({1-{{\delta}_{1+2,0}}}\right){J_{{\bm{\delta}_{2}}}}\alpha{c_{1+2}}+{{\delta}_{1+2,0}}{J_{{\bm{\delta}_{2}}}}}\right]{{\cal
G}_{j-1}}\left({i{\omega_{n}}}\right)$
$\displaystyle+\frac{1}{2}\left({1-{{\delta}_{1+2,0}}}\right){J_{{\bm{\delta}_{2}}}}\alpha{c_{1}}{{\cal
G}_{j-1-2}\left({i{\omega_{n}}}\right)}-\frac{1}{2}\left({1-{{\delta}_{1,2}}}\right){J_{{\bm{\delta}_{2}}}}\alpha{c_{1}}{{\cal
G}_{j-2}\left({i{\omega_{n}}}\right)}$
$\displaystyle-\frac{1}{2}{{\delta}_{1+2,0}}{J_{{\bm{\delta}_{2}}}}{{\cal
F}_{j}}\left({i{\omega_{n}}}\right)+\frac{1}{2}{{\delta}_{1,2}}{J_{{\bm{\delta}_{1}}}}{{\cal
F}_{j}}\left({i{\omega_{n}}}\right)+\left({{{\delta}_{1,j}}-{{\delta}_{0,j}}}\right){c_{1}},$

where

${{\cal
F}_{j}}\left({i{\omega_{n}}}\right)={\left\langle{{S_{0}^{z}S_{1}^{+}}}\mathrel{\left|{\vphantom{{S_{0}^{z}S_{1}^{+}}{S_{j}^{-}}}}\right.\kern-1.2pt}{{S_{j}^{-}}}\right\rangle_{i{\omega_{n}}}}-{\left\langle{{S_{0}^{+}S_{1}^{z}}}\mathrel{\left|{\vphantom{{S_{0}^{+}S_{1}^{z}}{S_{j}^{-}}}}\right.\kern-1.2pt}{{S_{j}^{-}}}\right\rangle_{i{\omega_{n}}}}.$
(16)

Here, we defined the spin correlation function by the following equation:

$c_{i-j}=c_{j-i}\equiv 2\langle S_{i}^{+}S_{j}^{-}\rangle=4\langle
S_{i}^{z}S_{j}^{z}\rangle.$ (17)

After the Fourier transformation, we obtain

${{\cal
G}_{\bm{k}}}\left({i{\omega_{n}}}\right)=\sum\limits_{j}{{e^{-i{\bm{k}}\cdot{{\bm{R}}_{j}}}}{{\cal
G}_{j}}\left({i{\omega_{n}}}\right)},$ (18) ${{\cal
F}_{\bm{k}}}\left({i{\omega_{n}}}\right)=\sum\limits_{j}{{e^{-i{\bm{k}}\cdot{{\bm{R}}_{j}}}}{{\cal
F}_{j}}\left({i{\omega_{n}}}\right)},$ (19)

where ${\bm{R}}_{j}$ represents the coordinate of site $j$, we obtain

$\displaystyle i{\omega_{n}}\left({\begin{array}[]{*{20}{c}}{{{\cal
G}_{\bm{k}}}\left({i{\omega_{n}}}\right)}\\\ {{{\cal
F}_{\bm{k}}}\left({i{\omega_{n}}}\right)}\end{array}}\right)$
$\displaystyle\simeq$
$\displaystyle\left({\begin{array}[]{*{20}{c}}0&{{J_{{\bm{\delta}_{1}}}}}\\\
{{K_{\bm{k}}}}&0\end{array}}\right)\left({\begin{array}[]{*{20}{c}}{{{\cal
G}_{\bm{k}}}\left({i{\omega_{n}}}\right)}\\\ {{{\cal
F}_{\bm{k}}}\left({i{\omega_{n}}}\right)}\end{array}}\right)$ (29)
$\displaystyle+\left({\begin{array}[]{*{20}{c}}0\\\
{\left({{e^{i{\bm{k}}\cdot{\bm{\delta}_{1}}}}-1}\right){c_{1}}}\end{array}}\right),$

where

$\displaystyle{K_{\bm{k}}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[{\left({1-{{\delta}_{1,2}}}\right){J_{{\bm{\delta}_{2}}}}\alpha{c_{1-2}}+{{\delta}_{1,2}}{J_{{\bm{\delta}_{1}}}}}\right]$
(30)
$\displaystyle-\frac{1}{2}\left[{\left({1-{{\delta}_{1+2,0}}}\right){J_{{\bm{\delta}_{2}}}}\alpha{c_{1+2}}+{{\delta}_{1+2,0}}{J_{{\bm{\delta}_{2}}}}}\right]{e^{i{\bm{k}}\cdot{\bm{\delta}_{1}}}}$
$\displaystyle-\frac{1}{2}\left({1-{{\delta}_{1,2}}}\right){J_{{\bm{\delta}_{2}}}}\alpha{c_{1}}{e^{i{\bm{k}}\cdot{\bm{\delta}_{2}}}}$
$\displaystyle+\frac{1}{2}\left({1-{{\delta}_{1+2,0}}}\right){J_{{\bm{\delta}_{2}}}}\alpha{c_{1}}{e^{i{\bm{k}}\cdot\left({{\bm{\delta}_{1}}+{\bm{\delta}_{2}}}\right)}}.$

From this equation, we obtain

${\cal
G}_{\bm{k}}(i\omega_{n})=\frac{J_{\bm{\delta_{1}}}\left(e^{i\bm{k}\cdot\bm{\delta_{1}}}-1\right)c_{1}}{\left(i\omega_{n}\right)^{2}-\omega_{\bm{k}}^{2}},$
(31)

where

$\displaystyle\omega_{\bm{k}}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}J_{\bm{\delta_{1}}}\left[(1-\delta_{\bm{\delta_{1}},\bm{\delta_{2}}})J_{\bm{\delta_{2}}}\alpha
c_{1-2}\right]$ (32) $\displaystyle-$
$\displaystyle\frac{1}{2}J_{\bm{\delta_{1}}}\left[\left(1-\delta_{1+2,0}\right)J_{\bm{\delta_{2}}}\alpha
c_{1+2}+\delta_{1+2,0}J_{\bm{\delta_{2}}}\right]e^{i\bm{k}\cdot\bm{\delta_{1}}}$
$\displaystyle-$
$\displaystyle\frac{1}{2}\left(1-\delta_{1,2}\right)J_{\bm{\delta_{1}}}J_{\bm{\delta_{2}}}\alpha
c_{1}e^{i\bm{k}\cdot\bm{\delta_{2}}}$ $\displaystyle+$
$\displaystyle\frac{1}{2}\left(1-\delta_{1+2,0}\right)J_{\bm{\delta_{1}}}J_{\bm{\delta_{2}}}\alpha
c_{1}e^{i\bm{k}\cdot\left(\bm{\delta_{1}+\bm{\delta}_{2}}\right)}.$

Equations (31) and (32) provide us with the general formula. When applying
this formula to a specific system, we only need to perform the necessary
summations with respect to 1 and 2, which are implicit in the equations above.
This allows us to efficiently compute the Green’s function and the
corresponding frequency for the given spin system without the need to derive
the equations from scratch each time.

The correlation functions $c_{i-j}$ are determined through a self-consistent
calculation. The Fourier transform of ${{{\cal
G}_{\bm{k}}}\left({i{\omega_{n}}}\right)}$ leads to

$\displaystyle{{\cal G}_{\bm{k}}}\left(\tau\right)$ $\displaystyle=$
$\displaystyle\frac{1}{\beta}\sum\limits_{i{\omega_{n}}}{{e^{-i{\omega_{n}}\tau}}{{\cal
G}_{\bm{k}}}\left({i{\omega_{n}}}\right)}$ (33) $\displaystyle=$
$\displaystyle\frac{{{A_{\bm{k}}}}}{{2{\omega_{\bm{k}}}}}\left({\frac{{{e^{-\tau{\omega_{\bm{k}}}}}}}{{{e^{-\beta{\omega_{\bm{k}}}}}-1}}-\frac{{{e^{\tau{\omega_{\bm{k}}}}}}}{{{e^{\beta{\omega_{\bm{k}}}}}-1}}}\right),$

where

${A_{\bm{k}}}={J_{{\bm{\delta}_{1}}}}\left({{e^{i{\bm{k}}\cdot{\bm{\delta}_{1}}}}-1}\right){c_{1}}$
(34)

The correlation function is given by

${c_{j}}=2\left\langle{S_{0}^{+}S_{j}^{-}}\right\rangle=\frac{1}{N}\sum\limits_{\bm{k}}{{e^{i{\bm{k}}\cdot{{\bm{R}}_{j}}}}\frac{{{A_{\bm{k}}}}}{{{\omega_{\bm{k}}}}}\coth\left({\frac{{\beta{\omega_{\bm{k}}}}}{2}}\right)}.$
(35)

If we introduce a single decoupling parameter $\alpha$ at Eq. (13), the total
number of parameters to be determined, including $\alpha$, is equal to the
number of self-consistent equations that need to be solved. This can be
achieved by utilizing the identity $c_{0}=1$ in the absence of long-range
order. However, if additional decoupling parameters are introduced, we must
impose extra conditions or constraints on the system to guarantee a unique
solution. These additional conditions can be other theoretical results and/or
experimental findings. We do not introduce additional decoupling parameters,
and employ a single decoupling parameter. Consequently, the system of
equations can be solved independently without requiring any extra conditions.
This self-contained nature of the formalism streamlines the solution process,
making it more efficient and eliminating the need for additional constraints
or conditions, which is particularly advantageous in practical applications.

##  III $d$-Dimensional Hypercubic Lattice

In this section, we shall derive the formula for the hypercubic lattice with a
spatial dimension $d$. We make the assumption that $J_{ij}=J$ if sites $i$ and
$j$ are nearest neighbors, and $J_{ij}=0$ otherwise. From Eq. (32), we obtain

$\omega_{\bm{k}}^{2}=\frac{1}{2}{J^{2}}z\left({1-{\gamma_{\bm{k}}}}\right)\left[{1-\left({1+z{\gamma_{\bm{k}}}}\right){a_{1}}+\left({z-2}\right){a_{11}}+{a_{2}}}\right],$
(36)

where $z=2d$, $a_{1}=\alpha c_{1}$ etc., and

${\gamma_{\bm{k}}}=\frac{1}{d}\sum\limits_{\mu=1}^{d}{\cos{k_{\mu}}}.$ (37)

Here, we set the lattice constant to unity. The Green’s function is given by

${{\cal
G}_{\bm{k}}}\left({i{\omega_{n}}}\right)=-\frac{{Jz\left({1-{\gamma_{\bm{k}}}}\right){c_{1}}}}{{{{\left({i{\omega_{n}}}\right)}^{2}}-\omega_{\bm{k}}^{2}}}.$
(38)

We note that this result includes the one-dimensional chain Kondo and Yamaji
(1972) and the two-dimensional square lattice Shimahara and Takada (1991).

Now we focus on the case of $d=3$. The self-consistent equations are given by

$1=-\frac{{z{c_{1}}J}}{N}\sum\limits_{\bm{k}}{\frac{{1-{\gamma_{\bm{k}}}}}{{{\omega_{\bm{k}}}}}\coth\left({\frac{{\beta{\omega_{\bm{k}}}}}{2}}\right)},$
(39)
${c_{1}}=-\frac{{z{c_{1}}J}}{N}\sum\limits_{\bm{k}}{{\gamma_{\bm{k}}}\frac{{1-{\gamma_{\bm{k}}}}}{{{\omega_{\bm{k}}}}}\coth\left({\frac{{\beta{\omega_{\bm{k}}}}}{2}}\right)},$
(40) $\displaystyle\left({z-2}\right){c_{11}}+{c_{2}}$ $\displaystyle=$
$\displaystyle-\frac{{z{c_{1}}J}}{N}\sum\limits_{\bm{k}}\left({z\gamma_{\bm{k}}^{2}-1}\right)$
(41)
$\displaystyle\times\frac{{1-{\gamma_{\bm{k}}}}}{{{\omega_{\bm{k}}}}}\coth\left({\frac{{\beta{\omega_{\bm{k}}}}}{2}}\right),$

where $c_{11}$ represents the correlation function between sites with a
displacement vector of $(1,1,0)$. For the cubic lattice, $c_{11}$ are the same
for $(0,1,1)$, $(1,0,1)$, $(1,-1,0)$, and so on.

Antiferromagnetic long-range order occurs when $\omega_{\mathbf{Q}}$ vanishes,
where $\mathbf{Q}=(\pi,\pi,\pi)$. We observe that Eqs. (39) to (41) exhibit
convergence when we replace summation with an integral over the Brillouin
zone. Conversely, divergence occurs for $d=1$ and $d=2$, indicating the
absence of long-range order at $T=0$ in these dimensions. To enhance the
efficiency of evaluating Eqs. (39) to (41), we apply the integral formula
primarily in the vicinity of $\mathbf{q}=\mathbf{Q}$, employing a simple
summation for other points in the Brillouin zone. Our result yields a
transition temperature for antiferromagnetic long-range order of $k_{\rm
B}T_{c}/J=1.0689$, in good agreement with the quantum Monte Carlo
resultSandvik (1998) of $k_{\rm B}T_{c}/J=0.946\pm 0.001$. We note that our
estimation of the transition temperature slightly exceeds the previous
resultBarabanov and Zhasinas (1994) of $k_{\rm B}T_{c}/J=1.039$, which was
derived from the point of disappearance of the sublattice magnetization.

##  IV The $J_{1}$-$J_{2}$ model

In the investigation of quantum spin liquid states, the influence of
frustration effects is significant. To assess the viability of the Green’s
function approach for addressing frustrated spin systems, we turn our
attention to the $J_{1}$-$J_{2}$ model on the square latticeGelfand  _et al._
(1989); Dagotto and Moreo (1989); Figueirido  _et al._ (1990); Jiang  _et al._
(2012); Hu  _et al._ (2013); Gong  _et al._ (2014); Wang and Sandvik (2018);
Ferrari and Becca (2020). In this model, $J_{1}$ denotes the interaction
between nearest neighbors, while $J_{2}$ denotes the interaction between next-
nearest neighbors. The system exhibits strong frustration when $J_{1}\sim
J_{2}$. By employing the derived formula above, we obtain the following
results:

$\displaystyle\omega_{\bm{k}}^{2}$ $\displaystyle=$ $\displaystyle
2J_{1}^{2}\left[{\left({1+{a_{2}}+2{a_{11}}}\right)-\left({1+4{\gamma_{\bm{k}}}}\right){a_{1s}}}\right]\left({1-{\gamma_{\bm{k}}}}\right)+2J_{1}^{2}\gamma_{\bm{k}}^{a}\left({1+4{\gamma_{\bm{k}}}}\right){a_{1a}}$
$\displaystyle+2J_{2}^{2}\left({1-{\gamma^{\prime}_{\bm{k}}}}\right)\left[{\left({1+{a_{22}}+2{a_{2}}}\right)-{a_{11}}\left({1+4{\gamma^{\prime}_{\bm{k}}}}\right)}\right]$
$\displaystyle+4{J_{1}}{J_{2}}\left[{\left({1-{\gamma_{\bm{k}}}}\right)+\left({1-{\gamma^{\prime}_{\bm{k}}}}\right)-2{\gamma^{\prime}_{\bm{k}}}\left({1-{\gamma_{\bm{k}}}}\right)}\right]{a_{1s}}+4{J_{1}}{J_{2}}\gamma_{\bm{k}}^{a}\left({1+2{\gamma^{\prime}_{\bm{k}}}}\right){a_{1a}}$
$\displaystyle+4{J_{1}}{J_{2}}\left[{\left({1-{\gamma_{\bm{k}}}}\right)+\left({1-{\gamma^{\prime}_{\bm{k}}}}\right)}\right]{a_{21s}}-4{J_{1}}{J_{2}}\gamma_{\bm{k}}^{a}{a_{21a}}-8{J_{1}}{J_{2}}{\gamma_{\bm{k}}}\left({1-{\gamma^{\prime}_{\bm{k}}}}\right){a_{11}},$

where ${\gamma_{\bm{k}}}=\left({\cos{k_{x}}+\cos{k_{y}}}\right)/2$,

${\gamma^{\prime}_{\bm{k}}}=\cos{k_{x}}\cos{k_{y}},$ (43)
$\gamma_{\bm{k}}^{a}=\frac{1}{2}\left({\cos{k_{x}}-\cos{k_{y}}}\right),$ (44)

and ${a_{1s}}=\left({{a_{1x}}+{a_{1y}}}\right)/2$,
${a_{1a}}=\left({{a_{1x}}-{a_{1y}}}\right)/2$,
${a_{21s}}=\left({{a_{21x}}+{a_{21y}}}\right)/2$,
${a_{21a}}=\left({{a_{21x}}-{a_{21y}}}\right)/2$. $a_{1x}$ ($a_{1y}$) refers
to the nearest neighbor correlation function in the $x$ ($y$) direction.
Similarly, $a_{21x}$ ($a_{21y}$) denotes the correlation function between the
sites separated by the displacement vector $(2,1)$ ($(1,2)$). Here, we
consider the possibility of nematic correlation Gauthé and Mila (2022) through
the variables $a_{1a}$ and $a_{21a}$. When $a_{1a}$ and $a_{21a}$ are non-
vanishing, that indicates the presence of nematic correlations in the system.
We note that Eq. (IV), when excluding these nematic correlations, agrees with
the previous resultSiurakshina  _et al._ (2001). The self-consistent equations
are given by

$\displaystyle\alpha$ $\displaystyle=$
$\displaystyle{a_{1s}}I_{00}^{\left(0\right)}+{a_{1a}}I_{00}^{\left(1\right)}+{a_{11}}I_{00}^{\left(2\right)},$
(45) $\displaystyle{a_{1s}}$ $\displaystyle=$
$\displaystyle{a_{1s}}I_{1s}^{\left(0\right)}+{a_{1a}}I_{1s}^{\left(1\right)}+{a_{11}}I_{1s}^{\left(2\right)},$
(46) $\displaystyle{a_{1a}}$ $\displaystyle=$
$\displaystyle{a_{1s}}I_{1a}^{\left(0\right)}+{a_{1a}}I_{1a}^{\left(1\right)}+{a_{11}}I_{1a}^{\left(2\right)},$
(47) $\displaystyle{a_{11}}$ $\displaystyle=$
$\displaystyle{a_{1s}}I_{11}^{\left(0\right)}+{a_{1a}}I_{11}^{\left(1\right)}+{a_{11}}I_{11}^{\left(2\right)},$
(48) $\displaystyle{a_{2}}$ $\displaystyle=$
$\displaystyle{a_{1s}}I_{20}^{\left(0\right)}+{a_{1a}}I_{20}^{\left(1\right)}+{a_{11}}I_{20}^{\left(2\right)},$
(49) $\displaystyle{a_{21s}}$ $\displaystyle=$
$\displaystyle{a_{1s}}I_{21s}^{\left(0\right)}+{a_{1a}}I_{21s}^{\left(1\right)}+{a_{11}}I_{21s}^{\left(2\right)},$
(50) $\displaystyle{a_{21a}}$ $\displaystyle=$
$\displaystyle{a_{1s}}I_{21a}^{\left(0\right)}+{a_{1a}}I_{21a}^{\left(1\right)}+{a_{11}}I_{21a}^{\left(2\right)},$
(51) $\displaystyle{a_{22}}$ $\displaystyle=$
$\displaystyle{a_{1s}}I_{22}^{\left(0\right)}+{a_{1a}}I_{22}^{\left(1\right)}+{a_{11}}I_{22}^{\left(2\right)},$
(52)

where

$I_{\eta}^{\left(\ell\right)}=-\frac{{8{J^{\left(\ell\right)}}}}{{\beta
N}}\sum\limits_{\bm{k}}{{f_{\eta}}\left({\bm{k}}\right)\sigma_{\bm{k}}^{\left(\ell\right)}g\left({\frac{{\beta{\omega_{\bm{k}}}}}{2}}\right)},$
(53)

with ${J^{\left(0\right)}}={J^{\left(1\right)}}={J_{1}}$,
${J^{\left(2\right)}}={J_{2}}$, $g(x)=x\coth x$, and

$\displaystyle\sigma_{\bm{k}}^{\left(0\right)}$ $\displaystyle=$
$\displaystyle\frac{{1-{\gamma_{\bm{k}}}}}{{\omega_{\bm{k}}^{2}}},$ (54)
$\displaystyle\sigma_{\bm{k}}^{\left(1\right)}$ $\displaystyle=$
$\displaystyle-\frac{{\gamma_{\bm{k}}^{a}}}{{\omega_{\bm{k}}^{2}}},$ (55)
$\displaystyle\sigma_{\bm{k}}^{\left(2\right)}$ $\displaystyle=$
$\displaystyle\frac{{1-{\gamma^{\prime}_{\bm{k}}}}}{{\omega_{\bm{k}}^{2}}}.$
(56)

The function ${f_{\eta}}\left({\bm{k}}\right)$ are defined by

$\displaystyle{f_{00}}\left({\bm{k}}\right)$ $\displaystyle=$ $\displaystyle
1,$ (57) $\displaystyle{f_{1s}}\left({\bm{k}}\right)$ $\displaystyle=$
$\displaystyle{\gamma_{\bm{k}}},$ (58)
$\displaystyle{f_{1a}}\left({\bm{k}}\right)$ $\displaystyle=$
$\displaystyle\gamma_{\bm{k}}^{a},$ (59)
$\displaystyle{f_{11}}\left({\bm{k}}\right)$ $\displaystyle=$
$\displaystyle{\gamma^{\prime}_{\bm{k}}},$ (60)
$\displaystyle{f_{20}}\left({\bm{k}}\right)$ $\displaystyle=$ $\displaystyle
4\gamma_{\bm{k}}^{2}-2{\gamma^{\prime}_{\bm{k}}}-1,$ (61)
$\displaystyle{f_{21s}}\left({\bm{k}}\right)$ $\displaystyle=$
$\displaystyle{\gamma_{\bm{k}}}\left({2{\gamma^{\prime}_{\bm{k}}}-1}\right),$
(62) $\displaystyle{f_{21a}}\left({\bm{k}}\right)$ $\displaystyle=$
$\displaystyle\gamma_{\bm{k}}^{a}\left({2{\gamma^{\prime}_{\bm{k}}}+1}\right),$
(63) $\displaystyle{f_{22}}\left({\bm{k}}\right)$ $\displaystyle=$
$\displaystyle
4{\gamma^{\prime}_{\bm{k}}}^{2}-8\gamma_{\bm{k}}^{2}+4{\gamma^{\prime}_{\bm{k}}}+1.$
(64)

We solve the self-consistent equations (45) to (52) as follows: Initially, we
solve the equations for $r=0$, resulting in a reduced set of equations that
simplifies to a single equation. This equation can be solved using the
bisection method. Subsequently, we solve the differential equations for
$a_{ij}$, employing either the parameter $r$ or the temperature $T$ as the
variable. Notably, it is unnecessary to compute $\alpha$ at intermediary
steps. The change in $\alpha$ concerning $r$ or $T$ can be calculated at the
end of the computation.

Figure 1 presents the results at temperature $T=0$. The correlation functions
are plotted as a function of the ratio $r=J_{2}/J_{1}$. It is worth noting
that there exists a symmetry between the cases $r=0$ and $r=1$, as previously
observed Siurakshina  _et al._ (2001), due to the exchange between $J_{1}$ and
$J_{2}$. Additionally, we observe that the nematic correlations
$c_{1a}=a_{1a}/\alpha$ and $c_{21a}=a_{21a}/\alpha$ are both zero within the
numerical error. The formula derived in this study was obtained by employing
the decoupling scheme at the second-order equation of motion. It is important
to note that the resulting self-consistent equations at $T=0$ do not directly
allow us to discuss the presence of long-range order. However, within this
framework, we can investigate the short-range order. Based on our analysis of
the self-consistent equations, we find no indication of short-range nematic
correlation in the system.

Figure 1:  (Color online) Spin correlation functions as a function of
$r=J_{2}/J_{1}$ at $T=0$. There is symmetry between the correlation functions
at $r=0$ and those at $r=1$. Specifically, $c_{1}$, $c_{11}$, and $c_{2}$ at
$r=0$ correspond to $c_{11}$, $c_{2}$, and $c_{22}$ at $r=1$, respectively.
The results confirm this symmetry. Additionally, the nematic correlations,
$c_{1a}$ and $c_{21a}$, are found to be zero within the numerical error.

The spin-wave dispersion at $T=0$ is shown in Fig. 2. There is a systematic
change in the energy dispersion by varying the values of $J_{1}$ and $J_{2}$.
It is important to note that there is no need to assume any long-range order
to obtain the spin-wave dispersion. However, a discrepancy arises when
considering the gap at $(\pi,\pi)$ and $(\pi,0)$. For the case when $J_{1}$ is
finite and $J_{2}=0$, the gap at $(\pi,\pi)$ should vanish, and similarly, for
$J_{1}=0$ and $J_{2}$ is finite, the gap at $(\pi,0)$ should also vanish.
However, the decoupling approximation at the second-order equation of motion,
as mentioned earlier, leads to a finite gap at these points, contradicting the
expected behavior. Employing the decoupling scheme at the third-order equation
of motion has the potential to yield improvements and could resolve the
discrepancy observed in the gap at $(\pi,\pi)$ and $(\pi,0)$. Such
investigations are left for future research. One can also compute the energy
dispersion at finite temperature without long-range order but with short-range
order. The advantage of this formula is that it enables the computation of
dynamical quantities, which can be experimentally verified.

Figure 2:  (Color online) Spin-wave dispersion along the high-symmetric lines
in the Brillouin zone for different values of $J_{1}$ and $J_{2}$. Here, the
values of $J_{1}$, $J_{2}$, and $\omega_{\bm{k}}$ are expressed in units of
$J$, which is taken as the unit of energy.

We expect the Green’s function approach to become accurate at high
temperatures due to the short spin-spin correlation length. Therefore, in
principle, we can utilize the high-temperature expansion resultRosner  _et
al._ (2003) to set the initial values of the correlation functions, $c_{ij}$,
when solving the self-consistent equations. This procedure works very well
when the frustration effect is weak. However, extremely precise numerical
computations are requiredHutak  _et al._ (2022) when the frustration effect is
significant.

To obtain reliable results for the $J_{1}$-$J_{2}$ model, there are two
possible approaches. One option is to perform a three-dimensional calculation
by considering a quasi-two-dimensional system. Such a calculation was carried
outSchmalfuß  _et al._ (2006) using several decoupling parameters and
incorporating input from exact diagonalization while introducing an
approximate expression for the ground state energy. If we introduce one
decoupling parameter, there is no need to consider inputs from other methods.
Another approach involves implementing the decoupling approximation at the
third-order equation of motion. These aspects remain to be explored in future
research.

##  V Summary

To summarize, we have presented a generalized formulation of the Green’s
function method that can be applied to diverse antiferromagnetic Heisenberg
spin systems. We have discussed the hypercubic lattice and the $J_{1}$-$J_{2}$
model as specific applications. We have derived the spin-wave dispersion
formula for the hypercubic lattice in any spatial dimension. For the cubic
lattice case, the Green’s function approach provides a good estimation for the
transition temperature. For the case of the $J_{1}$-$J_{2}$ model, we computed
the correlation functions at zero temperature and spin-wave dispersion.
Although we have included the effect of nematic correlation, there is no
signature of the nematic order. We note that the Green’s function approach can
be generalized to incorporate other interactions and higher spin values. The
key features of this approach are that it does not need to assume specific
correlations and enables the investigation of finite temperature and dynamical
properties.

Exploring higher-order approximations in the decoupling scheme is promising,
and it may provide more accurate results and a better understanding of the
physical behavior of the spin system. Such investigation can open up new
insights into the spin system and pave the way for more accurate and
comprehensive theoretical descriptions.

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# Free weakly Novikov metabelian algebra
of infinite rank

Iritan Ferreira dos Santos, Alexey M. Kuz’min

###### Abstract.

An explicit base with multiplication table is obtained for the free weakly
Novikov metabelian algebra of infinite rank over an arbitrary field of
characteristic $\neq 2$.

Keywords: Right symmetric algebra, metabelian algebra, Novikov algebra, free
algebra of variety.

MSC 2020: 17A30, 17A50, 17D25, 17D99.

##  1\. Introduction

Weakly Novikov rings appeared in papers by E. Kleinfeld and H. F. Smith [2, 3]
as the rings satisfying

(1) $\displaystyle(x,y,z)$ $\displaystyle=(x,z,y)\quad\textit{(the right
symmetry identity)},$ (2) $\displaystyle x(y,z,t)$
$\displaystyle=(y,z,xt)\quad\textit{(the weakly Novikov identity)},$

where $(x,y,z)=(xy)z-x(yz)$ stands, as usual, for the associator in variables
$x,y,z$. Note that (2) generalizes

(3) $x(yz)=y(xz)\quad\textit{(the left commutativity identity)}.$

For more results on nonmultilinear generalizations of identity (2), see recent
papers by Samanta and Hentzel [12, 13].

Let $F$ be a field of characteristic distinct from $2$,
$\operatorname{\mathsf{WN}}$ be the variety of algebras over $F$ defined by
(1) and (2), and $\operatorname{\mathsf{N}}$ be the Novikov subvariety of
$\operatorname{\mathsf{WN}}$, i. e. the subvariety distinguished by (3).
Following [5], we set $\mathcal{V}^{(2)}$ to be the metabelian subvartiety of
a given variety $\mathcal{V}$, i. e. the subvariety distinguished by

(4) $(xy)(zt)=0\quad\textit{(the metabelian identity)}.$

It’s known from the paper by Shestakov and Zhang [14] that
$\operatorname{\mathsf{N}}^{(2)}$ is left nilpotent of index not more then
$9$. In the present paper, we obtain an explicit base with multiplication
table for the free algebra of $\operatorname{\mathsf{WN}}^{(2)}$ on the
countable set of generators. As a corollary, we prove that
$\operatorname{\mathsf{WN}}^{(2)}$ and $\operatorname{\mathsf{N}}^{(2)}$ are
both left nilpotent of exact index $5$.

We stress that questions on finding certain effective bases for free algebras
in given varieties are strongly motivated by a number of open problems dealing
with varieties of all alternative, Jordan, and Maltsev algebras (see, for ex.
[15, 16, 17, 18]).

##  2\. Preliminary results

We set, as usual, $[a,b]=ab-ba$, $a\circ b=ab+ba$, and $ab=aR_{b}=bL_{a}$. In
what follows, $X=\\{x_{1},x_{2},\dots\\}$ stands for a countable set of
generators of given free algebra. Unless otherwise stated, all algebras are
considered over the field $F$ of characteristic $\neq 2$. Let $\mathfrak{M}$
be the variety of algebras defined by (2).

###  2.1. Monomial base for the free algebra of
$\operatorname{\mathfrak{M}}^{(2)}$

######  Theorem 1.

The free algebra
${\mathfrak{A}}=F_{\operatorname{\mathfrak{M}}^{(2)}}\left<X\right>$ possesses
a basis formed by all monomials of the form

(5) $x_{i}L_{x_{j_{1}}}\dots L_{x_{j_{n}}}R_{x_{k_{1}}}\dots R_{x_{k_{t}}},$

such that the variables $x_{j_{1}},\dots,x_{j_{n}}$, for $n\geqslant 4$, are
invariant with respect to even permutations and the multiplication in
$\left({\mathfrak{A}},\cdot\right)$ is defined by the table

$\displaystyle x_{j}\cdot x_{i}$ $\displaystyle=x_{i}L_{x_{j}},$
$\displaystyle\left(x_{i}L_{x_{j}}\right)\cdot x_{k}$
$\displaystyle=x_{i}L_{x_{j}}R_{x_{k}},$ $\displaystyle
x_{q}\cdot\left(x_{i}L_{x_{j}}R_{x_{k}}\right)$
$\displaystyle=x_{k}L_{x_{i}}L_{x_{j}}L_{x_{q}}-x_{k}L_{x_{q}}L_{x_{i}}L_{x_{j}},$
$\displaystyle\left(x_{i}L_{x_{j_{1}}}\dots L_{x_{j_{n}}}\right)\cdot x_{q}$
$\displaystyle=x_{i}L_{x_{j_{1}}}\dots L_{x_{j_{n}}}R_{x_{q}},$ $\displaystyle
x_{q}\cdot\left(x_{i}L_{x_{j_{1}}}\dots L_{x_{j_{n}}}\right)$
$\displaystyle=x_{i}L_{x_{j_{1}}}\dots L_{x_{j_{n}}}L_{x_{q}},$
$\displaystyle\left(x_{i}L_{x_{j_{1}}}\dots L_{x_{j_{n}}}R_{x_{k_{1}}}\dots
R_{x_{k_{t}}}\right)\cdot x_{q}$ $\displaystyle=x_{i}L_{x_{j_{1}}}\dots
L_{x_{j_{n}}}R_{x_{k_{1}}}\dots R_{x_{k_{t}}}R_{x_{q}},$

where all possible nonzero products of base monomials are given.

###### Proof.

Throughout this proof, we use metabelian identity (4) with no comments.
Consider all possible restrictions of left commutativity identity (2) such
that one of its variables is assumed to be taken form ${\mathfrak{A}}^{2}$.
First, by $x\in{\mathfrak{A}}^{2}$, we have

(6) ${\mathfrak{A}}^{2}R_{t}L_{z}L_{y}=0.$

Further, by setting $y\in{\mathfrak{A}}^{2}$, we get

${\mathfrak{A}}^{2}R_{z}R_{t}L_{x}=0.$

Moreover, in view of (6), $z\in{\mathfrak{A}}^{2}$, yields

${\mathfrak{A}}^{2}[L_{y},R_{t}]L_{x}={\mathfrak{A}}^{2}L_{y}R_{t}L_{x}=0.$

Consequently,

(7) ${\mathfrak{A}}^{3}R_{t}L_{x}=0.$

Furthermore, $t\in{\mathfrak{A}}^{2}$ implies

(8) ${\mathfrak{A}}^{2}\left[L_{z}L_{y},L_{x}\right]=0.$

Finally, by (6) and (2), we obtain

$xL_{y}L_{z}L_{t}L_{u}=-(z,y,x)L_{t}L_{u}=-(z,y,tx)L_{u}=xL_{t}L_{y}L_{z}L_{u},$

i. e.

(9) ${\mathfrak{A}}\left[L_{y}L_{z},L_{t}\right]L_{u}=0.$

Therefore, by virute of (6)–(9), ${\mathfrak{A}}$ is spanned by the monomials
of type (5).

To complete the proof it remains to check that identity (2) holds in
$\left({\mathfrak{A}},\cdot\right)$. Indeed, on generators of
${\mathfrak{A}}$, we have

$x_{q}\cdot\left(x_{j},x_{i},x_{k}\right)=x_{q}\cdot\left(x_{i}L_{x_{j}}R_{x_{k}}-x_{k}L_{x_{i}}L_{x_{j}}\right)=-x_{k}L_{x_{q}}L_{x_{i}}L_{x_{j}}=\left(x_{j},x_{i},x_{q}\cdot
x_{k}\right);$

otherwise, (2) is an immediate consequence of relations (6)-(8). ∎

######  Corollary 1.1.

Every Lie-nilpotent or Jordan-nilpotent subvariety of index $n$ in
$\operatorname{\mathfrak{M}}^{(2)}$ is nilpotent of index not more then
$2n+1$.

###### Proof.

Let us prove that, for ${\mathfrak{B}}\in\operatorname{\mathfrak{M}}^{(2)}$,
$H_{x}=R_{x}-L_{x}$, and $\varTheta_{x}=R_{x}+L_{x}$, each of the relations

${\mathfrak{B}}^{2}H_{x_{1}}\dots
H_{x_{n-1}}=0,\quad{\mathfrak{B}}^{2}\varTheta_{x_{1}}\dots\varTheta_{x_{n}-1}=0$

yields ${\mathfrak{B}}^{2n+1}=0$. In the case $n=1$, the affirmation is
trivial in view of (6). Further, we set

$R^{n}=R_{x_{1}}\dots R_{x_{n}},\quad L^{n}=R_{x_{1}}\dots L_{x_{n}},\quad
H^{n}=H_{x_{1}}\dots H_{x_{n}},\quad n\geqslant 2.$

By virtue of (6) and (7), the relations

${\mathfrak{B}}^{2}H^{n}L_{y}=0,\quad{\mathfrak{B}}^{3}R_{y}H^{n}=0$

yield, respectively,

${\mathfrak{B}}^{2}L^{n+1}=0,\quad{\mathfrak{B}}^{3}R^{n+1}=0.$

Therefore, by Theorem 1,

${\mathfrak{B}}^{3}T_{x_{1}}\dots T_{x_{2n}}=0,$

for all $T_{x_{i}}\in\left\\{R_{x_{i}},L_{x_{i}}\right\\}$. ∎

######  Corollary 1.2.

The flexible and the antiflexible subvarieties of
$\operatorname{\mathfrak{M}}^{(2)}$ are both nilpotent of index not more than
$5$. Moreover, for ${\mathfrak{B}}\in\operatorname{\mathfrak{M}}^{(2)}$ and
$n\geqslant 2$, the identity

(10)
$\left({\mathfrak{B}}^{n},x,y\right)=\pm\left(y,x,{\mathfrak{B}}^{n}\right)$

implies ${\mathfrak{B}}^{n+3}=0$.

###### Proof.

By virtue of (4), identity (10) yields

${\mathfrak{B}}^{n}R_{x}R_{y}=\mp{\mathfrak{B}}^{n}L_{x}L_{y}.$

Thus, in view of (6), we have

${\mathfrak{B}}^{n}L_{x}L_{y}R_{z}=\mp{\mathfrak{B}}^{n}R_{x}R_{y}R_{z}={\mathfrak{B}}^{n}R_{x}L_{y}L_{z}=0.$

Similarly, by (7), we obtain

${\mathfrak{B}}^{n}L_{x}R_{y}R_{z}=\mp{\mathfrak{B}}^{n}L_{x}L_{y}L_{z}={\mathfrak{B}}^{n}R_{x}R_{y}L_{z}=0.$

Therefore, by Theorem 1, ${\mathfrak{B}}^{n+3}=0$. ∎

##  3\. Main Theorem

###  3.1. Auxiliary identities

Let ${\mathfrak{A}}=F_{\operatorname{\mathsf{WN}}^{(2)}}\left<X\right>$ be the
free algebra of the variety $\operatorname{\mathsf{WN}}^{(2)}$ on the
countable set $X=\\{x_{1},x_{2},\dots\\}$ of generators; the symbol
$T_{x}\in\\{R_{x},L_{x}\\}$ is used as a common denotation for the operators
$R_{x}$ and $L_{x}$ of right and left multiplication by $x\in X$,
respectively.

######  Lemma 3.1.

The free algebra ${\mathfrak{A}}$ satisfies the identities

(11) $\displaystyle x\cdot(y,z,t)$ $\displaystyle=\left(y,xz,t\right),$ (12)
$\displaystyle(x,y,zt)$ $\displaystyle=(x,zy,t),$ (13) $\displaystyle x\cdot
y(zt)$ $\displaystyle=-(x,zy,t),$ (14) $\displaystyle x(yz)\cdot t$
$\displaystyle=\bigl{(}x,[y,z],t\bigr{)}+(y,xz,t),$ (15)
$\displaystyle\left(x,y,z\right)\cdot t$
$\displaystyle=\operatorname{\mathit{h}}(x,y,z,t)+\bigl{(}x,y\circ
z,t\bigr{)},$

where

$\operatorname{\mathit{h}}(x,y,z,t)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}(xy,z,t)-(y,xz,t)-2(x,yz,t).$

###### Proof.

By virtue of (2) and (1), we have

$x\cdot(y,z,t)=x\cdot(y,t,z)=(y,t,xz)=(y,xz,t).$

Thus, (11) is proved. Further, combining (2) and (11), we get (12):

$(x,y,zt)=z\cdot(x,y,t)=(x,zy,t).$

Furthermore, by (4) and (12), we obtain (13):

$x\cdot y(zt)=-(x,y,zt)=-(x,zy,t).$

Now, recall that every nonassociative ring satisfies the Teichmüller’s
identity

$x\cdot(y,z,t)+\left(x,y,z\right)\cdot t=(xy,z,t)-(x,yz,t)+(x,y,zt).$

Hence, using (11) and (12), we have

$\left(x,y,z\right)\cdot
t=(xy,z,t)-(y,xz,t)-\bigl{(}x,\left[y,z\right],t\bigr{)}.$

Consequently, to prove (14) it remains to observe that

$\left(x,y,z\right)\cdot t-\left(xy,z,t\right)=-x(yz)\cdot t,$

in view of (4). Moreover, taking into account the denotations, introduced
above,

$\left(x,y,z\right)\cdot
t=(xy,z,t)-(y,xz,t)+\bigl{(}x,\left[z,y\right],t\bigr{)}=\\\
=(xy,z,t)-(y,xz,t)-2(x,yz,t)+\bigl{(}x,y\circ z,t\bigr{)}=\\\
=\operatorname{\mathit{h}}(x,y,z,t)+\bigl{(}x,y\circ z,t\bigr{)},$

we complete the proof of (15). ∎

######  Lemma 3.2.

A monomial $\mu\in{\mathfrak{A}}^{5}$ can be nonzero only if it has the form
$\mu=wR_{x}R_{y}R_{z}$ for some $w\in{\mathfrak{A}}^{2}$ and $x,y,z\in X$

###### Proof.

Let us consider consecutively all possible cases for $\mu\neq
wR_{x}R_{y}R_{z}$. First, by (2) and (4), we have

(16) $\displaystyle wR_{x}R_{y}L_{z}$ $\displaystyle=z(w,x,y)-(w,x,zy)=0,$
(17) $\displaystyle wR_{x}L_{y}L_{z}$ $\displaystyle=w(z,y,x)-(z,y,wx)=0.$

Similarly, by (4) and (11), we get

(18) $wL_{x}L_{y}L_{z}=(y,zx,w)-z(y,x,w)=0.$

Further, combining (2), (4), (11) with (16) and (17), we proceed

(19) $\displaystyle wR_{x}L_{y}R_{z}$
$\displaystyle=wR_{x}R_{z}L_{y}+(y,wx,z)-w(y,x,z)=0,$ (20) $\displaystyle
wL_{x}R_{y}L_{z}$ $\displaystyle=wR_{y}L_{x}L_{z}+z(x,w,y)-(x,w,zy)=0.$

Now, by (11), (17), and (20), we obtain

(21)
$wL_{x}L_{y}R_{z}=wL_{x}R_{z}L_{y}+w[L_{y},R_{z}]L_{x}+(y,xw,z)-x(y,w,z)=0.$

Finally, using (19), (21), and taking into account (1), we prove

(22) $wL_{x}R_{y}R_{z}=wR_{y}L_{x}R_{z}-wL_{y}L_{x}R_{z}+(x,w,y)z-(x,y,w)z=0.$

∎

######  Lemma 3.3.

The T-ideal $\bigl{<}(a,bc,d)\bigr{>}^{\mathrm{T}}$ is spanned by the
associators $(x_{i},x_{j}x_{k},x_{\ell})$ taken only on generators
$x_{i},x_{j},x_{k},x_{\ell}\in X$ of ${\mathfrak{A}}$.

###### Proof.

It’s enough to notice that, by Lemma 3.2, the associators
$(x_{i},x_{j}x_{k},x_{\ell})$ lie in the annihilator of ${\mathfrak{A}}$ and
all associators of the form

$(w,x_{j}x_{k},x_{\ell}),\quad(x_{i},wx_{k},x_{\ell}),\quad(x_{i},x_{j}w,x_{\ell}),\quad(x_{i},x_{j}x_{k},w),$

for $w\in{{\mathfrak{A}}}^{2}$, are null ones. ∎

###  3.2. Pre-base

Recall that a function $f(x,y)$ is said to be symmetric w.r.t. $x,y$ if

$f(x,y)=f(y,x).$

######  Definition.

The base elements of ${\mathfrak{A}}^{3}$ are all elements of types (i)–(v)
defined by the list below, for $x,y,z,t_{1},\dots,t_{k}\in X$:

  * (i)

$x(yz)$,

  * (ii)

$(x,t_{1},t_{2})$,

  * (iii)

$\left(x,yt_{1},t_{2}\right)$,

  * (iv)

$\operatorname{\mathit{h}}(x,t_{1},t_{2},t_{3})$,

  * (v)

$\left(xt_{1}\right)R_{t_{2}}\ldots R_{t_{k}}$,

where the indices $t_{i}$ are used each time when the corresponding element
possesses, by definition, the symmetry property w.r.t. all the subset
$\left\\{t_{i}\right\\}$ of its variables.

######  Lemma 3.4.

The ideal ${\mathfrak{A}}^{3}$ is spanned by its base elements.

###### Proof.

First, taking into account the defining identity (1) for
$\operatorname{\mathsf{WN}}^{(2)}$, it’s not hard to see that the subspace of
polynomials of degree $3$ in ${\mathfrak{A}}$ is spanned by the elements of
types (i) and (ii), in view of the trivial equality

$(xy)z=(x,y,z)+x(yz).$

Further, identities (11), (13)–(15), yield that the subspace of polynomials of
degree $4$ in ${\mathfrak{A}}$ is spanned by the elements of types (iii) and
(iv). While that the symmetry property established for the elements of type
(iii) is a direct consequence of identity (11). At the same time, in the case
of elements of type (iv), the required symmetry property can be obtained as
follows. On one hand, the polynomial $\operatorname{\mathit{h}}(x,y,z,t)$ is
defined as an symmetric element of $z,t$. On the other hand, identity (15)
implies the symmetry of $\operatorname{\mathit{h}}(x,y,z,t)$ with respect to
$y,z$.

Furthermore, by Lemma 3.2, the subspace of polynomials of degree $\geqslant 5$
in ${\mathfrak{A}}$ is spanned by $R$-words. Moreover, combining the defining
identities (1) and (4) of $\operatorname{\mathsf{WN}}^{(2)}$, we have

$wR_{y}R_{z}=wR_{z}R_{y},\quad w\in{\mathfrak{A}}^{2}.$

Finally, Lemma 3.3 implies

$\operatorname{\mathit{h}}(x,t_{1},t_{2},t_{3})\cdot
y=\left(xt_{1}\right)R_{t_{2}}R_{t_{3}}R_{y}.$

This proves the required symmetry property of the elements of type (v). ∎

###  3.3. Base

######  Theorem 2.

The free algebra ${\mathfrak{A}}$ possesses a base formed by all elements from
$X$, $X^{2}$, and all base elements of ${\mathfrak{A}}^{3}$ equipped with the
multiplication $\cdot$ introduced by the following rules:

  * •

Every nonzero left action of a generator $x\in X$ on a base element is defined
by one of the following formulas:

$\displaystyle x\cdot y$ $\displaystyle=xy,$ $\displaystyle x\cdot yz$
$\displaystyle=x(yz),$ $\displaystyle x\cdot y(zt)$ $\displaystyle=-(x,zy,t),$
$\displaystyle x\cdot(y,t_{1},t_{2})$
$\displaystyle=\left(y,xt_{1},t_{2}\right),$

where $y,z,t,t_{1},t_{2}\in X$.

  * •

Every nonzero right action of a generator $y\in X$ on a base element of degree
$\geqslant 2$ is defined by one of the following formulas:

$\displaystyle xz\cdot y$ $\displaystyle=(x,z,y)+x(zy),$ $\displaystyle
x(zt)\cdot y$ $\displaystyle=\bigl{(}x,[z,t],y\bigr{)}+(z,xt,y),$
$\displaystyle\left(x,t_{1},t_{2}\right)\cdot y$
$\displaystyle=\operatorname{\mathit{h}}(x,t_{1},t_{2},y)+\bigl{(}x,t_{1}\circ
t_{2},y\bigr{)},$
$\displaystyle\operatorname{\mathit{h}}(x,t_{1},t_{2},t_{3})\cdot y$
$\displaystyle=\left(xt_{1}\right)R_{t_{2}}R_{t_{3}}R_{y},$
$\displaystyle\left(xt_{1}\right)R_{t_{2}}\ldots R_{t_{k}}\cdot y$
$\displaystyle=\left(xt_{1}\right)R_{t_{2}}\ldots R_{t_{k}}R_{y},\quad
k\geqslant 4,$

where $x,z,t,t_{1},\dots,t_{k}\in X$.

  * •

All the other products of base elements not listed in the formulas above are
null once by definition.

###### Proof.

First we stress that the multiplication $\cdot$ is well-defined. Indeed, by
the direct verification, one can confirm that all symmetry properties,
required for the elements of types (ii)–(v), are inherited by the
corresponding right parts in formulas of multiplication table.

Let ${\mathcal{B}}$ be the span of all introduced base elements over the field
$F$. Consider ${\mathcal{B}}$ as an algebra with respect to the multiplication
$\cdot$ defined by the rule of Theorem 2. Then, Lemma 3.4 yields that
${\mathfrak{A}}$ could be isomorphic to some quotient of ${\mathcal{B}}$. Let
us prove that actually ${\mathfrak{A}}$ is isomorphic to ${\mathcal{B}}$ under
the isomorphism induced by the identical mapping $X\mapsto X$.

Our proof is by formal checking of defining identities (2)–(4) of the variety
$\operatorname{\mathsf{WN}}^{(2)}$ for the algebra ${\mathcal{B}}$. First,
metabelian identity (4) holds in ${\mathcal{B}}$ immediately, by definition of
the multiplication $\cdot$. Further, let us set

$\bigl{(}a,b,c\bigr{)}_{{\mathcal{B}}}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}(a\cdot
b)\cdot c-a\cdot(b\cdot c).$

In what follows, we assume that $x,y,z,t,u$ are variables from $X$ and $w$ is
a common denotation for base elements of ${\mathcal{B}}$ such that neither $w$
nor $X\cdot w$, $w\cdot X$ lie in the annihilator

$\mathrm{Annh}({\mathcal{B}})=F\cdot(X,X^{2},X)$

of ${\mathcal{B}}$, i. e.

$w\in\bigl{\\{}x,\,xy,\,(x,y,z),\,\operatorname{\mathit{h}}(x,y,z,t),\,(xy)R_{z}R_{t}\dots
R_{u}\bigr{\\}}.$

Let us check the right symmetry of ${\mathcal{B}}$:

$\bigl{(}a,b,c\bigr{)}_{{\mathcal{B}}}=\bigl{(}a,c,b\bigr{)}_{{\mathcal{B}}}.$

First, we compute all nonzero associators of type
$\bigl{(}w,b,c\bigr{)}_{{\mathcal{B}}}$, for $b,c\in X$, and verify that the
obtained elements are symmetric in $b,c$. Indeed,

$\displaystyle\bigl{(}x,b,c\bigr{)}_{{\mathcal{B}}}$ $\displaystyle=xb\cdot
c-x\cdot bc=(x,b,c)+x(bc)-x(bc)=(x,b,c),$
$\displaystyle\bigl{(}xy,b,c\bigr{)}_{{\mathcal{B}}}$
$\displaystyle=\bigl{(}xy\cdot b\bigr{)}\cdot
c=\bigl{(}(x,y,b)+x(yb)\bigr{)}\cdot c=$
$\displaystyle=\operatorname{\mathit{h}}(x,y,b,c)+\bigl{(}x,y\circ
b,c\bigr{)}+\bigl{(}x,[y,b],c\bigr{)}+(y,xb,c)=$
$\displaystyle=\operatorname{\mathit{h}}(x,y,b,c)+\bigl{(}x,yb,c\bigr{)}+\bigl{(}x,yb,c\bigr{)}+(y,xb,c),$
$\displaystyle\bigl{(}(x,y,z),b,c\bigr{)}_{{\mathcal{B}}}$
$\displaystyle=\bigl{(}(x,y,z)\cdot b\bigr{)}\cdot c=$
$\displaystyle=\Bigl{(}\operatorname{\mathit{h}}(x,y,z,b)+\bigl{(}x,y\circ
z,b\bigr{)}\Bigr{)}\cdot c=(xy)R_{z}R_{b}R_{c},$
$\displaystyle\bigl{(}\operatorname{\mathit{h}}(x,y,z,t),b,c\bigr{)}_{{\mathcal{B}}}$
$\displaystyle=\bigl{(}\operatorname{\mathit{h}}(x,y,z,t)\cdot b\bigr{)}\cdot
c=(xy)R_{z}R_{t}R_{b}R_{c},$ $\displaystyle\bigl{(}(xy)R_{z}R_{t}\dots
R_{u},b,c\bigr{)}_{{\mathcal{B}}}$ $\displaystyle=\bigl{(}(xy)R_{z}R_{t}\dots
R_{u}\cdot b\bigr{)}\cdot c=(xy)R_{z}R_{t}\dots R_{u}R_{b}R_{c}.$

Further, we compare the values of associators in pairs of types

$\bigl{(}a,b,w\bigr{)}_{{\mathcal{B}}}\,,\quad\bigl{(}a,w,b\bigr{)}_{{\mathcal{B}}}\,,\quad
a,b\in X,\quad w\notin X.$

In the case $w\in X^{2}$, we have

$\displaystyle\bigl{(}a,b,xy\bigr{)}_{{\mathcal{B}}}$
$\displaystyle=-a\cdot(b\cdot xy)=-a\cdot b(xy)=(a,xb,y)=(a,xy,b),$
$\displaystyle\bigl{(}a,xy,b\bigr{)}_{{\mathcal{B}}}$ $\displaystyle=(a\cdot
xy)\cdot b-a\cdot(xy\cdot b)=a(xy)\cdot
b-a\cdot\bigl{(}(x,y,b)+x(yb)\bigr{)}=$
$\displaystyle=\bigl{(}a,[x,y],b\bigr{)}+(x,ay,b)-(x,ay,b)+(a,yx,b)=(a,xy,b).$

Furthermore, let ${{\mathcal{B}}}^{n}$ be the ideal of ${\mathcal{B}}$ spanned
by all base elements of degree $\geqslant n$. We observe that all terms of
associators of types

$\bigl{(}X,X,{{\mathcal{B}}}^{3}\bigr{)}_{{\mathcal{B}}},\quad\bigl{(}X,{{\mathcal{B}}}^{3},X\bigr{)}_{{\mathcal{B}}}$

are null, by virtue of the relations

$X\cdot{{\mathcal{B}}}^{4}=0,\quad
X\cdot(X^{2}X)\subseteq\mathrm{Annh}({\mathcal{B}}),\quad
X\cdot(X,X,X)\subseteq\mathrm{Annh}({\mathcal{B}}).$

Therefore, ${\mathcal{B}}$ is right symmetric.

Finally, to complete the proof, we verify that ${\mathcal{B}}$ is weakly
Novikov. Indeed, in view of the relations above, all terms of the identity

$a\cdot\bigl{(}b,c,d\bigr{)}_{{\mathcal{B}}}=\bigl{(}b,c,a\cdot
d\bigr{)}_{{\mathcal{B}}}\,$

are null whenever at least one of its variables $a,b,c,d$ takes a value that
doesn’t lie in $X$. Otherwise, we have,

$\displaystyle x\cdot\bigl{(}y,z,t\bigr{)}_{{\mathcal{B}}}$
$\displaystyle=x\cdot(y,z,t)=(y,xz,t),$
$\displaystyle\bigl{(}y,z,xt\bigr{)}_{{\mathcal{B}}}$
$\displaystyle=-y\cdot(z\cdot xt)=-y\cdot z(xt)=(y,xz,t).$

∎

##  4\. Corollaries

We stress that combining Lemma 3.2 and the established above symmetry property
for $R$-words

$wR_{y}R_{z}=wR_{z}R_{y},\quad w\in{\mathfrak{A}}^{2}$

with the Medvedev’s two-term identity theorem [7], one can prove that the
variety $\operatorname{\mathsf{WN}}^{(2)}$ is Spechtian (see, also, [6, 10, 9,
19]).

In this section, we describe defining identities for nilpotent and non-
nilpotent subvarieties of $\operatorname{\mathsf{WN}}^{(2)}$.

###  4.1. Nilpotent subvarieties

######  Lemma 4.1.

Every proper subvariety of $\operatorname{\mathsf{WN}}^{(2)}$ distinguished by
some multilinear identity of degree $n\geqslant 5$ is nilpotent of index not
more then $n+1$.

###### Proof.

Consider a multilinear polynomial $f=f(x_{1},\dots,x_{n})\in{\mathfrak{A}}$ on
$n\geqslant 5$ variables. By Theorem 2, $f$ can be written down as

$f=\sum_{i=1}^{n}\lambda_{i}\left(x_{i}x_{\sigma_{i}(2)}\right)R_{x_{\sigma_{i}(3)}}\ldots
R_{x_{\sigma_{i}(n)}},$

where all $\lambda_{i}\in F$ and every $\sigma_{i}$ is a permutation on the
set $\\{1,2,\dots,n\\}$ defined by the rule

(23) $\sigma_{i}(1)=i,\quad\sigma_{i}(2)<\sigma_{i}(3)<\dots<\sigma_{i}(n).$

In other words, rule (23) states that $\sigma_{1}=\mathrm{id}$ and

$\sigma_{i}={(1\,2\dots i)}^{-1},\text{ for }i=2,3,\dots,n.$

Suppose that $\mathcal{V}_{f}$ is a proper subvariety of
$\operatorname{\mathsf{WN}}^{(2)}$ distinguished by the identity $f$. Then,
with no loss of generality, one may assume $\lambda_{1}=1$ and set $x_{1}=w\in
X^{2}$. Hence, by Lemma 3.2,

$f(w,x_{2}\dots,x_{n})=wR_{x_{2}}\dots R_{x_{n}}.$

Therefore, by Theorem 2, $\mathcal{V}_{f}$ is nilpotent of index not more than
$n+1$. ∎

######  Remark.

In particular, Lemma 4.1 states that the variety
$\operatorname{\mathsf{WN}}^{(2)}$ has the topological rank111See [1, 4, 8,
11]. $r_{t}\bigl{(}\operatorname{\mathsf{WN}}^{(2)}\bigr{)}=2$.

######  Lemma 4.2.

Every proper subvariety of $\operatorname{\mathsf{WN}}^{(2)}$ distinguished by
some multilinear identity of degree $2$ is nilpotent of index not more then
$5$.

###### Proof.

Indeed, for

$f(x_{1},x_{2})=x_{1}x_{2}+\lambda\,x_{2}x_{1},\quad\lambda\in F,$

in view of (17), we have

$f\bigl{(}(x_{1}x_{3})R_{x_{4}}R_{x_{5}},x_{2}\bigr{)}=(x_{1}x_{3})R_{x_{4}}R_{x_{5}}R_{x_{2}}.$

Hence, Theorem 2 yield that a subvariety of $\operatorname{\mathsf{WN}}^{(2)}$
distinguished by $f$ should be nilpotent of index not more then $5$. ∎

###  4.2. Non-nilpotent subvarieties

######  Lemma 4.3.

If $\mathcal{V}_{f}$ is a proper non-nilpotent subvariety of
$\operatorname{\mathsf{WN}}^{(2)}$ distinguished by some multilinear identity
$f=f(x_{1},x_{2},x_{3},x_{4})$ of degree $4$, then $f$ should have the form

$f=\sum_{\delta\in\mathrm{A}_{4}}\lambda_{\delta}\left(x_{\delta(1)},x_{\delta(2)}x_{\delta(3)},x_{\delta(4)}\right),$

where all $\lambda_{\delta}\in F$ and $\mathrm{A}_{4}$ is the alternating
group on the set $\\{1,2,3,4\\}$.

###### Proof.

By Theorem 2, $f$ can be written down as

$f=\sum_{\delta\in\mathrm{A}_{4}}\lambda_{\delta}\left(x_{\delta(1)},x_{\delta(2)}x_{\delta(3)},x_{\delta(4)}\right)+\sum_{i=1}^{4}\mu_{i}\operatorname{\mathit{h}}(x_{i},x_{\sigma_{i}(2)},x_{\sigma_{i}(3)},x_{\sigma_{i}(4)}),\quad\lambda_{\delta},\mu_{i}\in
F,$

where $\sigma_{i}$ is the permutation defined by rule (23), for $n=4$. Suppose
that at least one of the scalars $\mu_{i}$ is nonzero. Then, with no loss of
generality, it’s enough to fix $\mu_{1}=1$ and set $x_{1}=w\in X^{2}$. Hence,
by Lemma 3.2, we get

$f(w,x_{2},x_{3},x_{4})=wR_{x_{2}}R_{x_{3}}R_{x_{4}}.$

However, in view of Theorem 2, this contradicts with the assumption of non-
nilpotency for $\mathcal{V}_{f}$. The obtained contradiction completes the
proof. ∎

######  Lemma 4.4.

If $\mathcal{V}_{f}$ is a proper non-nilpotent subvariety of
$\operatorname{\mathsf{WN}}^{(2)}$ distinguished by some multilinear identity
$f=f(x_{1},x_{2},x_{3})$ of degree $3$, then $f$ should have the form

$f=\sum_{\sigma\in\mathrm{S}_{3}}\lambda_{\sigma}\,x_{\sigma(1)}\left(x_{\sigma(2)}x_{\sigma(3)}\right),\quad\lambda_{\sigma}\in
F,$

where $\mathrm{S}_{3}$ is the symmetric group on the set $\\{1,2,3\\}$.

###### Proof.

Applying, as above, Theorem 2, we write down $f$ as

$f=\sum_{\sigma\in\mathrm{S}_{3}}\lambda_{\sigma}\,x_{\sigma(1)}\left(x_{\sigma(2)}x_{\sigma(3)}\right)+\sum_{\delta\in\mathrm{C}_{3}}\mu_{\delta}\left(x_{\delta(1)},x_{\delta(2)},x_{\delta(3)}\right),\quad\lambda_{\sigma},\mu_{\delta}\in
F,$

where $\mathrm{C}_{3}$ is the cyclic group on the set $\\{1,2,3\\}$. If there
is at least one of the scalars $\mu_{\delta}\neq 0$, then, similarly to the
above cases, we restrict with the assumption $\mu_{\mathrm{id}}=1$. Then,
applying Lemma 3.2, we obtain

$f\bigl{(}(x_{1}x_{4})x_{5},x_{2},x_{3}\bigr{)}=(x_{1}x_{4})R_{x_{5}}R_{x_{2}}R_{x_{3}}.$

Again by virtue of Theorem 2, we have the contradiction with the hypothesis of
non-nilpotency of $\mathcal{V}_{f}$. ∎

### Acknowledgments

The results of this paper were first presented at the scientific seminar of
the research group ”Algebra and the Universal Algebraic Geometry” of the
Federal University of the Rio Grande do Norte, Brazil. The authors are very
thankful to all participant of the seminar, especially to Nir Cohen, Arkady
Tsukov, Elena V. Aladova, Alexander S. Sivatski, Alan de Araújo Guimarães, Ana
Beatriz Gomez da Silva, and José Victor Gomes Teixeira for the creative work
atmosphere and the constantly useful discussions.

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while  _$\max_{1\leq j\leq J}w_{j}(t) <1-\ell$_ do

Increase $J$ by $1$;

Generate the variable $w_{J}(t)$;

while  _$J >1$_ do

Decrease $J$ by $1$ and $t$ by $1$;

Generate $w_{1}(t),\ldots w_{J}(t)$;

while  _$\max_{1\leq j\leq J}w_{j}(t) <1-\ell$_ do

Increase $J$ by $1$;

Generate the variable $w_{J}(t)$;

Decrease $t$ by $1$;

Generate $w_{1}(t)$;

Fix Stopping = $\\{w_{1}(t)\geq\ell\\}$;

Fix $\nu=\delta_{0}$;

for  _$s$ from $t+1$ to $-1$_ do

Generate the variables $w_{1}(s),\ldots,w_{||\nu||}(s)$ ;

Set $\mathfrak{m}=\max\\{\nu_{j}+w_{j}(s)\text{ for $1\leq j\leq\|\nu\|$}\\}$;

Add $\delta_{\mathfrak{m}}$ to $\nu$;

Set $\mathfrak{m}=\max\\{\nu_{j}+w_{j}(0)\text{ for }1\leq j\leq\|\nu\|\\}$;

Return: $\mathfrak{m}-\nu_{1}$;

Algorithm 1 Construction of a variable of law $\bar{\mathfrak{m}}(0)$.

We now describe more precisely the perfect simulation algorithm. Let $F$ be a
probability distribution satisfying (10.2), we fix $\ell\in[0,1)$ such that
$F([1-\ell,1])\in(0,1)$. The algorithm requires the construction of an array
of i.i.d. random variables with common distribution $F$ until the random
variable $T^{*}$ can be constructed.

To construct $T^{*}$ as well as $\bar{\mathfrak{m}}(0)$ from the sequence
$\\{w_{j}(t),j\in\mathbb{N},t\in\mathbb{Z}\\}$, one only needs to consider
a.s. finitely many elements of this set, as $\\{T^{*}=t\\}$ is a measurable
function of

$\\{w_{1}(t)\\}\cup\\{w_{j}(t+k),1\leq j\leq k\leq|t|\\}$

and $\bar{\mathfrak{m}}(0)$ is a measurable function of

$\\{w_{1}(T^{*})\\}\cup\\{w_{j}(T^{*}+k),1\leq j\leq k\leq|T^{*}|\\}.$

Therefore, we can explore triangular arrays of the form

$\\{w_{1}(t)\\}\cup\\{w_{j}(t+k),1\leq j\leq k\leq|t|\\},$

progressively decreasing $t$ until time $T^{*}$ is detected. Once this random
variable is known, we construct the random variable $\bar{\mathfrak{m}}(0)$
using the procedure described in Theorem 10.6 from the previously discovered
random variables. A possible implementation is described in Algorithm 1. We
show a graphical representation of a run of Algorithm 1 in Figure 9.

Figure 9: Illustration of the execution of Algorithm 1 on an example, in the
case where $\ell<1-\ell$. The variables sampled until the Boolean variable
Stopping becomes True are pictured by black/white squares and disks. One
searches for the first time $T^{*}$ such that every line of index $T^{*}+1\leq
t\leq 0$ has at least one black disk between columns $1$ and $t-T^{*}$ and
such that there is a black square in position $(T^{*},1)$. The full triangular
array of variables used in the construction of $\nu$ is enclosed by a red
boundary.

We observe that this algorithm has a complexity of $(T^{*})^{2}$, as it is the
number of steps needed to generate the variable $\bar{\mathfrak{m}}(0)$. It is
worth noting that $-T^{*}$ can be constructed as the first hitting time of $0$
of the Markov chain $(X_{n})$ with initial state

$X_{0}=\min\\{j\geq 1,w_{j}(0)\geq 1-\ell\\}$

and with transition probabilities defined for all $j\geq 2$ and $i\geq j$ by

$P(j,j-1)=1-(1-p)^{j-1}\text{ and }P(j,i)=p(1-p)^{i-1}$

where $p=\mathbb{P}(w_{1}(0)\geq 1-\ell)$, with

$P(1,0)=\mathbb{P}(w_{1}(0)\geq\ell),\quad\mathbb{P}(1,1)=\mathbb{P}(1-\ell\leq
w_{1}(0)<\ell),\quad P(1,j)=p(1-p)^{j-1}\text{ for }j\geq 2.$

The quantity $X_{n}$ corresponds to the value of the variable $J$ at the end
of the period when $t=-n$ in Algorithm 1. In the example shown in Figure 9, we
have

$(X_{0},X_{-1},X_{-2},X_{-3},X_{-4},X_{-5},X_{-6})=(4,3,2,1,2,1,0).$

Note that $T^{*}$ has exponential tails.

The choice of the parameter $\ell$ may have an important effect on the
behavior of the average complexity $\mathbb{E}((T^{*})^{2})$ of the algorithm.
We plotted $\ell\mapsto\mathbb{E}((T^{*})^{2})$ in Figure 10, when the charge
distribution is given by
$F(\mathrm{d}x)={\text{\Large$\mathfrak{1}$}_{\left\\{x\leq
1\right\\}}}e^{x-1}\mathrm{d}x$. Additionally, as $p\to 0$, the quantity
$\mathbb{E}((T^{*})^{2})$ grows to $\infty$. We estimated
$\mathbb{E}((T^{*})^{2})$ for $F=p\delta_{1}+(1-p)\delta_{-\infty}$ and
plotted this quantity as a function of $p$ in Figure 11.

Figure 10: Dependency in the parameter $\ell$ of the complexity of Algorithm 1
with a charge distribution
$F(\mathrm{d}x)={\text{\Large$\mathfrak{1}$}_{\left\\{x\leq
1\right\\}}}e^{x-1}\mathrm{d}x$. The figure was obtained with a Monte Carlo
simulation of $N=10^{4}$ copies of $T^{*}$ for $100$ different values of
$\ell$. For this charge distribution, the Monte Carlo simulations give
$C(F)=0.4432\pm 0.0006$.

We observe in Figure 10 that different choices of the value $\ell$ can have a
dramatic impact on the efficiency of Algorithm 1. Choosing a value $\ell$ too
small has the effect of making the first appearance of a triangular event too
late. On the other hand, if $\ell$ is too big then with high probability, one
will have $w_{1}(T)\leq\ell$, and thus the first “successful” triangular event
will appear much later. For the distribution $F$ we chose, it appears that an
optimal choice of $\ell$ seems to be around $\ell=0.7$, which balances between
these two extremes.

Figure 11: Dependency in the parameter $p$ of the complexity of Algorithm 1
applied to the detection of the longest path in the Barak-Erdős graph with
parameter $p$. Figure obtained through Monte Carlo simulation of $N=10^{5}$
copies of $T^{*}$ for 120 different values of $p$.

We observe in Figure 11 that if $F$ puts a large mass on the negative half-
line, the complexity of Algorithm 1 can become quite large. The function
$p\mapsto\mathbb{E}(T^{*})^{2}$ grows at least exponentially in $1/p$ as $p\to
0$ in the Barak-Erdős graph, but we were not able to obtain a good estimate of
this rate of increase.

######  Remark 10.8 (Perfect simulation when the charge distribution has an
infinite essential supremum).

We explained how to perfectly simulate the random variable
$\bar{\mathfrak{m}}(0)^{+}$ whose expectation is $C(F)$ under the assumption
that the supremum of the support of $F$ is $1$ or any finite number for that
matter.

To perfectly simulate a random variable whose expectation is $C(F)$ when the
supremum of the support of $F$ is $\infty$, we use an idea by Glynn and Rhee
[51]. Denote by $X_{n}$ the variable $\bar{\mathfrak{m}}(0)^{+}$ when
$F=F_{n}$ has $n$ as the supremum of its support and note that $X_{n}\to X$
a.s. and in $L^{1}$ for some random variable $X$. Then
$C(F_{n})=\mathbb{E}X_{n}\to\mathbb{E}X=C(F)$, as $n\to\infty$. We may not be
able to perfectly simulate $X$. However, if we let $\nu$ be a positive random
integer such that $p_{n}=\mathbb{P}(\nu=n)>0$ for all $n$ and set
$Y=(X_{\nu}-X_{\nu-1})/p_{\nu}$ then $Y$ can be perfectly simulated. Moreover,
$\mathbb{E}Y=C(F)$. This is because
$\mathbb{E}Y=\sum_{n=1}^{\infty}p_{n}\mathbb{E}(Y|\nu=n)=\sum_{n=1}^{\infty}\mathbb{E}(X_{n}-X_{n-1})=\lim_{n\to\infty}\mathbb{E}X_{n}=\mathbb{E}X$.

######  Remark 10.9.

For the special case of a Barak-Erdős graph, a simpler perfect simulation
algorithm is explained in [41].

##  11 Additional remarks and open problems

The last passage percolation constant $C(p)$ for
$\overrightarrow{G}(\mathbb{Z},p)$ has been one of the main concerns in this
survey. We showed how very good approximations can be obtained analytically by
relating $\overrightarrow{G}(\mathbb{Z},p)$ to the IBM$(p)$ model. The
function $C(p)$ is not convex. We conjecture however that $C(p)/p$ is.

The passage to the IBM enables using branching processes techniques that help
explain the slow rate of convergence of $C(p)/p$ to $e$ as $p\to 0$ and its
relation to the Brunet-Derrida behavior. Such phenomena should be present when
the geometric$(p)$, $0<p<1$, distributions in the IBM are replaced by a family
of more general distributions.

The CLT’s are generally available for the length of the longest or heaviest
path but, as explained, some moment conditions are needed in case random
weights are added to the edges. There is one thing that is not totally clear
in the CLT’s and this is the variance parameter. In, e.g., the case of
$\overrightarrow{G}(\mathbb{Z},p)$, we have that
$(L_{n}-Cnt)/\sqrt{\sigma^{2}\lambda n}$ converges in distribution to a
standard normal with $\sigma^{2}$ being equal to the variance of
$L_{\Gamma_{1},\Gamma_{2}}-C(\Gamma_{2}-\Gamma_{2})$. This quantity is unknown
and not even bounds are available. The situation becomes more complex in
presence of weights.

CLT’s are also available for the IBM and can take various forms. For example,
for the IBM$(p)$ model, it has been shown that
$\sqrt{n}\left(n^{-1}\sum_{k=0}^{[A_{n}t]}X_{n}(k)-t\right)$, $0\leq t\leq 1$,
converges in law to a Brownian motion if $A_{n}=nC$ and to a Brownian bridge
if $A_{n}=L_{n}$. See [41, Thm. 9.2].

When considering random directed graphs on partially ordered vertex sets, the
CLT limits are not necessarily Gaussian. The Brownian last passage percolation
process $Z(t)=\max_{0\leq s\leq t}(X(s)+Y(t)-Y(s))$ pops up in the case of the
vertex set $\mathbb{Z}\times\\{1,2\\}$. We know nothing about the random
variable $S=S(t)$ that achieves this maximum. This is of interest as it would
related to the path that achieves the maximum.

Concerning longest paths of $\overrightarrow{G}(\mathbb{Z}\times\mathbb{Z},p)$
on a window $[0,n]\times[0,m]$, as $n,m=m(n)\to\infty$, the limit theorem of
(7.6) is not optimal. We conjecture that the Tracy-Widom limit is also
possible when $m(n)$ grows with $n$ linearly. One can of course ask for
analogous results in the $\overrightarrow{G}(\mathbb{Z}^{d},p)$ case, when
$d\geq 3$. It is not clear at all what kind of laws would replace the Tracy-
Widom distribution here.

The recent paper of Terlat [90] makes progress in the behavior of $C(F)$ when
$F$ is a distribution whose support may include negative numbers with a
possible atom on $-\infty$. Concerning however the type of limit theorems that
one can obtain when $F$ is heavy-tailed, such as those of Section (8.2), the
field is open.

Concerning other types of behavior of maximal lengths or weights, e.g., in the
large deviations sense, we mention that, when weights have a distribution
supported on the integers, normal and moderate large deviations were obtained
in [65]. In addition local limit theorems were obtained when the weight
distribution is non-lattice.

Distributional results for maximal/heaviest weighted paths are not available
beyond results around CLTs. However, in the sparse case, one can use the
recursive nature of the (possibly weighted) PWIT to come up with functional
equations for such quantities. See e.g. [42, eq. (13)].

Recall that $C(p,x)$ of Section 9, is the last passage percolation constant
when the weight law is $p\delta_{1}+(1-p)\delta_{x}$ for all edges $(i,j)$
with $i<j$. There is only one solvable model we are aware of, that of $C(p,0)$
as in [38]. Can there be other ones?

If we replace $p\delta_{1}+(1-p)\delta_{x}$ above by
$F_{p,x}=pQ+(1-p)\delta_{x}$ where $Q$ is a probability measure on
$(0,\infty)$ we expect that the behavior of $p\mapsto C(F_{p,x})$ can be
obtained by techniques similar to those of [76, 77], so long as $\int
y^{2}Q(dy)<\infty$. As a function of $x$, the $C(F_{p,x})$ quantity is
continuous and convex and it can be seen that $C(F_{p,x})$ is not
differentiable at $x$ iff $Q$ has an atom $y$ such that
$x/y\in\mathbb{X}_{\text{\rm crit}}$. See [44, Sec. 5].

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