# Number of Independent Sets in Regular and Irregular Graphs: A 31 Year
Journey

Dev Chheda, Ram Goel, Eddie Qiao

###### Abstract

We review the progress made on bounding the number of independent sets in
$d$-regular and irregular graphs over the last 31 years. We particularly focus
on contributions from Kahn [5], Zhao [7], and Sah et al. [6] in incrementally
proving stronger and more general versions of the upper bound. We reproduce
the main results of these works, particularly focusing on the unweighted
special case (with fugacity $\lambda=1$), which allows us to provide more
intuitive and clear explanations of the key ideas that have been developed in
the field over three decades.

##  1 Introduction

Let $G$ be a graph on $n$ vertices. Denote the collection of independents sets
of $G$ by $\mathcal{I}(G)$, and let $i(G)=|\mathcal{I}(G)|$ be the number of
independent sets. What is the maximum value of $i(G)$ when $G$ has $n$
vertices, and which graphs achieve this maximum? This paper explores the key
ideas, developed over more than three decades, which resulted in the eventual
resolution of this question by Sah et al. in 2019 [6].

###  1.1 Historical Context

We begin with some historical context regarding this problem (also see [4] and
[7]). In the 1988 Number Theory Conference at Banff [2], Granville conjectured
that for $d$-regular graphs, $i(G)\leq 2^{(1/2+o(1))n}$. This conjecture was
resolved in the 1991 paper from Alon [1], who showed that $i(G)\leq
2^{(1/2+O(d^{-0.1}))n}$. Alon additionally conjectured that $i(G)$ is
maximized when $G$ is a disjoint union of $n/2d$ complete bipartite graphs
$K_{d,d}$. This conjecture was formalized by Kahn [5] in 2001:

######  Conjecture 1.1 (Alon [1] and Kahn [5]).

Let $G$ be a $d$-regular graph with $n$ vertices. Then,

$\displaystyle i(G)\leq i(K_{d,d})^{n/(2d)}=(2^{d+1}-1)^{n/(2d)}.$

Kahn [5] proved that 1.1 result holds when $G$ is bipartite by using entropy
methods. The next major improvement came in 2009 from Zhao [7] who used a
bipartite-swapping trick to extend Kahn’s result to all $d$-regular graphs
$G$, thereby resolving Conjecture 1.1.

Kahn also conjectured an extension to irregular graphs, which was finally
resolved by Sah et al. in 2019 [6]. Specifically, the result proven by Sah et
al. is:

######  Theorem 1.2.

Let $G$ be a graph without isolated vertices. Let $d_{v}$ be the degree of $v$
in $G$. Then,

$i(G)\leq\prod_{uv\in E(G)}i(K_{d_{u},d_{v}})^{1/(d_{u}d_{v})}.$

Equality holds if and only if $G$ is a disjoint union of complete bipartite
graphs.

###  1.2 Paper Overview

In our paper, we first cover the main results from Kahn [5] in Section 2 and
Zhao [7] in Section 3 which prove Conjecture 1.1, which is essentially a
specialized form of Theorem 1.2 for $d$-regular graphs. We then cover the key
methods and results from Sah et al. [6] in Section 4, of which the primary
result is Theorem 1.2. Sah et al. also prove some lower bounds on $i(G)$, but
we only focus on covering the upper bound. We briefly state the lower bound
results and some generalizations of the upper bound results in Section 5.

All of the results which we cover can be extended using the hard-core model.
The partition function of the hard-core model for a graph $G$ is
$P_{G}(\lambda)=\sum_{I\in\mathcal{I}(G)}\lambda^{|I|}$ where the parameter
$\lambda$ is often referred to as fugacity. Each of the key results which we
cover in Sections 2 \- 4 can be generalized with a weighted version using the
hard-core model.

For example, a weighted version of Theorem 1.2 using the partition function
is:

######  Theorem 1.3.

Let $G$ be a graph without isolated vertices. Let $d_{v}$ be the degree of $v$
in $G$. Let $\lambda>0$. Then,

$P_{G}(\lambda)\leq\prod_{uv\in
E(G)}P_{K_{d_{u},d_{v}}}(\lambda)^{1/(d_{u}d_{v})}.$

Sah et al. [6] prove this more general version (along with a double-weighted
version which we discuss in Section 5.1).

We only focus on the $\lambda=1$ case (e.g. which would reduce Theorem 1.3 to
Theorem 1.2), since this allows us to focus the key ideas, methods, and
insights, all of which remain the same even in the more general case.

##  2 Bipartite and Regular Graphs – Kahn

###  2.1 Overview of Kahn’s Result

Kahn proved the first result in this progression in 1999, by proving Theorem
1.2 for the case of $G$ being bipartite and $d$-regular.

######  Theorem 2.1 (Theorem 1.9 of Kahn [5]).

If $G$ is a $d$-regular bipartite graph on $n$ vertices, then

$i(G)\leq(2^{d+1}-1)^{n/(2d)}.$

Equality holds when $G$ is a disjoint union of $K_{d,d}$’s.

In Sah et al. 4, the upper bound above is stated also as
$i(K_{d,d})^{n/(2d)}$, since $i(K_{d,d})=2^{d+1}-1$. This is because each
independent set of a bipartite graph is a subset of one of the two parts, and
each part has $2^{d}$ subsets. We subtract 1 for overcounting the empty set
twice.

In Kahn [5], Theorem 2.1 is stated in log-form:

$\log i(G)\leq\frac{n}{2d}\log(2^{d+1}-1).$

The reason for this is that the argument used to prove this is based on
entropy, which is more conducive to the log-form. We will go into detail on
relevant background and notation in 2.2, and the proof in 2.3.

###  2.2 Entropy

Entropy is a fundamental concept in information theory, originally introduced
by Claude Shannon. It quantifies the uncertainty or the average amount of
information inherent in a random variable’s possible outcomes. The entropy of
a discrete random variable $X$ with pdf $p(x)=\text{Pr}(X=x)$ is defined as

$H(X)=\sum_{x}p(x)\log\frac{1}{p(x)}.$

The base of the logarithm is set to 2, in which case the entropy is measured
in bits.

Conditional entropy is given by

$H(X\mid Y)=\mathbb{E}H(X\mid Y=y)=\sum_{y}p(y)\sum_{x}p(x\mid
y)\log\frac{1}{p(x\mid y)}.$

For a random vector $X=(X_{1},\ldots,X_{n})$, define

$H(X)=\sum_{i=1}^{n}H(X_{i}\mid X_{i-1},\ldots,X_{1}).$

Some useful inequalities are used. These are well-known and often used in
literature on information theory and are used freely without proof.

$H(X)\leq\log|\text{range}(X)|.$ (1) $H(X\mid Y)\leq H(X).$ (2)

The above equation essentially says that given more information, entropy
decreases. This intuition is generalized by the fact that if $Y$ determines
$Z$, then $H(X\mid Y)\leq H(X\mid Z)$. Similarly the joint entropy of a
collection of r.v.’s is at most the sum of the individual entropies, which is
intuitively true since there could be correlation among the r.v.’s. A more
general version of this fact is the following:

######  Lemma 2.2 (Shearer’s Lemma).

Let $X=(X_{1},\ldots,X_{n})$ be a random vector and $\mathcal{A}$ a collection
of subsets of $[n]$, with each element of $[n]$ contained in at least $m$
members of $\mathcal{A}$. Then

$H(X)\leq\frac{1}{m}\sum_{A\in\mathcal{A}}H(X_{A}).$

The notation $X_{A}$ denotes the restriction of $X$ onto indices $A$, i.e.
$X_{A}=(X_{i}:i\in A)$.

###  2.3 Proof of Theorem 2.1

####  2.3.1 Notation

We first set up notation. Let $I$ be a random member of $\mathcal{I}(G)$,
chosen uniformly at random. This is one of the key ideas, since a uniform
distribution makes

$H(I)=\sum_{I\in\mathcal{I}(G)}\frac{1}{i(G)}\log i(G)=\log i(G).$

Hence we are interested in upper bounds on the entropy of $I$, which
intuitively quantifies the information on $I$.

Denote $\mathbf{1}_{v}$ to be the indicator of $v\in I$, and $p(v)$ for
$\text{Pr}(v\in I)$. Let

$X_{v}:=I\cap N(v).$

A key idea is that $I$ cannot contain $v$ and its neighbors. So only one of
$\mathbf{1}_{v}$ and $X_{v}$ can contribute towards the information provided
by $I$, which is in turn encoded by the entropy of $I$.

Denote

$q(v):=\text{Pr}(Q_{v})$

where the event $Q_{v}$ is defined as

$Q_{v}:=\\{X_{v}=\emptyset\\}=\\{I\cap N(v)=\emptyset\\}.$

That is, $Q_{v}$ is the event that no neighbors of $v$ are in the independent
set $I$. Similar intuition as the above implies $X_{v}$ and $Q_{v}$ are
uncorrelated in a sense, and $\mathbf{1}_{v}$ and $\overline{Q_{v}}$ are
uncorrelated in a sense, the reason being that regardless of whether $v$ is in
$I$ or not, if $X_{v}=\emptyset$, then the neighbors of $v$ provide no
information on $I$.

####  2.3.2 Proof

Recall that $G$ is bipartite. Let the two parts be $\mathcal{E}$ and
$\mathcal{O}$ (for even and odd, respectively). All sums will be over
$v\in\mathcal{E}$. Note

$\displaystyle H(I)$ $\displaystyle=H(I\cap\mathcal{O})+H(I\cap\mathcal{E}\mid
I\cap\mathcal{O})$ $\displaystyle\leq H(I\cap\mathcal{O})+\sum
H(\mathbf{1}_{v}\mid I\cap\mathcal{O})$ $\displaystyle\leq
H(I\cap\mathcal{O})+\sum H(\mathbf{1}_{v}\mid\mathbf{1}_{Q_{v}})$
$\displaystyle=H(I\cap\mathcal{O})+\sum q(v)$

The first equality follows since $G=\mathcal{E}\sqcup\mathcal{O}$. The second
inequality follows by splitting up joint entropy into individual sums, and the
last by containment of neighbors in the other bipartite part. The latter
aforementioned step is where the bipartite part is crucial. The last equality
follows since $H(\mathbf{1}_{v}\mid\mathbf{1}_{Q_{v}})=q(v)$ due to picking
$I$ uniformly at random.

We establish one more inequality chain. We utilize Shearer’s Lemma, noting
that the sets $N(v)$ cover each $x\in\mathcal{O}$ exactly $d$ times, due to
regularity. Each of the $d$ neighbors lies in the other bipartite part. Note

$\displaystyle H(I\cap\mathcal{O})$ $\displaystyle\leq\frac{1}{d}H(X_{v})$
$\displaystyle=\frac{1}{d}\sum
H(\mathbf{1}_{Q_{v}})+H(X_{v}\mid\mathbf{1}_{Q_{v}})$
$\displaystyle=\frac{1}{d}\sum H(q(v))+(1-q(v))H(X_{v}\mid\overline{Q_{v}})$
$\displaystyle\leq\frac{1}{d}\sum H(q(v))+(1-q(v))\log(2^{d}-1).$

The first step followed from Shearer’s Lemma. The final inequality above
follows from the crude bound $H(X)\leq\log|\text{range}(X)|$, since there are
$2^{d}-1$ nonempty subsets of $N(v)$ and $X_{v}=I\cap N(v)\subseteq N(v)$.

Combining these two inequality chains, we get

$\displaystyle H(I)$ $\displaystyle\leq\frac{1}{d}\sum
H(q(v))+(1-q(v))\log(2^{d}-1)+\sum q(v)$
$\displaystyle=\frac{n}{2d}\log(2^{d}-1)+\frac{1}{d}\sum
H(q(v))+q(v)\log\frac{2^{d}}{2^{d}-1}.$

By differentiation, the maximum of the function
$H(x)+x\log\frac{2^{d}}{2^{d}-1}$ occurs at $x_{0}=\frac{2^{d}}{2^{d+1}-1}$.
Now, plugging this back into the expression above and reducing algebraically
gives the desired result of $\frac{n}{2d}\log(2^{d+1}-1)$, as desired.

##  3 Extension to all Regular Graphs – Zhao

Kahn conjectured that the upper bound could be extended to all $d$-regular
graphs, not just bipartite. In 2009, Zhao [7] used a bipartite-swapping trick
(later expanded on in [8]) to resolve this conjecture. Specifically, Zhao
proves the following:

######  Theorem 3.1 (Theorem 2 of Zhao [7]).

If $G$ is a $d$-regular graph on $n$ vertices,

$\displaystyle i(G)\leq i(K_{d,d})^{n/(2d)}=(2^{d+1}-1)^{n/(2d)}.$

Equality holds if and only if $G$ is a disjoint union of $K_{d,d}$’s.

###  3.1 Bipartite swapping trick

For $A,B\in V(G)$, we say that $A$ and $B$ are independent from each other if
$G$ does not contain edge $ab$ for any $a\in A$, $b\in B$. Let $G[A]$ be the
subgraph of $G$ induced by $A$. We define $\mathcal{J}(G)$ as the set of pairs
$(A,B)$ such that $A$ and $B$ are independent and $G[A\cup B]$ is bipartite.

######  Lemma 3.2 (Bipartite-swapping trick of Zhao [7]).

Let $G$ be any graph. There exists a bijection between
$\mathcal{I}(G)\times\mathcal{I}(G)$ and $\mathcal{J}(G)$.

###### Proof.

For any $W\subset V(G)$, such that $G[W]$ is bipartite, let $W_{1},W_{2}$ be
the bipartition of $W$. Now, for $A,B\in\mathcal{I}(G)$, note that $G[A\cup
B]$ is bipartite since $(A,B\backslash A)$ is a valid partition (due to
independence).

Letting $W=A\cup B$, we define $S_{1}=A\cap W_{1}$, $S_{2}=A\cap W_{2}$,
$S_{3}=B\cap W_{1}$, and $S_{4}=B\cap W_{2}$, as shown in Figure 1. The lines
in the figure represent the existence of edges between subsets of vertices.
Since $W_{1},W_{2}$ is a bipartition, there are no edges within $W_{1}$ or
$W_{2}$, and, since $A,B$ are independent, there are no edges within $A$ or
$B$. This means that the only possible edges are between $S_{1}$ and $S_{4}$
or $S_{2}$ and $S_{3}$, as shown.

ABW1W2S1S2S3S4A’B’W1W2S1S4S3S2 Figure 1: Bijection between
$\mathcal{I}(G)\times\mathcal{I}(G)$ and $\mathcal{J}(G)$

As shown in Figure 1, we will map $(A,B)$ to
$(A^{\prime},B^{\prime}):=(S_{1}\cup S_{4},S_{2}\cup S_{3})$. We can see that
$(A^{\prime},B^{\prime})\in\mathcal{J}(G)$ since $A^{\prime}\cup
B^{\prime}=A\cup B=W$ is clearly bipartite, and $A^{\prime},B^{\prime}$ are
clearly independent from each other.

Now, consider the inverse mapping which starts with
$(A^{\prime},B^{\prime})\in\mathcal{J}(G)$. $W=A^{\prime}\cup B^{\prime}$ is
bipartite by definition, and so we let $S_{1}=A^{\prime}\cap W_{1}$,
$S_{4}=A^{\prime}\cap W_{2}$, $S_{3}=B^{\prime}\cap W_{1}$, and
$S_{2}=B^{\prime}\cap W_{2}$ as shown in Figure 1. Similar to before, since
$W$ is bipartite and $A^{\prime},B^{\prime}$ are independent from each other,
the only edges in $G[W]$ are those between $S_{1}$ and $S_{4}$ and those
between $S_{2}$ and $S_{3}$. So, if we map $(A^{\prime},B^{\prime})$ to
$(A,B)=(S_{1}\cup S_{2},S_{3}\cup S_{4})$, we see that $A,B\in\mathcal{I}(G)$.

So, we have clearly shown a bijection between
$\mathcal{I}(G)\times\mathcal{I}(G)$ and $\mathcal{J}(G)$. ∎

###  3.2 Proof of Theorem 3.1

Now, let us see why this swapping trick extends the upper bound to non-
bipartite graph. We will consider $G\times K_{2}$, the bipartite double cover
of $G$, which contains vertices $(v,i)$ for $v\in V(G)$ and $i\in\\{0,1\\}$
and edges $(u,0),(v,1)$ if and only if $uv\in E(G)$. Then, any independent set
in $G\times K_{2}$ can be separated into its $i=0$ side $(A,0)$ and $i=1$ side
$(B,1)$ where $A,B\subset V(G)$. Note that
$(A,0)\cup(B,1)\in\mathcal{I}(G\times K_{2})$ holds precisely when $A$ and $B$
are independent from each other in $G$. Then, we can see

$\displaystyle i(G\times K_{2})$ $\displaystyle=|\\{A,B\subset V(G):A\text{
independent from }B\\}|$ $\displaystyle\geq|\mathcal{J}(G)|$
$\displaystyle=|\mathcal{I}(G)\times\mathcal{I}(G)|$ $\displaystyle=i(G)^{2}$

The second inequality is since $\mathcal{J}(G)$ is a subset of $\\{A,B\subset
V(G):A\text{ independent from }B\\}$ since it has the additional bipartite
condition. The third equality is from Lemma 3.2, since a bijection between
$\mathcal{J}(G)$ and $\mathcal{I}(G)\times\mathcal{I}(G)$ means they have the
same size.

Note that if $G$ is $d$-regular, then so is $G\times K_{2}$ since degrees
don’t change. Since $G\times K_{2}$ is bipartite and has $2n$ vertices,
Theorem 2.1 gives

$\displaystyle i(G\times K_{2})\leq i(K_{d,d})^{(2n)/(2d)}=i(K_{d,d})^{n/d}$

and combining with the previous inequality gives

$\displaystyle i(G)\leq i(G\times K_{2})^{1/2}\leq i(K_{d,d})^{n/(2d)}$

as desired.

##  4 Generalization to Irregular Graphs – Sah et al.

Despite the previous progress from Section 3, the graphs $G$ that we deal with
are all $d$-regular. Sah [6] generalizes this result to deal with all graphs.
To do so, techniques from the literature, particularly Section 3, are key to
this generalization. However, the entropy approach taken in Section [5] is not
sufficient, and Sah et al. find another approach to the problem. Specifically,
they show the following theorem.

######  Theorem 4.1 (Sah, Sawhney, Stoner, and Zhao [6]).

Let $G$ be a graph, and let $d_{v}$ denote the degree of vertex $v$ in $G$.
Then

$\displaystyle i(G)$ $\displaystyle\leq 2^{\mathrm{iso}(G)}\prod_{uv\in
E(G)}i(K_{d_{u},d_{v}})^{1/(d_{u}d_{v})}$
$\displaystyle=2^{\mathrm{iso}(G)}\prod_{uv\in
E(G)}(2^{d_{u}}+2^{d_{v}}-1)^{1/(d_{u}d_{v})}.$

Equality holds if and only if $G$ is a disjoint union of complete bipartite
graphs and isolated vertices.

In the paper, Sah et al. state the theorem for graphs $G$ without any isolated
vertices (i.e. $\mathrm{iso}(G)=0$). However, it is easy to see that proving
this theorem is equivalent. Since each of the isolated vertices have a binary
choice of whether or not they are in an independent set, they contribute a
factor of $2^{\mathrm{iso}(G)}$ to $i(G)$. The paper from Sah et. al. also
differs because they provide a proof of a generalized version of this problem
involving the hard-core model, which will be discussed in Section 5. We will
now provide a sketch of the proof, specifically proving Theorem 4.1 rather
than proving the generalized version. The hope is that the proof of this
theorem rather than the generalized version will yield more intuition about
what the inequalities and techniques mean. We will take the following steps to
solve the problem.

  1. 1.

Reduce the problem to assuming $G$ is bipartite using the bipartite swapping
trick covered in Section 3.1.

  2. 2.

Use a recursion involving $i(G)$ to reduce the problem to another inequality.

  3. 3.

Apply Hölder’s inequality and computation to prove this inequality.

We first reduce the problem to assuming $G$ is bipartite. First, define $j(G)$
to be

$j(G)\coloneqq 2^{\mathrm{iso}(G)}\prod_{uv\in
E(G)}i(K_{d_{u},d_{v}})^{1/(d_{u}d_{v})}.$

We claim that $j(G)^{2}=j(G\times K_{2})$. Recall that for each edge $(u,v)\in
G$, we have the edges $((u,0),(v,1)),((u,1),(v,0))\in G\times K_{2}$. This
means that the degree of $(u,i)$ in $G\times K_{2}$ is equal to that of $u$ in
$G$ for $i=0,1$. Thus, when computing the product for $j(G\times K_{2})$, each
of the edges within $G$ is counted twice, giving $j(G)^{2}=j(G\times K_{2})$.

Recall from the bipartite swapping trick in Lemma 3.2 that $i(G)^{2}\leq
i(G\times K_{2})$. Combining these two inequalities, in order to prove
$i(G)\leq j(G)$, it suffices to show that $i(G\times K_{2})\leq j(G\times
K_{2})$. As a result, it suffices to show that $i(G)\leq j(G)$ for all
bipartite graphs $G$.

Next, we claim that we can assume $G$ is connected. Observe that if $G$ can be
divided into two connected components $G_{1}$ and $G_{2}$, then
$i(G)=i(G_{1})i(G_{2})$ and $j(G)=j(G_{1})j(G_{2})$. Thus, the problem reduces
down to proving the inequality for each of the connected components, so we can
assume $G$ is connected. The proof so far has been the same technique used by
Zhao to solve the problem for all $d$-regular graphs. However, the proof
differs from here because we no longer have the $d$-regular assumption.

Now, we induct on the number of vertices in $G$. It is simple to check the
base case, which is when $G$ is an empty graph or an isolated vertex. Note
that for any vertex $w$, we have

$i(G)=i(G-w)+i(G-w-N(w)),$

where $G-w$ denotes $G$ with the vertex $w$ and the edges with $w$ deleted,
and $G-w-N(w)$ is the same with the neighbors of $w$ deleted as well. In this
equation, $i(G-w)$ represent the independent sets that don’t contain $w$, and
$i(G-w-N(w))$ contain the independent sets that contain $w$. Since any vertex
$w$ works, we select $w$ to be the one with maximum degree. By using
induction, we have

$\displaystyle i(G-w)$ $\displaystyle\leq j(G-w)$ $\displaystyle i(G-w-N(w))$
$\displaystyle\leq j(G-w-N(w)).$

Combining these, we have

$i(G)=i(G-w)+i(G-w-N(w))\leq j(G-w)+j(G-w-N(w)).$

Thus, it suffices to prove that

$\displaystyle j(G-w)+j(G-w-N(w))\leq j(G).$ (3)

To tackle this inequality, define $V_{k}$ to be the set of vertices with
distance exactly $k$ to $w$. Intuitively, this works well with the inequality
we have because

$\displaystyle G$ $\displaystyle=V_{0}\cup V_{1}\cup\dots$ $\displaystyle G-w$
$\displaystyle=V_{1}\cup V_{2}\cup\dots$ $\displaystyle G-w-N(w)$
$\displaystyle=V_{2}\cup V_{3}\cup\dots$

With nice expressions for each of these three values, there is a lot of
potential to prove (3). Let $E_{k}$ be the edges between $V_{k-1}$ and
$V_{k}$. Define $E_{\geq k}\coloneqq\bigcup_{i\geq k}E_{i}$ and $E_{\leq k}$
analogously. Furthermore, for each $i=1,2,\dots$, we let $I_{i}$ be the
vertices that are not connected to any vertices in $V_{i+1}$. Finally, let
$\Delta$ be the degree of $w$. These definitions can be visualized in Figure
2. From this visualization, it is evident why it is important for $G$ to be
bipartite.

Figure 2: Setup for proof of Theorem 4.1

With this setup, we return to 3. For each $u\in V_{k}$, we let $d_{u}^{+}$
denote the number of neighbors of $u$ in $V_{k+1}$. We have

$\displaystyle j(G)$ $\displaystyle=2^{\mathrm{iso}(G)}\prod_{(u,v)\in
E}i(K_{d_{u},d_{v}}),$ $\displaystyle j(G-w)$
$\displaystyle=2^{\mathrm{iso}(G-w)}\prod_{\begin{subarray}{c}(u,v)\in
E_{2}\\\ v\in
V_{1}\end{subarray}}i(K_{d_{u},d_{v}})^{1/(d_{u}d_{v})}\prod_{(u,v)\in E_{\geq
3}}i(K_{d_{u},d_{v}})^{1/(d_{u}d_{v})},$ $\displaystyle j(G-w-N(w))$
$\displaystyle=2^{\mathrm{iso}(G-w-N(w))}\prod_{\begin{subarray}{c}(u,v)\in
E_{3}\\\ v\in
V_{2}\end{subarray}}i(K_{d_{u}^{+},d_{v}})^{1/(d_{u}d_{v})}\prod_{(u,v)\in
E_{\geq 4}}i(K_{d_{u},d_{v}})^{1/(d_{u}d_{v})}.$

The first equation is simply the definition of $j(G)$. For the second
equation, each edge $(u,v)\in E$ is either in $E_{2}$ with $v\in V_{1}$ or in
$E_{\geq 3}$. The same logic is used for the third equation. Thus, when
considering (3), we have a shared factor of $\prod_{(u,v)\in E_{\geq
3}}i(K_{d_{u},d_{v}})^{1/(d_{u}d_{v})}$. The first two explicitly contain this
term, while removing the term from the third will give a nicer expression.
Removing this factor from the first term, we are left with the product over
all edges $(u,v)\in E_{\leq 2}$. Removing this factor from the second term, we
have the product over all edges $(u,v)\in E_{2}$. Finally, the third term
becomes the product over all edges $(u,v)\in E_{3}$ divided by a factor that
will be written out. Considering the factors of the form
$2^{\mathrm{iso}(\cdot)}$, we have

$\displaystyle\mathrm{iso}(G)$ $\displaystyle=0$
$\displaystyle\mathrm{iso}(G-w)$ $\displaystyle=|I_{1}|$
$\displaystyle\mathrm{iso}(G-w-N(w))$ $\displaystyle=|I_{2}|.$

This is evident from the diagram. Furthermore, we are using the fact that $G$
is connected. Removing this shared factor and replacing the terms of the form
$2^{\mathrm{iso}(\cdot)}$, we are left with the equation

$\displaystyle 2^{|I_{1}|}\prod_{(u,v)\in
E_{2}}i(K_{d_{u},d_{v}})^{\frac{1}{d_{u}(d_{v}-1)}}+2^{|I_{2}|}$
$\displaystyle\prod_{(u,v)\in
E_{3}}\frac{i(K_{d_{u}^{+},d_{v}})^{\frac{1}{d_{u}^{+}d_{v}}}}{i(K_{d_{u},d_{v}})^{\frac{1}{d_{u}d_{v}}}}$
$\displaystyle\leq\prod_{(u,v)\in E_{1}\cup
E_{2}}i(K_{d_{u},d_{v}})^{\frac{1}{d_{u}d_{v}}}$

At this point, we rewrite the expressions using the fact that

$i(K_{d_{u},d_{v}})=2^{d_{u}}+2^{d_{v}}-1.$

This is equivalent to

$\displaystyle 2^{|I_{1}|}\prod_{(u,v)\in
E_{2}}\left(2^{d_{u}}+2^{d_{v}-1}-1\right)^{\frac{1}{d_{u}(d_{v}-1)}}$
$\displaystyle+2^{|I_{2}|}\prod_{(u,v)\in
E_{3}}\left(\frac{(2^{d_{u}^{+}}+2^{d_{v}}-1)^{\frac{1}{d_{u}^{+}d_{v}}}}{(2^{d_{u}}+2^{d_{v}}-1)^{\frac{1}{d_{u}d_{v}}}}\right)$
$\displaystyle\leq\prod_{(u,v)\in E_{1}\cup
E_{2}}\left(2^{d_{u}}+2^{d_{v}}-1\right)^{\frac{1}{d_{u}d_{v}}}$

This problem is significantly simpler — we only have to points in $G$ that are
at most distance $3$ away from $w$. Furthermore, since the graph is bipartite,
there are many properties with which we can characterize these points distance
$3$ away. The rest of the proof involves two applications of Hölder’s
inequality as well as careful computation. Since this computation does not
reveal too much insight on the problem and the proof strategy, the details are
omitted.

##  5 Further Remarks

###  5.1 Generalized Results

In Section 4, we proved an upper bound on the number of independent sets
$i(G)$ for a general graph. In [6], Sah et al. also discuss a lower bound on
the number of independent sets $i(G)$.

######  Theorem 5.1.

Let $G$ be a graph and $d_{u}$ be the degree of vertex $v$ in $G$. Then, we
have

$i(G)\geq\prod_{v\in V(G)}(d_{v}+2)^{1/(d_{v}+1)},$

where equality holds if and only if $G$ is a disjoint union of cliques.

This builds on the previous literature from Cutler and Radcliffe [3]. In
particular, Cutler and Radcliffe provide the same theorem for all $d$-regular
graphs. The proof strategy is very similar to the one discussed in Section 4.
In particular, we divide into sets $V_{0},V_{1},V_{2},\dots$ again and use the
same recursion. The key difference lies in the computation afterwards – the
proof of the lower bound does not use any techniques more than computation and
algebra. Furthermore, Sah et al. generalize this bound to one involving the
partition function $P_{G}(\lambda)=\sum_{I\in\mathcal{I}(G)}\lambda^{|I|}$. In
particular, we have

######  Theorem 5.2.

Let $G$ be a graph. Let $d_{v}$ be the degree of vertex $v$ in $G$. Let
$\lambda>0$. Then

$P_{G}(\lambda)\geq\prod_{v\in V(G)}((d_{v}+1)\lambda+1)^{1/(d_{v}+1)},$

and equality holds if and only if $G$ is a disjoint union of cliques.

This is a lower bound on the partition function of $G$, and it is a
generalization on the lower bound on the number of independent sets. Setting
$\lambda=1$, we arrive at Theorem 5.1. Beyond the lower and upper bound on the
number of independent sets $i(G)$, Sah et al. extend the result to the
independent set polynomial.

######  Definition 5.3.

A _bigraph_ $G=(A,B,E)$ is a bipartite graph with bipartition $V(G)=A\sqcup B$
and edge set $E\subseteq A\times B$. Then, the two-variable independent set
polynomial of $G$ is

$P_{G}(\lambda,\mu)=\sum_{I\in\mathcal{I}(G)}\lambda^{|I\cap A|}\mu^{|I\cap
B|}.$

Note that setting $\mu=1$ and letting $A=G$, we get the _hard-core model_ with
parameter $\lambda$ discussed in class:

$P_{G}(\lambda,1)=\sum_{I\in\mathcal{I}(G)}\lambda^{|I|}.$

Furthermore, setting $\lambda=1$ and $\mu=1$, we get the number of independent
sets $i(G)$. Sah et al. prove the result

######  Theorem 5.4.

Let $G=(A,B,E)$ be a bigraph without isolated vertices. Let $d_{v}$ denote the
degree of $v$ in $G$. Let $\lambda,\mu>0$. Then

$P_{G}(\lambda,\mu)\leq\prod_{\begin{subarray}{c}uv\in E\\\ u\in A,v\in
B\end{subarray}}\left((1+\lambda)^{d_{v}}+(1+\mu)^{d_{u}}-1\right)^{1/(d_{u}d_{v})},$

and equality holds if and only if $G$ is a disjoint union of complete
bipartite graphs.

First, note that setting $\lambda=\mu=1$ gives Theorem 4.1. Furthermore, we
can easily extend this result to all bigraphs by adding a factor of
$2^{\mathrm{iso}(G)}$. The proof technique is exactly the same as the one
described in Section 4.

## References

  * [1] Noga Alon.  Independent sets in regular graphs and sum-free subsets of finite groups.  Israel journal of mathematics, 73(2):247–256, 1991.
  * [2] Peter J. Cameron and Paul Erdős.  On the number of sets of integers with various properties.  Number Theory (Banff, AB, 1988), de Gruyter, Berlin, pages 61–79, 1990.
  * [3] Jonathan Cutler and A. J. Radcliffe.  The maximum number of complete subgraphs in a graph with given maximum degree, 2013.
  * [4] David Galvin.  An upper bound for the number of independent sets in regular graphs.  Discrete mathematics, 309(23-24):6635–6640, 2009.
  * [5] Jeff Kahn.  An entropy approach to the hard-core model on bipartite graphs.  Combinatorics, Probability and Computing, 10(3):219–237, 2001.
  * [6] Ashwin Sah, Mehtaab Sawhney, David Stoner, and Yufei Zhao.  The number of independent sets in an irregular graph.  Journal of Combinatorial Theory, Series B, 138:172–195, 2019.
  * [7] Yufei Zhao.  The number of independent sets in a regular graph.  Combinatorics, Probability and Computing, 19(2):315–320, November 2009.
  * [8] Yufei Zhao.  The bipartite swapping trick on graph homomorphisms.  SIAM Journal on Discrete Mathematics, 25(2):660–680, 2011.


# Inertial longitudinal magnetization reversal for non-Heisenberg ferromagnets

E. G. Galkina  Institute of Physics, 03028 Kiev, Ukraine  CEMS, RIKEN,
Saitama, 351-0198, Japan  V. I. Butrim  Vernadsky Taurida National University,
Simferopol, 95007 Ukraine  Yu. A. Fridman  Vernadsky Taurida National
University, Simferopol, 95007 Ukraine  B. A. Ivanov<EMAIL_ADDRESS>Institute of Magnetism, 03142, Kiev, Ukraine  Taras Shevchenko National
University of Kiev, 01601, Ukraine  CEMS, RIKEN, Saitama, 351-0198, Japan
Franco Nori  CEMS, RIKEN, Saitama, 351-0198, Japan  Department of Physics, The
University of Michigan, Ann Arbor, MI 48109-1040, USA.

###### Abstract

We analyze theoretically the novel pathway of ultrafast spin dynamics for
ferromagnets with high enough single-ion anisotropy (non-Heisenberg
ferromagnets). This longitudinal spin dynamics includes the coupled
oscillations of the modulus of the magnetization together with the quadrupolar
spin variables, which are expressed through quantum expectation values of
operators bilinear on the spin components. Even for a simple single-element
ferromagnet, such a dynamics can lead to an inertial magnetization reversal
under the action of an ultrashort laser pulse.

###### pacs:

75.10.Jm, 75.10.Hk, 78.47.J-, 05.45.-a

##  I Introduction

Which is the fastest way to reverse the magnetization of either a magnetic
particle or a small region of a magnetic film? This question has attracted
significant interest, both fundamental and practical, for magnetic information
storage. [1]  Intense laser pulses, with durations less than a hundred
femtoseconds, are able to excite the ultrafast evolution of the total spin of
a magnetically-ordered system on a picosecond time scale, see e.g. the
reviews. [2] ; [3] ; [4]  The limitations for the time of magnetization
reversal come from the characteristic features of the spin evolution for a
magnet with a concrete type of magnetic order. The dynamical time cannot be
shorter than the characteristic period of spin oscillations $T$,
$T=2\pi/\omega_{0}$, where $\omega_{0}$ is the magnetic resonance frequency.
For ferromagnets, the frequency of standard spin oscillations (precession) is
$\omega_{0,\mathrm{FM}}=\gamma H_{\mathrm{r}}$, where $\gamma$ is the
gyromagnetic ratio, and $H_{\mathrm{r}}$ is an effective field of relativistic
origin, like the anisotropy field, which is usually less than a few Tesla.
Thus, the dynamical time for Heisenberg ferromagnets cannot be much shorter
than one nanosecond. For antiferromagnets, all the dynamical characteristics
are exchange enhanced, and
$\omega_{0,\mathrm{AFM}}=\gamma\sqrt{H_{\mathrm{ex}}H_{\mathrm{r}}}$, where
$H_{\mathrm{ex}}$ is the exchange field,
$H_{\mathrm{ex}}=J/2\mu_{\mathrm{B}}$, $J$ is the exchange integral, and
$\mu_{\mathrm{B}}$ is the Bohr magneton, see Ref. [5], . The excitation of
terahertz spin oscillations has been experimentally demonstrated for
transparent antiferromagnets using the inverse Faraday effect or the inverse
Cotton-Mouton effect. [6] ; [7] ; [8] ; [9] ; [10] ; [11]  The non-linear
regimes of such dynamics include the inertia-driven dynamical reorientation of
spins on a picosecond time scale, which was observed in orthoferrites. [10] ;
[11]

The exchange interaction is the strongest force in magnetism, and the exchange
field $H_{\mathrm{ex}}$ can be as strong as $10^{3}$ Tesla. The modulus of the
magnetization is determined by the exchange interaction, and the direction of
the magnetization is governed by relativistic interactions. It would be very
tempting to produce a magnetization reversal by changing the modulus of the
magnetization vector, i.e., via the longitudinal dynamics of ${\rm{\bf M}}$.
For such a process, dictated by the exchange interaction, the characteristic
times could be of the order of the exchange time $\tau_{\mathrm{ex}}=1/\gamma
H_{\mathrm{ex}}$, which is shorter than one picosecond. However, within the
standard approach such dynamics is impossible. The evolution of the modulus of
the magnetization, $M=|{\rm{\bf M}}|$, within the closed Landau-Lifshitz
equation for the magnetization only (or the set of such equations for the
sublattice magnetizations, ${\rm{\bf M}}_{\alpha})$, is purely dissipative.
[12]  This feature could be explained as follows: two angular variables,
$\theta$ and $\varphi$, describing the direction of the vector ${\rm{\bf M}}$,
within the Landau-Lifshitz equation determine the pair of conjugated Hamilton
variables $(\cos\theta$ and $\varphi$ are the momentum and coordinate,
respectively). Also, the evolution of the single remaining variable
$M=|{\rm{\bf M}}|$, governed by a first-order equation can be only
dissipative; see a more detailed discussion below. Moreover, the exchange
interaction conserves the total spin of the system, and the relaxation of the
total magnetic moment of any magnet can be present only when accounting for
relativistic effects. Thus the relaxation time for the total magnetic moment
is relativistic but it is exchange-enhanced, as was demonstrated within the
irreversible thermodynamics of the magnon gas. [12]  Note here that the
relaxation of the magnetization of a single sublattice for multi-sublattice
magnets can be of purely exchange origin. [13]  Recently, magnetization
reversal on a picosecond time scale has been experimentally demonstrated for
the ferrimagnetic alloy GdFeCo, see Refs. [13], ; [14], . These results can be
explained within the concept of exchange relaxation, developed by Baryakhtar,
[15]  accounting for the purely exchange evolution of the sublattice
magnetization. [16]  Such an exchange relaxation can be quite fast, but its
characteristic time is again longer than the expected “exchange time”
$\tau_{\mathrm{ex}}=1/\gamma H_{\mathrm{ex}}$.

Thus, the ultrafast mechanisms of magnetization reversal implemented so far
are: the dynamical (inertial) switching possible for antiferromagnets, [10] ;
[11]  and the exchange longitudinal evolution for ferrimagnets. [13] ; [14] ;
[16]  These are both quite fast, with a characteristic time of the order of
picoseconds; but their characteristic times are longer than the “ideal
estimate”: the exchange time $\tau_{\mathrm{ex}}$.

In this work, we present a theoretical study of the possibility of the
dynamical evolution of the modulus of the magnetization for non-Heisenberg
ferromagnets with high enough single-ion anisotropy that can be called
longitudinal spin dynamics. For such a dynamics, an inertial magnetization
reversal is possible even for a simple single-element ferromagnet.
Longitudinal dynamics does not exist in Heisenberg magnets, and this dynamics
cannot be described in terms of the Landau-Lifshitz equation, or using the
Heisenberg Hamiltonian, which is bilinear over the components of spin
operators for different spins, see more details below in Sec. II. The key
ingredient of our theory is the inclusion of higher-order spin quadrupole
variables. It is known that for magnets with atomic spin $S>1/2$, allowing the
presence of single-ion anisotropy, the spin dynamics is not described by a
closed equation for spin dipolar variable $\left\langle{\rm{\bf
S}}\right\rangle$ alone (or magnetization ${\rm{\bf
M}}=-2\mu_{\mathrm{B}}\left\langle{\rm{\bf S}}\right\rangle$). [17] ; [18] ;
[19] ; [20] ; [21] ; Chubuk90 ; Matveev ; AndrGrishchuk  Here and below
$\left\langle{...}\right\rangle$ means quantum and (at finite temperature)
thermal averaging. To be specific, we choose the spin-one ferromagnet with
single-ion anisotropy, the simplest system allowing this effect. The full
description of these magnets requires taking into account the dynamics of
quadrupolar variables,
$S_{ik}=(1/2)\left\langle{S_{i}S_{k}+S_{k}S_{i}}\right\rangle$, that represent
the quantum averages of the operators, bilinear in the spin components. Our
theory is based on the consistent semiclassical description of a full set of
spin quantum expectation values (dipolar and quadrupolar) for the spin-one
system, which was investigated by many authors from different viewpoints. [17]
; [18] ; [19] ; [20] ; [21] ; Chubuk90 ; Matveev ; AndrGrishchuk  As we will
show, the longitudinal dynamics of spin, including nonlinear regimes, can be
excited by a femtosecond laser pulse. With natural accounting for the
dissipation, the longitudinal spin dynamics can lead to changing the sign of
the total spin of the system (longitudinal magnetization reversal).

##  II  MODEL DESCRIPTION

The Landau-Lifshitz equation was proposed many years ago as a phenomenological
equation, and it is widely used for the description of various properties of
ferromagnets. Concerning its quantum and microscopic basis, it is worth noting
that this equation naturally arises using the so-called spin coherent states.
[22] ; [23]  These states can be introduced for any spin $S$ as the state with
the maximum value of the spin projection on an arbitrary axis ${\rm{\bf n}}$.
Such states can be parameterized by a unit vector ${\rm{\bf n}}$; the
direction of the latter coincides with the quantum mean values for the spin
operator $\left\langle{\rm{\bf S}}\right\rangle=S{\rm{\bf n}}$ (dipolar
variables). This property is quite convenient for linking the quantum physics
of spins to a phenomenological Landau-Lifshitz equation. The use of spin
coherent states is most efficient when the Hamiltonian of the system is linear
with respect to the operators of the spin components. If an initial state is
described by a certain spin coherent state, its quantum evolution will reduce
to a variation of the parameters of the state (namely, the direction of the
unit vector ${\rm{\bf n}})$, which are described by the classical Landau-
Lifshitz equation. Thus, spin coherent states are a convenient tool for the
analysis of spin Hamiltonians containing only operators linear on the spin
components or their products on different sites. An important example is the
bilinear Heisenberg exchange interaction, described by the first term in Eq.
(1) below.

In contrast to the cases above, for the full description of spin-$S$ states,
one needs to introduce $SU(2S+1)$ generalized coherent states. [17] ; [19] ;
[20] ; [21]  The analysis shows that spin coherent states are less natural for
the description of magnets whose Hamiltonian contains products of the spin
component operators at a single site. Such terms are present for magnets with
single-ion anisotropy or a biquadratic exchange interaction. Magnets with non-
small interaction of this type are often called non-Heisenberg. For such
magnets, some non-trivial features, absent for Heisenberg magnets, are known.
Among them we note the possibility of so-called quantum spin reduction; namely
the possibility to have the value of $|\left\langle{\rm{\bf S}}\right\rangle|$
less than its nominal value, $|\left\langle{\rm{\bf S}}\right\rangle|<S$, even
for pure states at zero temperature. This was first mentioned by Moriya, [24]
as early as 1960. As an extreme realization of the effect of quantum spin
reduction, we note the existence of the so-called spin nematic phases with a
zero mean value of the spin in the ground state at zero temperature. In the
last two decades, the interest on such states has been considerable, motivated
by studies of multicomponent Bose-Einstein condensates of atoms with non-zero
spin. [25] ; [26] ; [27] ; [28]

A significant manifestation of quantum spin reduction is the appearance of an
additional branch of the spin oscillations, which is characterized by the
dynamics (oscillations) of the length of the mean value of spin without spin
precession. [17] ; [18] ; [19] ; [20] ; [21] ; Chubuk90 ; Matveev  The
characteristic frequency of this mode can be quite high (of the order of the
exchange integral). For this reason, for a description of resonance properties
or thermodynamic behavior of magnets, this mode is usually neglected, and the
common impression is that the dynamics of magnetic materials with constant
single-ion anisotropy $K<$(0.2-0.3)$J$ is fully described by the standard
phenomenological theory. However, for an ultrafast evolution of the spin
system under a femtosecond laser pulse, one can expect a lively demonstration
of this longitudinal high-frequency mode. Thus, it is important to explore the
possible manifestations of the effects of quantum spin reduction in the
dynamic properties of ferromagnets.

The simplest model allowing spin dynamics with effects of quantum spin
reduction is described by the Hamiltonian

$H=-\frac{1}{2}\sum\limits_{\mathbf{n},\boldsymbol{\ell}}{\bar{J}{\rm{\bf
S}}_{\mathbf{n}}{\rm{\bf
S}}_{\mathbf{n}+\boldsymbol{\ell}}}+\frac{K}{2}\sum\limits_{\mathbf{n}}{\left({S_{\mathbf{n},x}}\right)^{2}},$
(1)

where ${\rm{\bf S}}_{\mathbf{n}}$ is the spin-one operator at the site
$\mathbf{n}$; $\bar{J}>0$ is the exchange constant for nearest-neighbors
$\boldsymbol{\ell}$, and $K>0$ is the constant of the easy-plane anisotropy
with the plane $yz$ as the easy plane. The quantization axis can be chosen
parallel to the $z$-axis and $\left\langle{\rm{\bf
S}}\right\rangle=\left\langle{S_{z}}\right\rangle{\rm{\bf e}}_{z}$. For the
full description of spin $S=1$ states, let us introduce $SU(3)$ coherent
states [17] ; [19] ; [20] ; [21]

$\left|{\mathbf{u},\mathbf{v}}\right\rangle=\sum\limits_{j=x,y,z}{\left({u_{j}+iv_{j}}\right)\left|{\psi_{j}}\right\rangle},$
(2)

where the states $\left|{\psi_{j}}\right\rangle$ determine the Cartesian
states for $S=1$ and are expressed in terms of the ordinary states
{$\left|{\pm 1}\right\rangle,\;\left|0\right\rangle$} with given projections
$\pm 1,\;0$ of the operator $S_{z}$ by means of the relations
$\left|{\psi_{x}}\right\rangle=\left({\left|{-1}\right\rangle-\left|{+1}\right\rangle}\right)/\sqrt{2}$,
$\left|{\psi_{y}}\right\rangle=i\left({\left|{-1}\right\rangle+\left|{+1}\right\rangle}\right)/\sqrt{2}$,
$\left|{\psi_{z}}\right\rangle=\left|0\right\rangle$, with the real vectors
$\mathbf{u}$ and $\mathbf{v}$ subject to the constraints
$\mathbf{u}^{2}+\mathbf{v}^{2}=1$, $\mathbf{u}\cdot\mathbf{v}=0$. All
irreducible spin averages, which include the dipolar variable
$\left\langle\mathbf{S}\right\rangle$ (average value of the spin) and
quadrupole averages $S_{ik}$, bilinear over the spin components, can be
written through $\mathbf{u}$ and $\mathbf{v}$ as follows

$\displaystyle\left\langle\mathbf{S}\right\rangle$ $\displaystyle=$
$\displaystyle 2(\mathbf{u}\times\mathbf{v}),$ $\displaystyle
S_{ik}=\frac{1}{2}\left\langle{S_{i}S_{k}+S_{k}S_{i}}\right\rangle$
$\displaystyle=$ $\displaystyle\delta_{ik}-u_{i}u_{k}-v_{i}v_{k}.$ (3)

At zero temperature and within the mean-field approximation, the spin dynamics
is described by the Lagrangian [21]

$L=-2\hbar\sum\limits_{n}{\mathbf{v}_{n}(\partial\mathbf{u}_{n}/\partial
t})-W\left({\mathbf{u},\mathbf{v}}\right),$ (4)

where
$W\left({\mathbf{u},\mathbf{v}}\right)=\left\langle{\mathbf{u},\mathbf{v}}\right|H\left|{\mathbf{u},\mathbf{v}}\right\rangle$
is the energy of the system.

We are interested in spin oscillations which are uniform in space, and hence
we assume that the discrete variables $\mathbf{u}$ and $\mathbf{v}$ have the
same values for all spins and are only dependent on time. The frequency
spectrum of linear excitations, which consists of two branches, can be easily
obtained on the basis of the linearized version of the Lagrangian (4). In the
general case, the system of independent equations for $\mathbf{u}$ and
$\mathbf{v}$, taking into account the aforementioned constrains
$\mathbf{u}^{2}+\mathbf{v}^{2}=1$, $\mathbf{u}\cdot\mathbf{v}=0$, consists of
four nonlinear equations, describing two different regimes of spin dynamics.
One regime is similar to that for an ordinary spin dynamics treated on the
basis of the Landau-Lifshitz equation; it corresponds to oscillations of the
spin direction. The second regime corresponds to oscillations of the modulus
of the magnetization $\left\langle{\rm{\bf S}}\right\rangle=S(t){\rm{\bf
e}}_{z}$, with the vectors $\mathbf{u}$ and $\mathbf{v}$ rotating in the
$xy$-plane perpendicular to $\left\langle{\rm{\bf S}}\right\rangle$. This mode
of the spin oscillations corresponds to the longitudinal spin dynamics. It is
convenient to consider these two types of dynamics separately. Particular non-
linear longitudinal solutions, with $\left\langle{\rm{\bf
S}}\right\rangle=s(t){\rm{\bf e}}_{z}$ and $u_{z}=0,\;v_{z}=0$, were found in
Refs. [29], ; [30], . Note here that the longitudinal dynamics is much faster
than the standard transversal one, and the standard spin precession (described
by the Landau-Lifshitz equation) at a picosecond time scale just cannot
develop. Therefore, these two regimes, longitudinal and transverse, can be
treated independently, and we limit ourselves only to the longitudinal
dynamics with $\left\langle{\rm{\bf S}}\right\rangle=s(t){\rm{\bf e}}_{z}$ and
$u_{z}=0,\;v_{z}=0$.

##  III LONGITUDINAL SPIN DYNAMICS

To describe the longitudinal spin dynamics, it is convenient to introduce new
variables: the spin modulus $s=2|{\rm{\bf u}}||{\rm{\bf v}}|=2uv$ and angular
variable $\gamma$, with

${\rm{\bf u}}=u\left({{\rm{\bf e}}_{x}\cos\gamma-{\rm{\bf
e}}_{y}\sin\gamma}\right),\;{\rm{\bf v}}=v\left({{\rm{\bf
e}}_{x}\sin\gamma+{\rm{\bf e}}_{y}\cos\gamma}\right),$ (5)

In this representation $\left\langle{S_{z}}\right\rangle=s$, and the non-
trivial quadrupolar variables are
$\left\langle{S_{x}S_{y}+S_{y}S_{x}}\right\rangle=\sqrt{1-s^{2}}\sin 2\gamma$
and $\left\langle{S_{y}^{2}-S_{x}^{2}}\right\rangle=\sqrt{1-s^{2}}\cos
2\gamma$, with all other quantum averages being either zero (as the transverse
spin components $\left\langle{S_{x,y}}\right\rangle$ or $S_{xz}$, $S_{yz}$) or
trivial, independent on $s$ and $\gamma$, as
$\left\langle{S_{z}^{2}}\right\rangle=1$. The mean-field energy, written per
one spin through the variables $s,\;\gamma$, takes the form

$W(s,\gamma)=-\frac{J}{2}s^{2}-\frac{K}{4}\sqrt{1-s^{2}}\cos 2\gamma,$ (6)

where $J=\bar{J}Z$, $Z$ is the number of nearest neighbors. The ground state
at $\sqrt{1-s^{2}}>0$ corresponds to $\cos 2\gamma=1$, with the mean value of
the spin $s=\pm\bar{s}$, $\bar{s}=\sqrt{1-\kappa^{2}}<1$, that is a
manifestation of quantum spin reduction at non-zero anisotropy. Here we
introduce the dimensionless parameter $\kappa=K/4J$. For these variables, the
Lagrangian can be written as

$L=\hbar s\frac{\partial\gamma}{\partial t}-W(s,\gamma),$ (7)

and $\hbar s$ and $\gamma$ play the role of canonical momentum and coordinate,
respectively, with the Hamilton function $W(s,\gamma)$. The physical meaning
of the above formal definitions is quite clear: the angular variable $\gamma$
describes the transformation of quadrupolar variables under rotation around
the $z$-axis, with $\hbar s$ as the projection of the angular momentum on this
axis.

###  III.1 Small oscillations

Let us now start with the description of the dynamics of small-amplitude
oscillations. After linearization around the ground state, the equation leads
to a simple formula for the frequency of longitudinal oscillations

$\hbar\omega_{l}=2J\bar{s}=2J\sqrt{1-\kappa^{2}},$ (8)

which are in fact coupled oscillations of the projection of the spin and
quadrupolar variables, see Fig. 1.

One can see that, for a wide range of values of the anisotropy constant, like
$\kappa<0.2$-0.8, this frequency $\omega_{l}$ is of the order of
(1.8-1.2)$J/\hbar$, i.e., $\omega_{l}$ is comparable to the exchange frequency
$J/\hbar$. Thus the longitudinal spin dynamics is expected to be quite fast.
In contrast, standard transversal oscillations for a purely easy-plane model
(1) are gapless (they acquire a finite gap when accounting for a magnetic
anisotropy in the easy plane, which is usually small). Thus the essential
difference in the frequencies of these two dynamical regimes is clearly seen.

Figure 1:  (color online) Graphic presentation of the variables $s$ and
$\gamma$ and their evolution. The thick red arrow represents the mean value of
the spin. The quadrupolar variables are shown by the blue three-axial
ellipsoid with the directions of the main axis (chosen to have
$\left\langle{S_{1}S_{2}}\right\rangle=0)$: ${\rm{\bf e}}_{3}={\rm{\bf
e}}_{z}$ and ${\rm{\bf e}}_{1}$, ${\rm{\bf e}}_{2}$. The half-axes of the
ellipsoid are equal to $\left\langle{S_{1}^{2}}\right\rangle$,
$\left\langle{S_{2}^{2}}\right\rangle$ and
$\left\langle{S_{3}^{2}}\right\rangle=\left\langle{S_{z}^{2}}\right\rangle=1$.
(a) the ground state, (b) the standard transverse dynamics, i.e. the spin
precession. The other frames (c)-(e) present the transient values of the
variables in longitudinal oscillations. (c) and (e) correspond to the longest
and shortest length of the spin, and at the moment depicted in (d) the spin
length equals to its equilibrium value, but the quadrupolar ellipsoid is
turned on the angle $\gamma$ with respect to the $x$-axis. On the frames
(c)-(e), the shape of the unperturbed ellipsoid is shown by light grey.

At a first glance, there is a contradiction between the concept of
longitudinal dynamics caused by single-ion anisotropy and the result present
in equation (8): the value of $\omega_{l}$ is still finite for vanishing
anisotropy constant $K$; and it is even growing to the value $2J/\hbar$ when
$\kappa\to 0$. This can be explained as follows: for a given energy, the ratio
of amplitudes for the oscillations of the spin variable $s$ and quadrupolar
variable $\gamma$ vanish at $\kappa\to 0$ as $\kappa$. In fact, for extremely
low anisotropy the spin oscillations are not present in this mode, which
becomes just a free rotation of the quadrupole ellipsoid of the form
$\gamma=2Jt/\hbar$, with $s=\bar{s}=\mathrm{const}$. On a phase plane with
coordinates $(s,\gamma)$ this dynamics is depicted by vertical straight lines
parallel to the $\gamma$-axis, see Fig. 2(c) below. We will discuss this
feature in more detail with the analysis of non-linear oscillations.

###  III.2 Nonlinear dynamics and phase plane analysis.

Before considering damped oscillations, it is instructive to discuss
dissipationless non-linear longitudinal oscillations. It is convenient to
present an image of the dynamics as a “phase portrait” on the plane momentum-
coordinate $(s,\;\gamma)$, which shows the behavior of the system for
arbitrary initial conditions. The phase trajectories in the plane without
dissipation can be found from the condition $W(s,\gamma)=\mathrm{const}$.

The energy (6) has an infinite set of minima, with $s=\pm\bar{s}$ and
$\gamma=\pi n,$ with equal energies (green ellipses on the Fig.2), and an
infinite set of maxima at $s=0$ and $\gamma=\pi/2+\pi n$ (red ellipses on the
Fig.2), here $n$ is an integer. Only the minima with $s=\bar{s}$ and
$s=-\bar{s}$ are physically different; equivalent extremes with different
values of $n$ correspond to the equal values of the observables and are
completely equivalent. The minima on the phase plane correspond to foci with
two physically different equilibrium states with antiparallel orientation of
spin $s=\pm\bar{s}$, and
$\left\langle{S_{y}^{2}-S_{x}^{2}}\right\rangle=\sqrt{1-\bar{s}^{2}}$,
$\left\langle S_{z}^{2}\right\rangle=1$,
$\left\langle{S_{x}S_{y}+S_{y}S_{x}}\right\rangle=0$. The saddle points are
located at the values $s=0$ and $\gamma=\pi n$. The lines with $s=\pm 1$ are
singular; these correspond to degenerate motion with $\gamma$ linear in time
$\gamma=\pm tJ/\hbar$; the points at these lines where $d\gamma/dt$ change
sign, can be treated as some non-standard saddle points.

The shape of the phase trajectories, i.e., the characteristic features of
oscillations, varies with the change of the anisotropy parameter $\kappa$.
Note first the general trend, the relative amplitude of the changes of the
spin and $\gamma$ depends on $\kappa$: the bigger $\kappa$ is, the larger
values of the change of spin are observed. The topology of the phase
trajectories change at the critical value of the anisotropy parameter
$\kappa$. At small $\kappa<1/2$, the trajectories with infinite growing
$\gamma$ are present, and the standard separatrix trajectories connect
together different saddle points, see Fig. 2 (c). As mentioned above; only
such trajectories are present at the limit $\kappa\to 0$, but, in fact, even
for the small value of $\kappa=0.2$ used in Fig.2 (c), the main part of the
plane is occupied by the trajectories that change the spin. At the critical
value $\kappa=1/2,$ the separatrix trajectories connect the saddle points at
$s=0$, $\gamma=0$ and the degenerated saddle points at the singular lines with
$s=\pm 1$. At larger anisotropy $\kappa>1/2$, as in Fig. 2(a), the only
separatrix loops entering the same saddle point are present.

The minima, saddle point and the separatrix are the key ingredients for the
switching between the ground states with $s=\bar{s}$ and $s=-\bar{s}$. The
case of large anisotropy $\kappa>1/2$ looks like the standard one for the
switching phenomena, whereas for small anisotropy the situation is more
complicated.

(a) $\ \kappa=$0.6 (b) $\ \kappa=$0.5

(c) $\ \kappa=$0.2

Figure 2: (color online) Phase plane representations for dissipation-free non-
linear longitudinal spin oscillations for different values of the parameter
$\kappa$; $\kappa=$0.6, 0.5 and 0.2 for panels (a), (b), and (c),
respectively. The green and red ellipses present the minima and maxima,
respectively; the standard saddle points are depicted by red rectangles, while
the standard separatrix trajectories are drawn by red lines. The singular
trajectories with $s=\pm 1$ and the separatrix trajectories entering the non-
standard saddle points on these lines $s=\pm 1$ are shown by blue lines on the
frames (a) and (c). For the critical value $\kappa=0.5$, all the separatrix
trajectories and the singular trajectories with $s=\pm 1$ organize a common
net; and on the corresponding frame (b) all of them are presented by red
lines.

###  III.3 Damped longitudinal motion

Damping is a crucial ingredient for the dynamical switching between different,
but equivalent in energy, states. The high-frequency mode of longitudinal
oscillations have high-enough relative damping; as was found from microscopic
calculations, [29]  the decrement of longitudinal mode
$\Gamma=\lambda\omega_{l}$, where $\lambda\sim 0.2$. To account for the
damping in the dynamic equations for $s$ and $\gamma$, it is useful to
consider a different parametrization of the longitudinal dynamics. Let us now
introduce a unit vector, $\boldsymbol{\sigma}=\sigma_{1}{\rm{\bf
e}}_{1}+\sigma_{2}{\rm{\bf e}}_{2}+\sigma_{3}{\rm{\bf e}}_{3}$, with
components $\sigma_{3}=s$, $\sigma_{1}=\langle S_{y}^{2}-S_{x}^{2}\rangle,$
and $\sigma_{2}=\langle S_{1}S_{2}+S_{2}S_{1}\rangle$. Being written through
$\boldsymbol{\sigma}$, the equation of motion takes the form of the familiar
Landau-Lifshitz equation

$\hbar\frac{\partial\boldsymbol{\sigma}}{\partial
t}=[\boldsymbol{\sigma}\times{\rm{\bf h}}_{\mathrm{eff}}]+{\rm{\bf
R}},\quad{\rm{\bf h}}_{\mathrm{eff}}=-\frac{\partial
W}{\partial\boldsymbol{\sigma}},$ (9)

where ${\rm{\bf h}}_{\mathrm{eff}}$ can be treated as an effective field for
longitudinal dynamics, and the relaxation term ${\rm{\bf R}}$ is added. The
equation of motion with ${\rm{\bf R}}=0$ is fully equivalent to the Hamilton
form of the equation found from (7), but the form of the dissipation is more
straightforward in unit-vector presentation. The choice of the damping term in
a standard equation for the motion of the transverse spin is still under
debate, see. [15] ; [31]  But here the damping term can be written in the
simplest form, as in the original paper of Landau and Lifshitz, ${\rm{\bf
R}}=\lambda[{\rm{\bf h}}_{\mathrm{eff}}-\boldsymbol{\sigma}({\rm{\bf
h}}_{\mathrm{eff}}\boldsymbol{\sigma})]$. The arguments are as follows: (i)
this form gives the correct value of the decrement of linear oscillations,
$\Gamma=\lambda\omega_{l}$; (ii) it is convenient for analysis, because it
keeps the condition $\boldsymbol{\sigma}^{2}=1$. Finally, the equations of
motion with the dissipation term of the aforementioned form are:

$\hbar\frac{ds}{dt}=-\frac{\partial
W}{\partial\gamma}-\lambda(1-s^{2})\frac{\partial W}{\partial
s},\;\hbar\frac{d\gamma}{dt}=\frac{\partial W}{\partial
s}-\frac{\lambda}{(1-s^{2})}\frac{\partial W}{\partial\gamma}.$ (10)

These equations describe the damped counterpart of the non-linear longitudinal
oscillations discussed in the previous subsection and present as phase
portraits on Fig. 2. The character of the motion at not-too-large $\lambda$
can be qualitatively understood from energy arguments. The trajectories of
damped oscillations in any point of the phase plane approximately follow the
non-damped (described by equation $W(s,\gamma)=\mathrm{const})$ ones, but
cross them passing from larger to smaller values of $W$, see Figs. 3 and 4. It
happens that for the case of interest, the dynamics is caused by the time-
dependent stimulus. An action of the stimulus on the system can be described
by adding the corresponding time-dependent interaction energy $\Delta W$ to
the system Hamiltonian, $W\to W(s,\gamma)+\Delta W(s,\gamma,t)$. Within this
dynamical picture, $\Delta W$ produces an “external force” driving the system
far from equilibrium.

The analysis is essentially simplified for a pulse-like stimulus of a short
duration $\Delta t$ (much shorter than the period of motion, $\omega_{l}\Delta
t\ll 1)$. In this case, the role of the pulse is reduced to the creation of
some non-equilibrium state, which then evolves as some damped nonlinear
oscillations described by the “free” equations (10) with $\Delta W=0$. The
phase plane method, which shows the behavior of the system for arbitrary
initial conditions, is the best tool for the description of such an evolution.

Figure 3:  (color online) Phase plane representation of damped longitudinal
spin oscillations for $\kappa=$0.6. Here and in Fig. 4, the dashed lines
(obtained analytically before) represent the phase trajectories without
dissipation, while the full lines are trajectories for dissipation constant
$\lambda=0.2$, found numerically. The separatrix lines are drawn by red
curves.

It is worth noting that the asymptotic behavior of the separatrix trajectories
at $\gamma,s\to 0$, is important for this analysis, and can be easily found
analytically as

$\left({\frac{\gamma}{s}}\right)_{\mathrm{separ}}=R_{\mathrm{separ}}=\\\
=\frac{1}{8\kappa}\left[{\lambda(1+3\kappa)+\sqrt{\lambda^{2}(1+3\kappa)^{2}+16\kappa(1-\kappa)}}\right]\quad.$
(11)

First let us start with the analysis for high-enough anisotropy. The
corresponding phase portrait is present in Fig. 3. The general property of the
phase plane is that the phase trajectories cannot cross each other; they can
only merge at the saddle points. Thus the trajectories coming to different
minima are stretched between two separatrix lines entering the same saddle
point from different directions, as shown in Fig. 3. From this it follows that
any initial state with arbitrary non-equilibrium values of spin $s(+0)$, but
without deviation of $\gamma$ from its equilibrium value, evolve to the state
with the same sign of the spin as for $s(+0)$, and no switching occurs. On the
other hand, if the initial condition is above the separatrix trajectory,
entering the saddle point, the evolution will move the system to the
equivalent minimum with the sign of the spin opposite to the initial one,
$s(+0)$, realizing the switching.

Figure 4 shows the phase plane for equations (10) for the more complicated
case of low anisotropy, demonstrating possible scenarios of the switching of
the sign of the spin value during such dynamics. Here the separatrix
trajectories for the damped motion can be monitored from their maxima, and the
full picture of the behavior can be understood only when including a few
equivalent foci with $\gamma=0,\,\pm\pi,\,\pm 2\pi$, ect., with different, but
equivalent in energy, values of the spin, $s=\pm\bar{s}$,
$\bar{s}=\sqrt{1-\kappa^{2}}$, and different saddle points, located at
$\gamma=0,\,\pm\pi,\,\pm 2\pi$. As for small anisotropy, the trajectories
coming to different minima are located between two branches of the separatrix
lines, but now this “separatrix corridor” is organized by separatrix lines
entering different saddle points. The switching phenomena is also possible,
but the process involves a few full turns of the variable $\gamma$.

The general regulation for any anisotropy can be formulated as follows: for
realizing spin switching, one needs to have the initial deviation (reduction)
of the spin value, and, simultaneously, a non-zero deviation of the
quadrupolar variable $\gamma$. To switch the positive spin value to negative,
one needs to start from the states just above the separatrix line entering the
saddle point from positive values of $s$. The smaller the initial value of the
spin, the smaller value of $\gamma(0)$ would realize the switching. From the
asymptotic equation (11), the corresponding ratio
$R_{\mathrm{separ}}=\gamma(0)/m(0)$ is smaller for small values of $\lambda$;
but even when $\lambda\to 0,$ it exceeds the value
$R_{\mathrm{separ}}(\lambda=0)=0.5\sqrt{(1-\kappa)/\kappa}$. Thus, the
switching could occur for non-zero values of $\kappa$.

Figure 4: (color online) Phase plane for the damped spin evolutions for low
anisotropy, $\kappa=0.2$. The central frame shows the full diagram; left and
right panels demonstrate the details of the behavior near the equilibrium
values $s=-\bar{s}$ and $s=\bar{s}$, respectively. On this frame, the regions
colored by green and yellow correspond to different basins of attraction with
initial values leading to the equilibrium states with $s=\bar{s}$ and
$s=-\bar{s}$, respectively.

##  IV INTERACTION OF THE LIGHT PULSE ON THE SPIN SYSTEM: CREATION OF THE
INITIAL STATE FOR SWITCHING.

Let us now consider the longitudinal spin evolution caused by a specific
stimulus: a femtosecond laser pulse. The reduction of the spin to small values
was observed in many experiments, and the only non-trivial remaining question
is: how can we create a deviation of the quadrupolar variable $\gamma$ from
its equilibrium value $\gamma=0$. To find this, we now consider possible
mechanisms of light interaction with quadrupolar variables of non-Heisenberg
magnets.

The interaction of the spin system of magnetically-ordered media and light is
described by the Hamiltonian (as above, written per spin) $\Delta
W=\bar{\varepsilon}_{ij}v_{0}E_{i}(t)E_{j}^{\ast}(t)/16\pi$, where $v_{0}$ is
the volume per spin, $E_{i}(t)$ is the time-dependent amplitude of the light
in the pulse,
$\bar{\varepsilon}_{ij}=d(\omega\varepsilon_{ij}^{(\mathrm{spin})})/d\omega$,
$\varepsilon_{ij}^{(\mathrm{spin})}$ is the spin-dependent part of the
dielectric permittivity tensor, and $\omega$ is the frequency of light. For
the longitudinal dynamics considered here, circularly-polarized light
propagating along the $z$-axis acts on the $z$-component of the spin via the
standard inverse Faraday effect, with the antisymmetric part of
$\bar{\varepsilon}_{ij}^{(a)}$ as,
$\bar{\varepsilon}_{xy}^{(a)}=-\bar{\varepsilon}_{xy}^{(a)}=s\alpha_{\mathrm{F}}$,
giving an interaction of the form

$\Delta
W_{\mathrm{circular}}=s\frac{\alpha_{\mathrm{F}}v_{0}}{16\pi}|E_{\mathrm{circ}}|^{2}\sigma,$
(12)

where $E_{\mathrm{circ}}$ is the (complex) amplitude, $\sigma$ describes the
pulse helicity: $\sigma=\pm 1$ for right-handed and left-handed circularly
polarized laser pulses. To describe qualitatively the result of the action of
the light pulse, let us now assume that the pulse duration
$\tau_{\mathrm{pulse}}$ is the shortest time of the problem. If the pulse
duration is shorter than the period of spin oscillations, the real pulse shape
can be replaced by the Dirac delta function, $|E_{\mathrm{circ}}|^{2}\to
E_{\mathrm{p}}^{2}\tau_{\mathrm{pulse}}\delta(t),$ where
$E_{\mathrm{p}}^{2}=\int{|E_{\mathrm{circ}}|^{2}dt}/\tau_{\mathrm{pulse}}$
characterizes the pulse intensity. (Note that this approximation is still
qualitatively valid even for any comparable values of $\tau_{\mathrm{pulse}}$
and $2\pi/\omega)$ Then, using equations (10) one can find the effect produced
by the pulse. Within this approximation, the action of a pulse leads to an
instantaneous deviation of the variable $\gamma$ from its equilibrium value,
which then evolves following the non-perturbed equations of motion (10).
Keeping in mind that before the pulse action the system is in equilibrium,
$s(-0)=\bar{s}$ and $\gamma(-0)=0$, it is straightforward to find the values
of these variables $(s$ and $\gamma)$ after the action of the pulse, $s(+0)$
and $\gamma(+0)$. For our purposes, the non-equilibrium value of the
quadrupolar variable $\gamma$ is important:

$\gamma(+0)=-\frac{\alpha_{\mathrm{F}}}{16\pi\hbar}E_{\mathrm{p}}^{2}v_{0}\tau_{\mathrm{pulse}}\sigma.$
(13)

The cumulative action of the circularly-polarized pulse, including an
essential reduction of the spin value (caused either by thermal or non-thermal
mechanisms) and the deviation of $\gamma$ described by (13) could lead to the
evolution we are interested here, switching the spin of the system. Note here
that for standard spin reduction the polarization of the light pulse is not
essential, [32] ; [33] ; [34]  whereas the values of $\gamma(+0)$ are opposite
for right- and left-handed circularly polarized pulses. These features are
characteristic of the effect described here. Note the recent experiment where
the role of circular polarization in spin switching for GdFeCo alloy was
mentioned, but the authors have attributed it to magnetic circular dichroism.
[35]

##  V CONCLUDING REMARKS.

Let us now compare the approach developed in this article with previous
results on subpicosecond spin evolution. The first experimental observation of
demagnetization for ferromagnetic metals under femtosecond laser pulses shows
that the magnetic moment can be quenched very fast to small values, much
faster than one picosecond. [32] ; [33] ; [34]  These effects are associated
with a new domain of the physics of magnets, femtomagnetism, [36]  and its
analysis is based on the microscopic consideration of spins of atomic
electrons, [37] ; [38]  or itinerant electrons. [39]  Not discussing this
fairly promising and fruitful domain of magnetism, note that, to the best of
our knowledge, no effects of magnetization reversal during this “femtomagnetic
stage” has been reported in the literature. For example, the subpicosecond
quenching processes for the ferromagnetic alloy GdFeCo are responsible for the
creation of a far-from-equilibrium state, but the evolution of this state,
giving the spin reversal, can be described within the standard set of
equations for the sublattice magnetizations. [16]

In contrast, here we propose some pathway to switch the sign of the magnetic
moment during extremely short times, of order of the exchange time. It is
shown here that the spin dynamics for magnets with non-small single-ion
anisotropy can lead to the switching of the sign of the magnetic moment via
the longitudinal evolution of the spin modulus together with quadrupolar
variables, i.e., quantum expectation values of operators bilinear over the
spin components $S_{x}$ and $S_{y}$. It is worth to stress here that the
“restoring force” for this dynamics is the _exchange interaction_ , and the
characteristic time is the exchange time. On the other hand, to realize this
scenario, one needs to a have non-Heisenberg interaction, e.g., single-ion
anisotropy, which couples the spin dipole and quadrupole variables.

Obviously, this effect is beyond the standard picture of spin dynamics based
on any closed set of equations for the spin dipolar variables (i.e., the
quantum expectation values linear on the spin components) alone. Note that our
approach based on the full set of variables for the atomic spin is “more
macroscopic” than the “femtomagnetic” approach, [37] ; [38] ; [39]  dealing
with electronic states. To realize this type of switching, it is necessary to
have a significant coupling between dipolar and quadrupolar spin variables,
which is present in magnets with strong single-ion anisotropy. Such anisotropy
is known for the numerous magnets based on anisotropic ions of transition
elements such as Ni2+, Cr2+, Fe2+. As the classic example, note nickel
fluosilicate hexahydrate NiSiF${}_{6}\cdot$6H2O, with spin-one Ni2+ ions,
coupled by isotropic ferromagnetic exchange interaction and subject to high
single-ion anisotropy. For this compound, the strong effect of quantum spin
reduction is known, with its strength dependent on the pressure: the value of
$K/J$ is growing with the pressure $P$ resulting in the value $\langle
S\rangle=0.6$ at $P=6$ kbar and leading to the transition to the non-magnetic
state with $\langle S\rangle=0$ at $P\sim 10$ kbar. Bar+NiSiF ; Dyakonov+jetp

A number of recent experiments were done with rare-earth transition-metals
compounds. [13] ; [14] ; [35] ; [40]  Note here a rich variety of non-linear
spin dynamics observed for thin films of the FeTb alloy under the action of
femtosecond laser pulses. [40]  However, the theory developed here for simple
one-sublattice ferromagnet cannot be directly applied for the description of
such compounds. Ferromagnetic order with high easy-plane anisotropy is present
for many heavy rare-earth elements, such as Tb and Dy at low temperatures.
REbook  This feature is known both for bulk monocrystals, REbook  and in thin
layers and superlattices, see Ref. Dy_Ysuperlattices  and references wherein.
Strictly speaking, in our article only spin-one ions were considered. Rare-
earth metals have non-zero values of both spin and orbital momentum, forming
the total angular momentum of the ion, and for their description the theory
needs some modifications. However, we believe that the effects of spin
switching caused by quadrupolar spin dynamics will be present as well for such
magnets with high values of atomic angular momentum.

The scenario proposed here includes inertial features, with the evolution of
an initial deviation from one equilibrium state to the other, located far from
the initial one. The initial deviations should include both deviation of the
magnetization and of the quadrupolar variable, $\gamma$. Thus the effect is
based on standard magnetization reduction, but it is helicity dependent as
well. The necessary initial deviations can be created by a light pulse of
circular polarization and the possibility of switching depends on the
connection of the initial spin direction and the pulse helicity. The possible
materials should satisfy a number of general conditions: they should be very
susceptible to magnetization quenching, which is typical for many materials,
as well as a sizeable Faraday effect, and they should also have spin one and a
high enough easy-plane anisotropy.

This work is partly supported by the Presidium of the National Academy of
Sciences of Ukraine via projects no.VTs/157 (EGG) and No.0113U001823 (BAI) and
by the grants from State Foundation of Fundamental Research of Ukraine No.
F33.2/002 (VIB and YuAF) and No. F53.2/045 (BAI). FN acknowledges partial
support from the ARO, RIKEN iTHES Project, MURI Center for Dynamic Magneto-
Optics, JSPS-RFBR Contract No. 12-02-92100, Grant-in-Aid for Scientific
Research (S), MEXT Kakenhi on Quantum Cybernetics, and Funding Program for
Innovative R&D on S&T (FIRST).

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