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# A short proof of a conjecture of Matsushita

Benjamin Bakker  B. Bakker: Dept. of Mathematics, Statistics, and Computer
Science, University of Illinois at Chicago, Chicago, USA.
<EMAIL_ADDRESS>
###### Abstract.

In this note we build on the arguments of van Geemen and Voisin [18] to prove
a conjecture of Matsushita that a Lagrangian fibration of an irreducible
hyperkähler manifold is either isotrivial or of maximal variation. We also
complete a partial result of Voisin [20] regarding the density of torsion
points of sections of Lagrangian fibrations.

Let $X$ be an irreducible compact hyperkähler manifold, that is, a simply-
connected compact Kähler manifold $X$ for which
$H^{0}(X,\Omega^{2}_{X})=\mathbb{C}\sigma$ for a nowhere-degenerate
holomorphic two-form $\sigma$. A Lagrangian fibration of $X$ is a proper
morphism $f:X\to B$ to a normal compact analytic variety $B$ whose generic
fiber is smooth, connected, and Lagrangian (see [11] for a recent survey). It
follows that every smooth fiber is an abelian variety. We let
$B^{\circ}\subset B$ be a dense Zariski open smooth subset over which the
base-change $f^{\circ}:X^{\circ}\to B^{\circ}$ is smooth. By the period map of
$f$ we mean the period map $\varphi:B^{\circ}\to S$ to an appropriate moduli
space $S$ of polarized abelian varieties associated to the natural variation
of (polarized) weight one integral Hodge structures on $B^{\circ}$ with
underlying local system $R^{1}f^{\circ}_{*}\mathbb{Z}_{X^{\circ}}$. We say $f$
is _isotrivial_ if the period map is trivial (equivalently if
$R^{1}f_{*}^{\circ}\mathbb{Z}_{X^{\circ}}$ has finite monodromy) and _of
maximal variation_ if the period map is generically finite.

Our main result is to resolve a conjecture of Matsushita:

######  Theorem 1.

Let $X$ be an irreducible hyperkähler manifold (or more generally a primitive
symplectic variety in the sense of [2]). Then any Lagrangian fibration $f:X\to
B$ is either isotrivial or of maximal variation.

Both possibilities in Theorem 1 occur for K3 surfaces $S$—see for example [10,
Chapter 11]—and therefore also for their Hilbert schemes $S^{[g]}$ in each
(even) dimension. Primitive symplectic varieties are the natural singular
analog (as far as deformation theory is concerned) of irreducible hyperkähler
manifolds; see below for the definition and the precise meaning of a
Lagrangian fibration in this context.

Let $T_{0}\subset H^{2}(X,\mathbb{Q})$ be the rational transcendental lattice,
namely, the smallest rational Hodge substructure containing $[\sigma]\in
H^{2,0}(X)$. Theorem 1 was proven by van Geemen and Voisin [18, Theorem 5]
assuming $X$ is projective, that $T_{0}$ has generic (special) Mumford–Tate
group (namely $\operatorname{SO}(T_{0},q_{X})$, where $q_{X}$ is the
Beauville–Bogomolov–Fujiki form), and that $\operatorname{rk}T_{0}\geq 5$, by
showing that under these conditions any fiber of a Lagrangian fibration that
is not of maximal variation must be a factor of the Kuga–Satake variety of
$T_{0}$. Their result in particular applies to the generic projective
deformation of $f:X\to B$ assuming $b_{2}(X)\geq 7$, which includes all known
deformation types. The Beauville–Mukai system of a nonprimitve basepoint-free
ample divisor on a K3 surface has recently been shown to be of maximal
variation in many cases by Dutta–Huybrechts [7] (see Corollary 7 below) by
understanding the derivative of the period map.

We will instead prove Theorem 1 by considering the complex variation of Hodge
structures on $R^{1}f_{*}^{\circ}\mathbb{C}_{X^{\circ}}$. We first recall the
basic properties of complex variations. A complex variation of Hodge
structures on a Zariski open subset of a compact analytic variety (see for
example [5]) consists of a $\mathbb{C}$-local system $V$ and a holomorphic
(resp. antiholomorphic) descending filtration $F^{\bullet}$ (resp.
$\overline{F}^{\bullet}$) such that we have a splitting of the sheaf of
$C^{\infty}$ sections $A^{0}(V)=\bigoplus_{p}A^{0}(V^{p})$ where
$V^{p}=F^{p}\cap\overline{F}^{-p}$ and the flat connection maps $A^{0}(V^{p})$
to $A^{1,0}(V^{p-1})\oplus A^{1}(V^{p})\oplus A^{0,1}(V^{p+1})$. We refer to
the grading $V^{p}$ as the Hodge grading and we say the level of the variation
is the difference $p_{\mathrm{max}}-p_{\mathrm{min}}$ where $p_{\mathrm{max}}$
(resp. $p_{\mathrm{min}}$) is the maximum (resp. minimum) Hodge degree $p$ for
which $V^{p}\neq 0$. Observe that the level of a tensor product $V\otimes W$
is the sum of the levels of $V$ and $W$. A polarization of the variation is a
flat hermitian form $h$ for which the splitting is orthogonal and $(-1)^{p}h$
is positive definite on $V^{p}$. In this case
$\overline{F}^{-p}=(F^{p+1})^{\perp}$. A variation which admits a polarization
is said to be polarizable. We define $\mathbb{C}(-d)$ to be the polarizable
complex Hodge structure on $V=\mathbb{C}$ with $V^{d}=V$.

Recall that the category of polarizable complex variations of Hodge structures
is semi-simple. The theorem of the fixed part [17] says that for two
polarizable complex variations $V,W$, the group $\operatorname{Hom}(V,W)$ of
morphisms of local systems has a natural complex Hodge structure whose degree
$p$ part is exactly the morphisms of complex variations $V\to W(p)$. We have
the following further consequence due to Deligne:

######  Theorem 2 ([5, 1.13 Proposition]).

Suppose $V$ is a $\mathbb{C}$-local system underlying a polarizable complex
variation of Hodge structures and that we have a splitting of
$\mathbb{C}$-local systems

(1) $V=\bigoplus_{i}N_{i}\otimes A_{i}$

where the $N_{i}$ are irreducible and pairwise non-isomorphic and the $A_{i}$
are nonzero complex vector spaces. Then

  1. (1)

Each $N_{i}$ underlies a polarizable complex variation of Hodge structures,
unique up to shifting the Hodge grading.

  2. (2)

Each polarizable complex variation of Hodge structures with underlying local
system $V$ arises from (1) by equipping each $N_{i}$ with its unique (up to
shifts) polarizable complex variation of Hodge structures and each $A_{i}$
with a uniquely determined polarizable complex Hodge structure, namely
$A_{i}=\operatorname{Hom}(N_{i},V)$.

In particular, the theorem implies a polarizable complex variation is
irreducible if and only if the underlying local system is.

######  Remark 3.

The $\mathbb{C}$-local system $V$ underlying a polarizable complex variation
of Hodge structures on an algebraic space is semi-simple. This is a
consequence of an unpublished result of Nori (see [5, §1.12]) and uses that
the orthogonal complement of flat subbundle with respect to the hodge metric
(which is a harmonic metric) is flat. In our setting, we will only need to
apply this to $V$ underlying a polarizable integral variation of Hodge
structures. In this case, the semi-simplicity of the underlying
$\mathbb{Q}$-local system is a result of Deligne [6, 4.2.6]. Note that for a
perfect field $K$ and a field extension $K\subset L$, a $K$-local system $V$
is semi-simple if and only if $V_{L}$ is, as semi-simplicity is equivalent to
being generated by simple sub-local systems.

Given an $\mathbb{R}$-local system $V$, a polarizable real variation of Hodge
structures111Throughout, by a real variation we mean a pure real variation,
unless otherwise specified. on $V$ in the usual sense naturally induces a
polarizable complex variation on $V_{\mathbb{C}}$. Conversely, a polarizable
complex variation on $V_{\mathbb{C}}$ comes from a polarizable real variation
on $V$ if complex conjugation flips the Hodge grading, or more precisely if
for some (hence any) polarization $h$ the isomorphism of local systems
$V_{\mathbb{C}}\to V_{\mathbb{C}}^{\vee}$ given by $y\mapsto
h(-,\overline{y})$ induces an isomorphism of complex variations
$V_{\mathbb{C}}\to V_{\mathbb{C}}^{\vee}(-w)$ for some (uniquely determined)
$w$. Indeed, if this is the case then
$V^{p}\xrightarrow{\cong}(V^{w-p})^{\vee}$ so $\overline{V^{p}}=V^{w-p}$.
Moreover, for even $w$ (resp. odd $w$) a real polarization is provided by the
symmetric (resp. antisymmetric) real form
$q(x,y)=h(x,\overline{y})+h(y,\overline{x})$ (resp.
$q(x,y)=i(h(x,\overline{y})-h(y,\overline{x}))$), since
$q(x,\overline{x})=h(x,x)+h(\overline{x},\overline{x})$ (resp.
$-iq(x,\overline{x})=h(x,x)-h(\overline{x},\overline{x})$) is definite of
alternating sign on $V^{p}$.

The category of polarizable real variations is also semi-simple. Observe that
by Theorem 2 any isotypic component $W$ of an $\mathbb{R}$-local system $V$
underlying a polarizable real variation is canonically a real subvariation, as
the same is true over $\mathbb{C}$ and the isomorphism $V_{\mathbb{C}}\to
V_{\mathbb{C}}^{\vee}(-w)$ coming from $h$ restricts to an isomorphism
$W_{\mathbb{C}}\to W_{\mathbb{C}}^{\vee}(-w)$. If $V$ is a single isotypic
factor, then $V_{\mathbb{C}}$ either has one self-conjugate irreducible factor
$N$ or has two non-isomorphic conjugate irreducible factors $N,\overline{N}$.
Note that $N^{\vee}\cong\overline{N}$ via the polarization. Note also that the
level of a polarizable real variation $V$ is at least as large as the level of
any of the irreducible factors of $V_{\mathbb{C}}$.

We say that a real or complex variation is _isotrivial_ if the Hodge
filtration is flat, or equivalently if the irreducible factors of the
complexification are level zero222Or equivalently, if the monodromy is unitary
(by Theorem 2); since there may not be an integral structure, this does not
necessarily mean the monodromy is finite.. To summarize the above discussion:

######  Lemma 4.

Let $V$ be an irreducible polarizable real variation of Hodge structures of
level one. Then $V$ is either isotrivial, or every irreducible factor of
$V_{\mathbb{C}}$ is level one.

Before turning to the proof of Theorem 1 we recall the definition of a
primitive symplectic variety. Let $X$ be a symplectic variety in the sense of
Beauville333This definition is equivalent to Beauville’s original one. [3],
that is, a compact Kähler variety with rational singularities and a nowhere
degenerate 2-form $\sigma$ on its regular locus $X^{\mathrm{reg}}$. We say
that $X$ is primitive symplectic if $H^{1}(X,\mathcal{O}_{X})=0$ and
$H^{0}(X^{\mathrm{reg}},\Omega^{2}_{X^{\mathrm{reg}}})=\mathbb{C}\sigma$. As
the singularities are rational, for any resolution $\pi:Y\to X$ the form
$\sigma$ extends to a two-form on $Y$ [12, Corollary 1.7]. Moreover,
$\pi^{*}:H^{2}(X,\mathbb{Q})\to H^{2}(Y,\mathbb{Q})$ is injective, so the
Hodge structure on $H^{2}(X,\mathbb{Q})$ is pure, and we have an induced
isomorphism $\pi^{*}:H^{2,0}(X)\to H^{2,0}(Y)$ (see [2] for details). In
particular we have a well defined class $[\sigma]\in H^{2,0}(X)$.

By a Lagrangian fibration of a primitive symplectic variety we still mean a
proper morphism $f:X\to B$ to a normal compact analytic variety $B$ whose
generic fiber is smooth, connected, and Lagrangian. Each smooth fiber will
still be an abelian variety. Moreover, $B$ is in fact Kähler and Moishezon
[19] (and in particular an algebraic space) since $f$ is equidimensional as in
[11, Lemma 1.17], now using functorial pullback of reflexive forms [12,
Theoreom 1.11] and the fact that
$R\pi_{*}\omega_{Y}\cong\omega_{X}\cong\mathcal{O}_{X}$ by the rationality of
the singularities of $X$ [13, §5.1].

We use the same notation as above: $B^{\circ}\subset B$ is a dense Zariski
open smooth subset over which the restriction $f^{\circ}:X^{\circ}\to
B^{\circ}$ is smooth and $\varphi:B^{\circ}\to S$ is the period map associated
to the variation of (polarized) weight one integral Hodge structures on
$B^{\circ}$ with underlying local system
$R^{1}f^{\circ}_{*}\mathbb{Z}_{X^{\circ}}$.

###### Proof of Theorem 1.

Let $V_{\mathbb{Z}}:=R^{1}f^{\circ}_{*}\mathbb{Z}_{X^{\circ}}$. We start with
the following result of Voisin, whose proof we give for convenience (and to
extend it slightly).

######  Lemma 5 ([20, Lemma 5.5]).

$V_{\mathbb{R}}$ is irreducible as a polarizable real variation of Hodge
structures.

###### Proof.

First assume $X$ is smooth. By a result of Matsushita [14, Lemma 2.2] the
restriction map $H^{2}(X,\mathbb{R})\to H^{2}(X_{b},\mathbb{R})$ to a generic
fiber of $f^{\circ}$ is rank one and by Deligne’s global invariant cycles
theorem $H^{2}(X,\mathbb{R})\to
H^{0}(B^{\circ},R^{2}f^{\circ}_{*}\mathbb{R}_{X^{\circ}})$ is surjective [6].
If $V_{\mathbb{R}}$ splits as a variation then the polarizations of the
factors would yield a larger than one-dimensional space of sections of
$R^{2}f^{\circ}_{*}\mathbb{R}_{X^{\circ}}=\wedge^{2}V_{\mathbb{R}}$, which is
a contradiction.

Now if $X$ is a primitive symplectic variety, one easily checks using the
results of [2] that Matsushita’s proof carries through verbatim and that
$H^{2}(X,\mathbb{R})\to
H^{0}(B^{\circ},R^{2}f^{\circ}_{*}\mathbb{R}_{X^{\circ}})$ is still
surjective, since the cokernel of $\pi^{*}:H^{2}(X,\mathbb{R})\to
H^{2}(Y,\mathbb{R})$ is generated by exceptional divisors for a log resolution
$\pi:Y\to X$ since $X$ has rational singularities. ∎

Suppose now that $f$ is not of maximal variation. Define the real
transcendental lattice $T\subset H^{2}(X,\mathbb{R})$ to be the polarizable
real Hodge substructure spanned by $[\sigma]$ and $[\overline{\sigma}]$. We
next claim that the polarizable real variation of Hodge structures
$V_{\mathbb{R}}\otimes T^{\vee}$ has a nontrivial subvariation of level at
most one after a finite base-change; the argument below is that of [18] at its
core, with some mild modifications.

Let $\nu:B^{\circ\prime}\to B^{\circ}$ be a finite Galois étale cover so that
the base-change $V^{\prime}_{\mathbb{Z}}:=\nu^{*}V_{\mathbb{Z}}$ is pulled
back along its period map $\varphi^{\prime}:B^{\circ\prime}\to S^{\prime}$,
where $S^{\prime}$ is a level cover of $S$. Note that up to replacing
$B^{\circ\prime}$ with a further finite cover, we may assume
$\varphi^{\prime}$ can be embedded in a proper map
$\overline{\varphi}^{\prime}:\overline{B}^{\circ\prime}\to S^{\prime}$ [9].
Denote by $Z\subset S^{\prime}$ the image of $\overline{\varphi}^{\prime}$, by
$\psi:B^{\circ\prime}\to Z$ the resulting map, and by
$V^{\prime\prime}_{\mathbb{Z}}$ the variation on $Z$ so that
$V^{\prime}_{\mathbb{Z}}=\psi^{*}V_{\mathbb{Z}}^{\prime\prime}$. The map
$\overline{\varphi}^{\prime}$ and its image $Z$ are in fact algebraic [4,
Theorem 3.1]. We shrink $Z$ (and $B^{\circ},B^{\circ\prime},X^{\circ}$) so
that it is smooth and so that $R^{1}\psi_{*}\mathbb{R}_{B^{\circ\prime}}$ is a
local system, naturally underlying a graded polarizable real variation of
mixed Hodge structures whose only nonzero Hodge components are
$(0,0),(1,0),(0,1),(1,1)$ (for example using Saito’s theory of mixed Hodge
modules [15, 16]). Let $f^{\circ\prime}:X^{\circ\prime}\to B^{\circ\prime}$ be
the base-change of $f$. The natural map $H^{2}(X,\mathbb{C})\to
H^{0}(B^{\circ\prime},R^{2}f^{\circ\prime}_{*}\mathbb{C}_{X^{\circ\prime}})$
sends $[\sigma]$ and hence $T_{\mathbb{C}}$ to zero, since the fibers of $f$
are Lagrangian. By the Leray spectral sequence we have a natural morphism
$\operatorname{pt}_{Z}^{*}T\to R^{1}\psi_{*}V_{\mathbb{R}}^{\prime}\cong
V_{\mathbb{R}}^{\prime\prime}\otimes
R^{1}\psi_{*}\mathbb{R}_{B^{\circ\prime}}$ in the category of real variations
of mixed Hodge structures. This map is nonzero from the following geometric
description as in [18].

Through a very general point $b\in B^{\circ\prime}$, say above a point $z\in
Z$, let $F$ be the positive-dimensional fiber of $\psi$ through $b$. The
restricted family $X_{F}\to F$ has trivial monodromy so the following natural
diagram commutes

$\textstyle{T_{\mathbb{C}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(V^{\prime\prime}_{\mathbb{C}}\otimes
R^{1}\psi_{*}\mathbb{C}_{B^{\circ\prime}})_{z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{2}(X_{F},\mathbb{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}(X_{b},\mathbb{C})\otimes
H^{1}(F,\mathbb{C})}$

where the bottom arrow comes from the degeneration of the Leray spectral
sequence for $X_{F}\to F$. In the projective case we have $X_{F}\cong
X_{b}\times F$ (possibly after a further base-change) and this map is just the
Künneth projection. The image of $[\sigma]$ is nonzero in the bottom right
corner since: (i) $\sigma$ is nonzero when restricted to $X_{F}$ since $\dim
X_{F}>\frac{1}{2}\dim X$; (ii) $\sigma|_{X_{F}}$ extends to a smooth
compactification since $\sigma$ extends to a smooth compactification of
$X^{\circ}$, so $[\sigma]\neq 0\in H^{2}(X_{F},\mathbb{C})$; (iii) the image
of $[\sigma]$ in $H^{2}(X_{b},\mathbb{C})\otimes H^{0}(F,\mathbb{C})$ vanishes
and $[\sigma]$ is not in the image of $H^{0}(X_{b},\mathbb{C})\otimes
H^{2}(F,\mathbb{C})$, as it is not pulled back from $F$.

Thus, there is a nonzero morphism of real variations

(2)
$\operatorname{gr}_{-1}^{W}\psi^{*}(R^{1}\psi_{*}\mathbb{R}_{B^{\circ\prime}})^{\vee}\to
V_{\mathbb{R}}^{\prime}\otimes T^{\vee}.$

As the category of polarizable real variations of (pure) Hodge structures is
semi-simple, we therefore have a splitting

$V_{\mathbb{R}}^{\prime}\otimes T^{\vee}=U\oplus W$

of real variations, where $U\neq 0$ is the image of (2). In particular, $U$
has level at most one and weight -1.

Now by Lemma 5 the Galois group of $\nu$ acts transitively on the isotypic
factors of $V^{\prime}_{\mathbb{R}}$. In particular, if $f$ (and therefore
$V_{\mathbb{R}}$) is not isotrivial, no factor of $V^{\prime}_{\mathbb{R}}$
(as a variation) is isotrivial, or else its entire isotypic component would
be, and so would $V_{\mathbb{R}}$. But then there can be no nonzero morphism
of variations $V^{\prime}_{\mathbb{R}}\otimes T^{\vee}\to U$. Indeed, by Lemma
4, an irreducible factor $N$ of $V^{\prime}_{\mathbb{C}}$ has level one of
degrees 0,1, and $N\otimes T^{\vee}_{\mathbb{C}}$ can only map nontrivially to
an irreducible factor of $U_{\mathbb{C}}$ of the form $N(1)$, while
$\operatorname{Hom}(N\otimes
T^{\vee}_{\mathbb{C}},N(1))=T_{\mathbb{C}}(1)\cong\mathbb{C}(-1)\oplus\mathbb{C}(1)$
has no degree 0 elements. Thus, $f$ must be isotrivial. ∎

######  Example 6.

We revisit the example from [18, §4]. Let $p\geq 5$ be a prime and $\lambda$ a
$p$th root of unity. Consider a family of abelian varieties $f:X\to B$ with a
cyclic automorphism such that the induced automorphism $\alpha$ of
$V_{\mathbb{R}}=R^{1}f_{*}\mathbb{R}_{X}$ has $\lambda$ as an eigenvalue on
$V^{1,0}$ but not on $V^{0,1}$. Let $\alpha^{\prime}$ be the automorphism of
$T^{\vee}$ with eigenvalue $\lambda^{-1}$ on $(T^{\vee})^{-2,0}$ and
eigenvalue $\lambda$ on $(T^{\vee})^{0,-2}$. Then $V_{\mathbb{R}}\otimes
T^{\vee}$ has a level one factor, namely the $1$ eigenspace
$(V_{\mathbb{R}}\otimes T^{\vee})^{1}$ of $\alpha\otimes\alpha^{\prime}$. But
the condition on the eigenvalues means the eigenspaces
$(V_{\mathbb{C}})^{\lambda}$ and $(V_{\mathbb{C}})^{\lambda^{-1}}$ are level
zero, and the real variation
$(V_{\mathbb{C}})^{\lambda}\oplus(V_{\mathbb{C}})^{\lambda^{-1}}$ is an
isotrivial real factor.

We now discuss a few applications of Theorem 1. Dutta–Huybrechts have proven
many cases [7, Theorem 1.1] of the following corollary, and have also shown
that Theorem 1 implies the general case:

######  Corollary 7.

Let $H$ be a basepoint-free ample divisor on a K3 surface $S$. Then the
complete linear system $|H|$ is of maximal variation.

###### Proof.

The genus 2 case is proven unconditionally in [7, Prop. 5.4], and the genus
$g\geq 3$ case in [7, Prop. 5.2] assuming Theorem 1. ∎

The next application uses a result of Gao, which is a simple application of
the Ax–Schanuel theorem for universal families of abelian varieties [8,
Theorem 1.1]. Recall that for a projective family $f:X\to B$ of
$g$-dimensional abelian varieties444Meaning $X$ is an abelian scheme over $B$,
so there is in particular a 0-section of $f$. equipped with a section $s$ and
letting $\tilde{B}\to B^{\mathrm{an}}$ be the universal cover, the Betti map
$\beta:\tilde{B}\to H_{1}(X_{b},\mathbb{R})$ is the real analytic map obtained
by taking the coordinates of the section $s$ with respect to the flat real-
analytic trivialization of $f$. Observe that
$\beta^{-1}(H_{1}(X_{b},\mathbb{Q}))$ is the set of points of $\tilde{B}$ at
which $s$ is torsion. If $\varphi:B\to S$ is the period map of $f$ and
$\mathcal{X}\to S$ the universal family of abelian varieties, then $s$
naturally yields a map $B\to\mathcal{X}$ lifting $\varphi$. We say that $s$ is
of maximal variation if $B\to\mathcal{X}$ is generically finite.

######  Proposition 8 ([8, Theorem 9.1]).

Let $f:X\to B$ be a projective family of $g$-dimensional abelian varieties
with $\dim B\geq g$ and whose very general fiber has no nontrivial
$\mathbb{Q}$-factor. Let $s:B\to X$ be a non-torsion section of $f$ which is
of maximal variation. Then the Betti map $\beta:\tilde{B}\to
H_{1}(X_{b},\mathbb{R})$ associated to $s$ is generically submersive.

######  Corollary 9.

Let $X$ be a primitive symplectic variety and $f:X\to B$ a Lagrangian
fibration. Let $L$ be a line bundle whose restriction to the smooth fibers is
topologically trivial. Then the set of points $b\in B^{\circ}(\mathbb{C})$ for
which $L|_{X_{b}}$ is torsion is analytically dense in $B$.

Corollary 9 was proven by Voisin [20, Theorem 1.3] assuming either $f$ is of
maximal variation and $\dim X\leq 8$ or isotrivial with no restriction on the
dimension. Our use of Proposition 8 replaces the results of
André–Corvaja–Zannier [1] in [20].

###### Proof of Corollary 9.

By Voisin’s result and Theorem 1 we may assume $f$ is of maximal variation.
Note that Voisin’s proof works equally well in the singular case; we leave the
details to the reader. Consider the family of abelian varieties
$h:\operatorname{Pic}^{0}(X^{\circ}/B^{\circ})\to B^{\circ}$ and the section
$s:b\mapsto L|_{X_{b}}$. Let $\nu:B^{\circ\prime}\to B^{\circ}$ be a Galois
finite base-change for which the $\mathbb{Q}$-factors of the very general
fiber of $h$ are defined over $B^{\circ\prime}$. As the Galois group of $\nu$
acts transitively on the factors by Lemma 5, the $d$ factors all have the same
dimension $g^{\prime}$, and the image of the period map of each factor must
have dimension $\geq g^{\prime}$, or else the image of the period map of $f$
would have dimension smaller than
$dg^{\prime}=\dim(X^{\circ}/B^{\circ})=\dim(B^{\circ})$. The base-change of
the section $s$ is also Galois invariant, so it suffices to prove the density
statement for its projection to a single factor $Y^{\circ}\to
B^{\circ\prime}$. Applying Proposition 8, the Betti map $\beta:\tilde{B}\to
H_{1}(Y_{b},\mathbb{R})$ is submersive, so
$\beta^{-1}(H_{1}(Y_{b},\mathbb{Q}))$ is analytically dense in $\tilde{B}$ as
claimed. ∎

Corollary 9 has an interesting interpretation in terms of the Beauville
conjecture for irreducible hyperkähler manifolds $X$, see the discussion in
[20, §1.2]. There it is also shown how corollaries 7 and 9 can be used to
construct constant cycle curves on K3 surfaces.

## Acknowledgements

The author would like to thank C. Voisin for conversations related to Theorem
1 and Corollary 9, D. Maulik for bringing the paper [20] to his attention, and
C. Lehn for conversations related to Lagrangian fibrations in the singular
case. This work was partially supported by NSF grant DMS-2131688.

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# Dynamical Heterogeneity in Lattice Glass Models

Richard K. Darst and David R. Reichman  Department of Chemistry, Columbia
University, 3000 Broadway, New York, NY 10027, USA.  Giulio Biroli  Institut
de Physique Théorique, CEA, IPhT, F-91191 Gif-sur-Yvette, France and CNRS, URA
2306

###### Abstract

In this paper we consider in detail the properties of dynamical heterogeneity
in lattice glass models (LGMs). LGMs are lattice models whose dynamical rules
are based on thermodynamic, as opposed to purely kinetic, considerations. We
devise a LGM that is not prone to crystallization and displays properties of a
fragile glass-forming liquid. Particle motion in this model tends to be
locally anisotropic on intermediate time scales even though the rules
governing the model are isotropic. The model demonstrates violations of the
Stokes-Einstein relation and the growth of various length scales associated
with dynamical heterogeneity. We discuss future avenues of research comparing
the predictions of lattice glass models and kinetically constrained models to
atomistic systems.

###### pacs:

blah

##  I Introduction

The cause of the dramatic slowing of dynamics close to the empirically defined
glass transition is a subject of great continued interest and debate
Debenedetti and Stillinger (2001); Ediger et al. (1996). Different theoretical
proposals have been put forward aimed at describing some or all of the
phenomena commonly observed in experiments and computer simulations Garrahan
and Chandler (2002, 2003); Hedges et al. (2009); Lubchenko and Wolynes (2007);
Bouchaud and Biroli (2004); Kivelson et al. (2004); Gotze and Sjogren (1992);
Schweizer and Saltzman (2003); Dyre (2006); Langer (2006); Heuer (2008). While
these proposals are often based on completely divergent viewpoints, many of
them are able to rationalize the same observed behaviors. This fact stems from
the somewhat limited amount of information available from experiments and
simulations. Since the growth of relaxation times in glassy systems is
precipitous, it is very difficult, and in some cases impossible, to
distinguish models solely on the basis of different predictions of gross
temperature dependent relaxation behavior. In addition, computer simulations,
which are often more detailed than experiments, are limited by the range of
times scales and sizes of systems that can be studied. These difficulties have
hindered the search for a consensus on the microscopic underpinnings of
vitrification.

Despite the continued debate that revolves around the theoretical description
of supercooled liquids and glasses, little argument exists regarding the
importance of dynamical heterogeneity as a key feature of glassy behavior
Butler and Harrowell (1991); Hurley and Harrowell (1995); Kob et al. (1997);
Yamamoto and Onuki (1998); Schmidt-Rohr and Spiess (1991); Ediger (2000).
Dynamical heterogeneity refers to the fact that as a liquid is supercooled,
dynamics become starkly spatially heterogeneous, requiring the cooperative
motion of groups of particles for relaxation to occur. Dynamical heterogeneous
motion manifests in several ways, and leads to violations of the Stokes-
Einstein relation Chang and Sillescu (1997); Cicerone and Ediger (1996); Jung
et al. (2004); Xia et al. (2001); Tarjus and Kivelson (1995), cooperative
hopping motion reflected in nearly exponential tails in particle displacement
functions Berthier et al. (2005a); Stariolo and Fabricius (2006); Chaudhuri et
al. (2007); Saltzman and Schweizer (2008); Garrahan and Chandler (2002), and
growing length scales such as those associated with the recovery of Fickian
diffusion Swallen et al. (2009); Berthier (2004); Pan et al. (2005); Szamel
and Flenner (2006), growing multi-point correlation functions Lačević et al.
(2003); Dasgupta et al. (1991); Franz and Parisi (2000); Berthier et al.
(2007); Bouchaud and Biroli (2004); Franz and Montanari (2007); Biroli et al.
(2008); Berthier et al. (2005b); Biroli et al. (2006); Karmakar et al. (2009).
Indeed, the relatively recent explication of the phenomena of dynamical
heterogeneity has dramatically shifted the focus of the field and has placed
new constraints on the necessary ingredients for a successful theory of glass
formation.

Given the similarity of some aspects of dynamical heterogeneity to critical
fluctuations in standard critical phenomena, it is natural to investigate two
and three dimensional simplified coarse-grained models that encode the crucial
features of this heterogeneity. Currently, the most investigated class of
coarse-grained models are the “kinetically constrained models” (KCMs) Garrahan
and Chandler (2002, 2003); Fredrickson and Andersen (1984); Kob and Andersen
(1993); Ritort and Sollich (2003). KCMs are spin or lattice models that
generate slow, glassy relaxation via constraints on the dynamical moves that
are allowed. The slowing down of the dynamics is caused by rarefactions of
facilitating regions, also called defects. Importantly, although the dynamics
is complex the thermodynamics is trivial since the dynamical rules are such
that all configurations are equally likely. The philosophy of this viewpoint
is that thermodynamic quantities, such as the configurational entropy, are not
the fundamental underlying cause of the growing time scales in supercooled
liquids. It has been argued that the quantitative disagreement Biroli et al.
(2005) between thermodynamic features of KCMs and real experiments is of
little dynamical consequence Chandler and Garrahan (2005). In support of this
perspective is the fact that KCMs have been remarkably successful in
generating features of dynamical heterogeneity such as Stokes-Einstein
decoupling, growing dynamical length scales, and excess tails in the real-
space particle displacement function Jung et al. (2004); Pan et al. (2005);
Hedges and Garrahan (2008, 2007).

On the other hand, one may wonder if a deeper viewpoint would allow for an
understanding of the kinetic rules that govern particle motion in the
supercooled liquids. It is natural to speculate that such aspects might have
roots in the thermodynamics of configurations. Indeed, simple local Monte
Carlo “dynamics” can reproduce all features of dynamical heterogeneity seen in
Newtonian molecular dynamics simulations, and are based simply on making local
moves that are _configurationally_ allowed Berthier (2007). Lattice models
based on this concept are called “lattice glass models” (LGMs), and were first
considered by Biroli and Mézard Biroli and Mézard (2001). The rules for such
models seem at first sight like that of KCMs. For example in the simplest
versions of such models a particle may move if it is surrounded by no more
than a fixed number of nearest neighbors before and after the move McCullagh
et al. (2005); Dawson et al. (2003); Ciamarra et al. (2003); Pica Ciamarra et
al. (2003). Locally this is identical to the type of dynamical constraint that
appears in the KCMs introduced by Kob and Andersen Kob and Andersen (1993).
However, this constraint must be met globally: _all_ particles must have no
more than a fixed number of nearest neighbors. As the density of the system
increases, fewer and fewer configurations exist for which these constraints
may be satisfied. It is thus the entropy of configurations that governs the
slowing of dynamics, intimately connecting the non-trivial thermodynamic
weight of states accessible to the local dynamics. Indeed, LGMs can be solved
exactly within the Bethe approximation, or on Bethe lattices Biroli and Mézard
(2001); Ciamarra et al. (2003), and have been shown to have a glass transition
due to the vanishing of the configurational entropy. The distinction between
the KCM and LGM viewpoint is illustrated in Fig. 1.

Figure 1: Comparison and distinction of a caricature of a kinetically
constrained model with a lattice glass model. In the KCM any configuration is
allowed, but move may only be made if a particle has at least one missing
neighbor before and after the move. In the LGM, the global configuration is
defined such that all particles must have at least one missing neighbor, and
all dynamical moves must respect this rule. Note that the local environment
around the moving particle is identical in this example, while the global
configurations are distinct. Periodic boundary conditions are assumed for both
panels.

LGMs have been studied by a number of groups, but the focus has not generally
been on real-space aspects of dynamical heterogeneity. For example, Coniglio
and coworkers have developed a simple LGM that avoids crystallization and
displays many features of typical glass-forming materials, including a growing
multi-point susceptibility ($\chi_{4}(t)$) Ciamarra et al. (2003); Pica
Ciamarra et al. (2003). On the other hand, this system appears to behave as a
strong glass-former, with a stretching parameter close to one, and exhibits
essentially no Stokes-Einstein violation. Our goal in this work is to survey
in detail the dynamical behavior of a new LGM which bears similarity to the
original Biroli-Mézard model but is not prone to crystallization. The main
conclusion that we draw is that LGMs are at least as realistic as KCMs in
their description of all commonly studied features of dynamical heterogeneity.
In this regard, simple coarse-grained lattice models based on the
thermodynamic weight of states are no less viable as fundamental caricatures
of glassy liquids than are KCMs based on weights of trajectories. We conclude
our work by highlighting several key ways that LGMs and KCMs may be
distinguished. We reserve the investigation of these comparisons for a future
study. Our paper is organized as follows: Sec. II outlines the model. Sec. III
discusses both simple averaged dynamics as well as aspects of dynamical
heterogeneity. In Sec. IV we conclude with a discussion of the meaning of our
findings and the future directions to be pursued.

##  II Model

Here we define the LGM that forms the basis of our simulations. The original
model of Biroli-Mézard is quite prone to crystallization Biroli and Mézard
(2001). This fact makes its use problematic for the study of glassy behavior
since crystallization always intervenes before supercooling becomes
significant. The crystallization problem persists on a square lattice for all
binary mixtures we have studied. However, we have found that certain
generalizations of the Biroli-Mézard model with three species of particles are
stable against crystallization for the densities that are sufficiently high
that glassy dynamics may be clearly observed.

Our model follows the original rules of the Biroli-Mézard model. Particles
exist on a cubic periodic lattice of side $L=15$ and each lattice site can
contain only zero or one particle. All particles, at all times, must satisfy
the condition a particle of type “$m$” must have $m$ or fewer neighbors of any
type. A neighbor is considered any particle in one of the $2d$
($d$=dimensionality) closest lattice sites along the cubic coordinate axes
111It should be noted that LGMs of the type described here involve extreme
constraints that must be globally satisfied and are thus not realistic
translations of off-lattice particle-based models. Such constraints might
indeed induce artificial behavior, especially at higher densities. It would be
most interesting to investigate “soft” versions of such models where
constraints may be locally violated at the cost of an energy penalty. In this
regard, such models would be the configurational analog of KCMs where
dynamical constraints may be broken at the cost of an energy penalty, see
Chandler, D. and J.P. Garrahan, “Dynamics on the Way to Forming Glass: Bubbles
in Space-time” arXiv:0908.0418v1; Submitted to Annual Reviews of Physical
Chemistry (2010)..

The particular three species model we employ is defined by 10% type 1
particles, 50% type 2 particles, and 40% type 3 particles. We denote this
model the “t154” model to indicate its basis in thermodynamics and to specify
the types and percentages of each particle. The composition of t154 model was
determined via trial and error by picking particle types with clashing
crystallization motifs thereby frustrating crystallization. Crystallization
was monitored by inspection of the angle resolved static structure factor,
direct inspection of configurations, and by monitoring bulk thermodynamic
quantities.

As discussed in the introduction, there appear to be strong similarities
between the rules that govern KCMs such as the Kob-Andersen model and the t154
model Kob and Andersen (1993). For example both models employ constraints with
a maximum number of neighbors, but in the Kob-Andersen model this restriction
only applies to the mobile particles, while in the t154 model applies to all
particles. Our model does not require any special dynamics methods. We employ
local canonical Monte Carlo “dynamics” via primitive translational moves
Berthier (2007). Note that for the t154 model the energy can only be zero (no
packing violations) or infinite (packing violation or overlap), thus the
acceptance criteria reduces to rejection if there is a packing violation and
acceptance otherwise. This allows us to implement an event-driven algorithm
which accelerates the simulation of lattice dynamics Bortz et al. (1975).

For thermodynamic studies we employ grand-canonical Monte Carlo with both
translational moves as well as particle insertion/deletion. Fig. 2 contains a
plot of the density of the system as a function of the chemical potential of
type 1 particles. Models which crystallize (such as original binary model of
Biroli-Mézard) have a sharp jump in this curve at the crystallization point.
Clearly, this feature is absent in the t154 model. For comparison, both curves
are displayed 222A subtle issue arises in the nature of glassy behavior
observed in the t154 model outlined in this work. LGMs could have a dynamical
percolation-like transition, as in the spiral model Biroli and Toninelli
(2008). This has been indeed found in some LGMs on the Bethe lattice Rivoire
et al. (2003) and would slow down the dynamics for reasons completely
different from the diminishing of the configurational entropy. If there is a
low-lying crystal phase then one can show that this dynamical percolation-like
transition cannot take place in finite dimension. Although we have not found a
crystal phase for the model, the existence of such a transition seems unlikely
and irrelevant for our present work. First, it can be shown that blocked
structures, if they exist, have to verify much more constraints than in the
spiral model Biroli and Toninelli (2008). Second, we have found that the
relaxation time growth of the persistence functions with increasing density in
local canonical Monte Carlo simulations are similar to those under grand-
canonical dynamics, which cannot contain any blocked structure. The union of
these two facts render the dynamical blocking scenario highly unlikely..

Figure 2: Crystallization thermodynamics in LGM. Top: The t154 model.
$\mu_{1}$ refers to the chemical potential of the type 1 particles. The
maximum density observed for the 153 lattice is .5479 (exactly 1849 out of
3375 lattice sites occupied). The three plotted quenching rates vary between a
.01 and .05 increase of $\mu_{1}$ per 10000 cycles. Bottom: A close up of the
equivalent plot for the BM model. Note the clear discontinuity upon
crystallization. Slower $\mu$-increase rates produce a sharper discontinuity.

##  III Dynamical Behavior

###  III.1 Simple Bulk Dynamics

In this subsection we describe the behavior of a simple 2-point observable,
namely the self-intermediate scattering function Balucani and Zoppi (1994),
defined as

$F_{s}(k,t)=\left<\frac{1}{N}\sum_{i}e^{i\mathbf{k}\cdot\left[\mathbf{r}_{i}(t)-\mathbf{r}_{i}(0)\right]}\right>.$
(1)

We measure $F_{s}(k,t)$ only for the type-2 particles which are present in the
greatest fraction for the three distinct species. Throughout this paper, we
report $k$-vectors using $k^{\prime}$, where $k=\frac{2\pi}{L}k^{\prime}$. We
have checked that $F_{s}(k,t)$ is qualitatively similar for the other species
of particles. The relaxation of $F_{s}(k,t)$ of the system at the wavevector
$k^{\prime}=5$ ($k=\frac{2\pi}{3}$) for various densities is shown in Fig. 3.
The bulk of the decay may be fit to a stretched exponential function,
$F_{s}(k,t)=\exp(-(t/\tau_{\alpha}(k))^{\beta(k)})$. As is customary, the
alpha-relaxation time is found by the value $F_{s}(\tau_{\alpha})=1/e$ and the
$\beta(k)$ exponent is determined by a direct fit to the terminal decay. We
find that for densities below approximately $\rho=0.48$ the value of $\beta$
saturates at the expected value $\beta=1$ characteristic of simple non-glassy
dynamics, while for the highest density simulated, $\beta=0.7$. This behavior,
over a similar range of supercooling, is reminiscent of the behavior found in
atomistic models of glass-forming liquids Kob and Andersen (1995); Wahnström
(1991). In order to better reveal the relaxation behavior, $F_{s}(t)$ is also
displayed on a log-log vs. log-time scale. In this plot, the slope of the long
time growth is related to the exponent $\beta.$ We have found that the values
of $\beta$ extracted from the slopes of the long time portion of the log-log
vs. log plot indeed coincide with that found by a direct fit to a stretched
exponential form. At the highest densities a shoulder appears in the short
time relaxation. This feature is indicative of a secondary relaxation feature
perhaps akin to beta-relaxation in realistic glass-forming liquids. It should
be noted, however, that the amplitude of this feature is very close to unity.
This is quantitatively distinct from the plateau values expected in atomistic
off-lattice models Kob and Andersen (1995); Wahnström (1991) and even LGMs
with more complicated lattice degrees of freedom Ciamarra et al. (2003); Pica
Ciamarra et al. (2003), but is similar to that encountered in simple spin
models such as variants of the Random Orthogonal Model Sarlat et al. (2009).

As is typical of fragile glass-forming systems, the t154 model exhibits
relaxation times that do not follow the (generalized) Arrhenius form Berthier
and Witten (2009). This behavior is illustrated in Fig. 4. At low densities,
plots of $\log(\tau)$ versus $\rho$ indeed follow a straight line, however in
the vicinity of $\rho\sim 0.5$ the plot of $\tau$ versus $\rho$ deviates from
this straight line and the functional density dependence of the relaxation
time becomes much more precipitous. While we have not attempted to
quantitatively characterize this density dependence, it should be noted that
the onset of increased sensitivity to changes in density occurs is the same
narrow window that marks the noticeable decrease in the values of the
stretching exponent $\beta$.

Figure 3: Decay of the self-intermediate scattering function $F_{s}(k,t)$ for
$k^{\prime}=5$ ($k=\frac{2\pi}{L}k^{\prime}$). Densities are .3, .4, .45, .48,
.50, .51, .52, .53, .535, .5375, .5400, .5425 from fastest relaxation to
slowest relaxation. These densities are used in all plots in this paper unless
otherwise indicated. Top: Plotted on a linear-log scale. Bottom: Same data as
upper panel plotted on a $\log(-\log_{10}(F_{s}(k,t)))$ vs $\log(t)$ scale.
Lowest density curves are at the top left.

Figure 4: Top: $\tau_{\alpha}$ (time at which $F_{s}(k,t)=1/e$) as a function
of density, $\rho$. Plotted for $k^{\prime}=1,2,3,4,5,6,7$, with lowest $k$ at
the top. Center: Beta stretching exponent of $F_{s}(k,t)$ (from terminal fits
$F_{s}(k,t)\sim\exp(-(t/\tau_{\alpha})^{\beta})$). Lowest $k$ curve is at the
top of the plot. Bottom: Plot of log scale $\tau_{\alpha}$ against chemical
potential $\mu$ of type 2 particles. The behavior is consistent with
$\tau_{\alpha}=5.7\exp(-21\mu_{2}/(\mu_{2}-24))$.

###  III.2 Motion on the Atomic Scale

We begin our discussion of the nature of heterogeneous dynamical behavior in
the t154 LGM by observing the qualitative details of particle motion under
supercooled conditions. This will set the stage for analysis of quantitative
measures of dynamical heterogeneity in the model. For the sake of comparison,
we also investigate the analogous behavior in the Kob-Andersen model. This
comparison is useful because it suggests how models with similar local rules
but different global rules (rooted in either the purely kinetic or
thermodynamic basis of the particular model) may give rise to distinct
dynamics at the particle scale.

We start by simply observing the patterns of mobility in real space starting
from a set initial condition of the t154 model found at a given density after
equilibration. A similar analysis has been performed recently by Chaudhuri et
al. for the Kob-Andersen model, where no equilibration is required since all
initial configurations with a set density of defects are allowed Chaudhuri et
al. (2008). For a theoretical description of the dynamics of the Kob-Andersen
model, see Toninelli et al. (2004). We note that, as expected, the t154 model
exhibits regions of spatially localized particle activity against a backdrop
of transiently immobilized particles. A rather remarkable feature of the
patterns of mobility in this model is that we find evidence of string-like
motion, where a group of particles moves over a short distance, each taking
the place of the previous particle in the string Donati et al. (1998, 1999).
This motif can be seen mostly on timescales less than the $\alpha$-relaxation
time, but occasionally string-like motion may be seen to persist on longer
timescales. This behavior is demonstrated in Fig. 5.

The behavior of particle motion observed in the Kob-Andersen model is somewhat
different than that seen in the t154 model as described above. As in the t154
model, and as observed by Chaudhuri et al., motion in the Kob-Andersen model
shows similar activity regions in the vicinity of defect sites giving rise to
heterogeneous motion. However, the boundaries between active and inactive
regions at comparable timescales appear to be more distinct in the Kob-
Andersen model. Furthermore, the particle scale motion in the Kob-Andersen is
much more isotropic, exhibiting much fewer cases of directional mobility
compared with the t154 model. It would be interesting to compare the two
models by quantifying this difference via the type of directional multi-point
correlators devised by Doliwa and Heuer Doliwa and Heuer (1999). It is not
clear if the difference between the models is related to the fundamental
distinction between LGMs and KCMs or just the specifics of the particular
models considered. In particular, the t154 is a multi-component model, unlike
the Kob-Andersen model. The string-like motion on short time scales seems to
occur predominantly on the rather rough boundaries of slow clustersAppignanesi
et al. (2006). This behavior, reminiscent of the picture of dynamic
heterogeneity that put forward by Stillinger Stillinger (1988), might be
strongly influenced by compositional heterogeneity. A useful way to address
general issues related to how the initial configuration constrains subsequent
dynamics would be a systematic iso-configurational ensemble analysis comparing
LGMs and KCMs Widmer-Cooper et al. (2004). This will be the topic of a future
publication Darst et al. .

Figure 5: Examples of string-like motion apparent in the t154 model. (a) An
example of a string with all neighboring particles removed. (b) A similar
string in the context of other particles. Note that here the string is truly
isolated in space, away from other mobile particles. In these figures, type 1
particles are white, type 2 particles are blue, and type 3 are green. Sites
occupied at the initial time but vacated at the final time are shown in red.
These pictures show only the differences in position of particles between the
origin of time and the final time, not the path the particles took to achieve
that displacement. All figures are at a density of .5400, with $\Delta t$
times in (a) 251, (b) 199526. The $\alpha$-relaxation time for $k^{\prime}=5$
at this density is about $7.8\times 10^{6}$

Figure 6: Examples cluster shapes in the (a) the t154, model, density
$\rho=.5400$ and (b) the Kob-Andersen model, density $\rho=.8500$. Arrows
indicate motion between initial and final times. Time separation is 1/10th of
the $\alpha$-relaxation time. In the t154 model, we see more fractal and
disconnected clusters, while in the KA model, mobile domains tend to be
smoother clusters.

In the next few sections, we discuss how some of the most important indicators
of dynamical heterogeneity in supercooled liquids manifest in the t154 model.
The quantities that we discuss are the magnitude of violations of the Stokes-
Einstein relation, exponential tails (indicative of hopping transport) in the
van Hove function, the existence of a Fickian length scale and the development
of a dynamical length scale quantified by the multi-point function
$S_{4}(q,t)$. Unless otherwise stated, specific correlation functions and
transport coefficients are calculated with respect to type-2 particles.

###  III.3 Stokes-Einstein Violation

In typical fluids a mean-field linear-response relationship asserts that the
product of the tracer particle diffusion constant and the fluid viscosity
divided by the temperature is a constant Balucani and Zoppi (1994). This
connection between diffusion and dissipation is known as the Stokes-Einstein
relationship, and empirically is known to hold even at the atomic scale in
liquids over a wide range of densities and temperatures. In supercooled
liquids, the Stokes-Einstein relation generally does not hold Chang and
Sillescu (1997); Cicerone and Ediger (1996); Jung et al. (2004); Xia et al.
(2001); Tarjus and Kivelson (1995); Berthier et al. (2005a); Stillinger and
Hodgdon (1994); Liu and Oppenheim (1996). In fact, the product of the
diffusion constant and the viscosity of a liquid may exceed that expected from
the Stokes-Einstein relation by several orders of magnitude close to the glass
transition. There are many theoretical explanations for Stokes-Einstein
violations in supercooled liquids, which essentially all invoke dynamical
heterogeneity as the fundamental factor leading to the breakdown of the simple
relationship between diffusion and viscosity. It should be noted that similar
relationships hold between the diffusion constant and the self and collective
time constants associated with the decay of density fluctuations. In this work
we focus on the relaxation time of the self-intermediate scattering function
defined above as our proxy for the fluid viscosity.

It is well known that the product $D\tau_{\alpha}$, where $\tau_{\alpha}$ is
the $\alpha$-relaxation time of the self-intermediate scattering function
shows a strong temperature/density dependence in both realistic atomic
simulations as well as in the class of KCMs that describe fragile glass-
forming liquids. No direct studies of this quantity have been made in LGMs.
The LGM of Coniglio and coworkers would appear to show essentially no Stokes-
Einstein violations because the diffusion constant and the relaxation time may
both be fit to power laws with exponents that have, within numerical accuracy,
the same magnitude Ciamarra et al. (2003); Pica Ciamarra et al. (2003). This,
however is not surprising since many of the features of the model resemble
those of a strong glass-forming system, where violations of the Stokes-
Einstein relation are, at most, weak. The features of the t154 model with
regard to non-exponential relaxation and the density dependence of the
relaxation time $\tau_{\alpha}$ indicate that this model behaves more like a
fragile glass former. Thus, we expect clear violations of the Stokes-Einstein
relation. Indeed, as shown in Fig. 7, $D\tau_{\alpha}$ increases markedly as
density is increased. Over the range densities that we can access, the
magnitude of the violation is very similar to that seen in the canonical Kob-
Andersen Lennard Jones mixture over a comparable range of changes in
relaxation time Chaudhuri et al. (2007). Interestingly, violations begin to
become pronounced at densities similar to where the relaxation times and
stretching exponents become strongly sensitive to increased density. Thus, a
consistent onset density is observed as in more realistic atomistic systems.

Figure 7: Violation of the Stokes-Einstein relation,
$D\tau_{\alpha}\sim\mathbf{constant}$, using $\tau_{\alpha}$ at
$k^{\prime}$=5. Data has been normalized to $D{\tau_{\alpha}}=1$ at the lowest
density.

###  III.4 van Hove Function

It is now rather well established that an additional “quasi-universal” feature
of dynamical heterogeneity near the glass transition is contained in the shape
of the real-space van Hove function Stariolo and Fabricius (2006); Chaudhuri
et al. (2007); Saltzman and Schweizer (2008); Swallen et al. (2009). In
particular it has been argued the tails of the self van Hove function should
be approximately exponential in form. These “fat tails” imply that the rare
particles that do undergo large displacements exist in populations in excess
of what would be expected in a purely Gaussian displacement distribution.
While non-Gaussian tails should be expected of any distribution for the wings
that fall outside of limits of bounds set by the Central Limit Theorem, the
palpable exponential tails in supercooled liquids imply large non-Gaussian
effects indicative of transport that is strongly effected by heterogeneous
hopping motion.

Here, we demonstrate that such effects occur in the t154 model in a manner
similar to that seen both in experiments in colloidal and granular systems as
well as in computer simulations of atomic systems. Fig. 8 shows the self part
of the real-space van Hove function,

$G_{s}(x,t)=\left<\delta\left(x-\left|\mathbf{\hat{x}}\cdot\left(\mathbf{r}_{i}(t)-\mathbf{r}_{i}(0)\right)\right|\right)\right>,$
(2)

for the type two particles in the t154 model. Because we are on a lattice, we
restrict our distances along the three coordinate axes $\mathbf{\hat{x}}$
individually in our calculation. We see that for times of the order of the
$\alpha$-relaxation time, these tails are clearly visible. For very long or
short time scales, the shape of the tail deviates somewhat from the more
exponential form exhibited at intermediate times. This behavior is quite
similar to that seen in simulations of atomistic systems Chaudhuri et al.
(2007); Saltzman and Schweizer (2008), and is fully consistent with the
behavior found in KCMs Berthier et al. (2005a).

Figure 8: van Hove function for $\rho=.5375$ and various times. Distances are
measured independently along each coordinate axis. The times plotted, from
left to right, are $10^{5}$, 316227 (approx. the $\alpha$-relaxation time),
and $10^{6}$. An exponential fit to the tail of the $t=316227$ case is shown
by a dotted line.

###  III.5 Fickian Length

Related to the existence of excess tails in the van Hove function is the
existence of a length scale that characterizes the anomalous transport. More
specifically, the exponential tails in the van Hove function are distinguished
from the Gaussian form of the displacement distribution obtained at relatively
short distances for fixed times. The crossover from Fickian to non-Fickian
behavior should be characterized by time scales as well as length scales over
which this crossover occurs. A non-Fickian length scale may be defined by
examining the $k$-dependent diffusion constant
$D(k)=\frac{1}{\tau_{\alpha}k^{2}}$ Berthier (2004); Pan et al. (2005); Szamel
and Flenner (2006). The wavevector that characterizes the crossover from the
expected diffusive behavior to an anomalous regime is inversely related to
such a length scale. In Fig. 9 we plot $D(k^{\prime})$. Clearly, as the
density is increased, the length scale separating the Fickian and non-Fickian
regimes increases. This behavior is consistent with that found in KCMs and
simulations of atomistic glass-forming liquids. It should be noted that
Stokes-Einstein violations, the development of exponential tails in the self
van Hove function, and a well-developed Fickian length scale are all
manifestations of related aspects of dynamically heterogeneous motion in
supercooled liquids Chaudhuri et al. (2007).

Figure 9: $k$-dependent diffusion constant $=1/\left(k^{\prime
2}\tau_{\alpha}(k)\right)$. Densities of .3000 (upper) and .5425 (lower). The
higher density curve is multiplied by a scale factor of $2.992\times 10^{5}$
for ease of comparison. A dotted flat line is included for reference of
behavior expected in the purely Fickian case.

###  III.6 $\chi_{4}$ and $S_{4}$ Fluctuation Measures

The Fickian length scale is merely one length scale that arises naturally in
systems where dynamics become increasingly heterogeneous. Perhaps more
fundamental is the growth of dynamical length scales associated with multi-
point correlations of the dynamics. Supercooled liquids do not show simple
static correlations that would indicate a growing correlation length. It
should be noted that this does not exclude growing static correlations of a
more complex kind, for example point-to-set correlations Bouchaud and Biroli
(2004); Mézard and Montanari (2006); Biroli et al. (2008). Regardless,
cooperativity in dynamics may be measured via first defining a local overlap
function Garrahan and Chandler (2002); Yamamoto and Onuki (1998); Lačević et
al. (2003); Franz and Parisi (2000); Berthier et al. (2007)

$\delta
f_{k}(q,t)=\frac{1}{N}\sum_{i}e^{i\mathbf{q}\cdot\mathbf{r}_{i}(0)}\left[\cos\left(\mathbf{k}\cdot\left(\Delta\mathbf{r}_{i}(t)\right)\right)-F_{s}(k,t)\right]$
(3)

where $\Delta\mathbf{r}_{i}(t)=\mathbf{r}_{i}(0)-\mathbf{r}_{i}(t)$.
$f_{k}(q,t)$ is defined for one configuration, and the average is over all
$\mathbf{k}$ and $\mathbf{q}$ with the magnitudes $k$ and $q$. Then,
$S_{4}(q)$ is defined as

$S_{4}(q)=N\left<\left|\delta f_{k}(q,t)\right|^{2}\right>$ (4)

where this average is over the most general ensemble of configurations
Berthier et al. (2007). The $\chi_{4}$ value is defined as the limit
$S_{4}(q\to 0)$. $\chi_{4}(t)$ may be calculated strictly at $q=0$ from

$\chi_{4}(t)=N\left<\left|\delta f_{k}(q=0,t)\right|^{2}\right>$ (5)

where the average is over the entire ensemble and all $\mathbf{k}$ consistent
with the magnitude of k. Note that, as discussed in Berthier et al. (2007),
the value of $\chi_{4}(t)$ computed in this manner is a lower bound for the
extrapolation of $S_{4}(q\to 0,t)$.

The quantity $S_{4}(q,t)$ is a multi-point dynamical analog of $S(q)$. Just as
the low $q$ behavior of $S(q)$ indicates a growing (static) length scale in
systems approaching a second order phase transition, scattering from
dynamically heterogeneous regions undergoing cooperative motion will manifest
growth in the amplitude of the low $q$ region of $S_{4}^{ol}(q,t)$, indicative
of the size scale of the dynamical correlations for systems approaching the
glass transition.

The behavior of the quantity $S_{4}^{ol}(q,t)$ is shown in Fig. 10. Only
type-2 particles have been used in the calculation. As can clearly be seen,
for densities above $\rho\sim 0.5$ which constitutes the onset density of this
system, the low $q$ behavior shows a marked upturn as $q\rightarrow 0$. The
growth of $S_{4}^{ol}(q,t)$ as $q\rightarrow 0$ as density is increased
suggests a growing length scale as supercooling progresses. This non-trivial
behavior is what is found in atomistic simulated systems. Future work will be
devoted to a precise characterization of the length scale that may be
extracted from $S_{4}^{ol}(q,t)$ in the t154 model so that a comparison may be
made with recent work detailing the behavior of this length in realistic off-
lattice systems Stein and Andersen (2008); Karmakar et al. (2009).

Figure 10:  Top: Top:Plot of $S_{4}(q,t)$ at $\tau_{\alpha}$ for densities
0.51, 0.52, 0.53 and 0.54. Bottom: Plot of $\chi_{4}(t)$ for the same
densities. Peak values correspond to lower bounds for of the value of
$S_{4}(q,t)$ in the upper panel at $q=0$.

##  IV Conclusion

In this paper we have presented a new LGM based on the original Biroli-Mézard
model Biroli and Mézard (2001). Via the introduction of an additional species
of particle, we have demonstrated that our model is stable against
crystallization. This fact allows us to study sufficiently high density
configurations that manifest features of dynamical heterogeneity. Unlike some
previous LGMs, our model exhibits the canonical features of a fragile glass-
former. In terms of the gross features of relaxation behavior, our LGM shows
behavior similar to the standard Kob-Andersen Lennard-Jones (KALJ) mixture. In
particular, we find that the degree of violation of the Stokes-Einstein
relation and the magnitude of stretching in the decay of the self-intermediate
scattering function track the relaxation times at densities above the onset of
supercooling in a manner consistent with that seen in the KALJ system.
Features of dynamical heterogeneity such as exponential tails in the van Hove
function, the growth of a dynamical length scale as quantified by the function
$S_{4}(q,t)$, Stokes-Einstein violations and the emergence of a Fickian length
scale all occur in a manner expected from experiments and simulations of
fragile glass-forming liquids.

The similarity between the description of dynamic heterogeneity found in KCMs
and LGMs stands in stark contrast to the underlying foundations of the models
themselves. As emphasized in the introduction, KCMs are based on a constrained
dynamics for which the number of available dynamical paths leading to
relaxation becomes increasingly rare as the density increases and the number
of defects decrease. In KCMs all real-space configurations at a fixed number
of defects (excluding rare blocked configurations) are equally likely. On the
other hand LGMs are based on transitions between real-space configurations
that become increasingly scarce as the density is increased. This is not to
say that there is not a facilitated-like dynamics in LGMs. On the contrary, as
we have demonstrated in sec. III, local and sometimes anisotropic dynamics may
be generated naturally in LGMs without the explicit introduction of
facilitating defects. An important message that emerges from this study is
that the phenomenology of dynamic heterogeneity is not sufficient to
distinguish pictures or validate models based on transitions between sets of
states in configuration space from those based on sets of paths in space-time.

How then might these pictures be differentiated? While contrasting competing
models that generate seemingly similar dynamical behavior is a difficult
endeavor, several possible studies might be useful for this task. Here we
outline four avenues that could provide key information that distinguish the
purely dynamical picture from one based on transitions thermodynamic states.

a) _The mosaic length scale:_ The Random First Order Theory (RFOT) of Wolynes
and coworkers posits the existence of a static length scale which is defined
by the region over which particles are pinned by the surrounding self-
generated amorphous configuration Lubchenko and Wolynes (2007); Bouchaud and
Biroli (2004); Biroli et al. (2008). This length scale also exists in KCMs,
but it is decoupled from the relaxation dynamics of the system Jack and
Garrahan (2005). Recent atomistic computer simulations have successfully
located the mosaic length scale Biroli et al. (2008). It would be quite useful
to perform an analysis similar to that devised by Jack and Garrahan for LGMs
Jack and Garrahan (2005). Since LGMs are based on the entropy of real-space
configurations, it is expected that here the mosaic length does couple to the
glassy dynamics. Since LGMs are much simpler than atomistic off-lattice
models, the direct study of the mosaic length (and point-to-set correlations
in general) in LGMs might provide key avenues for the testing of the putative
coupling between relaxation and such length scales in simulated atomistic
systems.

b) _Correlations between configurational entropy and dynamics:_ Empirical
correlations between the configurational entropy and the $\alpha$-relaxation
time have been noted for many years, and this correlation lies at the heart of
several prominent theories. Such correlations are still widely debated, but
seem to hold at least crudely in many glass-forming systems Richert and Angell
(1998); Martinez and Angell (2001). LGMs should be expected to exhibit such
correlations, while it is known that KCMs do not exhibit such correlations.
Recently Karmakar et al. purported to show that finite-size effects of the
$\alpha$ relaxation time follow precisely the Adam-Gibbs relation between the
configurational entropy and the $\alpha$-relaxation time in the KALJ system
Karmakar et al. (2009). If true, such correlations would be a challenge to
KCMs, since it is difficult to envision how the configurational entropy would
track the $\alpha$-relaxation time for different system sizes if it were not a
crucial component of relaxation phenomena. Such correlations, however, are
subtle to measure since the Adam-Gibbs relationship is an exponential one and
the apparent correlation could depend on the somewhat indirect computational
method used in Karmakar et al. (2009) to define the configurational entropy.
It would be most useful to investigate such effects in the simpler LGMs, which
might provide a cleaner means of isolating the configurational entropy. It
should be noted that finite size effects do appear to follow an approximate
Adam-Gibbs relationship in at least one other lattice model Crisanti and
Ritort (2000). Such studies might spur more detailed investigations in
simulated atomistic systems thus allowing for a clear comparison between LGMs,
KCMs and more realistic systems.

c) _Single-particle and collective predictability ratios:_ In an important
piece of work, Jack and Berthier devised metrics that access the degree to
which single particle and collective dynamics are deterministically predicted
by a set initial configuration over a given time scale Berthier and Jack
(2007). KCMs and LGMs differ in how allowed configurations are constructed.
KCMs have explicit defects, while configurations in LGMs are determined by
global constraints, and thus do not contain explicit defects. Since the very
composition of initial conditions differ markedly in these models, one expects
that the metrics defined by Jack and Berthier would behave differently in KCMs
and LGMs. Thus, it would be very profitable to examine the density and
temperature dependence of the single particle and collective predictability
ratios in KCMs and LGMs as a possible means of distinguishing between state-
based, and dynamical constraint-based pictures Darst et al. .

d) _Evolution of the facilitation mechanism approaching the glass transition:_
Although in both KCM and LGM pictures facilitation plays an important role in
the relaxation of the system, a peculiar and different temperature and density
evolution is expected. In particular, in the KCM picture, facilitation is due
to the motion of mobility regions or defects. Dynamics slows down, and
concomitantly dynamic heterogeneity increases, because these regions become
rarer approaching the glass transition. A crucial assumption is that these
defects are conserved or at least that non-conservation is a rare event that
becomes rarer at lower temperature/high density. These assumptions impose
important constraints on the evolution of the facilitation mechanism. Thus, it
would be very interesting to examine this issue for example using the cluster
analysis developed in Candelier et al. (2009) to study the relaxation dynamics
of granular systems.

Investigation of these and other studies aimed at distinguishing the
underlying pictures that LGMs and KCMs are based on will be the subject of
future work.

###### Acknowledgements.

RKD would like to thank the John and Fannie Hertz Foundation for research
support via a Hertz Foundation Graduate Fellowship. RKD and DRR would like to
thank the NSF for financial support. We would like to thank Ludovic Berthier,
Joel Eaves, Peter Harrowell, Robert Jack, Peter Mayer and Marco Tarzia for
useful discussions.

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