# The salient crossings of a crown diagram

Jonathan D. Williams  Department of Mathematical Sciences, Binghamton
University<EMAIL_ADDRESS>
###### Abstract.

A crown diagram of a smooth, closed oriented 4-manifold can be thought of as
the projection of a link in the product of a closed surface and the circle,
with chords in the circle direction connecting the strands of each crossing.
This paper uses a straightforward assignment of integers to these chords to
show that the smooth structures on the topological 4-manifold underlying a
pair of Fintushel-Stern knot surgery 4-manifolds which are known to have the
same Seiberg-Witten invariant are not isotopic. A natural question is to
determine if the argument may be strengthened to show these manifolds are not
diffeomorphic.

##  1\. Introduction

Let $M$ be a smooth, closed, oriented, connected 4-manifold, and let $\Sigma$
be a closed oriented surface of genus $g>2$. The construction of this paper
begins with a certain type of smooth map $\alpha\colon\thinspace M\to S^{2}$
of which $\Sigma$ is a regular fiber, a _crown map_ , discussed in Section
2.1. The surface $\Sigma$ is decorated with a cyclically ordered collection of
simple closed curves $\Gamma=\\{\gamma_{n}\\}_{n\in\mathbb{Z}/k\mathbb{Z}}$
coming from $f$, which has a natural interpretation as a $k$-component link
$L$ in $S^{1}\times\Sigma$ using the natural embedding of additive groups
$\mathbb{Z}/k\mathbb{Z}\to S^{1}$, $n\mapsto\theta_{n}\in S^{1}$:

$L=\left\\{\\{\theta_{n}\\}\times\gamma_{n}\ |\
n\in\mathbb{Z}/k\mathbb{Z}\right\\}.$

The condition that each component of $L$ inhabits its own slice
$\\{\theta_{n}\\}\times\Sigma$ is a rather useful one; such links are called
_cyclically stacked_ links in this paper. Analogously, a _linearly stacked_
link lies in $[0,1]\times\Sigma$, with each component inhabiting its own
subset $\\{t\\}\times\Sigma$. There are a few interesting sources of stacked
links, such as Heegaard diagrams of closed 3-manifolds, yielding links which
are linearly stacked with two levels of $g$-component links in
$[0,1]\times\Sigma$ (or $2g$ levels, after a slight perturbation), and
trisection diagrams of smooth closed 4-manifolds, yielding cyclically stacked
links with three levels in $S^{1}\times\Sigma$. These links, however, do not
present an obvious distinguished collection of crossings as crown diagrams do.
For crown diagrams, this collection leads to a basic smooth 4-manifold
invariant, the collection of so-called _salient sequences_ defined in Section
2.2 and established as an invariant in Section 3. Section 4 uses this to show
that distinguish the smooth structures of the knot surgery manifolds
$E(1)_{7_{6}}$ and $E(1)_{10_{133}}$ cannot be isotopic in the sense of
smoothing theory, where two smooth manifolds $A$ and $B$ are isotopic
smoothings of the topological 4-manifold $X$ if there is a smooth manifold $Y$
and a homeomorphism to $X\times[0,1]\to Y$ such that the projection to $[0,1]$
is regular, and diffeomorphisms sending the two ends of $Y$ to $A$ and $B$.
Supposing $E(1)_{K}$ and $E(1)_{K^{\prime}}$ are diffeomorphic, the author
knows of no other way to distinguish the isotopy classes of their smooth
structures.

### Acknowledgements

The author would like to thank José Romàn Aranda Cuevas, Yasha Eliashberg, Bob
Gompf, Rob Kirby, Robert Lewis, Robert Lipshitz, Cary Malkiewich, Jeff Nye,
Tim Perutz, and Lisa Piccirillo for helpful conversations during the course of
this work. The author would especially like to thank Umur Çiftçi for his
crucial contribution of the software EQMaker.

##  2\. Definitions

###  2.1. Topological setting

A _Morse 2-function_ is a smooth map $\alpha\colon\thinspace M\to S$, where
$S$ is a smooth 2-manifold, such that each critical point of $f$ is locally
modeled on a _fold_ critical point,

$(x_{1},x_{2},x_{3},x_{4})\mapsto(x_{1},x_{2}^{2}+x_{3}^{2}\pm x_{4}^{2}),$
(1)

or a _cusp_ critical point,

$(x_{1},x_{2},x_{3},x_{4})\mapsto(x_{1},x_{2}^{3}-3x_{1}x_{2}+x_{3}^{2}\pm
x_{4}^{2}).$ (2)

The critical locus $\operatorname{crit}f$ forms a smooth 1-submanifold of $M$,
stratified into a 1-dimensional stratum of fold points and a finite stratum of
cusp points. Crown diagrams come from what this paper calls _crown maps_ ,
after a suggestion of R. Kirby (In previous work such as [B, BH, W1, W2], a
crown diagram was called a _surface diagram_). A crown map is a Morse
2-function $f$ for which $S=S^{2}$, the signs above are both chosen to be
negative (that is, the critical points are _indefinite_ folds and cusps), and
$\operatorname{crit}f$ is a single circle on which $f$ is injective (Figure
1(c)). A crown map induces a singular fiber bundle called a _crown fibration_
on $M$ as follows. The critical values of $f$ partition $S^{2}$ into a pair of
disks, the _higher-genus side_ whose regular fibers are diffeomorphic to a
closed oriented genus-$g$ surface $\Sigma$, and the _lower-genus side_ , which
has closed oriented genus $g-1$ fibers. The two disks meet along the critical
image of $f$, which is a circle with corners, where each corner is the image
of a cusp point.

(a) The image of a fold arc.

(b) The image of a cusp point connecting two fold arcs.

2pt $v_{3}$ at 67 80 $v_{2}$ at 82 53 $v_{1}$ at 68 25 $v_{k}$ at 34 25
$v_{4}$ at 32 80 $v_{0}$ at 42 53

(c) The critical image of a crown map, decorated with a graph $T$.

Figure 1. The critical image of an indefinite Morse 2-function. Arrows and
circles in depicted reference fibers indicate the attaching circles
corresponding to indicated fold arcs.

The local model for an indefinite fold point is
$(h,\operatorname{id}_{\mathbb{R}})\colon\thinspace\mathbb{R}^{3}\times\mathbb{R}\to\mathbb{R}\times\mathbb{R}$,
where $h$ is the local model for a 3-dimensional Morse critical point of index
1 or 2 (in Figure 1(a), it is index 1 with $h$ understood to increase from
left to right along any horizontal arc). Such a Morse function induces a
fibration of $R^{3}$ by the one- and two-sheeted hyperboloids
$x^{2}+y^{2}-z^{2}=const$, and each one-sheeted hyperboloid
$\\{x^{2}+y^{2}-z^{2}=const>0\\}$ intersects the $xy$ plane in a distinguished
circle. In the image of a Morse 2-function, the region of regular values
corresponding to $const>0$ is called the _higher genus side_ of the fold arc
and the distinguished circle is a simple closed curve in the regular fiber
called a _vanishing cycle_. All of the vanishing cycles are depicted as simple
closed curves in a fixed reference fiber $f^{-1}(p)$, measured by choosing a
reference path from $p$ to the image of a fold arc, and a connection for the
fibration, then marking the points in the regular fiber that flow to the fold
point. In general, the isotopy class of the vanishing cycle depends on these
choices; however, the crown condition implies that the vanishing cycles in
this paper are all non-separating and lie in a smooth, closed oriented surface
$\Sigma$ of genus at least 3, and different choices of higher-genus reference
fiber, connection and reference path yield isotopic vanishing cycles [BH,
Section 4].

With this understood, a _crown diagram_ of the smooth, closed oriented
4-manifold $M$ is the oriented diffeomorphism class of a pair
$(\Sigma,\Gamma)$, where $\Sigma$ is a closed, oriented surface of genus
greater than 2 and $\Gamma$ is a collection of simple closed curves in
$\Sigma$ which occurs as the counter-clockwise ordered sequence of vanishing
cycles of a crown fibration whose oriented higher-genus fiber is $\Sigma$. A
diffeomorphism of crown diagrams is required to preserve the cyclic ordering
of $\Gamma$. According to [W1, Corollary 2], a crown diagram specifies $M$ up
to orientation-preserving diffeomorphism.

The local model for an indefinite cusp is the same as the local model for the
introduction via homotopy of a canceling pair of index-1 and index-2 Morse
critical points in a 3-manifold (the homotopy parameter would increase from
left to right in Figure 1(b)). For this reason, the vanishing cycles of the
two fold arcs which meet at a cusp point can be depicted to have exactly one
transverse crossing in $\Sigma$.

######  Definition 2.1.

After canceling all crossings but one from each pair of consecutive vanishing
cycles in a crown diagram, each unique transverse crossing between consecutive
vanishing cycles is called a _salient crossing_.

At various points it will be useful to designate a vanishing cycle as the
“first” element of $\Gamma$, indexing with integers as
$(\gamma_{1},\ldots,\gamma_{k})$, where the ordering of any three consecutive
elements respects the cyclic order of $\Gamma$. Such a choice will be called a
_labeling_ of $\Gamma$.

######  Definition 2.2.

A _homotopy class_ of $(\Sigma,\Gamma)$ is the smooth homotopy class of any
crown map whose diagram is $(\Sigma,\Gamma)$ for some choice of reference
fiber, connection and reference paths.

It may be possible (and would be interesting to find) that two non-homotopic
crown maps $M\to S^{2}$ have the same crown diagram. In that case, both
homotopy classes of maps would be homotopy classes of the diagram.

######  Definition 2.3.

Two crown diagrams $(\Sigma_{i},\Gamma_{i})_{i=1,2}$ are _isotopic_ if there
is an isotopy of individual vanishing cycles in $\Gamma_{1}$ resulting in a
crown diagram which is diffeomorphic to $(\Sigma_{2},\Gamma_{2})$.

######  Definition 2.4.

A crown diagram $(\Sigma,\Gamma)$ is _allowable_ if $\Sigma\setminus\Gamma$ is
a disjoint union of open disks whose closures are closed disks, and
$|\gamma_{i}\cap\gamma_{i+1}|=1$ for every $i$.

######  Proposition 2.5.

Every crown diagram with four or more vanishing cycles is isotopic to an
allowable crown diagram, and two isotopic allowable crown diagrams can be made
diffeomorphic using an isotopy consisting of moves taken from Figure 3, such
that each move preserves allowability.

###### Proof.

One may apply what are generally called _finger moves_ or point-pushing maps
to individual vanishing cycles to subdivide regions until $(\Sigma,\Gamma)$ is
allowable, introducing crossings between non-consecutive vanishing cycles. To
convert into a different yet isotopic allowable diagram, one may add temporary
finger moves at various stages of the isotopy as needed to preserve
allowability at every stage.∎

###  2.2. The chords of a crown diagram

The cyclic ordering of $\Gamma$ gives data reminiscent of crossings in knot
diagrams as follows. For $i,j\in\mathbb{Z}/k\mathbb{Z}$, an intersection
$x\in\gamma_{i}\cap\gamma_{j}$ is associated with the two arcs
$[\theta_{i},\theta_{j}]\times\\{x\\}$ and
$[\theta_{j},\theta_{i}]\times\\{x\\}$ whose union is $S^{1}\times\\{x\\}$.
These are both oriented as subspaces of $S^{1}\times\Sigma$, connecting the
circles $\\{\theta_{i}\\}\times\gamma_{i}$ and
$\\{\theta_{j}\\}\times\gamma_{j}$. These arcs are called the _chords_ of $x$.

Any chord $[\theta_{i},\theta_{j}]\times\\{x\\}$ singles out the sublink of
$L$ contained in $[\theta_{i},\theta_{j}]\times\Sigma$ and the complimentary
chord $[\theta_{j},\theta_{i}]\times\\{x\\}$ singles out the sublink of $L$
contained in $[\theta_{j},\theta_{i}]\times\Sigma$, because the interiors of
the two intervals are disjoint positively oriented arcs in $S^{1}$. The
crossing $x$ appears in both sublinks, and if $\gamma_{i}$ passes over
$\gamma_{j}$ in one, then $\gamma_{i}$ passes under $\gamma_{j}$ in the other.
In either case, given a chord $a$, one may choose a labeling of $\Gamma$ so
that the under-crossing vanishing cycle at $x$ is $\gamma_{1}$. Then the over-
crossing vanishing cycle would be $\gamma_{n}$ for some $n\in\\{2,\ldots,k\\}$
and that corresponding sublink is

$L_{a}=\bigcup\limits_{i=1}^{n}\\{\theta_{i}\\}\times\gamma_{i}\subset[\theta_{1},\theta_{n}]\times\Sigma.$

For each labeling, the two strands of each crossing in the crown diagram
necessarily come from an intersection of two distinct circles, so this paper
adopts the visual convention that, having chosen a crossing and a labeling,
the under-crossing strand is the one with the lesser subscript. Using this
convention, the corners of the disks adjacent to each crossing in a sublink
$L_{a}$ can be given signs as in Figure 2.

2pt $-$ at 47 47 $-$ at 28 28 $+$ at 47 28 $+$ at 28 47

Figure 2. The signs of the quadrants near a crossing of vanishing cycles in a
crown diagram are defined as in this figure only after choosing a labeling for
$\Gamma$.

Since $\gamma_{i}$ intersects $\gamma_{i+1}$ at a unique point for each
$i\in\mathbb{Z}/k\mathbb{Z}$, there are $k$ distinguished chords whose
associated sublinks each have two components $\gamma_{i},\gamma_{i+1}$ and one
crossing, which is a salient crossing. These chords are the _salient chords_
of $\mathcal{A}(\Sigma,\Gamma)$.

Suppose $(\Sigma,\Gamma)$ is allowable, choose a labeling of $\Gamma$ and let
$U$ be the collection of disks forming the disjoint union
$\Sigma\setminus\cup_{i}\gamma_{i}=\sqcup D^{2}$. For any crossing there is a
unique chord which does not intersect $[\theta_{k},\theta_{1}]\times\Sigma$;
in other words, it is the chord which is contained in the link diagram whose
over-crossing strands are designated by the labeling. Thus, a labeling gives
signs to all the corners in the diagram. Given a labeling and a disk $u\in U$,
denote by $Q_{+}(u)$ the set of chords corresponding to positive corners of
$u$ and denote by $Q_{-}(u)$ the set of chords corresponding to negative
corners of $u$. Note that allowability implies $Q_{+}(u)$ and $Q_{-}(u)$ are
disjoint. Assign a variable $\operatorname{gr}(a)$ to each chord $a$ and
impose the following _grading equations_ for every labeling of $\Gamma$ and
for every region $u$:

$\sum\limits_{a\in Q_{+}(u)}\operatorname{gr}(a)-\sum\limits_{a\in
Q_{-}(u)}\operatorname{gr}(a)=|Q_{+}(u)|-|Q_{-}(u)|.$ (3)

The system of equations given by Equation 3 will be referred to as the
_grading system_. It is defined as a collection of equations associated to
disks, but Equation 3 also makes sense for any region $u\subseteq\Sigma$ whose
boundary is a union of arcs of vanishing cycles that meet at crossings. The
right hand side of Equation 3 assigns an integer to each $u\in U$, and it is
not hard to check that this assignment is additive in the sense that the
grading equation for a union of disks is the sum of the grading equations for
the disks.

It is now possible to define the main invariant of the paper.

######  Definition 2.6.

A _salient sequence_ of $(\Sigma,\Gamma)$ is a choice of values of $gr(x_{i})$
for the salient chords $x_{1},\ldots,x_{k}$ of the diagram, obtained from a
solution of the collection of all the grading equations for the disks in the
complement of the vanishing cycles. Let $\mathcal{S}(\Sigma,\Gamma)$ denote
the set of salient sequences for the diagram $(\Sigma,\Gamma)$.

The grading system is defined as a system of equations over the integers, but
it may only have solutions in a finite cyclic group, and this solution is not
necessarily unique.

##  3\. Invariance

This section contains the proof that $\mathcal{S}(\Sigma,\Gamma)$ is an
invariant of the collection of crown diagrams of $M$ with any fixed genus and
homotopy class.

######  Definition 3.1.

For two closed curves $a,b$ in $\Sigma$, a _slide_ of $a$ over $b$ is the
result of replacing $a$ with the connect sum $a\\#b$ using some choice of
connecting arc, which is a smooth embedded curve from one point of $a$ to one
point of $b$ whose interior is disjoint from $a$ and $b$. In this situation,
$a$ is the _nonstationary_ circle and $b$ is the _stationary_ circle.

######  Theorem 3.2.

If two allowable crown diagrams $(\Sigma_{i},\Gamma_{i})_{i=1,2}$ of $M$ have
the same genus and a common homotopy class, then
$\mathcal{S}(\Sigma_{1},\Gamma_{1})=\mathcal{S}(\Sigma_{2},\Gamma_{2})$.

######  Lemma 3.3.

If two crown diagrams $(\Sigma_{i},\Gamma_{i})_{i=1,2}$ of $M$ are isotopic
and allowable, then
$\mathcal{S}(\Sigma_{1},\Gamma_{1})=\mathcal{S}(\Sigma_{2},\Gamma_{2})$.

###### Proof.

For any labeling, any isotopy sending $(\Sigma_{1},\Gamma_{1})$ to
$(\Sigma_{2},\Gamma_{2})$ can be slightly perturbed to occur as a sequence of
Reidemeister moves, chosen from Figure 3.

2pt II  $\downarrow$ $\uparrow$ II-1 at 63 70 $\bullet\ p_{1}$ at 63 140
$\bullet\ p_{2}$ at 63 115 $\bullet\ p_{3}$ at 63 90 $\bullet\ p_{4}$ at 63 45
$\bullet\ p_{5}$ at 22 26 $\bullet\ p_{6}$ at 63 25 $\bullet\ p_{7}$ at 102 26
$\bullet\ p_{8}$ at 63 5 $a$ at 43 26 $b$ at 77 26

(a) Reidemeister-II.

2pt  $\downarrow$ at 60 70 $d$ at 58 135 $c$ at 80 108 $e$ at 39 108 $d$ at 63
8 $e$ at 80 35 $c$ at 42 35 $\bullet\ p_{t}$ at 62 107 $\bullet\ p_{9}$ at 85
120 $\bullet\ p_{10}$ at 95 92 $\bullet\ p_{11}$ at 95 48 $\bullet\ p_{12}$ at
85 22 $\bullet\ p_{t}$ at 62 32

(b) Reidemeister-IIIa.

2pt  $\downarrow$ at 60 70

(c) Reidemeister-IIIb.

Figure 3. Isotopies of $L$ corresponding to isotopies of a crown diagram.

Invariance of $\mathcal{S}$ comes from examining the grading equations for
those disks forming $\Sigma\setminus\bigcup_{i}\gamma_{i}$ that are affected
by such an isotopy, with an arbitrary fixed labeling. Except for the corners
$a$ and $b$ in Figure 3(a), the crossings have unchanging labels which follow
along with the isotopy, and outside of the pictured regions the disks, their
corners, and their grading equations are unchanged. In the following
arguments, let $E_{i}$ denote the grading equation for the disk containing the
point $p_{i}$ in Figure 3, for any labeling that yields the pictured crossing
data. Since relabeling yields the same pictures, possibly rotated, the
following arguments are sufficient.

For the Reidemeister-II move in Figure 3(a), $E_{1}=E_{4}+E_{6}$,
$E_{2}=E_{5}-E_{6}+E_{7}$, and $E_{3}=E_{6}+E_{8}$, while all other equations
for $(\Sigma_{1},\Gamma_{1})$ are also equations for
$(\Sigma_{2},\Gamma_{2})$. For this reason,
$\mathcal{S}(\Sigma_{2},\Gamma_{2})\subseteq\mathcal{S}(\Sigma_{1},\Gamma_{1})$.
On the other hand, the chords $c_{a}$ and $c_{b}$ corresponding to the
crossings $a$ and $b$ in the grading system for $(\Sigma_{2},\Gamma_{2})$ can
be removed from all equations except for that of $u_{6}$ by rearranging the
sums above for $u_{4}$ and $u_{8}$, and by substitution for $u_{5}$ and
$u_{7}$. For $u_{5}$, for example, the $-gr(c_{a})$ appearing in the equation
for $u_{5}$ can be replaced by $gr(b)$ using the equation for $u_{6}$, and
then $gr(b)$ can be replaced by the result of solving for $gr(b)$ in the
equation for $u_{7}$, yielding the equation for $u_{2}$. The argument for
$u_{7}$ runs similarly. For this reason, appending the two grading equations
in four variables coming from $u_{6}$ to the first grading system results in a
grading system which is equivalent to the second, and since the variables for
those two extra equations do not appear elsewhere in the first system,
$\mathcal{S}(\Sigma_{1},\Gamma_{1})\subseteq\mathcal{S}(\Sigma_{2},\Gamma_{2})$.

For a Reidemeister-II-1 move between consecutive vanishing cycles, it may be
the case that one of the crossings that disappear is a salient crossing. This
cannot happen if both diagrams are allowable, but it is possible that a
sequence of slides results in a diagram which is not allowable, but becomes an
allowable diagram after canceling some crossings between consecutive vanishing
cycles. In that case, the relevant figure is a pair of bigons which share a
corner, coming from three crossings between two vanishing cycles. In this case
the two grading equations for any chosen labeling set the three associated
$gr$ variables to be equal.

For the Reidemeister-III moves, referring to Figure 3(b), the central
triangles before and after a Reidemeister-IIIa move have the same equation
$E_{t}$. The regions opposite a corner of the triangle have equations
analogous to $E_{9}=E_{11}-E_{t}$, and regions opposite a side of the triangle
have equations analogous to $E_{10}=E_{12}+E_{t}$. There are also analogous
equations for the disks in Figure 3(c). Since the reverse of a Reidemeister-
IIIa move is another Reidemeister-IIIa move, and the same is true of a
Reidemeister-IIIb move, the systems of grading equations before and after a
Reidemeister-IIIa or Reidemeister-IIIb move are equivalent. Since the grading
equations are only defined after choosing a labeling, the cases in which the
central triangle has all positive or all negative corners do not occur.∎

According to [W3, Theorem 4.9], if two crown diagrams of $M$ have the same
genus and homotopy class, then they are related by the so-called _genus-
preserving moves_ , which are handleslide, multislide, and shift. According to
[W3, Theorem 4.3], these moves are all realized as sequences of slides of a
certain form determined by which move is applied. The following lemmas show
that these sequences of slides leave the salient sequence $\mathcal{S}$
unchanged.

######  Lemma 3.4.

If two crown diagrams $(\Sigma_{i},\Gamma_{i})_{i=1,2}$ of $M$ are related by
a multislide or a handleslide move, then
$\mathcal{S}(\Sigma_{1},\Gamma_{1})=\mathcal{S}(\Sigma_{2},\Gamma_{2})$.

###### Proof.

This follows from two facts. First, for any chord $x$, the set of values taken
by $\operatorname{gr}(x)$ in the collection of solutions to the grading system
is unchanged by any slide. Second, the handleslide and multislide moves can be
performed in such a way that the collection of salient crossings is left
unchanged.

2pt $\bullet\ p_{1}$ at 25 62 $\bullet\ p_{2}$ at 70 62 $\bullet\ p_{3}$ at
270 62 $\bullet\ p_{4}$ at 320 78 $\bullet\ p_{5}$ at 320 46 $\bullet\ p_{6}$
at 375 95 $\bullet\ p_{7}$ at 375 28  $\rightarrow$ at 205 62

Figure 4. A slide. The vertical arc on the left is part of a vanishing cycle
that slides over the vertical vanishing cycle on the right. The dotted parts
of the segments may contain crossings. The two ovals in each figure are the
ends of an annulus in $\Sigma$.

As in the proof of Lemma 3.3, $E_{i}$ is understood to be the grading equation
for the disk in the complement of the vanishing cycles containing $p_{i}$, for
some fixed labeling. To prove the first fact, in Figure 4 there are two arcs
crossing the vertical vanishing cycle on the right. These could come from an
arbitrary finger move, or, since this is coming from a crown diagram, they are
the crossings coming from the vanishing cycles which are listed in $\Gamma$
before and after that vanishing cycle. By additivity of the grading equations,
one may as well assume the regions containing $p_{1},\ldots,p_{7}$ are as
shown, possibly with corners hidden in the dotted parts of the segments, and
no other vanishing cycles passing though their interiors. With this
understood, the regions containing $p_{6}$ and $p_{7}$ can be interpreted as
two pieces of a single rectangle whose grading equation will be called
$E_{6}$. Then $E_{1}=E_{3}+E_{6}$, $E_{2}=E_{4}+E_{5}-E_{6}$, and the grading
equations for all other regions before the slide are either unchanged or can
be obtained by adding the grading equation for an appropriate rectangle in the
annulus containing $p_{6}$ and $p_{7}$. Since the inverse of any slide is
another slide up to isotopy, the first fact is proved.

For the second fact, [W3] presents the sequence of slides for both multislide
and handleslide as follows. Given a suitable pair of vanishing cycles for the
move, choose a labeling so that they are $\gamma_{1}$ and $\gamma_{n}$, and
such that the set of vanishing cycles affected by the move (that is, the
nonstationary set) is $\\{\gamma_{2},\ldots,\gamma_{n-1}\\}$. Then the move is
achieved by applying a diffeomorphism of $\Sigma$ to the nonstationary set
whose effect is a sequence of slides applied to the nonstationary set. This
fact alone is enough to show that all of the salient crossings of the crown
diagram except possibly $x_{1}$ and $x_{n-1}$ are preserved under handleslide
and multislide. As for $x_{1}$ and $x_{n-1}$, the effect of the move near
these crossings is depicted in Figure 8b of that paper, where the short
vertical arc, contained in $\gamma_{2}$ or in $\gamma_{n-1}$, crosses the
lower circle ($\gamma_{1}$ or $\gamma_{n}$ respectively) at the salient
crossing $x_{1}$ or $x_{n-1}$, and that crossing is unchanged by the move. ∎

######  Lemma 3.5.

If two crown diagrams $(\Sigma_{i},\Gamma_{i})_{i=1,2}$ of $M$ are related by
a shift move, then
$\mathcal{S}(\Sigma_{1},\Gamma_{1})=\mathcal{S}(\Sigma_{2},\Gamma_{2})$.

###### Proof.

According to [W3, Theorem 3.1], a shift move can be realized by a sequence of
slides. As in the proof of Lemma 3.4, any slide which does not change the
collection of salient crossings will preserve $\mathcal{S}$; however, the
argument in the proof of Lemma 3.4 that the collection of salient crossings is
unchanged does not apply to the shift move, so there is one further argument
to make.

Applying the sequence of slides that produces a shift move can introduce new
crossings between a consecutive pair of vanishing cycles $\gamma_{i}$ and
$\gamma_{i+1}$. One of those new crossings $x^{\prime}$ may be the only one
left after canceling all the other crossings between $\gamma_{i}$ and
$\gamma_{i+1}$ by Reidemeister-II-1 moves in the course of an isotopy to
obtain an allowable crown diagram. In this situation the original salient
crossing $x$ can be cancelled last, and since $x^{\prime}$ was not cancelled,
it was one of the three corners in the last remaining pair of bigons, and in
that situation it has already been shown that
$\operatorname{gr}(x)=\operatorname{gr}(x^{\prime})$. In this way, changing
the salient crossing from $x$ to $x^{\prime}$ does not change $\mathcal{S}$.∎

###### Proof of Theorem 3.2.

According to [W2, Theorem 1.1], any pair of crown diagrams in the same
homotopy class become diffeomorphic after applying an isotopy and a sequence
of moves chosen from a list of four moves called stabilization, handleslide,
multislide, and shift. According to [W3, Theorem 4.9], the stabilization move
can be omitted from this list because the diagrams have the same genus.
According to Lemmas 3.3, 3.4, and 3.5, $\mathcal{S}$ is unchanged by isotopy
and the remaining three moves.∎

##  4\. Examples

###  4.1. Relating crown diagrams of knot surgery manifolds

According to [FS, Theorem 1.9], if $K$ is a fibered knot, then $E(1)$ is
homeomorphic to the knot surgery manifold $E(1)_{K}$ and the Seiberg-Witten
invariant of $E(1)_{K}$ is the Alexander polynomial $\Delta_{K}(t)$. In the
same paper, Fintushel and Stern described a Lefschetz fibration structure on
$E(1)_{K}$. The current paper uses a monodromy factorization of this Lefschetz
fibration due to Yun [Y], and it refers to KnotInfo for the Alexander
polynomials and monodromy for the knots $7_{6}$ and $10_{133}$. These are
genus-2 fibered knots with the same Alexander polynomial. Genus-1 knots would
have simpler diagrams, but they are not used because the two fibered genus-1
knots are the trefoil $3_{1}$ and the figure-8 knot $4_{1}$, which are
distinguished by their Alexander polynomials.

It has been known since at least the publication of [L] that one may convert a
Lefschetz fibration over $S^{2}$ into a crown map by a homotopy of the
fibration in which one converts the Lefschetz critical points into 3-cusped
circles and performs cusp merges to unite the circles, resulting in a crown
map whose crown diagram is called a _coronation_ of the Lefschetz fibration
(see [W4, Figure 7] for an example of how this looks in a crown diagram).
Using the notion of _surgered diagrams_ from [W2] to deduce properties of the
resulting vanishing cycles, Behrens and Hayano gave a description of the full
collection of coronations of a Lefschetz fibration in terms of mapping class
groups of surfaces [BH]. In [W4] there is a fully diagrammatic interpretation
of the same homotopy, yielding a single, much more explicit, member of this
collection. Recording the vanishing cycles using the notation of train tracks,
the weights of such a diagram reach the tens of thousands for a diagram with
22 vanishing cycles, such as the ones for knot surgery on $E(1)$. Instead,
this paper will consider a simpler kind of diagram which is slide-equivalent
to any coronation of Yun’s Lefschetz fibration. It comes with a collection of
crossings that end up being the salient crossings of a coronation. Before
discussing this, though, recall that the invariance of $\mathcal{S}$ applies
to crown diagrams with a common homotopy class. The manifolds $E(1)_{7_{6}}$
and $E(1)_{10_{133}}$ are nominally different manifolds, so it is necessary to
explain how their coronations would have a common homotopy class if their
corresponding smoothings were isotopic.

######  Lemma 4.1.

Suppose $K$ and $K^{\prime}$ are fibered knots in $S^{3}$, and let
$f\colon\thinspace E(1)_{K}\to S^{2}$, $f^{\prime}\colon\thinspace
E(1)_{K^{\prime}}\to S^{2}$ be the Lefschetz fibrations studied in [Y].

  1. (1)

There is a topological manifold $M$ and homeomorphisms $h\colon\thinspace M\to
E(1)_{K}$, $h^{\prime}\colon\thinspace M\to E(1)_{K^{\prime}}$ such that
$f\circ h$ and $f^{\prime}\circ h^{\prime}$ are homotopic maps $M\to S^{2}$.

  2. (2)

If the two smoothings coming from these Lefschetz fibrations are isotopic,
then the salient sets of their coronations are equal.

###### Proof.

Let $M$ be a topological 4-manifold homeomorphic to
$\mathbb{CP}{}^{2}\\#9\overline{\mathbb{CP}}{}^{2}$ (and thus homeomorphic to
the total spaces of $E(1)_{K}$ and $E(1)_{K^{\prime}}$). From the Pontrjagin-
Thom construction, it is known that the collection of homotopy classes of maps
$M\to S^{2}$ correspond to framed cobordism classes of framed surfaces in $M$,
where the framed surface is obtained as the preimage of a neighborhood of a
point in $S^{2}$. The goal is to find homeomorphisms $h,h^{\prime}$ from $M$
to $E(1)_{K}$ and to $E(1)_{K^{\prime}}$ sending a cobordism class of surfaces
in $M$ to the cobordism class of the fiber in $E(1)_{K}$ and in
$E(1)_{K^{\prime}}$, and then to deal with the framing separately. Let
$h\colon\thinspace M\to E(1)_{K}$ and $\phi\colon\thinspace M\to
E(1)_{K^{\prime}}$ be homeomorphisms, and let $c\in H^{2}(E(1)_{K})$ and
$c^{\prime}\in H_{2}(E(1)_{K^{\prime}})$ be the fiber classes of $f$ and
$f^{\prime}$, respectively. Then neither $(\phi\circ h^{-1})_{*}(c)$ nor
$c^{\prime}$ are characteristic homology classes in $E(1)_{K^{\prime}}$, and
both are primitive, because $f$ and $f^{\prime}$ each have a section whose
self-intersection number is $-2$ [FS, Section 6], while its intersection with
the fiber class is 1. Thus, by [GS, Proposition 1.2.18], there is an
automorphism $\alpha$ of the intersection pairing of $E(1)_{K^{\prime}}$
sending $(\phi\circ h^{-1})_{*}(c)$ to $c^{\prime}$. According to the
topological $h$-cobordism theorem, $\alpha$ is realized as the induced map of
a self-homeomorphism $\psi$ of $E(1)_{K^{\prime}}$ (see for example [GS,
Chapter 9]). Thus, $h^{\prime}=\psi\circ\phi$ is a homeomorphism such that
$h^{-1}_{*}(c)=(h^{\prime})^{-1}_{*}(c^{\prime})$, which yields the required
correspondence of cobordism classes.

${M}$${E(1)_{K}}$${E(1)_{K^{\prime}}}$${S^{2}}$$\scriptstyle{h}$$\scriptstyle{h^{\prime}=\psi\circ\phi}$$\scriptstyle{f}$$\scriptstyle{f^{\prime}}$

Having found $h,h^{\prime}$ such that $(f\circ h)^{-1}(pt)$ is cobordant in
$M$ to $(f^{\prime}\circ h^{\prime})^{-1}(pt)$, let $\Sigma\subset M$ be a
surface in that cobordism class. According to Theorem 2 and Example 1.3 of
[KMT], there is a unique framing of $\Sigma$ up to framed cobordism because
$M$ is simply connected and its intersection form is odd. For this reason, $h$
and $h^{\prime}$ send the unique framed cobordism class of $\Sigma$ to that of
the fiber of $f$ and $f^{\prime}$, respectively. For this reason,
$f_{0}=f\circ h$ and $f_{1}=f^{\prime}\circ h^{\prime}$ are continuously
homotopic maps $M\to S^{2}$. This completes the proof of Item (1).

For Item (2), since the smoothings coming from $f_{0}$ and $f_{1}$ are
isotopic, there is an isotopy of $f_{1}$ converting it to be a smooth map on
the total space of $f_{0}$. Thus, $f_{0}$ and $f_{1}$ can be taken as smooth
maps on the ends of $E(1)_{K}\times[0,1]$ connected by a continuous homotopy.
That is, they share a homotopy class.

To the ends of this homotopy, append the further homotopies which convert the
Lefschetz fibrations $f_{0}$ and $f_{1}$ into crown maps $c_{0},c_{1}$ using
[W1, Section 4]. Then, according to [W3, Section 4], these can be connected by
a deformation corresponding to a sequence of genus-preserving moves on crown
maps, which have equal salient sets by Theorem 3.2.∎

######  Question 4.2.

It seems natural to ask whether two diffeomorphic smoothings realized as the
total spaces of homotopic fibration maps is enough to imply that the two
smoothings are isotopic. In that case, the main result of the paper could be
immediately strengthened to state that the manifolds are not diffeomorphic.

###  4.2. Diagrams for two knot surgery manifolds

Figure 5 depicts the relevant information to obtain Yun’s monodromy
factorizations for $E(1)_{7_{6}}$ and $E(1)_{10_{133}}$. Starting with the
five circles in Figure 5(a), cut the surface at the four circles shown in
Figure 5(b). Let $t_{x}$ denote the right Dehn twist about the circle $x$.
Then, according to KnotInfo, and using the circles shown in Figure 5(d), the
monodromy of $7_{6}$ can be written as the composition of Dehn twists

$\phi_{7_{6}}=t_{a}\cdot t_{b}\cdot t_{c}^{-1}\cdot t_{d},$

applied from left to right, while the monodromy of $10_{133}$ can be written

$\phi_{10_{133}}=t_{a}\cdot t_{b}\cdot t_{c}\cdot t_{d}^{-1}\cdot
t_{d}^{-1}\cdot t_{f}.$

According to [Y, Theorem 2.11], the Lefschetz monodromy factorization for
$E(1)_{K}$ is

$\left(\prod\limits_{i=0}^{4}t_{\phi_{K}(B_{i})}\right)^{2}\cdot\left(\prod\limits_{i=0}^{4}t_{B_{i}}\right)^{2}.$

The collection of circles that results for $7_{6}$ is a messy picture, which
can be simplified considerably by applying one Dehn twist and pushing the feet
of two of the handles around, switching their places in a clockwise rotation
as in Figure 6. The resulting collection of Lefschetz vanishing cycles can be
seen in Figures 7(a)–7(j) by deleting the pair of long rectangles and
connecting the resulting ends with vertical line segments. The simplifying
diffeomorphism is not used for $E(1)_{10_{133}}$, and the resulting collection
of circles appears analogously in Figure 8. Note that in those figures, the
reversal of ordering of the circles comes from the fact that the circles in
the above monodromy factorization are vanishing cycles of a Lefschetz
fibration, while the circles in Figures 7 and 8 get their ordering from the
merging algorithm depicted in [W4, Figure 6]. In short, the so-called
_pseudocoronation_ shown in Figure 7 or Figure 8 comes from choosing an arc in
the fiber of the Lefschetz fibration that crosses each of the fibration’s
Lefschetz vanishing cycles exactly once, replacing that arc with a tube over
which all of those vanishing cycles run, adding two more vanishing cycles in a
standard way as in Figures 7(k) and 8(k), then ordering the vanishing cycles
as required by [W4, Figure 6]: In that figure, $r_{i}$ or $r_{i}^{s}$
corresponds to the $i^{th}$ Lefschetz vanishing cycle. Circle 20 in Figures in
Figures 7(k) and 8(k) corresponds to $g_{1}$ in [W4, Figure 6], and circle 22
in Figures 7(k) and 8(k) corresponds to $b_{1}$ in [W4, Figure 6].

2pt $B_{0}$ at 100 250 $B_{1}$ at 135 230 $B_{2}$ at 135 204 $B_{3}$ at 135
184 $B_{4}$ at 135 160

(a) [Y, Figure 3] in the case where $g=2$ and $n=1$.

2pt 1 at 135 250 2 at 135 186 3 at 135 97 4 at 135 30

(b) Cuts used to produce the diagrams in Figures 5(c) and after.

2pt $B_{0}$ at 60 222 $B_{1}$ at 159 200 $B_{2}$ at 33 192 $B_{3}$ at 120 108
$B_{4}$ at 152 24 1 at 95 215 1 at 175 215 2 at 95 165 2 at 175 165 3 at 95
115 3 at 175 115 4 at 95 65 4 at 175 65

(c) These are circles of Figure 5(a) after cutting the surface.

2pt $f$ at 70 230 $a$ at 90 185 $c$ at 120 172 $d$ at 208 230 $b$ at 191 191

(d) The circles $a$–$f$ in KnotInfo’s factorization of the monodromy for
$7_{6}$ and $10_{133}$.

Figure 5. Figures for finding the Lefschetz vanishing cycles for
$E(1)_{7_{6}}$ and $E(1)_{10_{133}}$.

According to the algorithm in [W4] and the subsequent proofs that the
algorithm is realizable as a sequence of slides between vanishing cycles in
the pseudocoronation, a coronation of either of the Lefschetz fibrations of
$E(1)_{7_{6}}$ or $E(1)_{10_{133}}$ can be obtained from applying slides among
the vanishing cycles in Figures 7 and 8, while keeping the staircase-shaped
arrangement in Figures 7(l) and 8(l) unchanged. Since $\mathcal{S}$ is
invariant under such slides, one could instead use the grading system as given
by the depicted pseudocoronations, taking the crossings between consecutive
vanishing cycles in the staircase arrangement as the salient crossings: The
first step in the algorithm for finding the vanishing cycle $\gamma_{i}$ of
the coronation is to repeatedly slide $\gamma_{i}$ over $\gamma_{22}$ along
$\gamma_{i+1}$ to eliminate all crossings between $\gamma_{i}$ and
$\gamma_{i+1}$ other than the salient crossing in the staircase.

2pt 1 at 34 231 1 at 140 231 2 at 34 165 2 at 140 165 3 at 34 99 3 at 140 99 4
at 34 33 4 at 140 33 5 at 310 149 5 at 310 115 $c$ at 180 200 $\rightarrow$ at
430 132

(a)

2pt 1 at 34 231 1 at 140 231 2 at 34 165 2 at 140 165 3 at 75 84 3 at 140 99 4
at 34 33 4 at 140 33 5 at 310 149 5 at 310 115 $\rightarrow$ at 430 132

(b)

2pt 1 at 34 231 1 at 140 231 2 at 34 165 2 at 140 165 3 at 77 55 3 at 140 99 4
at 33 78 4 at 140 33 5 at 310 149 5 at 310 115 $\swarrow$ at -25 -25

(c)

2pt 1 at 34 231 1 at 140 231 2 at 34 165 2 at 140 165 4 at 34 99 3 at 140 99 3
at 34 33 4 at 140 33 5 at 310 149 5 at 310 115

(d)

Figure 6. Two diffeomorphisms which are used to simplify the depiction of the
Lefschetz vanishing cycles for $E(1)_{7_{6}}$. The first is a negative Dehn
twist about the circle $c$. The second moves the feet of the tubes 3 and 4 so
that they switch places. The marked points are meant to illustrate how the
squares do not rotate as they move.

2pt 6 at 300 195

(a) Circles 1 and 6.

2pt 7 at 270 200

(b) Circles 2 and 7.

2pt 8 at 270 205

(c) Circles 3 and 8.

2pt 9 at 270 210

(d) Circles 4 and 9.

2pt 10 at 270 215

(e) Circles 5 and 10.

2pt 11 at 330 175

(f) Circles 11 and 16.

2pt 12 at 325 175

(g) Circles 12 and 17.

2pt 13 at 320 175

(h) Circles 13 and 18.

2pt 14 at 315 175

(i) Circles 14 and 19.

2pt 15 at 310 175

(j) Circles 15 and 21.

2pt 20 at 30 30

(k) Circles 20 and 22.

(l) All circles.

Figure 7. A pseudocoronation of $E(1)_{7_{6}}$. One of the two circles is
labeled in each diagram except the last.

2pt 6 at 300 195

(a) Circles 1 and 6.

2pt 7 at 270 200

(b) Circles 2 and 7.

2pt 8 at 270 205

(c) Circles 3 and 8.

2pt 9 at 270 210

(d) Circles 4 and 9.

2pt 10 at 270 215

(e) Circles 5 and 10.

2pt 11 at 330 175

(f) Circles 11 and 16.

2pt 12 at 325 175

(g) Circles 12 and 17.

2pt 13 at 320 175

(h) Circles 13 and 18.

2pt 14 at 315 175

(i) Circles 14 and 19.

2pt 15 at 310 175

(j) Circles 15 and 21.

2pt 20 at 345 30

(k) Circles 20 and 22.

(l) All circles.

Figure 8. A pseudocoronation of $E(1)_{10_{133}}$. One of the two circles is
labeled in each diagram except the last.

###  4.3. Equations and calculations

The number of regions in the pseudocoronations given by Figures 7 and 8 is
large, and each region has multiple equations coming from the different
choices of labeling. For this reason, there is software written by Umur
Çiftçi, called EQMaker, which produces the grading system as a text file, and
there is a list of Mathematica commands for extracting from this system a new
system of equations which expresses the salient set as a sum of formal
parameters, all available at the author’s website. The grading systems for
these diagrams both have integer solutions, the rank of the system for
$E(1)_{7_{6}}$ is 21, and the rank of the system for $E(1)_{10_{133}}$ is 20.
For this reason, the salient sets are not equal, so the corresponding
smoothings are not isotopic.

## References

  * [B] S. Behrens, On 4-manifolds, folds and cusps, _Pac. J. Math_ 264(2) (2013), 257-306. doi:10.2140/pjm.2013.264.257 MR3089398
  * [BH] S. Behrens and K. Hayano, Elimination of cusps in dimension 4 and its applications, _Proc. London Math Soc._ 113 (5), 674-724. doi:10.1112/plms/pdw042 MR3570242
  * [FS] R. Fintushel and R. Stern, Knots, links and 4-manifolds, _Invent. Math._ 134 (1998), no. 2, 363-400. doi 10.1007/s002220050268 MR1650308
  * [GS] R. Gompf and A. Stipsicz, 4-manifolds and Kirby Calculus, Graduate Studies in Math. 20, Amer. Math. Soc., Providence, RI (1999). MR1707327
  * [KMT] R. Kirby, P. Melvin, and P. Teichner, Cohomotopy sets of 4-manifolds, _Geom. Topol. Monographs_ 18 (2012), 161-190.
  * [L] Y. Lekili, Wrinkled fibrations on near-symplectic manifolds, _Geom. Topol._ 13 (2009), 277-318. doi:10.2140/gt.2009.13.277 MR2469519
  * [Wa] G. Wassermann, Stability of unfoldings in space and time, _Acta. Math._ , 135 (1975), 58-128. doi:10.1007/BF02392016
  * [W1] J. Williams, The $h$-principle for broken Lefschetz fibrations, _Geom. Topol._ 14 no. 2 (2010), 1015-1061. doi:10.2140/gt.2010.14.1015 MR2629899
  * [W2] J. Williams, Uniqueness of crown diagrams of smooth $4$-manifolds, 2011 preprint (submitted).
  * [W3] J. Williams, Existence of 2-parameter crossings, with applications, Geom. Ded. 207, 265–286(2020). doi:10.1007/s10711-019-00499-1
  * [W4] J. Williams, Depicting a generalized shift move in crown diagrams, 2020 preprint.
  * [Y] K-H Yun, Twisted fiber sums of Fintushel-Stern’s knot surgered 4-manifolds, _Trans. AMS_ 360, no. 11 (2008), 5853-5868. doi: 0002-9947(08)04623-0


subsecref subsecname = thmref thmname = theorem lemref lemname = lemma

# Dynamical Mean Field Theory of the Bilayer Hubbard Model with Inchworm Monte
Carlo

Dolev Goldberger  School of Physics, Tel Aviv University, Tel Aviv 6997801,
Israel  Yehonatan Fridman  Department of Physics, NRCN, P.O. Box 9001, Beer
Sheva 84190, Israel  Department of Physics, Ben-Gurion University of the
Negev, Beer-Sheva 84105, Israel  Emanuel Gull  Department of Physics,
University of Michigan, Ann Arbor, Michigan 48109, USA  Eitan Eidelstein
Department of Physics, NRCN, P.O. Box 9001, Beer Sheva 84190, Israel  School
of Chemistry, Tel Aviv University, Tel Aviv 6997801, Israel  Guy Cohen
<EMAIL_ADDRESS>School of Chemistry, Tel Aviv University, Tel Aviv 6997801,
Israel  The Raymond and Beverley Sackler Center for Computational Molecular
and Materials Science, Tel Aviv University, Tel Aviv 6997801, Israel

###### Abstract

Dynamical mean-field theory allows access to the physics of strongly
correlated materials with nontrivial orbital structure, but relies on the
ability to solve auxiliary multi-orbital impurity problems. The most
successful approaches to date for solving these impurity problems are the
various continuous time quantum Monte Carlo algorithms. Here, we consider
perhaps the simplest realization of multi-orbital physics: the bilayer Hubbard
model on an infinite-coordination Bethe lattice. Despite its simplicity, the
majority of this model’s phase diagram cannot be predicted by using
traditional Monte Carlo methods. We show that these limitations can be largely
circumvented by recently introduced Inchworm Monte Carlo techniques. We then
explore the model’s phase diagram at a variety of interaction strengths,
temperatures and filling ratios.

##  I Introduction

In strongly correlated materials quantum many-body physics shapes the
electronic properties. Such materials exhibit a wide variety of unique and
interesting phases and transitions [1]. Many of these behaviors are thought to
be beyond the predictive power of standard simulation schemes like the density
functional theory, where electronic correlation is essentially treated as a
static mean field. Embedding methods like the dynamical mean field theory
(DMFT) [2]—which contains strong, but local, correlations—have therefore
emerged. For model systems, DMFT becomes exact in certain infinite
coordination number limits [3]. It can also be used to introduce approximate
strong correlation physics into ab initio calculations made by, e.g., density
functional theory [4].

DMFT can be phrased as a mapping between an extended model with many-body
interactions between orbitals in each unit cell; and a quantum impurity
problem representing the correlated orbitals in one unit cell, coupled to an
effective noninteracting bath that represents the rest of the system. The
coupling density between the impurity and bath, and the connection between the
impurity self-energy and the lattice self-energy, are determined by a self-
consistency condition [2]. This means that to solve systems with multiple
orbitals in each unit cell, a method for solving multiorbital quantum impurity
problems is required. In this context, “solving” refers to the calculation of
Green’s functions.

While several alternatives exist, the methods most often employed by DMFT
practitioners for general multiorbital problems are known collectively as
continuous-time quantum Monte Carlo (CT-QMC) algorithms [5]. These methods
take perturbative diagrammatic expansions and sum them to very high orders by
Monte Carlo integration over diagrams, with most calculations performed in
imaginary time on the Matsubara contour. Several different methods exist,
based on different expansions and with different regimes of applicability;
when these methods do break down, the breakdown typically takes the form of a
“sign problem” [6], such that the calculation becomes exponentially harder
with decreasing control parameter. For example, the interaction expansion (CT-
INT) [7] and auxiliary field (CT-AUX) [8] methods develop sign problems away
from half-filling. The hybridization expansion (CT-HYB) [9, 10] can work
better in the presence of large, complicated interactions [11], but often
develops sign problems when the hybridization in the impurity model has large
off-diagonal components. More recently, we proposed an inchworm Monte Carlo
method[12] based on the hybridization expansion, which can circumvent some of
the sign problems afflicting multiorbital DMFT calculations [13].

One of the earliest models used to investigate multiorbital physics within the
framework of DMFT is the bilayer Hubbard model on the infinite dimensional
Bethe lattice [14], which we present in detail in Sec. II.1 below. While
originally introduced to study the relationship between short-ranged spin
correlations and the Mott metal–insulator transition, the model has reemerged
over the years: for example, more realistic variations on the model have been
used to simulate the metal insulator transition in vanadium dioxide [15, 16,
17]. A two-dimensional version on a square lattice has been considered as a
toy model for unconventional superconductivity [18] and realized
experimentally using ultracold atoms in an optical lattice [19]. Finally,
extensions to hexagonal lattices are currently of great interest in the study
of layered 2D materials [20].

The infinite-dimensional bilayer Hubbard model on the Bethe lattice is exactly
solvable within DMFT, given an exact solution for the corresponding two-
orbital impurity problem. Its phase diagram at half-filling, which includes a
Mott metal–insulator transition and an antiferromagnetic regime, was
investigated first by Hirsch–Fye Monte Carlo[21] and shortly after by CT-INT
[22]. Notably, the 2D model on a square lattice, where DMFT is only
approximate, was also probed using an extension of DMFT with a discrete
approximation for the bath [23], and by a variational Monte Carlo technique
[24]. The paramagnetic regime of the doped system was recently investigated
using CT-HYB [25], by taking advantage of a specialized symmetry property to
remove the diagonal hybridizations; this suggested the existence of an
interesting pseudogap phase at intermediate doping.

Here, we show that the inchworm Monte Carlo method can be used as an impurity
solver for the bilayer Hubbard model within DMFT. The robustness of the
inchworm scheme to sign problems allows us to extend the phase diagram beyond
half-filling without any particular symmetry restrictions, enabling
exploration of ferromagnetic and antiferromagnetic response. We use this to
investigate the effect of doping on the metallic and magnetic properties of
the system, and analyze the results to reveal simple short-ranged mechanisms
behind much of the phenomenology. The rest of this work is structured as
follows: Sec. II briefly describes the model (II.1) and methodology (II.2). In
Sec. III we first compare the CT-HYB impurity solver with its inchworm
counterpart (III.1). We then show how we determine phase transitions from our
numerical data (III.2), and finally present and discuss the phase diagram
(III.3). In Sec. IV we conclude and discuss future prospects.

##  II Theory

###  II.1 Model

Figure 1: Illustration: a small region in the bilayer Hubbard model on a Bethe
lattice with $Z=3$ (we consider the limit $Z\rightarrow\infty$). Blue and red
spheres, respectively, represent orbitals in the first and second layer.
Orbitals on adjacent sites within a layer are coupled by hopping amplitudes
$t$. Within each dimer or unit cell, one of which is marked by a rectangular
box, particles can hop between layers with hopping amplitude.

We define the bilayer Hubbard model on the Bethe lattice in the infinite
coordination number limit $Z\rightarrow\infty$. In Fig. 1 we provide an
illustration with $Z=3$. The model comprises two identical Hubbard models, or
layers, on Bethe lattices with hopping $t$, chemical potential $\mu$ and
interaction strength $U$. Each site in the first layer is coupled to its
counterpart in the second with hopping $t^{\prime}$. The Hamiltonian is given
by:

$\displaystyle H$ $\displaystyle=-t\sum_{\left\langle
ij\right\rangle\sigma}\left(a_{i\sigma}^{\dagger}a_{j\sigma}+b_{i\sigma}^{\dagger}b_{j\sigma}\right)+t^{\prime}\sum_{i\sigma}\left(a_{i\sigma}^{\dagger}b_{i\sigma}+b_{i\sigma}^{\dagger}a_{i\sigma}\right)$
$\displaystyle-\sum_{i\sigma}\mu\left(n_{i\sigma}^{\left[a\right]}+n_{i\sigma}^{\left[b\right]}\right)+U\sum_{i}\left(n_{i\uparrow}^{a}n_{i\downarrow}^{a}+n_{i\uparrow}^{b}n_{i\downarrow}^{b}\right).$
(1)

The subscript $\sigma\in\left\\{\uparrow,\downarrow\right\\}$ denotes spin,
and $a^{\dagger}$($b^{\dagger}$) creates an electron in layer $a$($b$). In the
second line, $n_{i\sigma}^{a}\equiv a_{i\sigma}^{\dagger}a_{i\sigma}$ and
$n_{i\sigma}^{b}\equiv b_{i\sigma}^{\dagger}b_{i\sigma}$. We also define the
ratio between the inter-layer and intra-layer hopping, $\alpha\equiv
t^{\prime}/t$. To obtain a nontrivial infinite coordination number limit, we
rescale $t$ to $\frac{t}{\sqrt{Z-1}}$, then set this to $1$ to define our unit
of energy. This scaling is not applied to $t^{\prime}$.

###  II.2 Methodology

The DMFT method provides a self consistent mapping from Eq. (1) onto a quantum
impurity model [2]. In the present model and for antiferromagnetic order, the
self consistency requirement takes on a particularly simple form [14, 22]:

$\displaystyle\mathcal{\boldsymbol{G}}_{\sigma}^{-1}\left(i\omega_{n}\right)$
$\displaystyle\equiv\int_{0}^{\beta}\mathrm{d}\tau
e^{i\omega_{n}\tau}\mathcal{\boldsymbol{G}}_{\sigma}^{-1}\left(\tau\right)$
(2) $\displaystyle=\left(\begin{array}[]{cc}i\omega_{n}+\mu&-\alpha t\\\
-\alpha
t&i\omega_{n}+\mu\end{array}\right)-\boldsymbol{\Delta}_{\sigma}\left(i\omega_{n}\right),$

where $\mathcal{\boldsymbol{G}}_{\sigma}\left(i\omega_{n}\right)$ is the Weiss
field at Matsubara frequency $\omega_{n}$. Here the hybridization function,

$\boldsymbol{\Delta}_{\sigma}=t^{2}\boldsymbol{G}_{-\sigma}\left(i\omega_{n}\right),$
(3)

is obtained from the local Green’s function at the (arbitrarily chosen) site
$i=0$, such that $a_{\sigma}=a_{0\sigma}$ and $b_{\sigma}=b_{0\sigma}$:

$\boldsymbol{G}_{\sigma}\left(\tau\right)\equiv-\left(\begin{array}[]{cc}\left\langle\mathrm{T}_{\tau}a_{\sigma}\left(\tau\right)a_{\sigma}^{\dagger}\left(0\right)\right\rangle&\left\langle\mathrm{T}_{\tau}a_{\sigma}\left(\tau\right)b_{\sigma}^{\dagger}\left(0\right)\right\rangle\\\
\left\langle\mathrm{T}_{\tau}b_{\sigma}\left(\tau\right)a_{\sigma}^{\dagger}\left(0\right)\right\rangle&\left\langle\mathrm{T}_{\tau}b_{\sigma}\left(\tau\right)b_{\sigma}^{\dagger}\left(0\right)\right\rangle\end{array}\right).$
(4)

Note that this assumes the spins in adjacent sites can be flipped by
antiferromagnetism, forming a bipartite lattice. A small symmetry breaking
term was also applied to the Hamiltonian in the initial DMFT iteration when
seeking antiferromagnetic solutions.

The impurity model to be solved is then defined by the effective action

$\displaystyle S_{\mathrm{eff}}$
$\displaystyle=-\sum_{\sigma}\int_{0}^{\beta}\mathrm{d}\tau\mathrm{d}\tau^{\prime}\mathrm{d}\tau^{\prime}\mathbf{c}^{\dagger}\left(\tau\right)\cdot\boldsymbol{\mathcal{G}}_{\sigma}^{-1}\left(\tau-\tau^{\prime}\right)\cdot\mathbf{c}\left(\tau^{\prime}\right)$
(5)
$\displaystyle+U\int_{0}^{\beta}\mathrm{d}\tau\left(n_{\uparrow}^{a}\left(\tau\right)n_{\downarrow}^{a}\left(\tau\right)+n_{\uparrow}^{b}\left(\tau\right)n_{\downarrow}^{b}\left(\tau\right)\right),$

where
$\mathbf{c}^{\dagger}\left(\tau\right)\equiv\left(a^{\dagger}\left(\tau\right),b^{\dagger}\left(\tau\right)\right)$,
$n_{\sigma}^{a}\left(\tau\right)\equiv
a_{\sigma}^{\dagger}\left(\tau\right)a_{\sigma}\left(\tau\right)$ and
$n_{\sigma}^{b}\left(\tau\right)\equiv
b_{\sigma}^{\dagger}\left(\tau\right)b_{\sigma}\left(\tau\right)$. This is a
two-orbital impurity problem. For this particular problem, at $\mu=-U/2$ the
system becomes particle–hole symmetric, and CT-INT works without a sign
problem [22]. The off-diagonal elements in Eq. (3) lead to sign problems in
CT-HYB, which can be eliminated in the paramagnetic case by transforming to
the bonding/antibonding orbital basis, where $\boldsymbol{\Delta}_{\sigma}$ is
diagonal. Neither algorithm works robustly in the general case. Here, we
employed an inchworm Monte Carlo algorithm based on CT-HYB, but able to deal
with the multiorbital sign problem emerging from nondiagonal hybridization
functions [13]. At their core, inchworm algorithms are a resummation technique
for diagrammatic Monte Carlo methods. They were originally developed almost a
decade ago to address the dynamical sign problem for populations dynamics in
the real time hybridization expansion[12] and related expansions for the
spin–boson model [26, 27, 28, 29, 30, 31], but were soon extended to Green’s
functions[32] and used to perform real-time DMFT calculations [33]. In
addition to a variety of applications in nonequilibrium physics [34, 35, 36,
37, 38, 39, 40, 41, 42], several extensions and generalizations have been
aimed at making these methods more useful as imaginary time impurity solvers
[43, 44, 45]. Our present implementation includes several optimizations not
present in the previous work [13], including fast diagram summations[46] and a
sparse representation of local propagators that takes advantage of conserved
quantum numbers. We plan to discuss these technical aspects in detail in
future work along with a public release of our code.

We note in passing that recent advances in tensor train methods as a
substitute for Monte Carlo techniques can also circumvent sign problems, and
may offer an alternative route to solving the impurity problems appearing here
[47, 48]. Novel decomposition schemes for finite-order diagrammatic methods
are another interesting development in this regard [49].

##  III Results

###  III.1 Sign problem in CT-HYB and inchworm algorithm

Figure 2: Development of the sign problem in CT-HYB (solid red curves) as
temperature is decreased, and comparison to inchworm Monte Carlo (dashed blue
curves) at interaction strength $U=4$, $\alpha=1$ and half-filling. The same
computation time is used for each of the six simulations shown.

The DMFT algorithm consists of a sequence of iterations, in each of which the
impurity solver is used to compute the Green’s function of Eq. (5) for a
hybridization function obtained from the previous iteration or—in the first
iteration—from the noninteracting problem. In the CT-HYB impurity solver [9],
generally a very efficient choice at anything but the smallest interaction
strengths [11], sign problems appear when the hybridization function develops
large nondiagonal components. To demonstrate this, consider Fig. 2, where the
Green’s function is calculated for interaction strength $U=4$ and tunneling
amplitude ratio $\alpha=1$, at half-filling. All simulations shown in the
figure and in the rest of this subsection, with both methods, were obtained
with the same total amount of computer time ($\sim 150$ minutes on 56 CPU
cores). We used the implementation of CT-HYB [50] based on the ALPSCore
libraries [51]. While CT-HYB (solid red curves) performs well at the higher
temperatures $\beta=4$ and $\beta=8$, it breaks down at $\beta=16$. The
corresponding data from an inchworm calculation [13] based on the same
expansion (dashed blue curves) remains reliable at the entire temperature
range shown.

While the example in Fig. 2 is illustrative, it is difficult to ascertain how
typical it is without further exploring the parameter space. To facilitate
this it is convenient to define a measure of the error, and we choose the
jackknife variance over $n=5$ independent realizations of the calculation with
different random seeds, averaged over imaginary time:

$\Delta
G\equiv\frac{1}{\beta}\int_{0}^{\beta}\mathrm{d}\tau\frac{1}{n\left(n-1\right)}\sum_{i=1}^{n}\left(G_{i}\left(\tau\right)-\bar{G}\left(\tau\right)\right)^{2}.$
(6)

In Fig. 3 we plot this measure on a logarithmic scale as a function of
$\alpha$, for $U=3$ and two temperatures, still at half-filling. Results from
CT-HYB and inchworm are shown in red and blue, respectively; while results at
$\beta=16$ and $8$ are shown in solid and dashed curves, respectively. Once
again, every data point uses the same overall computer time as in Fig. (2).
CT-HYB is substantially more accurate at small $\alpha$, i.e. when there is
little or no dimerization. However, its accuracy rapidly deteriorates due to a
severe sign problem at larger values of $\alpha$, especially at low
temperatures. It recovers slightly at very large values of $\alpha$. The
inchworm method, while not as accurate at small $\alpha$ near the part of
parameter space corresponding to a normal Hubbard model, generally suffers
less from both an increase in both $\alpha$ and the lowering of temperature.
It becomes very accurate at large $\alpha$, and generally converges more
easily when the system is in an insulating state.

Figure 3: Stochastic error estimates from inchworm (blue curves) and CT-HYB
(red curves) as a function of $\alpha$ at two temperatures (solid and dashed
curves). The interaction strength is set to $U=3$ and the system is half-
filled.

Next, in Fig. (4), we map out the dependence of the sign problem on $\alpha$,
$U$ and $\beta$ at half-filling. We carry out calculations of $\Delta G$ as
before. The red and blue shaded regions denote parameters where CT-HYB and the
inchworm method, respectively, provide a value of $\Delta G$ lower than
$10^{-2}$ for the given runtime. Note that this criterion is arbitrary and the
results are implementation-dependent; nevertheless, this procedure provides a
useful and intuitive description of the limitations of the methods. In the
high temperature case $\beta=4$ (left panel), both methods converge for the
entire parameter regime. At $\beta=8$ (middle panel), a region forms at
$1<\alpha<3$ and $U<4$ where the performance of CT-HYB degrades, but the
inchworm method continues to work well. As we go even lower in temperature to
$\beta=16$ (right panel), CT-HYB fails for $\alpha\gtrsim 0.5$ except at large
$U$. Here a parameter region where inchworm begins to fail as well (white
region) is also visible. This appears at small $U$ and around $\alpha=1$; note
that this behavior is also seen in Fig. 3, where the highest peak in $\Delta
G$ for the inchworm method appears near $\alpha=1$. While the errors in CT-HYB
at these parameters make most of the parameter space completely intractable at
$\beta=16$, inchworm data could still be obtained here given a reasonable
investment in computational resources. We expect that if it becomes necessary
to perform simulations at much lower temperatures, further algorithmic
improvements will be required even for the inchworm method.

Figure 4: Parameter regions where methods converge easily, as a function of
$U$ and $\alpha$. The red and blue shaded regions show where the CT-HYB and
inchworm methods, respectively, achieved errors $\Delta G$ below a threshold
value of $10^{-2}$ for a given amount of computational resources. Panels show
different temperatures, decreasing from left to right.

###  III.2 Studying phase transitions

We will be interested in two types of order parameter. The first distinguishes
between paramagnetic (PM) and antiferromagnetic (AFM) states, and the second
between metallic and insulating ones. These are the same distinctions made in
previous work [22], where the entire phase diagram at half filling was
obtained for $0\leq U\leq 5$ at $\beta=10$ using CT-INT as the impurity
solver. As we noted, CT-INT has no sign problem at half filling, and converges
rapidly at weak interaction strengths. It is therefore an ideal choice for
this problem, and can serve as an excellent benchmark for the hybridization-
expansion-based inchworm method.

We will first discuss the PM/AFM transition. One interesting finding in Ref.
[22] was that at the limit of low temperature and large interaction strength
$U$, the system undergoes a transition from AFM to PM as $\alpha$ increases
past $\sqrt{2}$. It was suggested that this is due to the transition happening
when the effective intra-layer Heisenberg exchange coupling,
$J^{\prime}\sim\frac{t^{\prime 2}}{U}$, is twice the size of its inter-layer
counterpart $J\sim\frac{t^{2}}{U}$. This, because every dimer has one
$J^{\prime}$ coupling and two $J$ couplings. This low energy argument is
correct at low temperatures, but can be expected to break down at higher ones.
In particular, for larger values of $U$ we might expect this breakdown to
occur when the temperature is comparable to the larger of $J^{\prime}$ and
$J$.

In Fig. 5 we plot the zero-field magnetization predicted from our inchworm-
based DMFT calculations at $\beta=10$ and half filling, for a range of $U$
values. This is defined simply as

$M=\left\langle
n_{\uparrow}^{a}+n_{\downarrow}^{b}-n_{\downarrow}^{a}-n_{\uparrow}^{b}\right\rangle,$
(7)

and—being a static property—can be directly obtained from inchworm Monte Carlo
calculations [12]. In practice, we apply a small staggered magnetic field
during the first DMFT iteration in order to break symmetry; then we turn it
off, and allow the DMFT self-consistency to evolve to either a symmetry-broken
($M\neq 0$) or symmetric $\left(M=0\right)$ state. In practice, our numerical
criterion for this was that the zero field magnetization, Eq. (7), be over an
arbitrary threshold value, $M_{\mathrm{threshold}}\equiv 0.03$.

The $U=4$ result is consistent with results from Ref. [22], where similar data
is shown. The phase transition indeed occurs at $\alpha\approx\sqrt{2}$, as
expected from the low energy theory. We also present a sequence of curves at
higher values of $U$, still at the same temperature. It is clear that the
value of $\alpha$ at which the transition occurs at this temperature decreases
with $U$. At $\alpha=0$ the system is a normal Hubbard model, and the decrease
in magnetization at larger values of $U$ is related to the known decrease in
the Curie temperature, which is expected to shift inversely with $U$ [52, 53,
54]. At large values of $U\gtrsim 6$, magnetization increases with $\alpha$
before decreasing again, showing that an intermediate inter-layer coupling can
either suppress or enhance magnetic properties.

.

Figure 5: Zero-field magnetization at $\beta=10$ and various values of $U$,
plotted as a function of $\alpha$. The theoretical low temperature limit for
the AFM–PM transition [22] is shown as a vertical dashed line at
$\alpha=\sqrt{2}$. The line widths indicate a confidence interval given by the
absolute value of the difference between the last two DMFT iterations.

The second order parameter we will discuss is an approximation for the
spectral function at the chemical potential, and therefore a proxy for
metallicity:

$A\left(\omega=\mu\right)\simeq-\frac{\beta}{\pi}G\left(\tau=\beta/2\right).$
(8)

The approximation becomes increasingly accurate at the low temperature limit
$\beta\rightarrow\infty$, but is commonly used in the DMFT literature at
finite temperature [2]. One possible alternative is to perform analytical
continuation and define the transition according to the formation of a gap,
but this requires additional assumptions. A more rigorous, but challenging,
route is to solve the problem on the real axis using nonequilibrium dynamical
mean field theory [55, 56, 57, 58, 42]; we will explore this in future work.
Here, we chose to define a simple, but arbitrary, transition threshold. When
$-\frac{\beta}{\pi}G\left(\tau=\beta/2\right)<0.25$, we refer to the system as
insulating; otherwise, we refer to is as metallic. This definition should be
seen as qualitative.

Fig. 6 shows how this definition can be used to investigate the
metal–insulator transition that occurs when the system is doped. The metallic
order parameter from Eq. (8) is shown as a function of the chemical potential,
at two values of the interaction strength $U$ and two values of the hopping
ratio $\alpha$. In all cases, higher doping takes the system from the
insulating to the metallic regime. Increasing both $U$ and $\alpha$ drives the
transition to higher chemical potentials. When $U=2.5$ and $\alpha=1$, we can
observe a drop in the order parameter at larger chemical potentials and an
eventual reentrance into insulating behavior due to the reduction of free
charge carriers. The latter will happen at high enough chemical potential
shift for all parameters.

Figure 6: Proxy for metallic behavior as a function of the chemical potential
shift $\Delta\mu$, at inverse temperature $\beta=16$. The dotted horizontal
black line denotes the (arbitrarily chosen) threshold value we use to indicate
the transition from an insulating to a metallic state. The line widths
indicate a confidence interval given by the absolute value of the difference
between the last two DMFT iterations.

###  III.3 Phase diagram

Using the order parameters and thresholds discussed in III.2, we performed a
set of calculations at $U=2.5$ and $U=8$, mapping out the phase diagram of the
model at each of these interaction strengths and temperature $\beta=16$ with
respect to the ratio $\alpha$ between inter-layer and intra-layer hoppings;
and the chemical potential shift from half filling,
$\Delta\mu=\mu+\frac{U}{2}$. Due to symmetry considerations, it is sufficient
to examine the $\Delta\mu>0$ part of the diagram. The two maps are presented
in the left and right panels of Fig. 7. The shaded red region in each denotes
parameters where we found the system to be antiferromagnetic. Similarly, the
blue region denotes parameters where we predict metallic behavior. Four
combinations of order parameters are possible, embodying four distinct phases:
the AFM insulator, AFM metal, PM insulator and PM metal.

We begin by considering previously available results. The $\alpha=0$ case
(bottom edge of both panels in Fig. 7) corresponds to the normal Hubbard
model. The behavior there is consistent with that known in the literature
[59]: for $U=2.5$, the system is in the AFM state for $\Delta\mu\lesssim 0.5$.
The metallic regime begins when $\Delta\mu\approx 0.2$ and ends at
$\Delta\mu\approx 0.5$. At $U=8$, the AFM regime expands while the insulating
regime shifts to larger values of $\Delta\mu$.

Next, we consider the $\Delta\mu=0$ case (left edge of both panels in Fig. 7),
which was explored by Hafermann et al. in Ref. [22]. At $U=2.5$, the system is
an AFM Mott insulator at $\alpha=0$. Increasing $\alpha$ drives it first to an
AFM metallic phase, then to a PM band insulator state. Depending on the
thresholds and temperature, a small PM metallic region may exist at
$\alpha\approx 1.7$ [22]. At $U=8$, however, the system goes directly from an
AFM insulator to a band insulator at $\alpha\approx\sqrt{2}$; this behavior
was also found in Ref. [22] at $U\apprge 4$. Note also the difference from the
behavior in Fig. 5, where $\beta=10$.

The rest of the phase diagram (everywhere else in the two panels of Fig. 7,
where neither $\alpha=0$ nor $\Delta\mu=0$) is where both DMFT calculations
based on the standard CT-HYB, CT-INT or CT-AUX methods face sign problems
unless special symmetries can be taken advantage of, as in Refs. [16, 17]. Our
CT-HYB-based inchworm method works well in the entire parameter space, without
any need for symmetrization, and can be used to obtain both the metallic and
AFM order parameters. For both values of $U$, an AFM regime appears at small
$\alpha$ and $\Delta\mu$. Increasing $U$ extends this regime in $\Delta\mu$,
while simultaneously contracting it in $\alpha$. The metallic behavior appears
at intermediate values of $\Delta\mu$ and is shifted approximately linearly in
$\alpha$. The metallic region always appears to be simply connected. In
particular, at small $U$ the AFM metallic state on the $\Delta\mu=0$ line is
connected to its counterpart on the $\alpha=0$ line. As a result of this, at
small $U$ the metallic and AFM regimes overlap in a narrow band surrounding
the AFM insulator at the origin, but this overlap shrinks with increasing $U$
as the metallic phase recedes from the $\Delta\mu=0$ line.

We emphasize that the particular size and shape of the regions and overlaps
depends to some degree on our choice of order parameters and thresholds, but
we expect the general physical trends to be robust.

Figure 7: Phase diagram of the bilayer Hubbard model at $\beta=16$ as function
of the ratio between the hopping terms $\alpha=t^{\prime}/t$ and the doping
$\Delta\mu$. The left and the right panel has coulomb onsite interaction $U$
of $2.5$ and $8$, respectively. Dots indicate parameter combinations where a
calculation was made, and bilinear interpolation is used to construct the
curves as described in III.2.

Much of the dependence of physical behavior on doping can stem from local
mechanisms like energetics and population switching effects within a unit
cell. It can therefore be understood to some degree by considering a
substantially simplified model: two adjacent dimers (here a dimer refers to
two coupled orbitals, one of which is in each layer). This model contains only
four spin-half orbitals, and can easily be solve by exact diagonalization
(ED); it also becomes equivalent to the lattice model at the fully dimerized
limit $\alpha\rightarrow\infty$, and bears some resemblance to a DMFT
calculation with a minimal discrete bath, at least with paramagnetic boundary
conditions. Since no spontaneous symmetry breaking can occur in the ED
solution without a self-consistency condition, we solve the system in the
presence of a small staggered field $h=0.1$ that induces an AFM state. In Fig.
8 we present a detailed view of how several local observables depend on the
chemical potential, for $U=2.5$ and $U=8$ (red and blue curves, respectively);
and for the numerically exact solution on Bethe lattice and the two-dimer
model (solid and dashed lines, respectively).

The top row of Fig. 8 shows the mean occupancy of the orbitals within a unit
cell. At $\Delta\mu=0$ the system is half filled: there are, on average, two
electrons per unit cell. Naturally, the occupancy rises as the chemical
potential increases, and at $\Delta\mu=10$ the system becomes completely
filled (4 electrons per unit cell) at all the parameter sets shown here. In
the lattice model, the increase in occupation is continuous and mostly linear.
In the two-dimer model, it increases in a series of steps with a width set by
the temperature, but over a similar range of chemical potentials.

The second row of Fig. 8 shows the AFM order parameter $M$. It is clear that
as the occupation increases, magnetic behavior is suppressed. This is not
surprising when considering that the half-occupied local state
$\left|\uparrow\right\rangle_{a}\left|\downarrow\right\rangle_{b}$ has the
maximal AFM response. In the two-dimer model, the drop in magnetism is
associated with the first jump in population. However, it is clear that the
two-dimer model does not fully capture the physics of the AFM–PM transition,
even with the addition of a symmetry-breaking field.

We now consider the metal–insulator transition, which is captured by the order
parameter of Eq. (8). This is plotted in the third row of Fig. 8. While the
two-dimer always produces sharp peaks in the proxy for the density of states,
these peaks do appear in the approximate region where the broadened density of
states appears in the lattice model. This region appears when the system
begins to exit the half-occupied state, which characterizes a Mott insulator;
and disappears when it enters the fully-occupied state, which characterizes a
band insulator.

Finally, the bottom row of Fig. 8 shows the probability of occupying metal-
like local states with 3 electrons. In the lattice model, this probability
appears and disappears in tandem with the presence of a metallic order
parameter. In the two-dimer model, the association between these observables
is still apparent, especially at larger values of $\alpha$; but both the
magnitude and extent in chemical potential where this state is likely are
significantly underestimated. The large discrepancy between the two-dimer and
lattice models in all panels on the bottom row of Fig. 8 illustrates the fact
that metallic behavior, in particular where it is associated with Kondo
correlations, is where the analogy between these two models breaks down most
noticeably.

Figure 8: The average occupation (top row), staggered magnetization (second
row), metallic order parameter (third row), and probability of occupying
states with $3$ electrons in the unit cell (bottom row); all as a function of
the chemical potential shift $\Delta\mu$, for two values of the coulomb
interaction (red and blue curves) and for four values of the ratio $\alpha$
between inter-band and intra-band hopping amplitudes (in different columns).
Solid lines denote the full DMFT solution on the Bethe lattice with inchworm
Monte Carlo as the impurity solver. Dashed lines are ED calculations for the
two-dimer model (see text).

##  IV Conclusions

We presented a study of the infinite-dimensional bilayer Hubbard model on the
Bethe lattice. The dynamical mean field approximation is exact for this model,
given a procedure for solving the associated multiorbital quantum impurity
model. We showed that the inchworm quantum Monte Carlo method enables this
solution in parameter regimes where other methods fail due to severe sign
problems. This allowed us to obtain the model’s phase diagram outside of half-
filling.

Our results show that at combinations of temperature and interaction strength
where the half-filled model has a metallic regime, it merges with the
corresponding metallic regime in the normal Hubbard model, where the two
layers are isolated from each other. It can then overlap with the
antiferromagnetic regime that exists near half-filling and at weak inter-layer
coupling. At stronger interaction strengths the metallic regime is pushed
towards higher chemical potentials. We also showed that most, though not all,
of the qualitative properties of the phase diagram can be understood from the
local energetics and state population probabilities within a unit cell.

In addition to shedding light on the properties of the doped bilayer Hubbard
model, our work illustrates some of the limitations of standard quantum Monte
Carlo methods in the study of multiorbital strongly correlated electron
physics. It then highlights inchworm techniques as a way around such
limitations, showing that they enable access to parameter regimes that were
previously difficult to simulate. We expect the methodology introduced here
may be useful for many other situations where sign problems have limited our
ability to answer important scientific questions regarding complex quantum
materials.

###### Acknowledgements.

This research was supported by the ISRAEL SCIENCE FOUNDATION (Grants No.
2902/21 and No. 218/19) and by the PAZY foundation (Grant No. 318/78).

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