# Hybrid Symmetry Breaking in Classical Spin Models With Subsystem Symmetries

Giovanni Canossa  Arnold Sommerfeld Center for Theoretical Physics, University
of Munich, Theresienstr. 37, 80333 München, Germany  Munich Center for Quantum
Science and Technology (MCQST), Schellingstr. 4, 80799 München, Germany  Lode
Pollet  Arnold Sommerfeld Center for Theoretical Physics, University of
Munich, Theresienstr. 37, 80333 München, Germany  Munich Center for Quantum
Science and Technology (MCQST), Schellingstr. 4, 80799 München, Germany
Wilczek Quantum Center, School of Physics and Astronomy, Shanghai Jiao Tong
University, Shanghai 200240, China  Ke Liu<EMAIL_ADDRESS>Arnold Sommerfeld
Center for Theoretical Physics, University of Munich, Theresienstr. 37, 80333
München, Germany  Munich Center for Quantum Science and Technology (MCQST),
Schellingstr. 4, 80799 München, Germany

###### Abstract

We investigate two concrete cases of phase transitions breaking a subsystem
symmetry. The models are two classical compass models featuring line-flip and
plane-flip symmetries and correspond to special limits of a Heisenberg-Kitaev
Hamiltonian on a cubic lattice. We show that these models experience a hybrid
symmetry breaking by which the system display distinct symmetry broken
patterns in different submanifolds. For instance, the system may look magnetic
within a chain or plane but nematic-like when observing from one
dimensionality higher. We fully characterize the symmetry-broken phases by a
set of subdimensional order parameters and confirm numerically both cases
undergo a non-standard first-order phase transition. Our results provide new
insights into phase transitions involving subsystem symmetries and generalize
the notion of conventional spontaneous symmetry breaking.

##  I Introduction

Subsystem symmetries are symmetries acting on lower-dimensional manifolds of
the system, such as lines or planes of a cube. Their early studies in
condensed matter physics are in the context of compass models Nussinov and van
den Brink (2015) which represent a family of effective Hamiltonians for
direction-dependent spin-orbital couplings in correlated electron systems
Kugel and Khomskiĭ (1982); van den Brink (2004); Khaliullin (2005) and
ultracold quantum gases Duan  _et al._ (2003); Wu (2008). The interest is
recently renewed thanks to the developments of fracton topological orders
Chamon (2005); Haah (2011); Vijay  _et al._ (2016); Nandkishore and Hermele
(2019); Pretko  _et al._ (2020) that can be obtained by gauging certain
subsystem symmetries. These symmetries have fundamentally different physical
implications than global and gauge symmetries. A global symmetry has a
constant ground-state degeneracy (GSD), and the Landau order-parameter theory
of spontaneous symmetry breaking describes its phase transition Anderson
(1984). A gauge symmetry supports topological degeneracies on closed manifolds
and fractionalized excitations, while its phase transition is typically
formulated as topological field theories Kogut (1979); WEN (1990); Kitaev
(2003). In contrast, the GSD arising from subsystem symmetries scales
subextensively with the system’s size. Their field theories can mix IR and UV
physics Seiberg and Shao (2021a, 2020, b), while their ordering mechanisms and
phase transitions do not fully fit into any known scenarios.

In this work, we study two classical $3D$ lattice models from an order-
parameter perspective to explore the impacts of subsystem symmetries on phase
transitions. The models belong to the compass model family and exhibit $Z_{2}$
line-flip or plane-flip subsystem symmetries. They do not support fracton
excitations by construction, hence can untwist the influences of subsystem
symmetries and fractons.

We find that breaking these subsystem symmetries leads to distinct long-range
orders in different submanifolds of the system, which we refer to as _hybrid
symmetry breaking_. The phase transitions in both cases are first order;
nevertheless, their finite-size scalings differ from the conventional first-
order scaling.

The manuscript is organized as follows. In Section II, we define the models
and construct their ground states and order parameters. Section III is devoted
to a brief review of the standard and non-standard finite-size scalings for
first-order phase transitions. Section IV and Section V discuss the phase
transitions and the resulting orders breaking the planar and linear subsystem
symmetries, respectively. We conclude in Section VI with an outlook.

Figure 1: Illustration of the models and ground states. (a) In the anti-Kitaev
limit, interactions between $S^{\gamma}$ components are canceled on $\gamma$
bonds. The reduced Hamiltonian is invariant under flipping $S^{\gamma}$ for
all spins in a spatial plane perpendicular to $\hat{\mathbf{e}}_{\gamma}$. A
ground state is illustrated in the right panel, where the single-head arrows
show the spin orientation in a plane, and the double-head arrow indicates the
planes are free to flip. (b) In the Kitaev limit, spins interact solely
through $S^{\gamma}$ components in $\hat{\mathbf{e}}_{\gamma}$ direction. The
model accordingly gains a line-flip symmetry, indicated by the double-head
arrows in the right panel. Red, green, and blue colors distinguish
orientations of lines and planes, and $\hat{\mathbf{e}}_{\gamma}$ denotes a
unit lattice vector.

##  II Models and order parameters

One of the simplest models permitting subsystem symmetries is the classical
Heisenberg-Kitaev Hamiltonian on a $3D$ cubic lattice,

$\displaystyle H=\sum_{\langle
ij\rangle_{\gamma}}\left(J\vec{S}_{i}\cdot\vec{S}_{j}+KS_{i}^{\gamma}S_{j}^{\gamma}\right),$
(1)

where $\vec{S}_{i}$ is a classical $O(3)$ spin at lattice site $i$,
$\gamma=x,y,z$ labels its three components, and $\langle ij\rangle_{\gamma}$
denotes a nearest-neighboring bond along the lattice direction
$\hat{\mathbf{e}}_{\gamma}$. Hence, the interaction in the second term is
directional. This Hamiltonian generally has a global cubic symmetry and
magnetic ground states. The linear and planar subsystem symmetries of our
interest and supporting non-trivial ground states are present in the limiting
cases $J=0$ and $J+K=0$, respectively.

For simplicity, we consider a ferromagnetic $K=-1$ at the $J=0$ limit and a
ferromagnetic $J=-1$ at the $J+K=0$ limit. The opposite cases with anti-
ferromagnetic couplings can be analyzed analogously, and we expect similar
physics.

###  II.1 Anti-Kitaev limit

In the limit $J+K=0$ with $J=-1$, Eq. (1) becomes

$\displaystyle H_{\text{anti-K}}=-\sum_{\langle
ij\rangle_{\gamma}}\sum_{\alpha\neq\gamma}S_{i}^{\alpha}S_{j}^{\alpha},$ (2)

which eliminates couplings between $S^{\gamma}$ components in the lattice
direction $\hat{\mathbf{e}}_{\gamma}$: we refer to it as the anti-Kitaev
limit. This reduced Hamiltonian is invariant upon transforming
$S^{\gamma}\rightarrow-S^{\gamma}$ for all spins in a spatial
$\alpha\beta$-plane orthogonal to $\hat{\mathbf{e}}_{\gamma}$, as visualized
in Fig. 1. The symmetry acts on $d=2$ submanifolds and leads to a subextensive
GSD $\propto 2^{L^{D-d}}=2^{L}$ for linear system size $L$.

The precise ground states are found by examining the local environment of
individual spins. First consider two of the nearest-neighboring bonds
$\langle\mathbf{i},\mathbf{i}+\hat{\mathbf{e}}_{x}\rangle$ and
$\langle\mathbf{i},\mathbf{i}+\hat{\mathbf{e}}_{y}\rangle$ in the $xy$-plane
for an arbitrary site $\mathbf{i}$. The minimal energy is expressed via a
single spin orientation $\vec{S}$ as $E=-|\vec{S}|^{2}-(S^{z})^{2}$ with
$S^{z}=\pm 1$. Adding the
$\langle\mathbf{i},\mathbf{i}+\hat{\mathbf{e}}_{z}\rangle$ bond, the full
minimal local energy has the same value $E_{g}=-2|\vec{S}|^{2}=-2$ (double
counting removed). This means its ground state manifold has an accidental
$O(3)$ symmetry. Because this $O(3)$ is not a symmetry of $H_{\text{anti-K}}$,
it will be lifted by entropy effects via the order-by-disorder mechanism
Villain, J.  _et al._ (1980); Henley (1989). The selected states are very much
are what we just derived and illustrated in Fig. 1: spins form collinear
configurations by spontaneously picking up a common axis $S^{c}$.
Nevertheless, due to the subsystem symmetry, a long-range spin order is only
possible within a spatial plane orthogonal to $\hat{\mathbf{e}}_{c}$.

This model provides an example of hybrid symmetry breaking: the system will
appear magnetic if observing along directions within an ordered plane but non-
magnetic in the perpendicular direction. The corresponding order parameters
can be constructed as follows,

$\displaystyle m_{P,r_{c}}^{c}$
$\displaystyle\coloneqq\frac{1}{L^{2}}\sum_{(r_{a},r_{b})\in
P}S^{c}_{r_{a}\hat{\mathbf{e}}_{a}+r_{b}\hat{\mathbf{e}}_{b}+r_{c}\hat{\mathbf{e}}_{c}},$
(3) $\displaystyle Q^{cc}$
$\displaystyle\coloneqq\frac{1}{L}\sum_{r_{c}}\left(\,m_{P,r_{c}}^{c}m_{P,r_{c}}^{c}-\frac{1}{3}\,\right),$
(4)

where $S^{c}$ represents the ordered spin components, and $a,b,c=x,y,z$ are
mutually exclusive. The two order parameters $m_{P,r_{c}}^{c}$ and $Q^{cc}$
measure the ordering within and between planes, respectively, and their
codimensions ${\rm codim}\left(m_{P,r_{c}}^{c}\right)=2$ and ${\rm
codim}\left(Q^{cc}\right)=1$, add up to the global dimension $D=3$. The
quadratic quantity $Q^{cc}$ may be viewed as an analog of the nematic order in
a uniaxial liquid crystal de Gennes and Prost (1995). However, its building
blocks here are not local director fields but sub-dimensional macroscopic
objects $m_{P,r_{c}}^{c}$. This also indicates that the associated phase
transition will not follow a usual Landau theory of two order parameters.

###  II.2 Kitaev limit

We next discuss the $J=0,K=-1$ limit

$\displaystyle H_{\rm K}=-\sum_{\langle
ij\rangle_{\gamma}}S_{i}^{\gamma}S_{j}^{\gamma},$ (5)

where spins couple solely through the $\gamma$ component in a $\gamma$-bond.
This reduced Hamiltonian $H_{\rm K}$ is also known as a $t_{2g}$ compass model
van den Brink (2004), and its form is akin to Kitaev’s honeycomb model Kitaev
(2006). Due to the cubic geometry, it features a $d=1$ subsystem symmetry by
flipping $S^{\gamma}$ in an entire $\gamma$-line of the lattice and
consequently a subextensive GSD $\propto 2^{L^{2}}$.

The ground states of $H_{\rm K}$ can be analyzed in a similar way as in the
case of $H_{\text{anti-K}}$. The energy per site is $E_{g}=-|\vec{S}|^{2}=-1$
indicating an accidental $O(3)$ symmetry. Stable ground states are
magnetically ordered, decoupled chains as illustrated in Fig. 1. More
concretely, spins form line magnetizations (rank-$1$) thanks to a thermal
order-by-disorder. The $d=1$ subsystem symmetry ensures no magnetic order
beyond one chain. The magnetic chains have to pack into planes towards one of
their two orthogonal directions, leading to a planar nematicity (rank-$2$) for
the inter-chain ordering. The inter-plane order further induces a rank-$4$
quantity as the two orthogonal directions are picked randomly in different
planes.

Thus, three order parameters are demanded to characterize the states fully,

$\displaystyle
m^{a}_{L,(r_{b},r_{c})}\coloneqq\frac{1}{L}\sum_{r_{a}}S_{r_{a}\hat{\mathbf{e}}_{a}+r_{b}\hat{\mathbf{e}}_{b}+r_{c}\hat{\mathbf{e}}_{c}}^{a}$
(6) $\displaystyle
Q^{aa}_{L,r_{c}}\coloneqq\frac{1}{L}\sum_{r_{b}}\left(m^{a}_{L,(r_{b},r_{c})}m^{a}_{L,(r_{b},r_{c})}-\frac{1}{3}\right)$
(7) $\displaystyle
Q^{aabb}_{4}\coloneqq\frac{1}{L}\sum_{r_{c}}\left(Q^{aa}_{L,r_{c}}-Q^{bb}_{L,r_{c}}\right)^{2},$
(8)

which all have a codimension-$1$. The mutually exclusive $a,b,c$ are
determined spontaneously in the hybrid symmetry breaking, while
$(r_{b},r_{c})$ and $r_{c}$ index the selected lines and planes, respectively.
$Q^{aa}_{L}$ is also a nematic-like order but now made from chain
magnetizations $m^{a}_{L}$. Correspondingly, $Q^{aabb}_{4}$ is analogous to a
generalized biaxial nematic order Nissinen  _et al._ (2016).

The hybrid symmetry-breaking patterns are richer than in the previous example.
The rank-$4$ order $Q^{aabb}_{4}$ breaks the spin and spatial permutation
symmetries with a global plane orientation. The four-fold rotation symmetry of
each square plane is further broken down to a two-fold one by the direction of
$Q^{aa}_{L}$, whereas no nontrivial symmetries are left in a magnetic chain.
We emphasize that these are not three sequential phase transitions but the
same phase transition observed at different submanifolds of the system.

##  III First-order finite size scaling

First-order phase transitions may be routinely considered less interesting as
they do not incur criticality but reflect a sharp change in the system.
However, a recent work Ref. Mueller  _et al._ (2014) showed that first-order
phase transitions involving subsystem symmetries deviate from the conventional
$L^{-D}$ finite-size scaling. In this section, we briefly review the standard
Fisher and Berker (1982); Lee and Kosterlitz (1991); Janke (2003) and non-
standard Mueller  _et al._ (2014) first-order scalings. For simplicity, we
will follow the arguments in Ref. Janke (2003) and consider a periodic
boundary condition (PBC), but refer to Ref. Lee and Kosterlitz (1991) for a
more comprehensive discussion and Ref. Borgs and Kotecký (1995) for the
effects of open boundaries.

Assume the system undergoes a single order-disorder transition, where the
ordered phase has $q$ equivalent degenerate states. We denote the
thermodynamical free energy density of the ordered and disordered phases as
$f_{o}$ and $f_{d}$, respectively. The possibility of the system in a
particular ordered or disordered state can be estimated by Boltzmann factors

$\displaystyle p_{o}$ $\displaystyle\propto e^{-\beta L^{D}\hat{f}_{o}},$ (9)
$\displaystyle p_{d}$ $\displaystyle\propto e^{-\beta L^{D}\hat{f}_{d}}.$ (10)

Then the weights of the two phases are given by $W_{o}\propto qe^{-\beta
L^{D}\hat{f}_{o}}$ and $W_{d}\propto e^{-\beta L^{D}\hat{f}_{d}}$, with
$W_{o}+W_{d}=1$.

A first-order phase transition occurs at the inverse temperature $\beta$ where
$W_{o}=W_{d}$. This fact can be understood from the behaviors of the specific
heat $C_{V}$. For simplicity, we ignore fluctuations within a phase and denote
energy density of the order and disordered phases by $E_{o}$ and $E_{d}$,
respectively. The $n$-th moment of the energy density for an arbitrary state
can then be approximately estimated by an weighted average Janke (2003),
$\langle E^{n}\rangle\simeq W_{o}E^{n}_{o}+W_{d}E^{n}_{d}$. The specific heat
hence becomes

$\displaystyle C_{V}$ $\displaystyle=\beta^{2}L^{D}\left(\langle
E^{2}\rangle-\langle E\rangle^{2}\right)$
$\displaystyle=\beta^{2}L^{D}W_{o}(1-W_{o})(E_{d}-E_{o})^{2},$ (11)

which peaks at $W_{o}=W_{d}=\frac{1}{2}$.

Hence, at a first-order phase transition, one has

$\displaystyle 0=\ln{(W_{o}/W_{d})}\simeq\ln{q}+L^{D}\beta(f_{d}-f_{o}).$ (12)

Using $f=E-TS$, where $S$ is the entropy density, and Taylor expanding Eq.
(12) around the thermodynamical ($L\rightarrow\infty$) transition point
$\beta^{\infty}$, we obtain in leading order

$\displaystyle\beta_{c}(L)=\beta_{c}^{\infty}-\frac{\ln{q}}{L^{D}(E_{d}-E_{o})}+\mathcal{O}(\frac{1}{L^{D-1}}).$
(13)

Namely, for a constant GSD, which is the case when breaking a global symmetry,
the leading finite-size correction is $\propto L^{-D}$.

The above analyses can be straightforwardly generalized to subsystem symmetry
breaking by accordingly taking into account the GSD’s size dependence, which
is the central idea of the non-standard first-order scaling Mueller  _et al._
(2014). Specifically, for a $d$-dimensional subsystem symmetry, the number of
equivalent ground states $q$ grows exponentially with $L^{D-d}$. In
consequence, Eq. (13) should be modified as

$\displaystyle\beta_{c}(L)=\beta_{c}^{\infty}+\frac{b}{L^{-d}(E_{d}-E_{o})}+\mathcal{O}(\frac{1}{L^{-d-1}}),$
(14)

where the constant parameter $b$ depends on specific global symmetries and, up
to the leading order, affects only the slope of the scaling.

The non-standard scaling Eq. (14) has been verified for the $3D$ plaquette
Ising model and an anisotropically coupled $3D$ Askin-Teller model Mueller
_et al._ (2014); Johnston  _et al._ (2017); both have a $d=2$ plane-flip
symmetry. As the essential idea in the above discussion is the sub-extensive
degeneracy, we expect it should generally hold in lattice models with
subsystem symmetries. Nevertheless, the phase transitions can experience
stronger finite-size effects due to the slower vanishing of correction terms.

Figure 2: Correlation functions for $H_{\text{anti-K}}$ at various
temperatures below (blue) and above (red) the phase transition. (a) Spin-spin
correlation along one of the lattice basis vectors in the ordered planes. At
low temperatures, a long-range magnetic correlation between the $S^{c}$
components arises. The symmetric form of the curves comes from periodic
boundary conditions. (b) No magnetic correlation is seen between the planes.
(c) Nematic correlation in the direction orthogonal to the planes, with a long
nematic-like order at low temperatures. A $L=30$ system is taken for example.
$N_{T}=56$ temperatures are simulated in a range $T\in[0.610,0.683]$; not all
curves are shown for better visibility. Figure 3: Binder cumulant $B_{Q}$ (a)
and reweighted energy density histogram $P(E)$ (b) near the phase transition
of $H_{\text{anti-K}}$. The pronounced Binder dips and double histogram peaks
confirm a first-order transition. Equal peak heights are found by attaching an
additional weight $e^{-(\beta-\beta^{\text{eqh}})EV}$ to a raw histogram at
inverse temperature $\beta$. $P(E)$ is renormalized and shifted by its mean
energy density. The minima between peaks fall slightly away from zero,
indicating low- and high-temperature peaks in general have different weights
at finite lattice sizes. Figure 4: Fits of $H_{\text{anti-K}}$ using the non-
standard first-order scaling equation
$\beta_{c}(L)=\beta_{c}^{\infty}-a/L^{2}$. The finite-size inverse transition
temperatures $\beta_{c}(L)$ are estimated from the extrema of $\chi$ and
$B_{Q}$ and the equal-weight peaks of $P(E)$. The largest five sizes are
considered for the fits to reduce finite size effects.

##  IV Phase transition in the anti-Kitaev limit

We first discuss the order and phase transition in the anti-Kitaev limit. We
simulate $H_{\text{anti-K}}$ with Monte Carlo techniques jointly using
parallel tempering Hukushima and Nemoto (1996), heat-bath, and microcanonical
overrelaxation updates Brown and Woch (1987). $N_{T}=56$ temperatures are
simulated in the proximity of the phase transition, whose distribution is fine
tuned to maximize iterations of the replicas Katzgraber  _et al._ (2006).
Simulations are carried out for linear lattice sizes up to $L=46$ ($\sim
10^{5}$ spins) under PBC. We typically run $10^{7}$ Monte Carlo sweeps and
determine the equilibration by comparing results of uniform and random
initializations. The system is considered equilibrated when expectation values
of all observables using different initializations agree within error bars.

The hybrid symmetry breaking in $H_{\text{anti-K}}$ and the associated order
parameters are confirmed by measuring disconnected correlators

$\displaystyle G_{S^{c}}\left(\mathbf{r}_{\in
P}\right)=\frac{1}{L^{3}}\sum_{\\{\mathbf{i}\\}}\langle
S^{c}_{\mathbf{i}}S^{c}_{\mathbf{i}+\mathbf{r}_{\in P}}\rangle,$ (15)
$\displaystyle
G_{S^{c}}(\mathbf{r}_{c})=\frac{1}{L^{3}}\sum_{\\{\mathbf{i}\\}}\langle
S^{c}_{\mathbf{i}}S^{c}_{\mathbf{i}+\mathbf{r}_{c}}\rangle,$ (16)
$\displaystyle
G_{Q}(\mathbf{r}_{c})=\frac{1}{L}\sum_{i_{c}}\langle\left(m_{P,\mathbf{i}}^{c}\right)^{2}\left(m_{P,\mathbf{i}+\mathbf{r}_{c}}^{c}\right)^{2}\rangle.$
(17)

Here, the ordered spin component $S^{c}$ varies over samples but is easily
identified by selecting an arbitrary site $\mathbf{i}=(i_{x},i_{y},i_{z})$ and
comparing $m_{P}$ for all three intersecting planes. These correlators relate
to the order parameters as $\lim\limits_{|\mathbf{r}_{\in
P}|\rightarrow\infty}G_{S^{c}}\left(\mathbf{r}_{\in
P}\right)\rightarrow\langle m_{P}^{c}\rangle^{2}$ and
$\lim\limits_{|\mathbf{r}_{c}|\rightarrow\infty}G_{Q}(\mathbf{r}_{c})\rightarrow\langle
Q^{cc}\rangle^{2}$, while $G_{S^{c}}(\mathbf{r}_{c})$ is introduced for
reference.

Table 1: Comparison between the standard ($1/L^{3}$) and non-standard ($1/L^{2}$) first-order scaling. The largest five lattice sizes in Fig. 3 and Fig. 4 are considered for fitting. The validity of the fits is evaluated by the $\chi$-square test, where smaller values of $\chi^{2}_{\rm dof}$ in general indicate better goodness. The best fits compatible with a $95\%$ confidence interval are in bold. $\qquad\qquad\,1/L^{2}$ | $1/L^{3}$
---|---
Fit |  $\beta_{\infty}$ |  ${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}^{2}_{\text{dof=3}}$ |  $\beta_{\infty}$ |  ${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}^{2}_{\text{dof=3}}$
$\beta_{C_{V}^{\text{max}}}$ | 1.566532(33) | 5.28 | 1.563882(15) | 11.03
$\beta_{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}^{\text{max}}}$ | 1.566685(21) | 6.30 | 1.563987(10) | 13.84
$\beta_{B^{\text{min}}}$ | $\mathbf{1.566883(91)}$ | $\mathbf{1.60}$ | 1.563073(34) | 5.18
$\beta_{H^{\text{eqh}}}$ | 1.566609(31) | 5.52 | 1.563968(15) | 11.29
$\beta_{H^{\text{eqw}}}$ | $\mathbf{1.566787(61)}$ | $\mathbf{2.29}$ | 1.563927(25) | 6.38
Figure 5: Correlation functions in the Kitaev limit for $L=30$ system below
(blue) and above (red) the phase transition. The intra-chain spin correlation
$G_{S}$, inter-line nematic correlation $G_{m_{L}}$, and inter-plane rank-$4$
correlation $G_{Q_{4}}$ are measured along their respective ordering
directions. Each of them shows a long range order at low temperatures. The
simulated temperature interval is indicated by $T_{\rm max}$ and $T_{\rm
min}$.

We measure the three correlators for several temperatures around $T_{c}$ at a
lattice $L=30$. As shown in Fig. 2, the spin correlators $G_{S^{c}}$ only
exhibit a long-range correlation along directions $\mathbf{r}_{\in P}$. In the
perpendicular direction $\mathbf{r}_{c}\perp P$, the spin-spin correlation
vanishes, but the planar nematic correlator $G_{Q}$ is long-ranged. This
confirms the hybrid symmetry breaking, as one will observe distinct symmetry
breaking patterns when looking at different submanifolds of the system.

To further understand the associated phase transition, we examine behaviors of
the energy histogram $P(E)$, specific heat $C_{V}$, susceptibility $\chi$, and
Binder cumulants $B$, where

$\displaystyle P(E)=\langle\delta(E-E^{\prime})\rangle,$ (18)
$\displaystyle\chi_{\mathcal{O}}=\frac{L^{D}}{T}\left(\langle\mathcal{O}^{\,2}\rangle-\langle\mathcal{O}\rangle^{2}\right),$
(19) $\displaystyle
B_{\mathcal{O}}=1-\frac{\langle\mathcal{O}^{\,4}\rangle}{3\langle\mathcal{O}^{\,2}\rangle^{2}}.$
(20)

and $\mathcal{O}=m_{P}^{c},\,Q^{cc}$. Since there is only a single phase
transition, both order parameters can detect $T_{c}$ and its scaling.
Nonetheless, we observe a slightly weaker finite size effect in $Q^{cc}$, and
hence use it as the major order parameter.

These quantities reliably determine the order of a phase transition. In
particular, near a first-order transition, the histogram $P(E)$ will show two
peaks representing the weights, $W_{o}$ and $W_{d}$, of the ordered and
disordered phases Challa  _et al._ (1986); Lee and Kosterlitz (1990).
Correspondingly, the curve of $B_{\mathcal{O}}(T)$ non-monotonically depends
on temperatures with a minimum dipping at an effective transition point Binder
and Landau (1984). Such behaviors are observed for $H_{\text{anti-K}}$ as we
plot in Fig. 3. The double $P(E)$ peaks and Binder dip become evident from
$L\sim 30$ and evolve towards $\delta$ functions when increasing system sizes,
which unambiguously confirms the first-order nature of the transition.

Effective transition temperatures $T_{c}(L)$ at a given $L$ can be estimated
from locations of the $C_{V}$ and $\chi$ peaks and that of the Binder dip. The
energy histogram provides two additional estimators: $T_{c}^{\rm eqw}(L)$ and
$T_{c}^{\rm eqh}(L)$ where $P(E)$’s two peaks have the same weight (${\rm
eqw}$) and height (${\rm eqh}$), respectively. The equal-weight criterion,
$W_{o}=W_{d}$, is understood by its relation to the $C_{V}$ maximum as
discussed in Sec. III, and can be found by minimizing

$\displaystyle\left[\Delta W(T)\right]^{2}$
$\displaystyle=\left[W_{o}(T)-W_{d}(T)\right]^{2}$
$\displaystyle=\left[\sum_{E<E_{\text{min}}}P(E,T)-\sum_{E>E_{\text{min}}}P(E,T)\right]^{2},$
(21)

with $E_{\text{min}}$ being the valley minimum between the two peaks. The
estimator of $T_{c}^{\rm eqh}(L)$ is more empirical; its common use in the
literature may be because an equal-height behavior can be easily found through
reweighting methods. These effective temperatures typically suffer different
degrees of finite-size effects but shall converge when approaching
$L\rightarrow\infty$.

To determine the infinite-volume $T_{c}^{\infty}$, or equivalently its inverse
$\beta_{c}^{\infty}$, and the leading order finite-size correction, we fit
$\beta_{c}(L)$ obtained from different estimators against single term powers
$L^{-2}$ and $L^{-3}$. The five largest sizes ($L\geq 36$) are used to ensure
$L$ sufficiently exceeds the correlation length. As summarized in Table 1, we
observe that the $L^{-2}$ fitting leads to noticeably more consistent values
of $\beta_{c}^{\infty}$, consolidating the non-standard scaling relation Eq.
(14). Fits with the best goodness are plotted in Fig. 4, and we extrapolate
$\beta_{c}^{\infty}\approx 1.5667(2)$. We expect including $L^{-3}$ as a sub-
leading term can further improve the results, while such a two-term fitting
also requires more data points.

It is interesting to know whether the accidental symmetry and its entropic
lifting play a critical role in the nature of the phase transition. For this
purpose, we replace the $O(3)$ spins in Eq. (2) with discrete $S_{3}$ spins
$\widetilde{S}_{i}=(\pm 1,0,0),(0,\pm 1,0),(0,0,\pm 1)$. By doing so,
accidental symmetries and soft modes become irrelevant, and the resulting
Hamiltonian $\widetilde{H}_{\text{anti-K}}$ formally has the same symmetries
and order parameters as $H_{\text{anti-K}}$. It turns out
$\widetilde{H}_{\text{anti-K}}$ shows even stronger jumps in energy and order
parameters. This may imply that the observed first-order nature of the phase
transitions in the two Hamiltonians may relate to the presence of the planar
subsystem symmetry.

##  V Phase transition in the Kitaev limit

We now discuss the hybrid symmetry breaking in the $H_{\rm K}$ limit. This
Hamiltonian was also studied in Ref. Gerlach and Janke (2015) based on energy
related quantities and mostly under a screw PBC that explicitly breaks line-
flip symmetries and modifies the ground-state manifold. In this section, we
consider a regular PBC preserving the subsystem symmetry and characterize its
phase transition from the viewpoints of order parameters. Simulations are
performed utilizing the same Monte Carlo methods as in Sec. IV, with system
sizes up to $L=40$.

As analyzed in Sec. II.2, the robust ground states of $H_{\rm K}$ comprise
decoupled magnetic chains. Normally, a $1D$ long-range order is prohibited for
classical spins at finite temperature due to the Mermin-Wagner theorem Mermin
and Wagner (1966). The chain magnetization here is nevertheless possible
thanks to order-by-disorder and entropic lifting of the accidental $O(3)$
degeneracy. A similar entropy driven $1D$ magnetization was also reported in a
$2D$ classical compass model Mishra  _et al._ (2004); Wenzel and Janke (2008).
For a consistency check, we also verified with separate Monte Carlo
simulations the absence of a phase transition when replacing the $O(3)$ spins
with discrete $S_{3}$ spins.

The in-chain magnetization $m_{L}$, line nematicity $Q_{L}$, and inter-plane
higher-rank nematic order $Q_{4}$ defined in Eqs. (6)– (8) lead to a rich set
of (high-rank) disconnected correlators

$\displaystyle G_{S^{a}}(r_{a})=\frac{1}{L^{3}}\sum_{\mathbf{i}}\langle
S^{a}_{\mathbf{i}}{S^{a}}_{\mathbf{i}+r_{a}\hat{\mathbf{e}}_{a}}\rangle,$ (22)
$\displaystyle
G_{m_{L}}(r_{b})=\frac{1}{L^{2}}\sum_{(i_{b},i_{c})}\langle(m^{a}_{L,(i_{b},i_{c})})^{2}(m^{a}_{L,(i_{b},i_{c})+r_{b}\hat{\mathbf{e}}_{b}})^{2}\rangle$
(23) $\displaystyle
G_{Q_{4}}(r_{c})=\frac{1}{L}\sum_{i_{c}}\langle(Q^{aa}_{L,i_{c}}-Q^{bb}_{L,i_{c}})^{2}(Q^{aa}_{L,i_{c}+r_{c}}-Q^{bb}_{L,i_{c}+r_{c}})^{2}\rangle,$
(24)

along their respective ordering directions. To probe these directions in a
given sample, one can arbitrarily choose a site
$\mathbf{i}=(i_{x},i_{y},i_{z})$ and compute the three possible line
magnetizations passing through it. The one with the largest magnitude
determines the relevant $m^{a}_{L,(i_{b},i_{c})}$ for this site. One next
compares the potential $Q^{aa}_{L,i_{c}}$ along the two remaining directions
$\hat{\mathbf{e}}_{b}$ and $\hat{\mathbf{e}}_{c}$, whose result then fixes the
common orientation of ordered planes. Nevertheless, since planes are decoupled
in ground states, the line magnetizations in different planes remain to be
determined individually (see Fig. 1).

Figure 6: Binder cumulant $B_{Q_{4}}$ (a) and reweighted energy density
histogram $P(E)$ (b), detecting a first-order phase transition in $H_{\rm K}$.
The rank-$4$ order $Q_{4}$ is considered, and the in-line magnetization
$m_{L}$ and line nematicity $Q_{L}$ give consistent results.

In Fig. 5, we measure the three correlators around the phase transition for a
lattice $L=30$. At temperatures below $T_{c}$, they all lead to long-range
order. The faster convergence of $G_{Q}$ and $G_{m_{L}}$ with distance can be
understood from their higher ranks. We also verified no long-range
correlations if the correlators were measured in a wrong direction, as in Fig.
2 for the anti-Kitaev limit. This hence confirms three distinct symmetry-
breaking patterns in different submanifolds of the system.

The nature of the phase transition is again detected by the energy density
histogram $P(E)$ and Binder cumulant $B_{\mathcal{O}}$. As shown in Fig. 6,
$H_{\rm K}$ also experiences a first-order transition.

We further crosscheck the behaviors of $C_{V}$, $P(E)$, $\chi_{\mathcal{O}}$,
and $B_{\mathcal{O}}$, with
$\mathcal{O}=m_{L}^{a},\,Q_{L}^{aa},\,Q_{4}^{aabb}$, to find the
thermodynamical transition temperature and its scaling. According to the non-
standard scaling relation Eq. (14), the leading contribution here should be
given by $L^{-1}$ instead of $L^{-3}$. Nevertheless, we observe very strong
finite-size effects, and, with system sizes $L\leq 40$, none of the single
term scaling $L^{-d^{\prime}}$ with $d^{\prime}=1,2,3$ can give satisfying
goodness of fits. A significantly improved fit is achieved when considering a
two term scaling $\beta_{c}(L)=\beta_{c}^{\infty}+aL^{-1}+bL^{-2}$, whereas,
given the limited data points, this could also be an artifact of overfitting.
Combining the standard $L^{-3}$ scaling under a screw PBC reported in Ref.
Gerlach and Janke (2015), a most likely scenario is that the line-flip
subsystem symmetry indeed has deflected $H_{\rm K}$ from the standard first-
order scaling relation, and $L^{-1}$ may still be the leading order
correction. Nevertheless, a sub-leading $L^{-2}$ term is suppressed very
slowly, which makes the single-term scaling not a good approximation for
accessible system sizes.

##  VI Summary and outlook

Subsystem symmetries provide new possibilities to explore novel states of
matter beyond existing frameworks of global and gauge symmetries. In this work
we showed that breaking them can lead to unconventional phases where the
system breaks various symmetries in different submanifolds. We referred to
this phenomenon as hybrid symmetry breaking and demonstrated it with the
compass-model limits of a classical cubic Heisenberg-Kitaev Hamiltonian
(Section II). Because of a planar subsystem symmetry, the low-temperature
phase in the anti-Kitaev limit of this model is magnetic within a plane but
nematic-like between planes. Similarly, its Kitaev limit simultaneously
possesses an in-chain magnetic order, an inter-chain nematic order, and an
inter-plane higher-rank nematic order due to a line-flip symmetry.

We further investigated their phase transitions comprehensively utilizing
large-scale Monte Carlo simulations. Our results consolidated the non-standard
first-order scaling that takes subextensive degeneracies into account. We
found strong evidence that the leading finite-size correction for the anti-
Kitaev limit follows a $L^{-2}$ scaling, instead of a conventional $L^{-3}$
behavior (Section IV). We also observed a strong violation of the conventional
scaling in the Kitaev limit and argued that single-term fits are not favored
due to a prolonged sub-leading correction (Section V).

Our work enriches the scenarios of conventional spontaneous symmetry breaking
and sheds light on the influences of subsystem symmetries in phase
transitions. The results are also useful for distinguishing genuine effects of
fractons which require additional geometrical constraints besides the presence
of a subsystem symmetry Vijay  _et al._ (2016).

Several interesting issues deserve future studies. The first question is if
this hybrid symmetry breaking is relevant for quantum systems. This is
promising as its occurrence in classical models essentially originates from
symmetries. Another exciting direction is investigating its impact on
dynamical and transport behaviors of a system. It was recently shown that a
$U(1)$ subsystem symmetry can lead to anomalous subdiffusion and new
hydrodynamical universality classes Iaconis  _et al._ (2019); Gromov  _et al._
(2020). Those subdimensional orders arising from a hybrid symmetry breaking
naturally support different quasiparticles and can potentially have distinct
diffusion properties. Moreover, it is also important to understand if there is
a generic symmetry reason why known $3D$ subsystem symmetrical models commonly
undergo a first-order phase transition. Examples aside from the two cases
studied here include the two dual spin models of the X-cube code Song  _et
al._ (2021); Mueller  _et al._ (2014), dual models of the checkerboard and
Haah’s codes Vijay  _et al._ (2016); Haah (2011); not , and a breathing
pyrochlore magnet Yan  _et al._ (2020). The answer to this question can be
crucial for the future development of field theories of subsystem symmetry
breaking.

###### Acknowledgements.

G.C., L.P., and K.L. acknowledge support from FP7/ERC Consolidator Grant
QSIMCORR, No. 771891. This project is partially funded by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s
Excellence Strategy – EXC-2111 – 390814868. The research is also part of the
Munich Quantum Valley, which is supported by the Bavarian state government
with funds from the Hightech Agenda Bayern Plus. Our numerical simulations
make use of the TKSVM library Greitemann  _et al._ (2019); Liu  _et al._
(2019) and the ALPSCore library Gaenko  _et al._ (2017). The simulations were
performed on the KCS cluster at Leibniz-Rechenzentrum (LRZ) and the ASC
cluster at Arnold Sommerfeld Center. The data used in this work will be made
available.

## References

  * Nussinov and van den Brink (2015) Z. Nussinov and J. van den Brink, Rev. Mod. Phys. 87, 1 (2015).
  * Kugel and Khomskiĭ (1982) K. I. Kugel and D. Khomskiĭ, Soviet Physics Uspekhi 25, 231 (1982).
  * van den Brink (2004) J. van den Brink, New Journal of Physics 6, 201 (2004).
  * Khaliullin (2005) G. Khaliullin, Progress of Theoretical Physics Supplement 160, 155 (2005).
  * Duan  _et al._ (2003) L.-M. Duan, E. Demler, and M. D. Lukin, Phys. Rev. Lett. 91, 090402 (2003).
  * Wu (2008) C. Wu, Phys. Rev. Lett. 100, 200406 (2008).
  * Chamon (2005) C. Chamon, Phys. Rev. Lett. 94, 040402 (2005).
  * Haah (2011) J. Haah, Phys. Rev. A 83, 042330 (2011).
  * Vijay  _et al._ (2016) S. Vijay, J. Haah, and L. Fu, Phys. Rev. B 94, 235157 (2016).
  * Nandkishore and Hermele (2019) R. M. Nandkishore and M. Hermele, Annual Review of Condensed Matter Physics 10, 295 (2019).
  * Pretko  _et al._ (2020) M. Pretko, X. Chen, and Y. You, International Journal of Modern Physics A 35, 2030003 (2020).
  * Anderson (1984) W. Anderson,  _Basic Notions Of Condensed Matter Physics_, Basic Notions of Condensed Matter Physics Series (Basic Books, 1984).
  * Kogut (1979) J. B. Kogut, Rev. Mod. Phys. 51, 659 (1979).
  * WEN (1990) X. G. WEN, International Journal of Modern Physics B 04, 239 (1990).
  * Kitaev (2003) A. Kitaev, Annals of Physics 303, 2 (2003).
  * Seiberg and Shao (2021a) N. Seiberg and S.-H. Shao, SciPost Phys. 10, 27 (2021a).
  * Seiberg and Shao (2020) N. Seiberg and S.-H. Shao, SciPost Phys. 9, 46 (2020).
  * Seiberg and Shao (2021b) N. Seiberg and S.-H. Shao, SciPost Phys. 10, 3 (2021b).
  * Villain, J.  _et al._ (1980) Villain, J., Bidaux, R., Carton, J.-P., and Conte, R., J. Phys. France 41, 1263 (1980).
  * Henley (1989) C. L. Henley, Phys. Rev. Lett. 62, 2056 (1989).
  * de Gennes and Prost (1995) P. de Gennes and J. Prost,  _The Physics of Liquid Crystals_ , International Series of Monographs on Physics (Clarendon Press, 1995).
  * Kitaev (2006) A. Kitaev, Ann. Phys. (N. Y.) 321, 2 (2006), january Special Issue.
  * Nissinen  _et al._ (2016) J. Nissinen, K. Liu, R.-J. Slager, K. Wu, and J. Zaanen, Phys. Rev. E 94, 022701 (2016).
  * Mueller  _et al._ (2014) M. Mueller, W. Janke, and D. A. Johnston, Phys. Rev. Lett. 112, 200601 (2014).
  * Fisher and Berker (1982) M. E. Fisher and A. N. Berker, Phys. Rev. B 26, 2507 (1982).
  * Lee and Kosterlitz (1991) J. Lee and J. M. Kosterlitz, Phys. Rev. B 43, 3265 (1991).
  * Janke (2003) W. Janke, “First-order phase transitions,” in  _Computer Simulations of Surfaces and Interfaces_, edited by B. Dünweg, D. P. Landau, and A. I. Milchev (Springer Netherlands, Dordrecht, 2003) pp. 111–135.
  * Borgs and Kotecký (1995) C. Borgs and R. Kotecký, Journal of Statistical Physics 79, 43 (1995).
  * Johnston  _et al._ (2017) D. A. Johnston, M. Mueller, and W. Janke, The European Physical Journal Special Topics 226, 749 (2017).
  * Hukushima and Nemoto (1996) K. Hukushima and K. Nemoto, Journal of the Physical Society of Japan 65, 1604–1608 (1996).
  * Brown and Woch (1987) F. R. Brown and T. J. Woch, Phys. Rev. Lett. 58, 2394 (1987).
  * Katzgraber  _et al._ (2006) H. G. Katzgraber, S. Trebst, D. A. Huse, and M. Troyer, Journal of Statistical Mechanics: Theory and Experiment 2006, P03018 (2006).
  * Challa  _et al._ (1986) M. S. S. Challa, D. P. Landau, and K. Binder, Phys. Rev. B 34, 1841 (1986).
  * Lee and Kosterlitz (1990) J. Lee and J. M. Kosterlitz, Phys. Rev. Lett. 65, 137 (1990).
  * Binder and Landau (1984) K. Binder and D. P. Landau, Phys. Rev. B 30, 1477 (1984).
  * Gerlach and Janke (2015) M. H. Gerlach and W. Janke, Phys. Rev. B 91, 045119 (2015).
  * Mermin and Wagner (1966) N. D. Mermin and H. Wagner, Phys. Rev. Lett. 17, 1133 (1966).
  * Mishra  _et al._ (2004) A. Mishra, M. Ma, F.-C. Zhang, S. Guertler, L.-H. Tang, and S. Wan, Phys. Rev. Lett. 93, 207201 (2004).
  * Wenzel and Janke (2008) S. Wenzel and W. Janke, Phys. Rev. B 78, 064402 (2008).
  * Iaconis  _et al._ (2019) J. Iaconis, S. Vijay, and R. Nandkishore, Phys. Rev. B 100, 214301 (2019).
  * Gromov  _et al._ (2020) A. Gromov, A. Lucas, and R. M. Nandkishore, Phys. Rev. Research 2, 033124 (2020).
  * Song  _et al._ (2021) H. Song, J. Schönmeier-Kromer, K. Liu, O. Viyuela, L. Pollet, and M. A. Martin-Delgado, arXiv preprint arXiv:2112.05122 (2021).
  * (43) In preparation.
  * Yan  _et al._ (2020) H. Yan, O. Benton, L. D. C. Jaubert, and N. Shannon, Phys. Rev. Lett. 124, 127203 (2020).
  * Greitemann  _et al._ (2019) J. Greitemann, K. Liu, and L. Pollet, Phys. Rev. B 99, 060404(R) (2019).
  * Liu  _et al._ (2019) K. Liu, J. Greitemann, and L. Pollet, Phys. Rev. B 99, 104410 (2019).
  * Gaenko  _et al._ (2017) A. Gaenko, A. Antipov, G. Carcassi, T. Chen, X. Chen, Q. Dong, L. Gamper, J. Gukelberger, R. Igarashi, S. Iskakov, M. Könz, J. LeBlanc, R. Levy, P. Ma, J. Paki, H. Shinaoka, S. Todo, M. Troyer, and E. Gull, Comput. Phys. Commun. 213, 235 (2017).


* (22) K. Noui, A. Perez, and D. Pranzetti, “Non-commutative holonomies in 2+1 LQG and Kauffman’s brackets,” J. Phys. Conf. Ser. 360 (2012) 012040, arXiv:1112.1825.
  * (23) D. Pranzetti, “Turaev-Viro amplitudes from 2+1 Loop Quantum Gravity,” Phys. Rev. D 89 (2014), no. 8, 084058, arXiv:1402.2384.
  * (24) B. Dittrich, “Diffeomorphism symmetry in quantum gravity models,” Adv. Sci. Lett. 2 (10, 2008) 151, arXiv:0810.3594.
  * (25) B. Bahr and B. Dittrich, “(Broken) Gauge Symmetries and Constraints in Regge Calculus,” Class. Quant. Grav. 26 (2009) 225011, arXiv:0905.1670.
  * (26) B. Bahr and B. Dittrich, “Breaking and restoring of diffeomorphism symmetry in discrete gravity,” AIP Conf. Proc. 1196 (2009), no. 1, 10, arXiv:0909.5688.
  * (27) V. Bonzom, M. Dupuis, F. Girelli, and E. R. Livine, “Deformed phase space for 3d loop gravity and hyperbolic discrete geometries,” arXiv:1402.2323.
  * (28) M. Dupuis, L. Freidel, F. Girelli, A. Osumanu, and J. Rennert, “On the origin of the quantum group symmetry in 3d quantum gravity,” arXiv:2006.10105.
  * (29) V. Bonzom, M. Dupuis, and F. Girelli, “Towards the Turaev-Viro amplitudes from a Hamiltonian constraint,” Phys. Rev. D 90 (2014), no. 10, 104038, arXiv:1403.7121.
  * (30) V. Bonzom and B. Dittrich, “Dirac’s discrete hypersurface deformation algebras,” Class. Quant. Grav. 30 (2013) 205013, arXiv:1304.5983.
  * (31) V. Bonzom, M. Dupuis, F. Girelli, and Q. Pan in preparation.
  * (32) M. Dupuis, F. Girelli, and E. R. Livine, “Deformed Spinor Networks for Loop Gravity: Towards Hyperbolic Twisted Geometries,” Gen. Rel. Grav. 46 (2014), no. 11, 1802, arXiv:1403.7482.
  * (33) E. Livine and J. Tambornino, “Spinor representation for loop quantum gravity,” Journal of Mathematical Physics 53 (Jan, 2012) 012503.
  * (34) M. A. Semenov-Tian-Shansky, “Poisson Lie groups, quantum duality principle, and the quantum double,” Contemporary Mathematics 175 (1994) 219–219.
  * (35) F. Girelli, A. Osumanu, and W. Wieland, “Canonical transformations generated by the boundary volume: unimodular and non-abelian teleparallel gravity,” arXiv:2101.05347.


00footnotetext: 2010 Mathematics Subject Classification: Primary: 53C20;
Secondary: 53B20, 58D19

# Local symmetry rank bound for
positive intermediate Ricci curvatures

Lawrence Mouillé  900 University Avenue
Skye Hall Room 278
Riverside, CA 92521<EMAIL_ADDRESS>
###### Abstract.

We generalize the Grove-Searle symmetry rank bound to the setting of positive
intermediate Ricci curvature using a local argument, only requiring commuting
Killing fields around a point. We address ramifications of this result related
to the Maximal Symmetry Rank Conjecture for non-negative curvature. We also
prove that our symmetry rank bound is locally optimal: Every Riemannian metric
is arbitrarily close in the $C^{1}$-topology to one that achieves the upper
bound at a point.

##  1\. Introduction

The symmetry rank of a Riemannian manifold $(M,g)$, denoted
$\operatorname{symrank}(M,g)$, is the rank of its isometry group, i.e. the
maximal dimension of a torus that can act isometrically and effectively on
$(M,g)$. In this article, we establish a symmetry rank bound for manifolds
with positive intermediate Ricci curvature:

######  Theorem A.

If $M$ is a connected Riemannian $n$-manifold with $\operatorname{Ric}_{k}>0$
for some $k\in\\{1,\dots,n-2\\}$, then

$\operatorname{symrank}(M,g)\leq\left\lfloor\frac{n+k}{2}\right\rfloor.$

The same conclusion holds if instead we assume $M$ is closed, connected, and
contains a point at which $\operatorname{Ric}_{k}>0$ for some
$k\in\\{1,\dots,n-2\\}$.

Because $\operatorname{Ric}_{1}>0$ is equivalent to $\sec>0$, this result
generalizes the symmetry rank bound established by Grove and Searle for
positively curved manifolds in [8]:

######  Maximal Symmetry Rank Theorem.

Any closed, connected Riemannian $n$-manifold $M$ with positive sectional
curvature has

$\operatorname{symrank}(M,g)\leq\left\lfloor\frac{n+1}{2}\right\rfloor,$

and in the case of equality, $M$ is diffeomorphic to $S^{n}$,
$\mathbb{R}\mathrm{P}^{n}$, $\mathbb{C}\mathrm{P}^{n/2}$, or a lens space.

Theorem A is a consequence of Theorem B below, which we state after a
discussion about positive intermediate Ricci curvature.

###  1.1. Positive intermediate Ricci curvature

In this section, we define positive intermediate Ricci curvature, discuss some
structure results and examples, and present motivation for studying positive
intermediate curvature with symmetries.

######  Definition.

Let $(M,g)$ be an $n$-dimensional Riemannian manifold, and let
$k\in\\{1,\dots,n-1\\}$. Given a $(k+1)$-dimensional subspace
$\mathcal{V}\subseteq T_{p}M$, let $R^{\mathcal{V}}$ denote the type-$(1,3)$
curvature tensor $R$ restricted to $\mathcal{V}^{3}$ composed with orthogonal
projection to $\mathcal{V}$. Then given a unit vector $u\in\mathcal{V}$,
define the $k^{\mathrm{th}}$-intermediate Ricci curvature for the pair
$(u,\mathcal{V})$ to be

$\operatorname{Ric}_{k}(u,\mathcal{V})=\operatorname{trace}\left(x\mapsto
R^{\mathcal{V}}(x,u)u\right)=\sum_{i=1}^{k}\sec(u,e_{i}),$

where $\\{e_{1},\dots,e_{k}\\}$ is any orthonormal basis for the orthogonal
complement of $\mathrm{span}\\{u\\}$ in $\mathcal{V}$.

Notice that $\operatorname{Ric}_{1}(u,\mathcal{V})$ is the sectional curvature
of the $2$-plane $\mathcal{V}$, and
$\operatorname{Ric}_{n-1}(u,T_{p}M)$ is the Ricci curvature
$\operatorname{Ric}(u,u)$.

######  Definition.

We say $(M,g)$ has positive $k^{\mathrm{th}}$-intermediate Ricci curvature if
$\operatorname{Ric}_{k}(u,\mathcal{V})>0$ for all pairs $(u,\mathcal{V})$ in
the tangent bundle $TM$. We abbreviate this by writing
$\operatorname{Ric}_{k}(M,g)>0$, omitting $M$ or $g$ when they are understood.

Thus $\operatorname{Ric}_{1}>0$ is equivalent to $\sec>0$, and
$\operatorname{Ric}_{n-1}>0$ is equivalent to $\operatorname{Ric}>0$.
Furthermore, it is easy to check that if $\operatorname{Ric}_{k}>0$ for some
$k\in\\{1,\dots,n-1\\}$, then $\operatorname{Ric}_{l}>0$ for all $l\geq k$. In
particular, $\operatorname{Ric}_{k}>0$ implies that $\operatorname{Ric}>0$. So
if a compact manifold $M$ has $\operatorname{Ric}_{k}>0$, then by the Bonnet-
Myers theorem, $\pi_{1}(M)$ is finite. We also have the following structure
results for $\operatorname{Ric}_{k}>0$ that go beyond what has been
established for $\operatorname{Ric}>0$:

  * •

If $M$ is open and $\operatorname{Ric}_{k}(M)>0$, then $M$ has the homotopy
type of a CW-complex with cells each of dimension $\leq k-1$. [25]

  * •

Suppose $\operatorname{Ric}_{k}(M)\geq k$ and the first systole of $M$
$>\pi\sqrt{\frac{k-1}{k}}$.

    1. (1)

If $M$ is even-dimensional and orientable, then it is simply connected.

    2. (2)

If $M$ is odd-dimensional, then it is orientable. [26]

  * •

If $\operatorname{Ric}_{k}(M)\geq k$ and $M$ contains a point for which the
conjugate radius $>\frac{\pi}{2}$, then the universal cover of $M$ is
$(n-k)$-connected. [10]

For more, see [1], [9], [11], [12], [14, Theorem 6.1], [24], [26, Theorem 11],
[27, Remark 2.4], or [29]. It appears there have been no attempts to
systematically document examples of manifolds with positive intermediate Ricci
curvature which do not have positive sectional curvature. Here, we list two
sources of examples:

  * •

Consider any positively curved manifolds $M^{m}$ and $N^{n}$. It is an easy
exercise to show that with respect to the product metric,
$\operatorname{Ric}_{k}(M^{m}\times N^{n})>0$ only for $k\geq\max\\{m,n\\}+1$.
Improving upon this, the author has shown that products of positively curved
homogeneous spaces admit metrics with $\operatorname{Ric}_{2}>0$ [15].

  * •

It follows from [16] that a compact homogeneous space $G/H$ admits a metric
with positive Ricci curvature if and only if $\pi_{1}(G/H)$ is finite.
Furthermore, if the rank of $G/H$ is $r$, then
$\operatorname{Ric}_{r-1}(G/H)\not>0$. Thus $\operatorname{Ric}_{k}(G/H)>0$
for some $k\in\\{\operatorname{rank}(G/H),\dots,\dim(G/H)-1\\}$.

In the setting of positive curvature, the Grove symmetry program has been
effective in inspiring innovative results. The goal of this program is to
classify positively curved manifolds that have large isometry groups. This
initiative has led to advances in global Riemannian geometry, providing
symmetry obstructions for positive curvature like the Maximal Symmetry Rank
Theorem [8]. Attempts to uncover connections between curvature and symmetries
have also helped in creating new examples of positive or non-negative
curvature, a large area of research in its own right. Because the Grove
symmetry program has proven to be so fruitful, the author’s intention is to
initiate a study of interactions between positive intermediate Ricci curvature
and symmetries.

###  1.2. Local symmetry rank bound

In this section, we introduce the key results of the article, Theorem B and
Theorem C.

The symmetry rank bound from the Grove-Searle Maximal Symmetry Rank Theorem
[8] is proven using global arguments. It relies on globally defined torus
actions and a theorem of Berger stating that any Killing field on an even
dimensional positively curved manifold has a zero. Wilking obtained the same
bound for quasi-positive curvature using only the Gauss equation [28], and
Galaz-García included Wilking’s argument in [5]. Searle and Wilhelm noticed
that Wilking’s argument only requires commuting Killing fields and positive
sectional curvature for planes spanned by the Killing fields [23].

Informed by these observations, our first key result is a local symmetry rank
bound for positive intermediate Ricci curvature:

######  Theorem B.

Suppose that $M$ is a Riemannian $n$-manifold, $N\subset M$ is a submanifold
through a point $p$, the tangent space $T_{p}N$ is spanned by commuting
Killing fields of $N$, and $k\in\\{1,\dots,n-2\\}$. If
$\operatorname{Ric}_{k}(u,\mathcal{V})>0$ for all $(k+1)$-dimensional
subspaces $\mathcal{V}\subseteq T_{p}N$ and all $u\in\mathcal{V}$, then

$\dim(N)\leq\left\lfloor\frac{n+k}{2}\right\rfloor.$

Note that the submanifold $N$ in Theorem B is not assumed to be complete; it
is only required to be defined in a neighborhood of the point $p$. Also notice
the utility of Theorem B is that it is local in nature, and thus can be
applied in many scenarios. Now when a torus acts isometrically and effectively
on a Riemannian manifold, the principal isotropy groups are trivial, and the
tangent spaces to any principal orbit are spanned by commuting Killing fields.
Thus, the first statement in Theorem A follows immediately from Theorem B. The
second statement in Theorem A follows by applying Theorem B on the principal
orbits of a torus action with rank $>\frac{n+k}{2}$ and then using a
compactness argument to reach a contradiction.

Now, we establish terminology for when the upper bound in Theorem B is
achieved:

######  Definition.

Suppose $M$ has a submanifold $N$ such that for some point $p\in N$, $T_{p}N$
is spanned by commuting Killing fields of $N$, and
$\operatorname{Ric}_{k}(u,\mathcal{V})>0$ for all $(k+1)$-dimensional
subspaces $\mathcal{V}\subseteq T_{p}N$ and all $u\in\mathcal{V}$. If
$\dim(N)=\lfloor\frac{n+k}{2}\rfloor$, then we say $M$ has $k$-maximal local
symmetry rank at $p$.

Notice that no manifold admits $(n-1)$-maximal local symmetry rank: If
$k=n-1$, then $\left\lfloor\frac{n+k}{2}\right\rfloor=n-1<n=k+1$. So when
$k=n-1$, there can be no $(k+1)$-dimensional subspaces $\mathcal{V}$ of
$T_{p}N$ while $\dim(N)=\lfloor\frac{n+k}{2}\rfloor$.

Recall from the Maximal Symmetry Rank Theorem that the list of manifolds which
admit positive sectional curvature and maximal global symmetry rank is
restrictive, only consisting of $S^{n}$, $\mathbb{R}\mathrm{P}^{n}$,
$\mathbb{C}\mathrm{P}^{n/2}$, or lens spaces. In stark contrast, we have the
following density result for $k$-maximal local symmetry rank:

######  Theorem C.

Let $M$ be an $n$-manifold, $n\geq 3$, $k\in\\{1,\dots,n-2\\}$, and $p\in M$.
Every Riemannian metric $g$ on $M$ is arbitrarily close in the
$C^{1}$-topology to a metric $\tilde{g}$ such that $(M,\tilde{g})$ has
$k$-maximal local symmetry-rank at $p$.

Theorem C shows that the upper bound in Theorem B can be realized on any
manifold. We prove Theorem C by sewing in a model metric on a small ball
around the point $p$ and showing that the resulting metric can be made
arbitrarily close in the $C^{1}$-distance to the original. These model metrics
are constructed on $\mathbb{R}^{n}$ and have $k$-maximal local symmetry rank
at the origin.

###  1.3. Global symmetry rank

In this section, we discuss the global symmetry rank bound in Theorem A.
Comparing Theorem A to the Maximal Symmetry Rank Theorem leads naturally to
the following:

######  Questions.

Is the symmetry rank bound in Theorem A optimal? Can we classify $n$-manifolds
with $\operatorname{Ric}_{k}>0$ that have large symmetry-rank?

In order to make progress towards addressing these and other questions
concerning positive intermediate Ricci curvature, it would be beneficial to
find non-trivial examples of manifolds with $\operatorname{Ric}_{k}>0$. By
this, we suggest finding and cataloging examples where
$\operatorname{Ric}_{k}>0$ while $\operatorname{Ric}_{k-1}\not>0$ for $k\geq
2$. See [15] for new examples of manifolds with $\operatorname{Ric}_{2}>0$,
many of which do not admit metrics with $\sec>0$.

In the $k=n-2$ case, the symmetry rank bound from Theorem A states that
$\operatorname{symrank}(M,g)\leq\big{\lfloor}\frac{n+(n-2)}{2}\big{\rfloor}=n-1$.
Cohomogeneity-one torus actions actions may occur on Ricci-positive manifolds
in dimensions $2$ and $3$; e.g. the $T^{1}$-action on $S^{2}$ and the
$T^{2}$-action on $S^{3}$. However, it is known that in dimensions $\geq 4$,
closed manifolds that admit cohomogeneity-one torus actions must have infinite
fundamental group [17, 18]. Thus by the Bonnet-Myers theorem, such manifolds
cannot admit invariant metrics of positive Ricci curvature, and hence cannot
admit $\operatorname{Ric}_{n-2}>0$. Therefore, we have the following:

######  Remark 1.1.

If $M$ is a closed, connected $n$-manifold with $\operatorname{Ric}_{k}>0$ for
$n\geq 4$ and $k\geq n-2$, then

$\operatorname{symrank}(M,g)\leq n-2.$

Corro and Galaz-García show in [2] that for each dimension $n\geq 6$, there
exist examples of $n$-manifolds which admit a metric of positive Ricci
curvature with $\operatorname{symrank}(M,g)=n-4$. It remains to be shown if
this can be improved to give examples of positive Ricci curvature with
$\operatorname{symrank}(M,g)=n-2$.

###  1.4. Ramifications in non-negative curvature

In this section, we discuss a consequence of Theorem A in the context of non-
negative curvature. Specifically, Galaz-García and Searle state the Maximal
Symmetry Rank Conjecture for non-negatively curved manifolds in [7], which has
since been sharpened by Escher and Searle in [4]:

######  Maximal Symmetry Rank Conjecture.

Let $(M,g)$ be a closed, simply connected, $n$-dimensional Riemannian manifold
with non-negative sectional curvature. Then

  1. (1)

$\displaystyle\operatorname{symrank}(M,g)\leq\left\lfloor\frac{2n}{3}\right\rfloor$,
and

  2. (2)

in the case of equality, $M$ is equivariantly diffeomorphic to a product of
spheres or a quotient thereof by a free linear action of a torus of rank less
than or equal to $2n\;\mathrm{mod}\;3$.

It was shown that the Maximal Symmetry Rank Conjecture holds for $4\leq n\leq
6$ by Galaz-García and Searle in [7], it holds for $7\leq n\leq 9$ by Escher
and Searle in [4], and also Part (1) holds for $10\leq n\leq 12$ in [4].
Furthermore, the Maximal Symmetry Rank Conjecture has been confirmed in all
dimensions for torus actions that are isotropy-maximal in [4]. With the
assumption of non-negative curvature replaced with rational ellipticity, Part
(1) has been established in all dimensions along with a rational homotopy
theoretic version of Part (2) by Galaz-García, Kerin, and Radeschi in [6].
This result is relevant to the conjecture above because the Bott Conjecture
claims that any non-negatively curved manifold is rationally elliptic.

Notice that a non-negatively curved manifold can have positive
$k^{\mathrm{th}}$-intermediate Ricci curvature for $k\geq 2$ without being
positively curved. If $\operatorname{Ric}_{k}>0$ at a point for
$k\leq\frac{n}{3}$, then by Theorem A,
$\operatorname{symrank}(M,g)\leq\lfloor\frac{2n}{3}\rfloor$. In particular, we
have the following:

######  Corollary D.

Let $(M,g)$ be a closed, simply connected Riemannian manifold with non-
negative section curvature and dimension $n\geq 13$. If $M$ contains a point
at which $\operatorname{Ric}_{\lfloor n/3\rfloor}>0$, then the conclusion of
Part (1) of the Maximal Symmetry Rank Conjecture holds.

###  1.5. Structure of this article

In Section 2, we prove Theorem B. In Section 3, we construct metrics on
$\mathbb{R}^{n}$ that have $k$-maximal local symmetry rank at the origin for
$n\geq 3$ and $k\in\\{1,\dots,n-2\\}$. In Section 4, we prove Theorem C by
establishing a result about sewing metrics in the $C^{1}$-topology and
applying it to the metrics constructed in Section 3.

###  1.6. Acknowledgments

This article is part of my doctoral thesis at the University of California,
Riverside, and many of the results were completed while I was supported by the
university’s Dissertation Year Program Award. I thank my thesis advisor, Fred
Wilhelm, for his insights and valuable feedback during the preparation of this
article. I also thank Priyanka Rajan, Karsten Grove, and Guofang Wei for
helpful discussions concerning this project. Finally, I thank the referee,
whose detailed comments and suggestions led to significant improvements of the
exposition and presentation in this article.

##  2\. Symmetry rank bound

In this section, we prove Theorem B. Our argument is informed by Wilking’s
proof of the symmetry rank bound for quasi-positive curvature [28], which may
be found in Galaz-García’s article [5].

Assume that $M$ and $N$ are as in Theorem B. Let $\operatorname{I\\!I}$ denote
the second fundamental form for $N\subset M$, i.e. for $u,v\in T_{p}N$,
$\operatorname{I\\!I}(u,v)=(\nabla_{u}V)^{\perp}\in T_{p}N^{\perp}$, where $V$
is any extension of $v$ to a vector field. Given a unit vector $u\in T_{p}N$,
let $\mathcal{O}_{u}$ denote the orthogonal complement of
$\mathrm{span}\\{u\\}$ in $T_{p}N$. Now considering
$\operatorname{I\\!I}(u,\cdot)$ as a linear map $\mathcal{O}_{u}\to
T_{p}N^{\perp}$, we have the following:

######  Lemma 2.1.

If $\dim(N)>\frac{n+k}{2}$, then $\dim(\ker\operatorname{I\\!I}(u,\cdot))\geq
k$.

###### Proof.

If $\dim(N)>\frac{n+k}{2}$, then

$\displaystyle\dim(\ker\operatorname{I\\!I}(u,\cdot))=\dim(\mathcal{O}_{u})-\dim(\mathrm{Im}\operatorname{I\\!I}(u,\cdot))$
$\displaystyle\geq\left(\dim(N)-1\right)-\left(n-\dim(N)\right)$
$\displaystyle>\left(\frac{n+k}{2}-1\right)-\left(n-\frac{n+k}{2}\right).$

Therefore, $\dim(\ker\operatorname{I\\!I}(u,\cdot))>k-1$. ∎

We now prove our main theorem using Lemma 2.1 and the Gauss equation.

###### Proof of Theorem B.

Suppose $\dim(N)>\frac{n+k}{2}$, and fix a unit vector $u\in T_{p}N$ such that
$|\operatorname{I\\!I}(u,u)|\geq 0$ is minimal. By Lemma 2.1, we can choose
orthonormal $e_{1},\dots,e_{k}\in\mathcal{O}_{u}$ such that
$\operatorname{I\\!I}(u,e_{i})=0$. For $i=1,\dots,k$, define
$f_{i}:\mathbb{R}\to\mathbb{R}$ by

$\displaystyle f_{i}(\theta)$
$\displaystyle=\left|\operatorname{I\\!I}\left(\cos(\theta)\,u+\sin(\theta)\,e_{i},\cos(\theta)\,u+\sin(\theta)\,e_{i}\right)\right|^{2}$
$\displaystyle=\cos^{4}(\theta)|\operatorname{I\\!I}(u,u)|^{2}+2\cos^{2}(\theta)\sin^{2}(\theta)\langle\operatorname{I\\!I}(u,u),\operatorname{I\\!I}(e_{i},e_{i})\rangle+\sin^{4}(\theta)|\operatorname{I\\!I}(e_{i},e_{i})|^{2}.$

Then because $f_{i}(0)=|\operatorname{I\\!I}(u,u)|^{2}$ is minimal,

$f_{i}^{\prime\prime}(0)=4\left(\langle\operatorname{I\\!I}(u,u),\operatorname{I\\!I}(e_{i},e_{i})\rangle-|\operatorname{I\\!I}(u,u)|^{2}\right)\geq
0,$

and so we must have
$\langle\operatorname{I\\!I}(u,u),\operatorname{I\\!I}(e_{i},e_{i})\rangle\geq
0$. Now let $\mathcal{V}=\mathrm{span}\\{u,e_{1},\dots,e_{k}\\}\subseteq
T_{p}N$. Because $\operatorname{I\\!I}(u,e_{i})=0$ and
$\langle\operatorname{I\\!I}(u,u),\operatorname{I\\!I}(e_{i},e_{i})\rangle\geq
0$ for all $i$, we have by the Gauss equation that

$\displaystyle\operatorname{Ric}_{k}(u,\mathcal{V})$
$\displaystyle=\sum_{i=1}^{k}\sec_{M}(u,e_{i})$
$\displaystyle=\sum_{i=1}^{k}\left[\sec_{N}(u,e_{i})+|\operatorname{I\\!I}(u,e_{i})|^{2}-\langle\operatorname{I\\!I}(u,u),\operatorname{I\\!I}(e_{i},e_{i})\rangle\right]$
$\displaystyle\leq\sum_{i=1}^{k}\sec_{N}(u,e_{i})$

However, because $T_{p}N$ is spanned by commuting Killing fields,
$\sec_{N}\equiv 0$ on $T_{p}N$. Therefore we have shown that if
$\dim(N)>\frac{n+k}{2}$, then there exists a pair $(u,\mathcal{V})$ such that
$\operatorname{Ric}_{k}(u,\mathcal{V})\leq 0$, thus proving Theorem B by
contraposition. ∎

##  3\. Construction of $k$-maximal local symmetry rank

In this section, we prove the following:

######  Proposition 3.1.

There exist metrics $g^{\mathrm{model}}$ on $\mathbb{R}^{n}$ that have
$k$-maximal local symmetry rank at the origin for all $n\geq 3$ and
$k\in\\{1,\dots,n-2\\}$.

Throughout this section, let $n\geq 3$, $k\in\\{1,\dots,n-2\\}$, and
$d=\lfloor\frac{n+k}{2}\rfloor$. To prove Proposition 3.1, we will construct
metrics on $\mathbb{R}^{n-d}\times\mathbb{R}^{d}$ such that the coordinate
vector fields for the $\mathbb{R}^{d}$ factor are Killing fields and
$\operatorname{Ric}_{k}(u,\mathcal{V})>0$ for all $(k+1)$-dimensional
subspaces $\mathcal{V}\subseteq T_{0}\mathbb{R}^{d}$. We do not claim that
these metrics are complete as this will not be necessary for proving Theorem C
in Section 4. First, we establish the following computational simplification:

######  Lemma 3.2.

Given any Riemannian manifold $M$ and a natural number $d\leq\dim(M)-1$,
suppose that $\\{K_{i}\\}_{i=1}^{d}$ is an orthonormal basis of a subspace
$\mathcal{K}\subseteq T_{p}M$, $k\in\\{1,\dots,d-1\\}$, and there exist
$\mu,\nu\in[0,\infty)$ such that the following hold:

  1. (A)

$R(K_{i},K_{j})K_{l}=0$ when the indices $i,j,l$ are mutually distinct.

  2. (B)

$\sec(K_{i},K_{j})\in\\{-\nu,\mu\\}$ for all $i\neq j$.

  3. (C)

For each $i$, there exist at most $k-1$ indices $j\neq i$ such that
$\sec(K_{i},K_{j})=-\nu$.

  4. (D)

$\mu-(k-1)\nu>0$.

Then $\operatorname{Ric}_{k}(u,\mathcal{V})>0$ for all $(k+1)$-dimensional
subspaces $\mathcal{V}\subseteq\mathcal{K}$ and $u\in\mathcal{V}$.

###### Proof.

Let $\operatorname{Ric}^{\mathcal{K}}:\mathcal{K}\to\mathcal{K}$ denote the
Ricci $(1,1)$-tensor restricted to $\mathcal{K}$ and composed with projection
onto $\mathcal{K}$. Then by definition,
$\langle\operatorname{Ric}^{\mathcal{K}}(u),u\rangle=\operatorname{Ric}_{d-1}(u,\mathcal{K})$.
By Property (A), we get that $\operatorname{Ric}^{\mathcal{K}}$ is
diagonalized by $\\{K_{i}\\}_{i=1}^{d}$; see, for example, [19, Proposition
4.1.3]. Then by Properties (B) and (C), we have that

$\operatorname{Ric}_{d-1}(u,\mathcal{K})\geq\min_{i=1,\dots,d}\\{\operatorname{Ric}_{d-1}(K_{i},\mathcal{K})\\}\geq(d-k)\mu-(k-1)\nu,\\\
$

for all $u\in\mathcal{K}$. Applying Property (D), we get

$\operatorname{Ric}_{d-1}(u,\mathcal{K})>(d-k-1)\mu.$ (1)

Now, define the operator
$\mathfrak{R}^{\mathcal{K}}:\Lambda^{2}\mathcal{K}\to\Lambda^{2}\mathcal{K}$
by $\langle\mathfrak{R}^{\mathcal{K}}\left(\textstyle\sum_{i}X_{i}\wedge
Y_{i}\right),\textstyle\sum_{j}V_{j}\wedge
W_{j}\rangle=\sum_{i,j}R(X_{i},Y_{i},W_{j},V_{j})$, which is the curvature
operator restricted to the subspace $\mathcal{K}$. Then by Property (A),
$\mathfrak{R}^{\mathcal{K}}$ is diagonalized by $\\{K_{i}\wedge
K_{j}\\}_{i,j}$; see, for example, [19, Proposition 4.1.2]. Recall that by
Property (B), on $\mathcal{K}$ we have

$\sec\leq\mu.$ (2)

Thus, given a $(k+1)$-dimensional subspace $\mathcal{V}\subseteq\mathcal{K}$
and unit vector $u\in\mathcal{V}$, choose orthonormal vectors
$\\{e_{i}\\}_{i=1}^{d-1}$ such that $\\{u,e_{1},\dots,e_{k}\\}$ is an
orthonormal basis for $\mathcal{V}$ and
$\\{u,e_{1},\dots,e_{k},e_{k+1},\dots,e_{d-1}\\}$ is an orthonormal basis for
$\mathcal{K}$. Then by Inequalities 1 and 2, we have

$\displaystyle\operatorname{Ric}_{k}(u,\mathcal{V})$
$\displaystyle=\sum_{i=1}^{k}\sec(u,e_{i})$
$\displaystyle=\operatorname{Ric}_{d-1}(u,\mathcal{K})-\sum_{i=k+1}^{d-1}\sec(u,e_{i})$
$\displaystyle>\left[(d-k-1)\mu\right]-\left[(d-k-1)\mu\right]=0.$

∎

Now we use Lemma 3.2 to construct metrics with $k$-maximal local symmetry
rank:

###### Proof of Proposition 3.1.

Consider $\mathbb{R}^{n-d}\times\mathbb{R}^{d}$ with coordinates
$(x_{1},\dots,x_{n-d},y_{1},\\\ \dots,y_{d})$. Given positive smooth functions
$\phi_{i}:\mathbb{R}^{n-d}\to\mathbb{R}$, define the metric
$g^{\mathrm{model}}$ on $\mathbb{R}^{n-d}\times\mathbb{R}^{d}$ by

$g^{\mathrm{model}}={dx_{1}}^{2}+\dots+{dx_{n-d}}^{2}+{\phi_{1}}^{2}{dy_{1}}^{2}+\dots+{\phi_{d}}^{2}{dy_{d}}^{2}.$

Setting $K_{i}=\frac{1}{\phi_{i}(0)}\frac{\partial}{\partial y_{i}}$, the
fields $\\{K_{i}\\}_{i=1}^{d}$ are the desired commuting Killing fields under
this metric, and they are orthonormal at the origin. We will choose the
$\phi_{i}$ such that $\left(\mathbb{R}^{n},g^{\mathrm{model}}\right)$ and
$K_{i}$ together satisfy the hypotheses of Lemma 3.2.

Let $\mathcal{K}=\mathrm{span}\\{K_{i}|_{0}\\}_{i=1}^{d}=T_{0}\mathbb{R}^{d}$,
and denote the orthogonal complement by
$\mathcal{K}^{\perp}=T_{0}\mathbb{R}^{n-d}$. Let
$\operatorname{I\\!I}:\mathcal{K}\times\mathcal{K}\to\mathcal{K}^{\perp}$
denote the second fundamental form for the submanifold
$\\{(0,\dots,0)\\}\times\mathbb{R}^{d}$. Then
$\operatorname{I\\!I}(K_{i},K_{j})=0$ for $i\neq j$, and combining this with
the fact that $K_{i}$ are commuting Killing fields, we get that Property (A)
of Lemma 3.2 is satisfied. We will show that the $\phi_{i}$ can be chosen such
that Properties (B), (C), and (D) are satisfied at the origin. Notice that at
the origin,
$\operatorname{I\\!I}(K_{i},K_{i})=-\frac{\operatorname{grad}\phi_{i}}{\phi_{i}(0)}$.
Thus by the Gauss equation, because the submanifold
$\\{(0,\dots,0)\\}\times\mathbb{R}^{d}$ has zero intrinsic curvature, we have
that the extrinsic curvatures are

$\sec(K_{i},K_{j})=-\frac{\langle\operatorname{grad}\phi_{i},\operatorname{grad}\phi_{j}\rangle}{\phi_{i}(0)\phi_{j}(0)}$

when $i\neq j$. We now break the construction into two cases: $k=n-2$ and
$k\leq n-3$.

Case $\mathbf{k=n-2}$: If $k=n-2$, then $d=\lfloor\frac{n+k}{2}\rfloor=n-1$,
and so $\mathcal{K}^{\perp}$ is $1$-dimensional. Choose a unit vector $U$ in
$\mathcal{K}^{\perp}$. We will define the $\phi_{i}$ such that for some
constants $a,b\in(0,\infty)$,

  1. (1)

$\operatorname{grad}\phi_{1}=\dots=\operatorname{grad}\phi_{k}=U$,

  2. (2)

$\phi_{1}(0)=\dots=\phi_{k}(0)=a$,

  3. (3)

$\operatorname{grad}\phi_{d}=-U$, and

  4. (4)

$\phi_{d}(0)=b$

Then by construction, Properties (B) and (C) of Lemma 3.2 are satisfied with

$\displaystyle\mu$ $\displaystyle=\frac{|U|^{2}}{ab}=\frac{1}{ab},$
$\displaystyle-\nu$ $\displaystyle=-\frac{|U|^{2}}{a^{2}}=-\frac{1}{a^{2}}.$

Thus, to satisfy Property (D), we need

$\frac{1}{ab}-\frac{k-1}{a^{2}}>0.$

Therefore, choosing any values for $a$ and $b$ such that $a>(k-1)b$, the
$k=n-2$ case of Proposition 3.1 follows from Lemma 3.2.

Case $\mathbf{k\leq n-3}$: If $k\leq n-3$, then $\mathcal{K}^{\perp}$ has
dimension $n-d\geq 2$. In the unit sphere
$S^{n-d-1}\subset\mathcal{K}^{\perp}$, choose a vector $U$. Letting
$U^{\perp}$ denote the orthogonal complement of $\mathrm{span}\\{U\\}$ in
$\mathcal{K}^{\perp}$, consider the equatorial sphere $S^{n-d-2}=U^{\perp}\cap
S^{n-d-1}$. Now inscribe a regular $(n-d-1)$-simplex in $S^{n-d-2}$, and
define vectors $V_{1}(0),\dots,V_{n-d}(0)\in S^{n-d-2}$ to be the vertices of
this simplex. Hence

$\langle V_{i}(0),V_{j}(0)\rangle=-\frac{1}{n-d-1}$

for all distinct $i,j\in\\{1,\dots,n-d\\}$. Now for
$\theta\in[0,\frac{\pi}{2}]$ define

$V_{i}(\theta)=\cos(\theta)V_{i}(0)-\sin(\theta)U.$

Notice that

$\langle
V_{i}(\theta),V_{j}(\theta)\rangle=-\frac{\cos^{2}\theta}{n-d-1}+\sin^{2}\theta$

is strictly increasing on $[0,\frac{\pi}{2}]$ and takes the value $0$ at
$\theta=\xi\in(0,\frac{\pi}{2})$ given by

$\xi=\arctan\left(\sqrt{\frac{1}{n-d-1}}\right).$

Hence, $\langle V_{i}(\theta),V_{j}(\theta)\rangle$ is negative for
$\theta\in[0,\xi)$. We will now choose the $\phi_{i}$ such that

  1. (1)

$\operatorname{grad}\phi_{1}=\dots=\operatorname{grad}\phi_{k}=U$,

  2. (2)

$\phi_{1}(0)=\dots=\phi_{k}(0)=a$,

  3. (3)

$\operatorname{grad}\phi_{k+j}=V_{j}(\theta)$ for $j=1,\dots,n-d$, and

  4. (4)

$\phi_{k+1}(0)=\dots=\phi_{d}(0)=b$

for some values of $a,b\in(0,\infty)$ and $\theta\in(0,\xi)$. See Figure 1 for
an illustration of how the $\operatorname{grad}\phi_{i}$ may be arranged in
$S^{n-d-1}$.

Figure 1. $\operatorname{grad}\phi_{i}$ in $S^{n-d-1}$

Then because $\theta<\xi$, the negative curvatures among
$\\{\sec(K_{i},K_{j})\\}_{i,j=1}^{d}$ correspond to distinct values of
$i,j\in\\{1,\dots,k\\}$, and these curvatures all have the same value

$-\nu\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=-\frac{|U|^{2}}{a^{2}}=-\frac{1}{a^{2}}.$

So for each $i$, there exist at most $k-1$ indices $j\neq i$ such that
$\sec(K_{i},K_{j})=-\nu$, and thus Property (C) of Lemma 3.2 is satisfied. To
satisfy Property (B), we need to choose $a,b,\theta$ so that for all
$l\in\\{1,\dots,k\\}$ and for all distinct $i,j\in\\{1,\dots,n-d\\}$,

$\sec(K_{l},K_{k+i})=\sec(K_{k+i},K_{k+j}).$

This common value is given by

$\mu\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=-\frac{\langle
U,V_{i}(\theta)\rangle}{ab}=-\frac{\langle
V_{i}(\theta),V_{j}(\theta)\rangle}{b^{2}}.$ (3)

Furthermore, to satisfy Property (D), we need

$\mu-\frac{k-1}{a^{2}}>0.$ (4)

Now choose any value for $a>0$. Then, once $\theta\in(0,\xi)$ is chosen, we
will define $b$ by

$b=\frac{a\langle V_{i}(\theta),V_{j}(\theta)\rangle}{\langle
U,V_{i}(\theta)\rangle}.$

Thus Equation 3 will be satisfied with $\mu$ taking the value

$\mu=-\frac{\langle U,V_{i}(\theta)\rangle^{2}}{a^{2}\langle
V_{i}(\theta),V_{j}(\theta)\rangle}.$

Finally, Inequality 4 holds if $\theta$ is chosen such that

$\langle U,V_{i}(\theta)\rangle^{2}>-(k-1)\langle
V_{i}(\theta),V_{j}(\theta)\rangle.$

Notice that $\langle U,V_{i}(\theta)\rangle^{2}$ approaches a positive
constant dependent on $n$ and $k$ as $\theta$ approaches $\xi$, while
$-\langle V_{i}(\theta),V_{j}(\theta)\rangle$ approaches $0$ as $\theta$
approaches $\xi$. Therefore, there exists a value $\theta$ such that
Inequality 4 holds. Therefore, applying Lemma 3.2, Proposition 3.1 is proven.
∎

##  4\. Sewing and the $C^{1}$-topology

In this section, we establish a general Sewing Theorem for changing a
Riemannian metric within metric ball around a point while remaining close to
the original metric in the $C^{1}$-topology. We then use the Sewing Theorem to
Prove Theorem C. First, we discuss the $C^{l}$-norm for tensors.

###  4.1. $C^{l}$-norm for tensors

Recall that two smooth maps $F_{1},F_{2}:M\to N$ are $\epsilon$-close in the
weak $C^{l}$-topology if their values and partial derivatives up to order $l$
are $\epsilon$-close with respect to fixed atlases on $M$ and $N$; see Chapter
2 of [13]. If the atlases are finite, this leads to a notion of
$C^{l}$-distance.

Now, given vector bundles $E_{1}$ and $E_{2}$, Euclidean metrics on $E_{1}$
and $E_{2}$, and a bundle map $\phi:E_{1}\to E_{2}$, the $C^{l}$-norm of
$\phi$ is defined as follows: If $E_{1}^{1}$ denotes the unit sphere bundle
under the Euclidean metric on $E_{1}$, then $|\phi|_{C^{l}}$ is the
$C^{l}$-distance from $\phi|_{E_{1}^{1}}$ to the zero bundle map $E_{1}\to
E_{2}$. The $C^{l}$-norm of a tensor $\omega$ is the $C^{l}$-distance from
$\omega$ to the zero-section. Notice that these definitions depend on the
choice of Euclidean metrics.

Throughout this section, let $M$ be a fixed manifold, and let $g$ be a fixed
Riemannian metric on $M$. All $C^{l}$-norms will be defined in terms of the
fixed metric $g$ on $TM$.

###  4.2. Sewing theorem

We now establish the following:

######  Theorem 4.1 (Sewing Theorem).

Let $(M,g)$ be a Riemannian manifold, let $p$ be a point in $M$, and let
$g^{*}$ be a Riemannian metric defined on a neighborhood of $p$ such that
$g(\gamma^{\prime},\cdot)=g^{*}(\gamma^{\prime},\cdot)$ for all geodesic rays
$\gamma$ emanating from $p$. For every $\epsilon>0$, there exists a Riemannian
metric $\tilde{g}$ on $M$ that is $\epsilon$-close to $g$ in the
$C^{1}$-distance such that $\tilde{g}\equiv g^{*}$ on an open ball centered at
$p$.

The author has been informed that Searle, Solórzano, and Wilhelm have proven
if $\sec(M,g)\geq K$, then $\tilde{g}$ in Theorem 4.1 can be made to satisfy
$\sec(M,\tilde{g})\geq\tilde{K}$ for any $\tilde{K}<K$ [22].

Now define $t$ to be the value of $\operatorname{dist}_{g}(p,\cdot)$, and let
$\partial_{t}$ denote $\operatorname{grad}(\operatorname{dist}_{g}(p,\cdot))$.
We begin by proving the following:

######  Lemma 4.2 (Converse Gauss Lemma).

Let $(M,g)$ be a Riemannian manifold, let $p\in M$, and let $g^{*}$ be as in
Theorem 4.1.

  1. (1)

$\operatorname{dist}_{g}(p,\cdot)=\operatorname{dist}_{g^{*}}(p,\cdot)$, and
the integral curves of $\partial_{t}$ are geodesics emanating from $p$ with
respect to both metrics $g$ and $g^{*}$.

  2. (2)

Along geodesic rays emanating from $p$, the family of Jacobi fields that
vanish at $p$ are the same for both metrics $g$ and $g^{*}$.

For a more general version of Lemma 4.2, see [20, Proposition 2.2].

###### Proof.

Because $g^{*}(\partial_{t},\partial_{t})=g(\partial_{t},\partial_{t})=1$, the
integral curves of $\partial_{t}$ are also geodesics under $g^{*}$, and hence
Part (1) follows. Now notice that along a geodesic ray emanating from $p$, a
Jacobi field $J$ that vanishes at $p$ under $g$ is realized as the variation
field of a variation by $g$-geodesics emanating from $p$. Because these curves
are also geodesics under $g^{*}$, $J$ is also a Jacobi field under $g^{*}$. ∎

Now letting $\operatorname{inj}\operatorname{rad}_{p}$ denote the injectivity
radius of $(M,g)$ at $p$, choose $\delta\in\\\
(0,\frac{1}{2}\operatorname{inj}\operatorname{rad}_{p})$ small enough such
that $g^{*}$ is defined on the closed ball $\overline{B(p,2\delta)}\subset M$.

######  Lemma 4.3.

There is a symmetric $(0,2)$-tensor $r$ defined on $\overline{B(p,2\delta)}$
such that

$g-g^{*}=t^{2}r.$

Furthermore, for any $l\in\mathbb{N}$, there exists a constant $C$ such that
$|r|_{C^{l}}<C$ with respect to the metric $g$.

###### Proof.

Choose normal coordinates $(x^{1},\dots,x^{n})$ around $p$, and let
$\partial_{1},\dots,\partial_{n}$ denote the corresponding coordinate vector
fields. Consider the rotational fields
$J=x^{j}\partial_{i}-x^{i}\partial_{j}$. These fields are normal to
$\partial_{t}=\operatorname{grad}(\operatorname{dist}(p,\cdot))$, and they
vanish at $p$. Now consider any geodesic $\gamma:[0,2\delta]\to M$ emanating
from $p$. Then $J$ are also Jacobi fields along $\gamma$ because the Lie
derivative $\mathcal{L}_{\partial_{t}}J=[\partial_{t},J]=0$. Notice that for
every point $\gamma(t_{0})$ with $t_{0}>0$, there is a neighborhood around
$\gamma(t_{0})$ such that every tangent space is spanned by $\partial_{t}$ and
$n-1$ of the rotational fields $J$. We choose $n-1$ such rotational fields and
denote them by $J_{1},\dots,J_{n-1}$. We will construct the tensor $r$ in this
neighborhood of $\gamma(t_{0})$ by defining its values on
$\big{\\{}\frac{J_{1}}{|J_{1}|},\dots,\frac{J_{n-1}}{|J_{n-1}|},\partial_{t}\big{\\}}$.
By the Gauss Lemma, $(g-g^{*})(\partial_{t},\cdot)\equiv 0$, and so we set
$r(\partial_{t},\cdot)\equiv 0$. Informed by [3, Chapter 5 Proposition 2.7],
we now compute the Taylor expansion of $(g-g^{*})(J_{i},J_{j})_{\gamma(t)}$
centered at $t=0$, where $\gamma$ is a geodesic emanating from $p$. First
notice that because the $J_{i}$ are Jacobi and $J_{i}(0)=0$, we have

$J_{i}^{\prime\prime}(0)=-\left(R_{g}(J_{i},\partial_{t})\partial_{t}\right)(0)=0.$

Second, for any vector field $V$ along $\gamma$, we have at $t=0$

$\displaystyle
g\left(\nabla_{\partial_{t}}(R_{g}(J_{i},\partial_{t})\partial_{t}),V\right)$
$\displaystyle=\partial_{t}g\left(R_{g}(J_{i},\partial_{t})\partial_{t},V\right)-g\left(R_{g}(J_{i},\partial_{t})\partial_{t},V^{\prime}\right)$
$\displaystyle=\partial_{t}g\left(R_{g}(V,\partial_{t})\partial_{t},J_{i}\right)$
$\displaystyle=g\left(\nabla_{\partial_{t}}(R_{g}(V,\partial_{t})\partial_{t}),J_{i}\right)+g\left(R_{g}(V,\partial_{t})\partial_{t},J_{i}^{\prime}\right)$
$\displaystyle=g\left(R_{g}(J_{i}^{\prime},\partial_{t})\partial_{t},V\right).$

Using these facts and that $J_{i}(0)=0=J_{j}(0)$, we have the following at
$t=0$:

$\displaystyle g(J_{i},J_{j})$ $\displaystyle=0,$ $\displaystyle
g(J_{i},J_{j})^{\prime}$
$\displaystyle=g(J_{i}^{\prime},J_{j})+g(J_{i},J_{j}^{\prime})=0,$
$\displaystyle g(J_{i},J_{j})^{\prime\prime}$
$\displaystyle=g(J_{i}^{\prime\prime},J_{j})+2g(J_{i}^{\prime},J_{j}^{\prime})+g(J_{i},J_{j}^{\prime\prime})=2g(J_{i}^{\prime},J_{j}^{\prime}),$
$\displaystyle g(J_{i},J_{j})^{\prime\prime\prime}$
$\displaystyle=g(J_{i}^{\prime\prime\prime},J_{j})+3g(J_{i}^{\prime\prime},J_{j}^{\prime})+3g(J_{i}^{\prime},J_{j}^{\prime\prime})+g(J_{i},J_{j}^{\prime\prime\prime})=0,$
$\displaystyle g(J_{i},J_{j})^{\prime\prime\prime\prime}$
$\displaystyle=g(J_{i}^{\prime\prime\prime\prime},J_{j})+4g(J_{i}^{\prime\prime\prime},J_{j}^{\prime})+6g(J_{i}^{\prime\prime},J_{j}^{\prime\prime})+4g(J_{i}^{\prime},J_{j}^{\prime\prime\prime})+g(J_{i},J_{j}^{\prime\prime\prime\prime})$
$\displaystyle=-8R_{g}(J_{i}^{\prime},\partial_{t},\partial_{t},J_{j}^{\prime}).$

By Lemma 4.2, $\gamma$ is a geodesic for both $g$ and $g^{*}$, and
$J_{i},J_{j}$ are Jacobi fields along $\gamma$ for either metric. Hence, we
have that the equations above also hold with $g^{*}$ substituted for $g$.
Thus, by applying these calculations to the Taylor expansions of
$g(J_{i},J_{j})_{\gamma(t)}$ and $g^{*}(J_{i},J_{j})_{\gamma(t)}$ centered at
$t=0$, and using the fact that the inner products induced by $g$ and $g^{*}$
on $T_{p}M$ agree, we have

$(g-g^{*})(J_{i},J_{j})_{\gamma(t)}=-\frac{t^{4}}{3}(R_{g}-R_{g^{*}})(J_{i}^{\prime},\partial_{t},\partial_{t},J_{j}^{\prime})_{p}+O(t^{5})\qquad\text{as
}t\to 0$

By normalizing this equation, we get

$(g-g^{*})\left(\frac{J_{i}}{|J_{i}|},\frac{J_{j}}{|J_{j}|}\right)_{\gamma(t)}=\frac{t^{4}}{3|J_{i}(t)||J_{j}(t)|}(R_{g^{*}}-R_{g})(J_{i}^{\prime},\partial_{t},\partial_{t},J_{j}^{\prime})_{p}+O(t^{3}),$

where $|J_{i}|$ is measured using $g$. It follows from the computations of
derivatives $g(J_{i},J_{j})$ above that $|J_{i}(t)|=t+O(t^{2})=|J_{j}(t)|$ as
$t\to 0$. Thus, we have

$\lim_{t\to
0}\frac{d}{dt}\left[(g-g^{*})\left(\frac{J_{i}}{|J_{i}|},\frac{J_{j}}{|J_{j}|}\right)_{\gamma(t)}\right]=0.$
(5)

Now, recall that if a smooth function $f:\mathbb{R}\to\mathbb{R}$ satisfies
$f(0)=0=f^{\prime}(0)$, then there is a smooth function
$h:\mathbb{R}\to\mathbb{R}$ such that $f(t)=t^{2}h(t)$. Thus

$(g-g^{*})\left(\frac{J_{i}}{|J_{i}|},\frac{J_{j}}{|J_{j}|}\right)_{\gamma(t)}=t^{2}r_{i,j}(t)$

for some smooth function $r_{i,j}$. Thus we define
$r\big{(}\frac{J_{i}}{|J_{i}|},\frac{J_{j}}{|J_{j}|}\big{)}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=r_{i,j}$
so that $g-g^{*}=t^{2}r$. Because the inner products induced by $g$ and
$g^{*}$ on $T_{p}M$ agree, it follows from Equation 5 and Taylor’s theorem
that $r$ can be smoothly extended to $\overline{B(p,2\delta)}$. Finally, the
inequality $|r|_{C^{l}}<C$ follows from the fact that $r$ is smooth on the
compact set $\overline{B(p,2\delta)}$. ∎

We are now prepared to prove the Sewing Theorem.

###### Proof of Theorem 4.1.

Choose $\delta\in(0,\frac{1}{2}\operatorname{inj}\operatorname{rad}_{p})$ such
that $g^{*}$ is defined on the closed ball $\overline{B(p,2\delta)}\subset M$.
By Lemma 7.3 in [21], there exists a smooth function $\phi:M\to[0,1]$ such
that

$\displaystyle\phi\equiv 1\text{ where
}\operatorname{dist}(p,\cdot)\leq\delta,$ $\displaystyle\phi\equiv 0\text{
where }\operatorname{dist}(p,\cdot)\geq 2\delta,$ $\displaystyle|$
$\displaystyle\phi|_{C^{1}}\leq\frac{2}{\delta}.$

On $M$, define the Riemannian metric $\tilde{g}=(1-\phi)g+\phi g^{*}$. Because
$\tilde{g}\equiv g$ for $\operatorname{dist}(p,\cdot)\geq 2\delta$, assume
$t=\operatorname{dist}(p,\cdot)<2\delta$. Given unit vector fields $U$ and
$V$,

$\displaystyle(g-\tilde{g})(U,V)$ $\displaystyle=g(U,V)-(1-\phi)g(U,V)-\phi
g^{*}(U,V)$ $\displaystyle=\phi(g-g^{*})(U,V)$ $\displaystyle=\phi
t^{2}r(U,V),$

where $r$ is the tensor from Lemma 4.3. Furthermore, for a unit vector $X$, we
have

$\displaystyle X(g-\tilde{g})(U,V)$ $\displaystyle=(X\phi)t^{2}r(U,V)+\phi
X(t^{2})r(U,V)+\phi t^{2}Xr(U,V)$
$\displaystyle=(X\phi)t^{2}r(U,V)+\phi(2t)(Xt)r(U,V)+\phi t^{2}Xr(U,V).$

Thus, because $\phi\leq 1$, $|\phi|_{C^{1}}\leq\frac{2}{\delta}$, $t<2\delta$,
and $|r|_{C^{1}}<C$, we have

$\displaystyle|(g-\tilde{g})(U,V)|$ $\displaystyle<4\delta^{2}C,$
$\displaystyle|X(g-\tilde{g})(U,V)|$ $\displaystyle<12\delta C+4\delta^{2}C.$

Therefore, for any $\epsilon>0$, $\delta$ can be chosen so that
$|g-\tilde{g}|_{C^{1}}<\epsilon$. ∎

###  4.3. Density of $k$-maximal local symmetry rank

In this section, we prove Theorem C using Theorem 4.1 and the metrics
established in Proposition 3.1.

###### Proof of Theorem C.

Choose any Riemannian $n$-manifold $(M,g)$ with $n\geq 3$. Let $p\in M$ and
define $g^{*}$ on a neighborhood of $p$ to be the pull-back metric

$g^{*}=\left(\exp_{p}^{-1}\right)^{*}\left(g^{\mathrm{model}}\right),$

where $g^{\mathrm{model}}$ is one of the metrics from Proposition 3.1 in
Section 3. Then by the Gauss Lemma,
$g(\gamma^{\prime},\cdot)=g^{*}(\gamma^{\prime},\cdot)$ for all geodesic rays
$\gamma$ emanating from $p$, and thus Theorem 4.1 can be applied to obtain a
metric $\tilde{g}$ on $M$ that has $k$-maximal local symmetry rank at $p$.
Therefore, the space of metrics on $M$ which have $k$-maximal local symmetry
rank is dense under the $C^{1}$-topology. ∎

## References

  * [1] Y. K. Chahine.  Volume estimates for tubes around submanifolds using integral curvature bounds.  preprint, arXiv:1810.01935, 2018.
  * [2] D. Corro and F. Galaz-García.  Positive Ricci curvature on simply-connected manifolds with cohomogeneity-two torus actions.  preprint, arXiv:1609.06125, revision to appear.
  * [3] M. Do Carmo.  Riemannian Geometry.  Birkhäuser, Boston, Massachusetts, 1992.
  * [4] C. Escher and C. Searle.  Torus actions, maximality, and non-negative curvature.  preprint, arXiv:1506.08685, 2015.
  * [5] F. Galaz-García.  A Note on Maximal Symmetry Rank, Quasipositive Curvature, and Low Dimensional Manifolds. In: Geometry of Manifolds with Non-negative Sectional Curvature, volume 2110 of Lecture Notes in Mathematics, pages 45–55.  Springer International Publishing, 2014.
  * [6] F. Galaz-Garcia, M. Kerin, and M. Radeschi.  Torus actions on rationally elliptic manifolds.  preprint, arXiv:1511.08383, 2015.
  * [7] F. Galaz-García and C. Searle.  Low-dimensional manifolds with non-negative curvature and maximal symmetry rank.  Proc. Amer. Math. Soc., 139(7):2559–2564, 2010.
  * [8] K. Grove and C. Searle.  Positively curved manifolds with maximal symmetry-rank.  J. Pure Appl. Algebra, 91:137–142, 1994.
  * [9] L. Guijarro and F. Wilhelm.  Restrictions on submanifolds via focal radius bounds.  preprint, arXiv:1606.04121, to appear in Math. Res. Lett.
  * [10] L. Guijarro and F. Wilhelm.  A softer connectivity principle.  preprint, arXiv:1812.01021, to appear in Comm. Anal. Geom.
  * [11] L. Guijarro and F. Wilhelm.  Focal radius, rigidity, and lower curvature bounds.  Proc. Lond. Math. Soc., 116(6):1519–1552, 2018.
  * [12] D. Gumaer and F. Wilhelm.  On Jacobi field splitting theorems.  Differential Geom. Appl., 37:109–119, 2014.
  * [13] M. W. Hirsch.  Differential Topology.  Springer New York, 1976.
  * [14] C. Ketterer and A. Mondino.  Sectional and intermediate Ricci curvature lower bounds via optimal transport.  Adv. Math., 329:781–818, 2018.
  * [15] L. Mouillé.  Positive intermediate Ricci curvature on products of homogeneous spaces.  preprint, arXiv:1911.03491, 2019.
  * [16] J. C. Nash.  Positive Ricci curvature on fibre bundles.  J. Differential Geom., 14(2):241–254, 1979.
  * [17] J. Pak.  Actions of torus $T^{n}$ on $(n+1)$-manifolds $M^{n+1}$.  Pacific J. Math., 44(2):671–674, 1973.
  * [18] J. Parker.  $4$-dimensional $G$-manifolds with $3$-dimensional orbits.  Pacific J. Math., 125(1):187–204, 1986.
  * [19] P. Petersen.  Riemannian Geometry, volume 171 of Graduate Texts in Mathematics.  Springer International Publishing, $3^{\mathrm{rd}}$ edition, 2016.
  * [20] P. Petersen and F. Wilhelm.  Some principles for deforming nonnegative curvature.  preprint, arXiv:0908.3026v2, 2010.
  * [21] C. Pro and F. Wilhelm.  Diffeomorphism stability and codimension three.  preprint, arXiv:1606.01828.
  * [22] C. Searle, P. Solórzano, and F. Wilhelm.  Smoothing quotients with lower curvature bounds.  in preparation.
  * [23] C. Searle and F. Wilhelm.  personal communication.
  * [24] Z. Shen.  A sphere theorem for manifolds of positive Ricci curvature.  Indiana Univ. Math. J., 38(1):229–233, 1989.
  * [25] Z. Shen.  On complete manifolds of nonnegative kth-Ricci curvature.  Trans. Amer. Math. Soc., 338(1):289–310, 1993.
  * [26] F. Wilhelm.  On intermediate Ricci curvature and fundamental groups.  Illinois J. Math., 41(3):488–494, 1997.
  * [27] B. Wilking.  Torus actions on manifolds of positive sectional curvature.  Acta Math., 191(2):259–297, 2003.
  * [28] B. Wilking.  Torus actions on manifolds with quasi-positive curvature.  unpublished manuscript, prepared by M. Kerin, 2008.
  * [29] H. Wu.  Manifolds of partially positive curvature.  Indiana Univ. Math. J., 36(3):525–548, 1987.


# Extensions of the Bloch-Pólya theorem on the number of real zeros of
polynomials (II)

Tamás Erdélyi  Department of Mathematics, Texas A&M University, College
Station, Texas 77843  terdelyitamu.edu

We prove that there is an absolute constant $c>0$ such that for every

$a_{0},a_{1},\ldots,a_{n}\in[1,M]\,,\qquad 1\leq
M\leq\frac{1}{4}\exp\left(\frac{n}{9}\right)\,,$

there are

$b_{0},b_{1},\ldots,b_{n}\in\\{-1,0,1\\}$

such that the polynomial $\displaystyle{P(z)=\sum_{j=0}^{n}{b_{j}a_{j}z^{j}}}$
has at least $\displaystyle{c\left(\frac{n}{\log(4M)}\right)^{1/2}-1}$
distinct sign changes in $I_{a}:=(1-2a,1-a)$, where
$\displaystyle{a:=\left(\frac{\log(4M)}{n}\right)^{1/2}\leq 1/3}$. This
improves and extends earlier results of Bloch and Pólya and Erdélyi and, as a
special case, recaptures a special case of a more general recent result of
Jacob and Nazarov.

location of zeros, polynomials with constrained coefficients

††support: 2020 Mathematics Subject Classifications. 26C10, 11C08

## 1\. Introduction

In this note $c$ and $c_{j}$ will denote suitable positive absolute constants.
Let $\Cal{F}_{n}$ denote the set of polynomials of degree at most $n$ with
coefficients from $\\{-1,0,1\\}$. Let $\Cal{L}_{n}$ denote the set of
polynomials of degree $n$ with coefficients from $\\{-1,1\\}$. Let
$\Cal{A}_{n}$ denote the set of polynomials of degree at most $n$ with
coefficients from $\\{0,1\\}$. Updating what the authors write in [BEK99] and
[Er08] we have the following. The study of the location of zeros of the above
classes of polynomials starts with Bloch and Pólya [BP32]. They prove that the
average number of real zeros of a polynomial from $\Cal{F}_{n}$ is at most
$cn^{1/2}$. They also prove that if $n\geq 3$ then a polynomial from
$\Cal{F}_{n}$ cannot have more than

$\frac{cn\log\log n}{\log n}$

real zeros. This quite weak result appears to be the first on this subject.
Schur [Sch33] and by different methods Szegő [Sze34] and Erdős and Turán
[ET50] improve this to $c(n\log n)^{1/2}$ (see also [BE95]). (Their results
are more general, but in this specialization not sharp.) Our Theorem 4.1 in
[BEK99] gives the right upper bound of $cn^{1/2}$ for the number of real zeros
of polynomials from a much larger class, namely for all polynomials $P$ of the
form

$P(x)=\sum_{j=0}^{n}{a_{j}x^{j}}\,,\quad|a_{0}|=|a_{n}|=1\,,\quad|a_{j}|\leq
1\,,\quad a_{j}\in{\mathbb{C}}\,.$

Schur [Sch33] claims that Schmidt gives a version of part of this theorem.
However, it does not appear in the reference he gives, namely [Sch32], and we
have not been able to trace it to any other source. In [BE97] we prove that
$\eta n^{1/2}$ is an upper bound for the number of zeros of polynomials $P$ of
the form

$P(x)=\sum_{j=0}^{n}{a_{j}x^{j}}\,,\quad|a_{0}|=1\,,\quad|a_{j}|\leq 1\,,\quad
a_{j}\in{\mathbb{C}}\,,$

inside any polygon with vertices on the unit circle, where the constant $\eta$
depends only on the polygon. Bloch and Pólya [BP32] also proved that there are
polynomials $P\in\Cal{F}_{n}$ with at least

$\frac{cn^{1/4}}{(\log n)^{1/2}}$

sign changes on the real line ${\mathbb{R}}$. Schur [Sch33] claims they do it
for polynomials with coefficients only from $\\{-1,1\\}$, but this appears to
be incorrect.

In a seminal paper Littlewood and Offord [LO39] prove that the number of real
roots of a polynomial $P\in\Cal{L}_{n}$, on average, lies between

$\frac{c_{1}\log n}{\log\log\log n}\qquad\text{and}\qquad c_{2}(\log
n)^{2}\,,$

and it is proved by Boyd [Bo97] that every $P\in\Cal{L}_{n}$ has at most
$\displaystyle{\frac{c(\log n)^{2}}{\log\log n}}$ zeros at $1$ (in the sense
of multiplicity).

Kac [Ka48] prove that the expected number of real roots of a polynomial of
degree $n$ with random uniformly distributed independent coefficients is
asymptotically $(2/\pi)\log n$.

In [Er08] we prove that there are absolute constants $c_{1}>0$ and $c_{2}>0$
such that for every

$a_{0},a_{1},\ldots,a_{n}\in[1,M]\,,\qquad 1\leq M\leq\exp(c_{2}n^{1/4})\,,$

there are $b_{0},b_{1},\ldots,b_{n}\in\\{-1,0,1\\}$ such that
$\displaystyle{P(z)=\sum_{j=0}^{n}{b_{j}a_{j}z^{j}}}$ has at least
$c_{1}n^{1/4}$ distinct sign changes in $(0,1)$. This improves and extends the
earlier result of Bloch and Pólya [BP32].

Jacob and Nazarov [JN24] prove a general theorem including that there are
polynomials $P\in\Cal{L}_{n}$ having at least $cn^{1/2}$ distinct zeros in
$[0,1]$; and there are polynomials $0\not\equiv P\in\Cal{A}_{n}$ for which the
polynomials $Q$ defined by $Q(x):=P(-x)$ have at least $cn^{1/2}$ distinct
zeros in $[0,1]$.

## 2\. New Result

###### Theorem 2.1

There is an absolute constants $c>0$ such that for every

$a_{0},a_{1},\ldots,a_{n}\in[1,M]\,,\qquad 1\leq
M\leq\frac{1}{4}\exp\left(\frac{n}{9}\right)$ $None$

there are

$b_{0},b_{1},\ldots,b_{n}\in\\{-1,0,1\\}$

such that the polynomial $P$ of the form
$\displaystyle{P(z)=\sum_{j=0}^{n}{b_{j}a_{j}z^{j}}}$ has at least
$\displaystyle{c\left(\frac{n}{\log(4M)}\right)^{1/2}-1}$ distinct sign
changes in $I_{a}:=(1-2a,1-a)$, where
$\displaystyle{a:=\left(\frac{\log(4M)}{n}\right)^{1/2}\leq 1/3}$.

This improves and extends earlier results of Bloch and Pólya [BP32] and
Erdélyi [Er08], and, as a special case, recaptures a special case of a more
general recent result of Jacob and Nazarov [JN24].

## 3\. Lemmas

We use the notation $\log^{-}z:=\max\\{0,-\log z\\}$ and
$\log^{+}z:=\max\\{0,\log z\\}$. The geometric mean $L_{0}([a,b])$ of a
continuous function $P$ on a finite interval $[a,b]\subset{\mathbb{R}}$ is

$\|P\|_{L_{0}([a,b])}:=\exp\left(\frac{1}{b-a}\int_{a}^{b}{\log|P(x)|\,dx}\right)\,.$

In [JN24] Jacob and Nazarov showed the following.

###### Lemma 3.1

Let $a\in(0,1/3)$ and $I_{a}:=(1-2a,1-a)$. Then

$\int_{I_{a}}{\log^{-}P(x)\,dx}\leq c_{3}\log(4M)$

for every polynomial $P$ of the form

$P(x)=\sum_{j=0}^{n}{b_{j}a_{j}x^{j}}\,,\quad a_{j}\in[1,M]\,,\quad
b_{j}\in\\{-1,0,1\\}\,,\quad b_{0}=\pm 1\,.$ $None$

Moreover $c_{3}:=9\pi/\sqrt{2}$ is a suitable choice.

As a consequence we have the following.

###### Lemma 3.2

Let $a\in(0,1/3)$ and $I_{a}:=(1-2a,1-a)$. Then

$\|P\|_{L_{0}(I_{a})}\geq\exp\left(\frac{-c_{3}\log(4M)}{a}\right)$

for every polynomial $P$ of the form (3.1).

We will need the version of Lemma 3.2 below in the proof of Theorem 2.1.

###### Lemma 3.3

Let $a\in(0,1/3)$, $I_{a}:=(1-2a,1-a)$, and $0\leq k\leq n$. Then

$\|R\|_{L_{0}(I_{a})}\geq\exp\left(\frac{-c_{3}\log(4M)}{a}\right)$

for every polynomial $R$ of the form

$R(x)=\sum_{j=k}^{n}{b_{j}a_{j}x^{j-k}}\,,\quad a_{j}\in[1,M]\,,\quad
b_{j}\in\\{-1,0,1\\}\,,\quad b_{k}=\pm 1\,.$ $None$

## 4\. Proofs

###### Demonstration Proof of Lemma 3.2

Using Lemma 3.1 we have

$\begin{split}\|P\|_{L_{0}(I_{a})}=&\exp\left(\frac{1}{a}\int_{I_{a}}{\log^{+}{P(x)}\,dx}-\frac{1}{a}\int_{I_{a}}{\log^{-}{P(x)}\,dx}\right)\cr\geq&\exp\left(\frac{-1}{a}\int_{I_{a}}{\log^{-}{P(x)}\,dx}\right)\cr\geq&\exp\left(\frac{-c_{3}\log(4M)}{a}\right)\,.\cr\end{split}$

∎

###### Demonstration Proof of Theorem 2.1

We modify the outline given by Nazarov in MathOverflow where $a_{j}=1$ for
each $j=0,1,\ldots,n$, is assumed. Let $a\in(0,1/3)$ and $I_{a}:=(1-2a,1-a)$.
Associated with a polynomial $P$ we define the polynomial
$\widetilde{P}(x):=P(1-a-ax)$. The function $f(x):=1-a-ax$ maps the interval
$(0,1)$ onto the interval $(1-2a,1-a)$, hence the polynomial $P$ of the form

$P(x)=\sum_{j=0}^{n}{a_{j}x^{j}},\qquad a_{j}\in[1,M]\,,$

satisfies

$0\leq\widetilde{P}(x)<(n+1)M\,,\qquad x\in[0,1]\,.$

For every fixed $a_{0},a_{1},\ldots,a_{n}\in[1,M]$ there are $2^{n+1}$
different polynomials $P$ of the form

$P(x)=\sum_{j=0}^{n}{d_{j}a_{j}x^{j}},\qquad d_{j}\in\\{0,1\\}\,.$ $None$

By the pigeonhole principle there are two different polynomials $P_{1}$ and
$P_{2}$ of the form (4.1) such that the polynomial $0\not\equiv
P:=P_{1}-P_{2}$ is of the form (3.1) and

$\left|\int_{0}^{1}{\widetilde{P}(x)x^{j}\,dx}\right|\leq(n+1)M2^{-n/m}\,,\qquad
j=0,1,\ldots,m-1\,.$ $None$

We fix this $P$ and $b_{0},b_{1},\ldots,b_{n}\in\\{-1,0,1\\}$ in the rest of
the proof. There is a $0\leq k\leq n$ such that $b_{0}=b_{1}=\cdots=b_{k-1}=0$
and $b_{k}=\pm 1$. We have $P(x)=x^{k}R(x)$, where $R$ is a polynomial of the
form (3.2) and

$x^{k}\geq(1-2a)^{n}\geq\exp(-4an)\,,\qquad x\in I_{a}\,,\quad a\in(0,1/3)\,.$

Combining this with Lemma 3.3 gives

$\|P\|_{L_{0}(I_{a})}\geq\exp\left(\frac{-c_{3}\log(4M)}{a}-4an\right)\,.$

Now we choose $\displaystyle{a:=\left(\frac{\log(4M)}{n}\right)^{1/2}\leq
1/3}$ to get

$\|\widetilde{P}\|_{L_{0}([0,1])}=\|P\|_{L_{0}(I_{a})}\geq\exp\left(-c_{4}(\log(4M))^{1/2}n^{1/2}\right)\,.$
$None$

Assume that $P$ has exactly $m-1\geq 0$ sign changes in $I_{a}$. Then
$\widetilde{P}$ has exactly $m-1$ sign changes in $(0,1)$. We denote the sign
changes of $\widetilde{P}$ in $(0,1)$ by $x_{1},x_{2},\ldots,x_{m-1}$. We
define

$Q(x):=\prod_{j=1}^{m-1}{(x-x_{j})}$

(the empty product is defined to be $1$). Then $\widetilde{P}Q$ does not
change sign in $(0,1)$. Observe that

$\|x-x_{j}\|_{L_{0}([0,1])}\geq c_{5}\,,\qquad j=1,2,\ldots,m-1\,,$

hence

$\|Q\|_{L_{0}([0,1])}\geq\exp(-c_{6}m)\,.$ $None$

Combining (4.3), (4.4), the multiplicative property of the geometric mean, the
comparison between the $L_{0}([0,1])$ and $L_{1}([0,1])$ norms, the fact that
$\widetilde{P}Q$ does not change sign, (4.2), and the fact that the sum of the
absolute values of the coefficients of $Q$ is at most $2^{m}$, we obtain

$\begin{split}\exp\left(-c_{4}(\log(4M))^{1/2}n^{1/2}-c_{6}m\right)\leq&\|\widetilde{P}Q\|_{L_{0}([0,1])}\leq\|\widetilde{P}Q\|_{L_{1}([0,1])}=\left|\int_{0}^{1}{(\widetilde{P}Q)(x)\,dx}\right|\cr\leq&2^{m}(n+1)M2^{-n/m}\,.\cr\end{split}$

Taking the log of both sides above, we obtain

$-c_{4}(\log(4M))^{1/2}n^{1/2}-c_{6}m\leq m\log 2+\log(n+1)+\log
M-\frac{n}{m}\log 2\,,$

that is,

$\frac{n}{m}\log 2\leq(\log 2+c_{6})m+\log(n+1)+\log
M+c_{4}(\log(4M))^{1/2}n^{1/2}\,.$

Observe that the assumption (2.1) implies

$\log
M\leq(\log(4M))^{1/2}(\log(4M))^{1/2}\leq(\log(4M))^{1/2}\frac{1}{3}\,n^{1/2}\,,$

hence

$\frac{n}{m}\log 2\leq(\log
2+c_{6})m+\log(n+1)+(c_{4}+1/3)(\log(4M))^{1/2}n^{1/2}\,.$ $None$

Finally, assuming that $\displaystyle{1\leq m\leq
c_{7}\left(\frac{n}{\log(4M)}\right)^{1/2}}$ with a sufficiently small
absolute constant $c_{7}>0$ in (4.5), we get a contradiction. So
$\displaystyle{m\geq c_{7}\left(\frac{n}{\log(4M)}\right)^{1/2}}$ with a
sufficently small absolute constant $c_{7}>0$. ∎

## 5\. An open problem

###### Problem 5.1

What is the largest multiplicity of the zero a polynomial $0\not\equiv
P\in\Cal{F}_{n}$ can have at $1$?

As it is shown in [BEK99] a polynomial $0\not\equiv P\in\Cal{F}_{n}$ cannot
have more than $c_{1}n^{1/2}$ zeros at $1$. By using the pigeonhole principle
it is also shown in [BEK99] that there exists a polynomial $0\not\equiv
P\in\Cal{F}_{n}$ with at least $c_{2}(n/\log n)^{1/2}$ zeros at $1$. Closing
the gap between $c_{1}n^{1/2}$ and $c_{2}(n/\log n)^{1/2}$ looks difficult.

## 6\. Acknowledgment

The author thanks Fedor Nazarov for the discussions on the topic and for
presenting the main idea of the proof of Theorem 2.1 on MathOverflow.

## References

  * BP32 A. Bloch and G. Pólya, On the roots of certain algebraic equations, Proc. London Math. Soc. 33 (1932), 102–114.
  * BE95 P. Borwein and T. Erdélyi, Polynomials and Polynomial Inequalities, Springer-Verlag, 1995.
  * BE97 P. Borwein and T. Erdélyi, On the zeros of polynomials with restricted coefficients, Illinois J. Math. 41 (1997), 667–675.
  * BEK99 P. Borwein, T. Erdélyi, and G. Kós, Littlewood-type problems on $[0,1]$., Proc. London Math. Soc. 79 (1999), 22–46.
  * Bo97 D.W. Boyd, On a problem of Byrne’s concerning polynomials with restricted coefficients, Math. Comp. 66 (1997), 1697–1703.
  * Er08 T. Erdélyi,, Extensions of the Bloch-Pólya theorem on the number of real zeros of polynomials, Journal de théorie des nombres de Bordeaux 20, no. 2 (2008), 281–287.
  * ET50 P. Erdős and P. Turán, On the distribution of roots of polynomials, Ann. Math. 57 (1950), 105–119.
  * JN24 M. Jacob and F. Nazarov, The Newman algorithm for constructing polynomials with restricted coefficients and many real roots, https://arxiv.org/pdf/2404.07971.
  * Ka48 M. Kac, On the average number of real roots of a random algebraic equation (II), Proc. London Math. Soc. 50 (1948), 390–408.
  * LO39 J.E. Littlewood and A.C. Offord, On the number of real roots of a random algebraic equation. II, Proc. Cam. Phil. Soc. 35 , 133–148.
  * Sch32 E. Schmidt, Über algebraische Gleichungen vom Pólya-Bloch-Typos, Sitz. Preuss. Akad. Wiss., Phys.-Math. Kl. (1932), 321.
  * Sch33 I. Schur, Untersuchungen über algebraische Gleichungen., Sitz. Preuss. Akad. Wiss., Phys.-Math. Kl. (1933), 403–428.
  * Sze34 G. Szegő, Bemerkungen zu einem Satz von E. Schmidt über algebraische Gleichungen., Sitz. Preuss. Akad. Wiss., Phys.-Math. Kl. (1934), 86–98.