MnLargeSymbols’164 MnLargeSymbols’171

# Chiral Spin Liquid and Quantum Phase Transition in the Triangular Lattice
Hofstadter-Hubbard Model

Stefan Divic  Department of Physics, University of California, Berkeley, CA
94720, USA  Center for Computational Quantum Physics, Flatiron Institute, New
York, New York 10010, USA  Tomohiro Soejima  Department of Physics, Harvard
University, Cambridge, MA 02138, USA  Valentin Crépel  Center for
Computational Quantum Physics, Flatiron Institute, New York, New York 10010,
USA  Michael P. Zaletel  Department of Physics, University of California,
Berkeley, CA 94720, USA  Material Science Division, Lawrence Berkeley National
Laboratory, Berkeley, CA 94720, USA  Andrew Millis  Center for Computational
Quantum Physics, Flatiron Institute, New York, New York 10010, USA  Department
of Physics, Columbia University, New York, NY 10027, USA

(September 1, 2024)

###### Abstract

Recent advancements in moiré engineering motivate study of the behavior of
strongly-correlated electrons subject to substantial orbital magnetic flux. We
investigate the triangular lattice Hofstadter-Hubbard model at one-quarter
flux quantum per plaquette and a density of one electron per site, where
geometric frustration has been argued to stabilize a chiral spin liquid phase
intermediate between the weak-coupling integer quantum Hall and strong-
coupling 120∘ antiferromagnetic phases. In this work, we use Density Matrix
Renormalization Group methods and analytical arguments to analyze the
compactification of the Hofstadter-Hubbard model to cylinders of finite
radius. We introduce a glide particle-hole symmetry operation which for odd-
circumference cylinders, we show, is spontaneously broken at the quantum Hall
to spin liquid transition. We further demonstrate that the transition is
associated with a diverging correlation length of a charge-neutral operator.
For even-circumference cylinders the transition is associated with a dramatic
quantitative enhancement in the correlation length upon threading external
magnetic flux. Altogether, we argue that the 2+1D CSL-IQH transition is in
fact continuous and features critical correlations of the charge density and
other spin rotationally-invariant observables.

Introduction—The Kalmeyer-Laughlin chiral spin liquid (CSL) is one of the
earliest and most important examples of topological order [1, 2, 3] and has
been extensively studied in spin systems, both numerically [4] and through
partonic mean-field approaches [5, 6, 7]. Recently, considerable efforts have
been devoted to identifying electronic settings in which the CSL may emerge to
connect more directly to solid state experiments, in which electrons—rather
than spins—are the fundamental degrees of freedom and may be far from their
Mott limit. These systems enable the exploration of the Mott transition of
strongly-interacting electrons in strong magnetic fields, a regime that is
becoming experimentally accessible both at non-zero [8, 9] and zero Zeeman
coupling [10, 11]. They also set the stage for the study of charge
fluctuation-driven transitions out of the CSL phase into other phases such as
superconductors [12], exotic charge-density waves [13], or quantum Hall states
[14, 11].

Figure 1: (a) Upper panel: The Hofstadter-Hubbard model on the 2D triangular
lattice, with magnetic translations $T_{x,y}$, on-site Hubbard $U$
interaction, and flux per triangular plaquette $\Phi_{\triangle}$ indicated.
Lower left panel: “Hofstadter butterfly” representation of spectrum of non-
interacting model in the plane of single-particle energy $E$ and flux per
plaquette $\Phi_{\triangle}$. $\Phi_{\triangle}=\pi/2$ is the unique flux for
which the Hamiltonian enjoys a doubled “magnetic” unit cell and a particle-
hole symmetry $\mathcal{P}$ relating opposite $C=\pm 1$ bands [lower right
panel]. (b) A finite segment of the infinite-length, circumference-$L_{y}$
cylinder geometry, threaded by $\Phi_{\text{ext}}$ external magnetic flux.
With the chosen “YC” cylindrical geometry, the translation $T_{y}$ is
orthogonal to the cylinder axis and $x\in\mathbb{Z}$ indexes rings. (c)
Expected 2+1D phase diagram at half-filling. As interactions increase, the
spin-degenerate integer quantum Hall (IQH) ground state undergoes a quantum
phase transition into a Kalmeyer-Laughlin chiral spin liquid (CSL). By
studying both even and odd finite-circumference cylinders, we shed light on
the existence and nature of this transition.

A model electronic system thought to host both conventional and CSL electronic
phases is the triangular Hofstadter-Hubbard model (see Fig. 1a) in the regime
of strong orbital magnetic flux, where the CSL may be favored energetically by
both geometric frustration [15, 16] and the explicit breaking of time-reversal
symmetry [17, 7]. The Hofstadter-Hubbard model is parametrized by an on-site
interaction $U$ and a nearest neighbor hopping with Peierls phases
corresponding to a flux of $\Phi_{\triangle}$ per elementary plaquette. When
$\Phi_{\triangle}=\pi/2$ per triangle, this system realizes a spin-unpolarized
integer quantum Hall (IQH) insulator at small values of the Hubbard
interaction, with $\sigma_{xy}=2e^{2}/h$, obtained by filling the lower
topological sub-band in both spin sectors. This state possesses a gap to spin
and charge excitations, is topological and has a spin-degenerate charged edge
mode. At very large $U$ and density of one electron per site, the state is a
Mott insulator: the leading term in the $t/U$ expansion around the
infinite-$U$ Mott limit shows that the low-energy physics is governed by the
conventional antiferromagnetic nearest-neighbor $S=1/2$ Heisenberg model with
$J_{H}\sim t^{2}/U$, which has $120^{\circ}$ antiferromagnetic order at $T=0$,
so that the ground state is non-topological, with fully-gapped charge
excitations and soft (Goldstone) spin excitations.

Carrying out the $t/U$ expansion to higher order suggests the existence of a
CSL phase intermediate between the Mott/AF and IQH phases: the next correction
is a third-order chiral term $S\cdot S\times S$ with coefficient
$J_{\triangle}\sim t^{3}\sin\Phi_{\triangle}/U^{2}$. In the presence of the
magnetic flux this term explicitly breaks both time-reversal and parity
symmetry [17, 7]. Numerical studies of the $J_{H}$-$J_{\triangle}$ spin
Hamiltonian have indeed revealed a CSL ground state for large enough
$J_{\triangle}/J_{H}$ [18, 19], although at these values the accuracy of
truncating the $t/U$ expansion at third order may be questioned. Very recently
Kuhlenkamp et al. presented cylinder iDMRG evidence of an intermediate CSL
phase for $U\gtrsim 10t$ at fluxes $\Phi_{\triangle}=\pi/3,\pi/2$ [11]. The
CSL phase was identified by the absence of charge pumping as flux was threaded
through the cylinder and the presence of appropriate degeneracies in the
entanglement spectrum.

Supposing one accepts the existence of an intermediate CSL phase, the question
of the nature of the IQH-CSL transition arises. A parton theory in which one
decomposes the electron into a bosonic chargon $b$ carrying the charge and a
fermionic spinon $f$ carrying the spin index, $c_{\sigma}=bf_{\sigma}$ has
been proposed [14, 12]. This decomposition requires an emergent gauge field
$a$, corresponding to the relative phase between the partons, under which they
carry opposite unit $U(1)$ charge. In both the IQH and CSL phases the spinon
occupies both $C=1$ magnetic sub-bands descending from the electronic
bandstructure, while at the electronic IQH-CSL phase transition the chargon
and the gauge field together undergo a superconductor-Mott phase transition
[12]. The dynamical gauge field $a$ is responsible not only for producing the
correct topological response on both sides of the transition, but also for
gapping out the charge sector in the IQH phase through the Anderson-Higgs
mechanism [20, 21], and allowing for spinon deconfinement in the CSL [22, 23,
24]. The parton representation and related slave-particle numerical studies on
honeycomb and square lattices [14, 25] have provided a physically intuitive,
yet incomplete mean-field picture of the putative continuous transition. The
iDMRG investigation of Ref. [11] revealed the IQH-CSL transition only as a
crossover indicated by a weak maximum in a correlation length. Further
investigation of the CSL phase and the IQH-CSL transition is therefore needed.

In this work we use a compactification to an infinite-length, finite-radius
cylinder [see Fig. 1(b)] in combination with a symmetry analysis with large-
scale iDMRG simulations [26, 27] on cylinders of circumference $L_{y}=3$ to
$6$ to characterize the nature of the IQH and CSL states, as well as the
transition between them. Our symmetry analysis is based on an operator that is
a combination of particle-hole interchange and translation, which we show
explicitly is spontaneously broken for interactions above a critical
interaction strength for odd-circumference cylinders. We further show that the
broken symmetry occurs together with the degeneracy pattern in the
entanglement spectrum expected for a CSL phase. We present numerical evidence
that this transition is in fact continuous and argue that in the limit of
infinite cylinder radius, this transition evolves into a continuous 2+1D IQH-
CSL quantum phase transition featuring critical fluctuations in the charge
density and other spin rotationally-invariant observables.

Figure 2: CSL symmetry-breaking at odd circumference. (a) Graphical
illustration of the action of the particle-hole symmetry operator
$\mathcal{P}$ which transforms the gauge flux through a ring of an odd-$L_{y}$
cylinder, from $\phi$ to $\pi-\phi$. (b) Upper panel: In an odd-$L_{y}$ CSL
ground state, the alternating pattern $+\pi/2,-\pi/2,\dots$ of these gauge
fluxes spontaneously breaks $\mathcal{P}T_{x}$ symmetry, whose action is
equivalent to the threading of $\pi$ gauge flux. Lower panel: the low-lying
entanglement eigenvalues at cuts between two pairs of rings, which not only
exhibit the approximate degeneracy expected for the CSL at each $S_{z}$ [see
circled numerals], but also transform as half-integer spin representations
only at every other cut, signalling SSB of $\mathcal{P}T_{x}$. (c) In
contrast, the internal gauge flux configuration [upper panel] and entanglement
spectrum [lower panel] of the IQH state are consistent with it being a unique,
symmetry-preserving ground state.

Model, symmetry operator and cylinder compactification—We study the
Hofstadter-Hubbard model on the triangular lattice, defined pictorially in
Fig. 1(a). In this model, electrons hop between nearest-neighboring sites and
experience an Aharonov-Bohm phase of $\Phi_{\triangle}$ upon orbiting a
triangular plaquette. Note that this model focuses on the consequences of the
orbital motion, and sets the Zeeman coupling to zero. We fix
$\Phi_{\triangle}=\pi/2$, which endows the Hamiltonian with a unitary
particle-hole (PH) symmetry $\mathcal{P}$ acting as [28]

$\displaystyle\mathcal{P}c^{\dagger}_{x,y}\mathcal{P}^{-1}=(-1)^{x+y}(i\sigma_{y})c_{x,y},\quad\mathcal{P}i\mathcal{P}^{-1}=+i,$
(1)

on fermionic operators, where we have attached a conventional spin rotation
$\sigma_{y}$ to the naive definition of particle-hole transformation in order
for $\mathcal{P}$ to leave the spin operator
$\bm{S}=\frac{1}{2}c^{\dagger}\bm{\sigma}c$ invariant (see SM for details
[29]). The flux induces a magnetic translation algebra
$T_{x}T_{y}=-T_{y}T_{x}$ that we represent in a gauge where

$\displaystyle
T_{x}c^{\dagger}_{x,y}T_{x}^{-1}=(-1)^{y}c^{\dagger}_{x+1,y},\quad
T_{y}c^{\dagger}_{x,y}T_{y}^{-1}=c^{\dagger}_{x,y+1}.$ (2)

The commuting translations $(T_{x})^{2},T_{y}$ define a larger “magnetic” unit
cell, whose resulting two sub-bands are related by $\mathcal{P}$ and carry
opposite Chern number $C=\pm 1$. We fix the filling to $n=1$ electron per
lattice site, which in the absence of interactions corresponds to full filling
of the lowest spin-degenerate magnetic sub-band.

We now present the dimensional reduction of the problem to a cylinder. The key
results of our study rely on the different symmetry breaking patterns that
occur for even and odd circumferences $L_{y}$, which stem from generic
properties of magnetic fields (both external and internal) in the chosen
geometry. To illustrate how this arises, let us first consider the closed “tin
can”-shaped surface specified by a pair of adjacent rings at $x$ and $x+1$.
This surface is the union of $2L_{y}$ triangles on the outside of the
cylinder, as well as an $L_{y}$-sided polygonal surface at $x$ and another at
$x+1$ whose edges are the rings. Since $\pi/2$ external flux penetrates each
outer triangle, Gauss’s law requires that the external magnetic flux through
the two adjacent rings differ by $\pi L_{y}$ [30, 31]. These ring fluxes are
therefore uniform when $L_{y}$ is even, but are staggered when $L_{y}$ is odd,
which explicitly breaks $T_{x}$ symmetry.

Let us momentarily focus on the odd case and fix the sequence of ring fluxes
to alternate between $0$ and $\pi$. In this case, the system nonetheless
respects a combined translation-like symmetry, namely the product of
$\mathcal{P}$ and $T_{x}$—which we refer to as PH-glide—in spite of each being
individually broken by the compactification. The IQH state, being
adiabatically connected to the (unique) ground state of the non-interacting
Hamiltonian, is invariant under PH-glide symmetry.

We argue, however, that the CSL phase must break $\mathcal{P}T_{x}$
spontaneously when $L_{y}$ is odd. To see this, note that the CSL is a Mott
insulating phase. Thus, we may consider the limit where charge fluctuations
are frozen out, e.g. by increasing $U\to\infty$ while holding the three-spin
chiral term at a fixed favorable strength. Since the particle-hole operation
is transparent to the spin operator,
$\mathcal{P}\bm{S}\mathcal{P}^{-1}=\bm{S}$, this results in a spin model with
a pure translation symmetry $t_{x}$.00footnotetext: We note a similar
observation in a simpler 1+1D model [32]. Since each ring contains an odd
number of spin-$1/2$ moments, the Lieb-Schulz-Mattis theorem requires that any
gapped ground state be at least two-fold degenerate [33], e.g., by
spontaneously breaking $t_{x}$. One way to understand the translation symmetry
breaking of the CSL, in particular, is by leveraging its anyonic content.
Namely, the properties of the CSL under translations are captured by viewing
it has hosting a spinon in each unit cell of the triangular lattice [34, 35,
36]. Because these spinons have semionic statistics, the action of $t_{x}$
translation—which moves $L_{y}$ many spinons across any given cut—necessarily
permutes the two minimally-entangled CSL ground states [37] when $L_{y}$ is
odd [38, 39]. In other words, the only $t_{x}$-symmetric ground states are
long-range-entangled CSL “cat” states, implying that translation is
spontaneously broken [40].

Figure 3: Evidence for the IQH-CSL phase transition. (a) Upper panel: Sketch
of three rings of cylinder at indicated $x$ values, with solid and dashed
lines showing along which staggered-Heisenberg order parameter
$\mathcal{O}_{\delta\mathrm{Heis}}$ [defined in Eq. (3)] changes under glide
particle-hole symmetry $\mathcal{P}T_{x}$ on odd-circumference cylinders.
Lower panel: Magnitude of the staggered Heisenberg order parameter plotted
against $U/t$ for cylinder widths indicated and with $\mathcal{P}T_{x}$
preserving and broken phases indicated by the check and cross symbols.
Consistent with the recovery of this symmetry at the putative 2+1D critical
point, the magnitude of the order parameter [bottom left] decreases
substantially from $L_{y}=3$ to $5$. (b-d) Correlation length in the
$(Q,S_{z},k_{y})=\bm{0}$ charge sector, plotted against interaction strength,
for a range of MPS bond dimensions [increasing light to dark] and extrapolated
to infinite bond dimension [gray points]; the insets specify the external flux
threading each of the cylinder rings. (b) At circumference $L_{y}=5$, the
$\mathcal{P}T_{x}$ symmetry-breaking is associated with a diverging
correlation length. (c) While both the IQH and CSL states are fully-symmetric
on even-$L_{y}$ cylinders, the dominant correlation length is consistently
charge-neutral, peaking strongly for $\Phi_{\text{ext}}=\pi/2$ external flux
threading the cylinder (see inset). The maximum extrapolated value of the
correlation length increases from $L_{y}=4$ to $6$, consistent with an
approach to a continuous transition in the thermodynamic limit.

It is illustrative to phrase the above discussion in terms of operations on
the internal gauge field introduced in the parton construction. For
simplicity, let us focus on saddle point configurations of the gauge field on
each side of the transition, ignoring fluctuations. In both the CSL and IQH
ground states, the gauge flux through the triangular surface plaquettes should
match the external flux, $\phi_{\triangle}=\pi/2$ [6, 7, 12]. Thus, the only
remaining gauge-invariant degree of freedom is the gauge flux through the
cylinder rings; assuming $T_{x}^{2}$ is unbroken and $L_{y}$ is odd, we may
parameterize the two fluxes as $(\phi,\phi+\pi)$, differing by $\pi$ due to
Gauss’s law, as invoked above. Since the minimally-entangled CSL ground states
are adiabatically connected to the Mott limit in which $\mathcal{P}$ acts
trivially, their ring flux patterns are restricted to be $(\pi/2,-\pi/2)$ and
$(-\pi/2,\pi/2)$, which are insensitive to the action of $\mathcal{P}$ on ring
fluxes, namely $\phi\to\pi-\phi$ (Fig. 2a). We readily see that these two
configurations are related by $\mathcal{P}T_{x}$, exposing the two-fold
degeneracy and SSB of the CSL (Fig. 2b, top) [41]. In the IQH phase, on the
other hand, the Meissner effect pins together the internal and external gauge
fields [42]. The resulting flux pattern $(0,\pi)$ maps into itself under
$\mathcal{P}T_{x}$, consistent with a unique, gapped ground state (Fig. 2c,
top).

While the small $U$ IQH phase does not break the glide-PH symmetry and the
intermediate $U$ CSL phase has a two-fold symmetry breaking pattern on odd
circumference cylinders, the large $U$ $120^{\circ}$ phase breaks
translational symmetry (including the glide PH) in a threefold spatial
pattern.

Numerical observation of phase transition—We first comment on evidence for the
phases themselves. The primary diagnostic of chiral topological behavior is
the behavior of the charge-resolved entanglement spectrum (ES) [43, 44] at a
bipartition of the system into two semi-infinite halves, in this case between
any pair of adjacent rings [40, 11]. We exploit the relationship between the
physical and entanglement edges [45, 43, 46], namely that the low-lying states
in the entanglement spectrum correspond to those of the edge CFT [47, 48, 49,
50].

Within the bulk-boundary correspondence, the CSL is expected to host gapless
edges described at low energies by a free bosonic conformal field theory [51].
The counting of low-energy states as a function of momentum in each conserved
charge sector can be obtained combinatorially: for fixed $n\in\mathbb{Z}_{\geq
0}$ and associated momentum $q_{y}=2\pi n/L_{y}$, one considers a linearly
dispersing mode and imagines exciting states from below to above the Fermi
level. Note that these bosonic objects are charge and spin-neutral but carry
momentum. The counting sequence is namely the number of integer partitions of
$n$: $(1,1,2,3,5,\dots).$ On the other hand, the IQH phase has two independent
bosonic edge modes. Here, the total momentum must be the sum of those of the
two flavors. Working out the “two-color” integer partitions of $n$ yields the
counting $(1,2,5,10,20\dots).$ These expectations are borne out in our
numerics; see Fig. 2(b,c).

We now investigate the transition. Consistent with the expectations above, we
confirm that the CSL phase spontaneously breaks $\mathcal{P}T_{x}$ symmetry on
odd-$L_{y}$ cylinders. We first find that the bipartite ES in the CSL phase
exhibits a two-ring periodicity, which implies SSB. In contrast, the ES of the
IQH is fully invariant under the action of $\mathcal{P}T_{x}$, as shown in
Fig. 2(c). Second, by scanning $U/t$ and inspecting the transfer matrix
spectrum, we identify a diverging correlation length for the accessible odd-
circumference cylinder geometries, namely $L_{y}=3,5$. The divergence is found
to belong to the $(Q,S_{z},k_{y})=\bm{0}$ charge sector (see Fig. 3b for the
$L_{y}=5$ and the SM [29] for $L_{y}=3$). The spontaneous breaking of glide-PH
symmetry is the source of this quasi-(1+1)D criticality. To substantiate this,
we identified an order parameter
$\mathcal{O}_{\delta\mathrm{Heis}}=\sum_{m}\mathcal{O}^{\delta\mathrm{Heis}}_{2m}$,
where

$\displaystyle\mathcal{O}^{\delta\text{Heis}}_{x}=L_{y}^{-1}\sum_{y}\bm{S}_{x,y}\cdot\bm{S}_{x+1,y}-\bm{S}_{x+1,y}\cdot\bm{S}_{x+2,y}.$
(3)

The operator $\mathcal{O}_{\delta\mathrm{Heis}}$ is both charged under
$\mathcal{P}T_{x}$ and condenses in the CSL phase [see Fig. 3(a)], signaling
SSB. We remark that threading external flux $\Phi_{\text{ext}}=\pi/2$ through
an odd-circumference cylinder reduces the transition to a crossover, owing to
the fact that flux threading explicitly breaks glide-PH symmetry unless
$\Phi_{\text{ext}}\in\pi\mathbb{Z}$ (see SM [29]).

On the even-circumference geometries $L_{y}=4$ and $6$, we reproduce the weak
crossover, with a correlation length of $\sim 2$ lattice sites, found by
Kuhlenkamp et al. for $L_{y}=6$ and $\Phi_{\text{ext}}=0$ [11]. This is
consistent with the absence of translation symmetry-breaking in the even-
circumference Mott limit. Upon threading $\Phi_{\text{ext}}=\pi/2$ external
flux through each of these systems, we observe dramatically larger values of
the correlation length for $U$ near $U_{c}$ (Fig. 3), notably in the same
neutral sector in which the odd-circumference correlation length diverges [see
Fig. 3(c,d)]. We also perform an extrapolation to infinite bond dimension
following Ref. [52]; see SM for details [29]. The maximal extrapolated value
is $\xi=6.6$ at $L_{y}=4$, increasing substantially to $\xi=9.1$ at $L_{y}=6$.
Consistent with the view that accurate estimation of the 2+1D gap and other
properties requires consideration of all threaded cylinder fluxes [53], we
conjecture that the $\Phi_{\text{ext}}=\pi/2$ threaded system is more
reflective of the thermodynamic limit than the unthreaded system, and that
larger even-$L_{y}$ geometries will exhibit an increasingly-sharp sequence of
crossovers toward 2+1D critical behavior in the charge-neutral sector.

Figure 4: Density-density [left panel] and spin-spin [right panel] connected
correlations for odd-circumference cylinders with $U$ tuned to the respective
critical points, $U_{*}(L_{y}=3)=11.1$ and $U_{*}(L_{y}=5)=11.6$. Both
$L_{y}=5$ correlation functions are scaled down by 100 for ease of viewing.
(a) Density-density correlations decay nearly algebraically at the IQH-CSL
transition, pointing to the presence of gapless charge-neutral excitations in
the 2+1D limit. (b) In contrast, correlations of the spin density decay
exponentially with no apparent enhancement from $L_{y}=3$ to $5$, suggesting
that the excitations are spin-rotationally-invariant singlets.

Remarkably, the critical fluctuations at the odd-$L_{y}$ critical points
manifest not only in the order parameter, but also in the charge density. This
is seen manifestly in the connected density-density correlations $\langle
n(x)n(0)\rangle$, with $n(x)$ averaged over the ring $x$, which we find to
decay approximately algebraically, i.e., linearly on a log-log scale [see Fig.
4(a)] However, estimation of the 2+1D critical exponents from these data is
hampered by the fact that the critical features of the thermodynamic limit
need only show up in the regime $1\ll x\ll L_{y}$, which is inaccessible for
the presently-tractable system sizes. While PH-glide is no longer broken for
$L_{y}=4,6$ and the operator Eq. (3) no longer condenses (see Fig. 3a), we
also observe that their enhanced correlation length is accompanied by an
apparent algebraic plateau in $\langle n(x)n(0)\rangle$ over the scale of the
correlation length (see SM [29]).

Curiously, we do not observe critical behavior in the odd-$L_{y}$ connected
correlations of the spin density $S_{z}$ [Fig. 4(b)], which is also neutral
under the conserved charge and spin $U(1)$ symmetries. However, since $S_{z}$
may be rotated into $S_{x}$ by a global spin rotation, under which the CSL is
invariant, this is consistent with our not observing a diverging correlation
length in the $(Q,S_{z})=(0,\hbar)$ sector of the transfer matrix. Altogether,
under the assumption that this behavior persists for the entire sequence of
odd circumferences, our observations suggest that the 2+1D critical point
hosts gapless, charge-neutral spin-singlet excitations.

We remark that these data are consistent with the partonic view that the
transition into the IQH corresponds to chargon condensation. While the chargon
is not gauge invariant and may not develop conventional long-range connected
correlations, at the critical point the combination of the chargon, anti-
chargon and gauge string is a gapless neutral excitation [42]. Interestingly,
the power law correlations presented in Fig. 4 show that the physical charge
density operator, which has nonzero overlap with the gauge-invariant chargon
density $b^{\dagger}(x)b(x)$, produces a state with a nonzero overlap with the
critical excitations when applied to the ground state, raising the intriguing
possibility that the critical fluctuations can be experimentally observed.

Discussion—By combining numerical data on both even and odd-circumference
cylinder geometries, we have provided evidence that the CSL-IQH transition is
in fact continuous in the Hofstadter-Hubbard model at $\pi/2$ flux per
triangle and that critical properties are already manifest at the numerically-
accessible circumferences $3\leq L_{y}\leq 6$. Our finding that critical
fluctuations appear in the charge density correlation function, which may be
probed by experimental or numerical measurement of the structure factor, raise
the possibility of experimental and numerical investigation of this
transition.

This paradigmatic Hofstadter-Hubbard system provides a uniquely minimal and
experimentally-relevant setting [10, 11] in which to study a fundamental phase
transition at the interface of spin liquid and quantum Hall physics. As such,
a more complete understanding of this transition is an important task for
future work. Quantitatively, our investigation clarifies the possible
observable, numerical signatures that may be used to probe the phase
transition in a variety of numerical methods, such as AFQMC, VMC, or fermionic
PEPS, which may be able to access larger system sizes or spectral signatures
of spin-charge separation [54, 55, 56].

An important broader message of this work is that the ability to extrapolate
2+1D physics from dimensionally-reduced numerical data—namely on infinite
cylinder geometries—is greatly enhanced by carefully considering the manner in
which candidate ground states and their phase transitions respond to
dimensional reduction and to external fields that become boundary effects in
the thermodynamic limit. Applying such techniques more systematically, it
would be valuable to characterize the contexts in which UV features of Mott
phases, namely spin liquids, may be leveraged to better characterize their
charge fluctuation-induced phase transitions.

Acknowledgements—We acknowledge useful discussions with E. Altman, S.
Chatterjee, O. Gauthé, M. Knap, C. Kuhlenkamp, J.Y. Lee, C. Liu, F. Pollmann,
Z. Weinstein, and especially X.Y. Song. The Flatiron Institute is a division
of the Simons Foundation. AJM was supported in part by Programmable Quantum
Materials, an Energy Frontier Research Center funded by the U.S. Department of
Energy (DOE), Office of Science, Basic Energy Sciences (BES), under award DE-
SC0019443. S.D. acknowledges support from the NSERC PGSD fellowship. M.Z. and
S.D. were supported by the U.S. Department of Energy, Office of Science,
National Quantum Information Science Research Centers, Quantum Systems
Accelerator (QSA). This research is funded in part by the Gordon and Betty
Moore Foundation’s EPiQS Initiative, Grant GBMF8683 to T.S. Calculations were
performed using the TeNPy Library [57].

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oneΔ

Supplementary Material: “Chiral Spin Liquid and Quantum Phase Transition in
the Triangular Lattice Hofstadter-Hubbard Model”

###### Contents

  1. A Hamiltonian construction at all rational fluxes
    1. SI Choosing the magnetic unit cell
    2. SII Hamiltonian and choice of vector potential
  2. B Threading flux
    1. SI Affine-linear vector potential implementation
    2. SII Twisted boundary condition implementation
  3. C Two-band hopping Hamiltonian
    1. SI Hamiltonian at $(p,q)=(1,2)$
    2. SII Symmetries in the 2D plane
      1. SII.1 Magnetic translations
      2. SII.2 Particle-hole
      3. SII.3 Mirror composed with time-reversal
    3. SIII Symmetries on the cylinder
      1. SIII.1 Magnetic translations
      2. SIII.2 Particle-hole
  4. D Observable and Correlation Functions
    1. SI Heisenberg Operator in terms of Fermions
    2. SII Staggered Heisenberg Order Parameter
  5. E Correlation Lengths
  6. F Correlation Length Extrapolation
  7. G Density-density correlations at all circumferences

##  Appendix A Hamiltonian construction at all rational fluxes

For the underlying triangular lattice — with the “YC” orientation on the
cylinder (where the $\hat{y}$ site-to-site distance is the NN distance $a$) —
we take the following Bravais vectors:

$\displaystyle\bm{a}_{1}=a\left(\frac{\sqrt{3}}{2}\hat{x}+\frac{1}{2}\hat{y}\right),\qquad\bm{a}_{2}=a\hat{y}\qquad\text{(YC)},$
(1)

whose corresponding reciprocal lattice vectors are

$\displaystyle\bm{b}_{1}=\frac{4\pi}{\sqrt{3}a}\hat{x},\qquad\bm{b}_{2}=\frac{2\pi}{a}\left(-\frac{\hat{x}}{\sqrt{3}}+\hat{y}\right)\qquad\text{(YC)}.$
(2)

###  SI Choosing the magnetic unit cell

The following discussion applies to both the YC and XC lattice geometries. Let
the flux per _parallelogram_ be

$\displaystyle\Phi=\frac{p}{q}\Phi_{0},$ (3)

where $\Phi_{0}=\frac{2\pi\hbar}{e}\sim 2\pi$ in units where $\hbar/e=1.$ The
above definition of flux is convenient for defining the geometry, but we
briefly remark that the flux $\Phi/2$ per triangle — which is the smallest
traversable plaquette — is the quantity with respect to which physical
behavior (e.g., the Hofstadter spectrum) must be $\Phi_{0}$-periodic. In any
case, the corresponding magnetic field strength is

$\displaystyle
B=\frac{\Phi}{A_{\text{par}}}=\frac{p\Phi_{0}/q}{\frac{\sqrt{3}}{2}a^{2}}=\frac{2p\Phi_{0}}{\sqrt{3}q}a^{-2}.$
(4)

Regardless of the parity of $q$, there will be $q$ magnetic sublattice sites
in the magnetic unit cell, which we label by $\tau=0,\dots,q-1.$ When $q$ is
even, the magnetic Bravais vectors are

$\displaystyle\bm{R}_{1}=q\bm{a}_{1}-\frac{q}{2}\bm{a}_{2}\propto\hat{x},\qquad\bm{R}_{2}=\bm{a}_{2}\propto\hat{y},\qquad(q\
\text{even}),$ (5)

whose total area $A_{\text{muc}}$ (for either YC or XC) is precisely that of
$2q$ triangles of area $\frac{\sqrt{3}}{4}a^{2}.$ The associated reciprocal
lattice vectors are

$\displaystyle\bm{G}_{1}=\frac{1}{q}\bm{b}_{1}\propto\hat{x},\qquad\bm{G}_{2}=\frac{1}{2}\bm{b}_{1}+\bm{b}_{2}\propto\hat{y},\qquad(q\
\text{even}).$ (6)

As a sanity check, the total of the magnetic Brillouin zone (mBZ) is then

$\displaystyle
A_{\text{mBZ}}=|\bm{G}_{1}\times\bm{G}_{2}|=(2\pi)^{2}\frac{2}{q\sqrt{3}a^{2}}=\frac{(2\pi)^{2}}{A_{\text{muc}}}.$
(7)

When $q$ is odd, the enclosing magnetic unit cell again contains exactly $2q$
triangles, but we take

$\displaystyle\bm{R}_{1}=q\bm{a}_{1}-\frac{q-1}{2}\bm{a}_{2},\qquad\bm{R}_{2}=\bm{a}_{2}\propto\hat{y},,\qquad(q\
\text{odd}),$ (8)

whose total area is once again

$\displaystyle|\bm{R}_{1}\times\bm{R}_{2}|=q|\bm{a}_{1}\times\bm{a}_{2}|=\frac{q\sqrt{3}}{2}a^{2}=A_{\text{muc}}.$
(9)

The reciprocal lattice vectors are now

$\displaystyle\bm{G}_{1}=\frac{1}{q}\bm{b}_{1},\qquad\bm{G}_{2}=\frac{q-1}{2q}\bm{b}_{1}+\bm{b}_{2},\qquad(q\
\text{odd}),$ (10)

which also satisfy the inverse area relationship, Eq. (7).

In Fig. LABEL:fig:YC_c2zsymm_locations, we illustrate the choice of magnetic
unit cell for both odd and even $q$ for the YC triangular lattice boundary
conditions. For both even and odd denominator $q,$ note that the enclosing
magnetic unit cell contains exactly $2q$ triangles, so $2\pi p\in
2\pi\mathbb{Z}$ total flux. In addition, we position the sublattice sites
$\tau$ within the magnetic unit cell in such a way that all the sites within
magnetic unit cell $\bm{R}$ transform under $C_{2z}$ into sites in $-\bm{R}$.
In particular, the sites in the home magnetic unit cell transform into one
another in a clean way. We achieved this by further partitioning the odd-$q$
geometries into two subclasses. In Fig. LABEL:fig:XC_c2zsymm_locations, we
make the same specifications for the XC boundary conditions.

###  SII Hamiltonian and choice of vector potential

Placing the underlying triangular lattice on the cylinder, let the resulting
system have circumference $L_{y}$, i.e., the points
$\bm{r}\sim\bm{r}+L_{y}\bm{a}_{2}$ are geometrically identified. Let the
lattice sites be indexed by

$\displaystyle\bm{r}=x\bm{a}_{1}+y\bm{a}_{2},$ (11)

which defines our coordinate system, where $(x,y)\in\mathbb{Z}^{2}.$ Since
$\bm{a}_{2}$ is purely vertical in both geometries, then $x$ indexes the
rings, and $y$ is a integer spatial index within each ring. To generate
complex hoppings and modify the translation algebra to a magnetic translation
algebra, we introduce an external $U(1)$ vector potential. We assume that it
is linear and that its curl is the magnetic field:

$\displaystyle\bm{A}(\bm{r})=M\bm{r},\qquad\nabla\times\bm{A}(\bm{r})=B\hat{z},$
(12)

where $B$ is as in Eq. (4). The hoppings are obtained from the vector
potential via the Peierl’s substitution, for which we adopt the following sign
convention:

$\displaystyle
t_{\bm{r}^{\prime}\leftarrow\bm{r}}=t\exp\left(i\int^{\bm{r}^{\prime}}_{\bm{r}}\bm{A}(\bm{\ell})\cdot
d\bm{\ell}\right)=t\exp\left(i\bm{A}\left(\frac{\bm{r}^{\prime}+\bm{r}}{2}\right)\cdot(\bm{r}^{\prime}-\bm{r})\right),$
(13)

where the second equality is due to linearity of $\bm{A}(\bm{r})$. Our
convention is that the single-particle Hamiltonian is (note the minus sign):

$\displaystyle\hat{h}=-\sum_{\bm{r}^{\prime},\bm{r}}c^{\dagger}(\bm{r}^{\prime})t_{\bm{r}^{\prime}\leftarrow\bm{r}}c(\bm{r}),$
(14)

where we only include nearest-neighbour hopping on the triangular lattice, for
which

$\displaystyle
t_{\bm{r}^{\prime}\leftarrow\bm{r}}=te^{i\bm{A}\left(\frac{\bm{r}^{\prime}+\bm{r}}{2}\right)\cdot(\bm{r}^{\prime}-\bm{r})}.$
(15)

To constrain the form Eq. (12), we remark that the terms $x\hat{x}$ and
$y\hat{y}$ are curl-free, whereas $Bx\hat{y}$ and $-By\hat{x}$ — the standard
Landau gauges on the rectangular lattice — have the correct curl. We therefore
reason that the most general permissible $M$ is given by

$\displaystyle M=B\begin{pmatrix}\alpha&\gamma-1\\\
\gamma&\beta\end{pmatrix},$ (16)

where $\alpha,\beta,\gamma\in\mathbb{R}$. We will now demand that the choice
of vector potential is compatible with our chosen geometry of magnetic Bravais
vectors. Specifically, we ask that the Peierl’s hoppings generated by $\bm{A}$
are periodic in the chosen Bravais vectors. This allows us to place the system
easily on a torus or cylinder, provided the system tiles an integer number of
magnetic unit cells. Quantitatively, we demand that the RHS of Eq. (13) is
invariant under shifts of the argument of $\bm{A}$ by the magnetic Bravais
vectors $\bm{R}_{1}$ and $\bm{R}_{2}$, for all bond vectors
$\bm{r}^{\prime}-\bm{r}$. By linearity of the vector potential, and since the
bond vectors are linear combinations of $\bm{a}_{1}$ and $\bm{a}_{2}$, it
suffices to require that

$\displaystyle\bm{a}_{i}\cdot\bm{A}(\bm{R}_{j})\in 2\pi\mathbb{Z},$ (17)

for all $i,j$. Following Ref. [58], an elegant solution can be found in terms
of the reciprocal vectors of the underlying triangular lattice. We take the
following since $\bm{b}_{1}\propto\hat{x}$ for both YC and XC geometries, and
we want the vector potential to be independent of $y$:

$\displaystyle\bm{A}(\bm{r})=\frac{\Phi}{(2\pi)^{2}}(\bm{r}\cdot\bm{b}_{1})\bm{b}_{2}.$
(18)

where $\Phi$ is the flux per parallelogram. Noting the vector identity

$\displaystyle\left[\nabla\times\left((\bm{r}\cdot\bm{Q})\bm{P}\right)\right]_{i}$
$\displaystyle=\epsilon_{ijk}\partial_{j}\left(\bm{r}_{\ell}\bm{Q}_{\ell}\bm{P}_{k}\right)=\epsilon_{ijk}\bm{Q}_{j}\bm{P}_{k}=(\bm{Q}\times\bm{P})_{i},$
(19)

then

$\displaystyle\bm{B}(\bm{r})$
$\displaystyle=\nabla\times\bm{A}(\bm{r})=\frac{\Phi}{(2\pi)^{2}}\bm{b}_{1}\times\bm{b}_{2}=\frac{\Phi}{(2\pi)^{2}}\frac{(2\pi)^{2}}{A_{\text{par}}}\hat{z}=\frac{\Phi}{A_{\text{par}}}\hat{z}.$
(20)

Moreover, it can be shown to have the required periodicity property:

$\displaystyle\bm{a}_{i}\cdot\bm{A}(\bm{R}_{j})$
$\displaystyle=\frac{\Phi}{(2\pi)^{2}}(\bm{R}_{j}\cdot\bm{b}_{1})\bm{b}_{2}\cdot\bm{a}_{i}=\delta_{i,2}\frac{2\pi
p/q}{2\pi}(\bm{R}_{j}\cdot\bm{b}_{1})=\delta_{i,2}\frac{p}{q}\bm{R}_{j}\cdot\bm{b}_{1},$
(21)

which clearly vanishes when $j=2.$ Otherwise,

$\displaystyle\bm{a}_{i}\cdot\bm{A}(\bm{R}_{1})$
$\displaystyle=\delta_{i,2}\frac{p}{q}\left(\begin{cases}\left(q\bm{a}_{1}-\frac{q}{2}\bm{a}_{2}\right)\cdot\bm{b}_{1}&q\text{
even}\\\
\left(q\bm{a}_{1}-\frac{q-1}{2}\bm{a}_{2}\right)\cdot\bm{b}_{1}&q\text{
odd}\end{cases}\right)=2\pi p\delta_{i,2},$ (22)

so that, indeed, in either case:

$\displaystyle\bm{a}_{i}\cdot\bm{A}(\bm{R}_{j})\in 2\pi\mathbb{Z}.$ (23)

##  Appendix B Threading flux

###  SI Affine-linear vector potential implementation

When dimensionally-reducing the lattice system to the cylinder, the charge and
spin flux through the cross sections of the cylinder distinguish gauge-
inequivalent systems. This additional flux can be implemented by adding a
constant term to the vector potential:

$\displaystyle\bm{A}(\bm{r})\longrightarrow\bm{A}_{\phi}(\bm{r})=\bm{A}(\bm{r})+\phi\frac{\bm{b}_{2}}{2\pi
L_{y}},$ (24)

where

$\displaystyle\phi=\phi_{q}+\frac{\sigma_{z}}{2}\phi_{s}.$ (25)

This modifies the hopping elements by

$\displaystyle t_{\bm{r}+\bm{\delta}\leftarrow\bm{r}}\longrightarrow
t_{\bm{r}+\bm{\delta}\leftarrow\bm{r}}^{\phi}=t_{\bm{r}+\bm{\delta}\leftarrow\bm{r}}\exp\left(i\frac{\phi}{2\pi}\frac{\bm{b}_{2}\cdot\bm{\delta}}{L_{y}}\right).$
(26)

Since we made the choice $\bm{A}\propto\bm{b}_{2}$ for the original vector
potential (in the absence of flux threading), then this is equivalent to a
shift in the origin:

$\displaystyle\bm{A}_{\phi}(\bm{r})=\bm{A}(\bm{r}+\bm{X}_{\phi})=\bm{A}(\bm{r})+\frac{\Phi}{(2\pi)^{2}}(\bm{X}_{\phi}\cdot\bm{b}_{1})\bm{b}_{2},$
(27)

where

$\displaystyle\bm{X}_{\phi}=\frac{\phi\bm{a}_{1}}{\Phi L_{y}}.$ (28)

###  SII Twisted boundary condition implementation

For the purposes of numerical implementation, particularly in hybrid space
where we exchange $y$ for $k_{y}$, it is convenient to instead implement flux
threading via a twist in the boundary conditions. Starting with periodic
boundary conditions $c^{\dagger}(\bm{r}+L_{y}\bm{a}_{2})=c^{\dagger}(\bm{r})$
for $\phi=0$, define a new set of operators

$\displaystyle
c^{\dagger}_{\phi}(\bm{r})=c^{\dagger}(\bm{r})\exp(iy\phi/L_{y})=c^{\dagger}(\bm{r})\exp\left(i\phi\frac{\bm{b}_{2}\cdot\bm{r}}{2\pi
L_{y}}\right).$ (29)

Then instead of implementing Eq. (24), we can instead replace every instance
of $c$ with $c_{\phi}$. This does not affect terms that depend solely on
charge and spin densities, e.g., the Hubbard interaction. However, it will
modify the hoppings in a way that is equivalent to Eq. (24). Under the
operator replacement, the hoppings are replaced by

$\displaystyle
c^{\dagger}(\bm{r}+\bm{\delta})t_{\bm{r}+\bm{\delta}\leftarrow\bm{r}}c(\bm{r})$
$\displaystyle\longrightarrow
c_{\phi}^{\dagger}(\bm{r}+\bm{\delta})t_{\bm{r}+\bm{\delta}\leftarrow\bm{r}}c_{\phi}(\bm{r})$
(30)
$\displaystyle=c^{\dagger}(\bm{r}+\bm{\delta})\exp\left(i\phi\frac{\bm{b}_{2}\cdot(\bm{r}+\bm{\delta})}{2\pi
L_{y}}\right)t_{\bm{r}+\bm{\delta}\leftarrow\bm{r}}\exp\left(-i\phi\frac{\bm{b}_{2}\cdot\bm{r}}{2\pi
L_{y}}\right)c(\bm{r})$ (31)
$\displaystyle=c^{\dagger}(\bm{r}+\bm{\delta})\exp\left(i\frac{\phi}{2\pi}\frac{\bm{b}_{2}\cdot\bm{\delta}}{L_{y}}\right)t_{\bm{r}+\bm{\delta}\leftarrow\bm{r}}c(\bm{r}),$
(32)

which is exactly equivalent to Eq. (26). To see that Eq. (29) corresponds to a
twisting of the boundary condition on the fermion operators, we note that

$\displaystyle
c^{\dagger}_{\phi}(\bm{r}+L_{y}\bm{a}_{2})=c^{\dagger}(\bm{r}+L_{y}\bm{a}_{2})\exp\left(i\phi\frac{\bm{b}_{2}\cdot(\bm{r}+L_{y}\bm{a}_{2})}{2\pi
L_{y}}\right)=c_{\phi}^{\dagger}(\bm{r})\exp(i\phi).$ (33)

The utility of this alternative formulation is that it leads to a clean
numerical implementation upon fourier transforming $y\to k_{y}$. In
particular, recall that
$(x,y)=(\bm{r}\cdot\bm{b}_{1}/2\pi,\bm{r}\cdot\bm{b}_{2}/2\pi)$ and define

$\displaystyle
c_{\phi}^{\dagger}(x,k_{y})=\frac{1}{\sqrt{L_{y}}}\sum_{y}e^{-ik_{y}y}c_{\phi}^{\dagger}(x,y)\iff
c_{\phi}^{\dagger}(x,y)=\frac{1}{\sqrt{L_{y}}}\sum_{k_{y}}e^{ik_{y}y}c_{\phi}^{\dagger}(x,k_{y}).$
(34)

Note that this is the Fourier convention of TeNPy. When we express the
Hamiltonian in the hybrid $xk_{y}$ representation, we find that all that’s
required is to shift the allowed values of $k_{y}.$ To see this, we impose the
twisted boundary condition Eq. (33) on the Fourier-expanded operators:

$\displaystyle\frac{1}{\sqrt{L_{y}}}\sum_{k_{y}}e^{ik_{y}(y+L_{y})}c_{\theta}^{\dagger}(x,k_{y})=\frac{1}{\sqrt{L_{y}}}\sum_{k_{y}}e^{ik_{y}y}c_{\theta}^{\dagger}(x,k_{y})e^{i\phi},$
(35)

so that

$\displaystyle e^{ik_{y}L_{y}}=e^{i\phi}.$ (36)

The allowed values of momenta are therefore quantized to:

$\displaystyle
k_{y}=\frac{2\pi}{L_{y}}\left(n+\frac{\phi_{q}}{2\pi}+\frac{\sigma_{z}\phi_{s}}{4\pi}\right),\qquad
n\in\mathbb{Z},$ (37)

which are shifted relative to the $\phi=0$ case. When the original Hamiltonian
$\hat{H}_{\phi=0}$ is written in hybrid space, some coefficients will depend
on $k.$ The only affect of the replacement Eq. (29) is then for those
coefficients to instead be evaluated on these shifted set of allowed values.

##  Appendix C Two-band hopping Hamiltonian

###  SI Hamiltonian at $(p,q)=(1,2)$

Now let us specialize to the case $(p,q)=(1,2)$, which corresponds to
$\Phi_{\triangle}=\pi/2$. To specify the unit cells (which each contain two
sites), we often write $x=\tau+2u$, where $u\in\mathbb{Z}$, while
$\tau\in\\{0,1\\}$ specifies the even/odd rings. With periodic boundary
conditions and before flux insertion, the hoppings Eq. (15) take the
particular form

$\displaystyle t_{(x,y+1)\leftarrow(x,y)}$
$\displaystyle=t\exp\left(i\bm{A}\left(x\bm{a}_{1}+y\bm{a}_{2}+\frac{\bm{a}_{2}}{2}\right)\cdot\bm{a}_{2}\right)$
$\displaystyle=t\exp\left(\frac{i}{4\pi}\left(x\bm{a}_{1}+y\bm{a}_{2}+\frac{\bm{a}_{2}}{2}\right)\cdot\bm{b}_{1}\bm{b}_{2}\cdot\bm{a}_{2}\right)$
$\displaystyle=t\exp\left(i\pi x\right),$ (38) $\displaystyle
t_{(x+1,y)\leftarrow(x,y)}$
$\displaystyle=t\exp\left(i\bm{A}\left(\frac{\bm{a}_{1}}{2}+x\bm{a}_{1}+y\bm{a}_{2}\right)\cdot\bm{a}_{1}\right)$
$\displaystyle=t,$ (39) $\displaystyle t_{(x+1,y-1)\leftarrow(x,y)}$
$\displaystyle=t\exp\left(i\bm{A}\left(\frac{\bm{a}_{1}-\bm{a}_{2}}{2}+x\bm{a}_{1}+y\bm{a}_{2}\right)\cdot(\bm{a}_{1}-\bm{a}_{2})\right)$
$\displaystyle=t\exp\left(-i\bm{A}\left(\frac{\bm{a}_{1}-\bm{a}_{2}}{2}+x\bm{a}_{1}+y\bm{a}_{2}\right)\cdot\bm{a}_{2}\right)$
$\displaystyle=t\exp\left(-i\pi(x+1/2)\right).$ (40)

We depict the resulting hopping phases in Fig. S1(a) for the lattice
orientation where $\bm{a}_{2}=a\hat{y}$ is a nearest-neighbour bond; this
becomes becomes the YC geometry when the system is placed on a cylinder under
the identification $\bm{r}\sim\bm{r}+L_{y}\bm{a}_{2}$.

Figure S1: (a) Hopping phases for the $(p,q)=(1,2)$ two-band Hamiltonian
corresponding to $\pi/2$ flux per triangle. The two black circles indicate the
two sites in the magnetic unit cell. The phases indicated by the color bar are
accrued when hopping upward, right-upward, and right-downward; the complex-
conjugate phases are accrued when hopping against these directions. (b)
Graphical depiction of the fluxes through various plaquettes upon placing the
lattice on a cylinder. Every triangle on the surface of the infinite cylinder
is penetrated by $\Phi_{\triangle}=\pi/2$ external flux. As a consequence of
Gauss’s law, the external flux through adjacent rings—each of which is a
polygonal surface bounded by a ring of length $L_{y}$—must alternate by $\pi$
when $L_{y}$ is odd, and must be uniform when $L_{y}$ is even.

###  SII Symmetries in the 2D plane

####  SII.1 Magnetic translations

Recall that we designed the vector potential so that the Hamiltonian would be
invariant under $T_{y}$ translations, i.e., in the circumferential direction
$\bm{a}_{2},$ defined as

$\displaystyle
T_{y}c_{\bm{r}}^{\dagger}T_{y}^{-1}=c_{\bm{r}+\bm{a}_{2}}^{\dagger}.$ (41)

To see this, note that

$\displaystyle t_{\bm{r}+\bm{a}_{2}+\bm{\delta}\leftarrow\bm{r}+\bm{a}_{2}}$
$\displaystyle=t\exp\left(i\bm{A}\left(\bm{r}+\bm{a}_{2}+\bm{\delta}/2\right)\cdot\bm{\delta}\right)$
(42)
$\displaystyle=t\exp\left(i\bm{A}\left(\bm{r}+\bm{\delta}/2\right)\cdot\bm{\delta}\right)$
(43) $\displaystyle=t_{\bm{r}+\bm{\delta}\leftarrow\bm{r}},$ (44)

since $\bm{A}(\bm{a}_{2})=0$ by design. On the other hand, translation along
$\bm{a}_{1}$ introduces an additional term:

$\displaystyle t_{\bm{r}+\bm{a}_{1}+\bm{\delta}\leftarrow\bm{r}+\bm{a}_{1}}$
$\displaystyle=t\exp\left(i\bm{A}\left(\bm{a}_{1}+\bm{r}+\bm{\delta}/2\right)\cdot\bm{\delta}\right)$
(45)
$\displaystyle=\exp\left(i\bm{A}\left(\bm{a}_{1}\right)\cdot\bm{\delta}\right)t_{\bm{r}+\bm{\delta}\leftarrow\bm{r}}$
(46)
$\displaystyle=\exp\left(\frac{i}{2}\bm{b}_{2}\cdot\bm{\delta}\right)t_{\bm{r}+\bm{\delta}\leftarrow\bm{r}}.$
(47)

This affects the hopping directions with some weight along
$\bm{\delta}=\bm{a}_{2}$ and, by comparison to Eq. (28) above, corresponds to
threading $\phi=\pi L_{y}$ through the cylinder. In the 2D plane geometry —
and later on some even-$L_{y}$ geometries as we’ll show — the Hamiltonian can
be restored by composing the bare $T_{x}$ with a large gauge transformation.
In particular, define the magnetic translation operation by

$\displaystyle
T_{x}^{\text{mag}}c_{\bm{r}}^{\dagger}(T_{x}^{\text{mag}})^{-1}=c_{\bm{r}+\bm{a}_{1}}^{\dagger}\exp\left(\frac{i}{2}\bm{b}_{2}\cdot\bm{r}\right)=c_{\bm{r}+\bm{a}_{1}}^{\dagger}(-1)^{y},$
(48)

so that

$\displaystyle T_{x}^{\text{mag}}\hat{h}(T_{x}^{\text{mag}})^{-1}$
$\displaystyle=T_{x}^{\text{mag}}\left(\sum
c_{\bm{r}+\bm{\delta}}^{\dagger}t_{\bm{r}+\bm{\delta}\leftarrow\bm{r}}c_{\bm{r}}\right)(T_{x}^{\text{mag}})^{-1}$
(49)
$\displaystyle=\sum\exp\left(\frac{i}{2}\bm{b}_{2}\cdot(\bm{r}+\bm{\delta})\right)c_{\bm{r}+\bm{a}_{1}+\bm{\delta}}^{\dagger}t_{\bm{r}+\bm{\delta}\leftarrow\bm{r}}c_{\bm{r}+\bm{a}_{1}}\exp\left(-\frac{i}{2}\bm{b}_{2}\cdot\bm{r}\right)$
(50) $\displaystyle=\sum
c_{\bm{r}+\bm{a}_{1}+\bm{\delta}}^{\dagger}\left(\exp\left(\frac{i}{2}\bm{b}_{2}\cdot\bm{\delta}\right)t_{\bm{r}+\bm{\delta}\leftarrow\bm{r}}\right)c_{\bm{r}+\bm{a}_{1}}$
(51) $\displaystyle=\sum
c_{\bm{r}+\bm{a}_{1}+\bm{\delta}}^{\dagger}t_{\bm{r}+\bm{a}_{1}+\bm{\delta}\leftarrow\bm{r}+\bm{a}_{1}}c_{\bm{r}+\bm{a}_{1}}$
(52) $\displaystyle=\hat{h},$ (53)

where the second-last equality follows from Eq. (47) above.

####  SII.2 Particle-hole

Consider the naive definition of the particle-hole operation:

$\displaystyle\tilde{\mathcal{P}}c^{\dagger}(\bm{r})\tilde{\mathcal{P}}^{-1}=c(\bm{r}),\qquad\tilde{\mathcal{P}}i\tilde{\mathcal{P}}^{-1}=+i,$
(54)

where the condition specifies it is a linear (as opposed to anti-linear)
unitary many-body operator. If 1 and 2 are an adjacent pair of sites, then
particle-hole acts as

$\displaystyle\tilde{\mathcal{\mathcal{P}}}(t_{12}c_{1}^{\dagger}c_{2}+t_{12}^{*}c_{2}^{\dagger}c_{1})\tilde{\mathcal{\mathcal{P}}}^{-1}$
$\displaystyle=t_{12}c_{1}c_{2}^{\dagger}+t_{12}^{*}c_{2}c_{1}^{\dagger}$ (55)
$\displaystyle=-t_{12}c_{2}^{\dagger}c_{1}-t_{12}^{*}c_{1}^{\dagger}c_{2}$
(56)
$\displaystyle=-t_{12}^{*}c_{1}^{\dagger}c_{2}-t_{12}c_{2}^{\dagger}c_{1},$
(57)

so that it flips the Peierl’s phase and increases it by $\pi.$ On the
triangular lattice, this modifies the flux through each triangle as follows:

$\displaystyle\Phi_{\triangle}\to-\Phi_{\triangle}+\pi.$ (58)

Requiring this operation to be invariant mod $2\pi,$ we find that we need

$\displaystyle\Phi_{\text{par}}=2\Phi_{\triangle}=\pi\left(2n+1\right),\qquad
n\in\mathbb{Z}.$ (59)

This is consistent with the observation that the Hofstadter spectrum on the
triangular lattice only has bands with opposite energy for these values of
$\Phi_{\triangle}$. In particular, in this work we consider
$\Phi_{\triangle}=\pi/2$. We find at each momentum $\bm{k}$ that both the
energy and Berry curvature are exactly opposite, with the bands having Chern
number $\pm 1.$

In general, the naive particle-hole must be composed with a gauge
transformation in order to restore all the hopping elements. Given the
particular form of the hoppings Eq. (SI) to Eq. (SI) for the two-band model,
we obtain the following form of the particle-hole symmetry:

$\displaystyle\mathcal{P}c^{\dagger}(\bm{r})\mathcal{P}^{-1}=(-1)^{x+y}c^{\mathsf{T}}(\bm{r}),\qquad\mathcal{P}i\mathcal{P}^{-1}=+i,$
(60)

where $\mathsf{T}$ is the transposition operation on the row vector of
electronic annihilation operators.

####  SII.3 Mirror composed with time-reversal

The orbital magnetic field breaks the time-reversal and mirror symmetries
enjoyed by the $\Phi_{\triangle}\equiv 0$ (mod $\pi$) Hubbard models on the
triangular lattice. However, the combination of mirror with time-reversal
remains a symmetry and should not be spontaneously broken by the CSL or IQH
phases. In particular, we focus on $M_{y}\mathcal{T}$ since this symmetry acts
“on-site” (i.e., within each ring) when the system is placed on the cylinder.
Recall that $M_{y}$ should send $\bm{r}\to M_{y}\bm{r}$ (flipping the
Cartesian $y$-coordinate) while also mapping $\bm{S}\to(-S_{x},S_{y},-S_{z}).$
On the other hand, time-reversal should flip all components of spin. Their
combination should therefore map $\bm{S}\to(S_{x},-S_{y},S_{z})$, which is
easy to implement due to $\mathcal{T}$ being anti-unitary. We claim that the
desired symmetry is

$\displaystyle(M_{y}\mathcal{T})c^{\dagger}(\bm{r})(M_{y}\mathcal{T})^{-1}=c^{\dagger}(M_{y}\bm{r})e^{i(-1)^{x}\pi/4},$
(61)

where the exponential factor is chosen to restore the phases of the hoppings.
We confirm this by explicit calculation:

$\displaystyle(M_{y}\mathcal{T})\hat{h}(M_{y}\mathcal{T})^{-1}$
$\displaystyle=(M_{y}\mathcal{T})\left(\sum
c_{\bm{r}+\bm{\delta}}^{\dagger}t_{\bm{r}+\bm{\delta}\leftarrow\bm{r}}c_{\bm{r}}\right)(M_{y}\mathcal{T})$
(62) $\displaystyle=\sum
c_{M_{y}(\bm{r}+\bm{\delta})}^{\dagger}e^{i(-1)^{\delta_{x}+x}\pi/4}t^{*}_{\bm{r}+\bm{\delta}\leftarrow\bm{r}}e^{-i(-1)^{x}\pi/4}c_{M_{y}\bm{r}}$
(63) $\displaystyle=\sum
c_{\bm{r}+\bm{\delta}}^{\dagger}e^{i(-1)^{x}\left((-1)^{\delta_{x}}-1\right)\pi/4}t^{*}_{M_{y}(\bm{r}+\bm{\delta})\leftarrow
M_{y}\bm{r}}c_{\bm{r}},$ (64)

which can be shown to be respected for Eqs. (SI) to (SI) above.

It is easy to verify that $M_{y}\mathcal{T}$ is not Kramers. However, it can
be convenient to define a Kramers symmetry by performing an additional $\pi$
spin rotation of about the $y$ axis:

$\displaystyle(M_{y}\mathcal{T}^{\prime})c^{\dagger}(\bm{r})(M_{y}\mathcal{T}^{\prime})^{-1}=c^{\dagger}(M_{y}\bm{r})(i\sigma_{y})e^{i(-1)^{x}\pi/4}.$
(65)

One then has that

$\displaystyle(M_{y}\mathcal{T})^{2}c^{\dagger}(\bm{r})(M_{y}\mathcal{T})^{-2}$
$\displaystyle=+c^{\dagger}(\bm{r}),$ (66)
$\displaystyle(M_{y}\mathcal{T}^{\prime})^{2}c^{\dagger}(\bm{r})(M_{y}\mathcal{T}^{\prime})^{-2}$
$\displaystyle=-c^{\dagger}(\bm{r}).$ (67)

Furthermore, we note that

$\displaystyle(M_{y}\mathcal{T})\bm{S}(\bm{r})(M_{y}\mathcal{T})^{-1}$
$\displaystyle=\left(S_{x},-S_{y},S_{z}\right)_{M_{y}\bm{r}},$ (68)
$\displaystyle(M_{y}\mathcal{T}^{\prime})\bm{S}(\bm{r})(M_{y}\mathcal{T}^{\prime})^{-1}$
$\displaystyle=-\bm{S}(M_{y}\bm{r}).$ (69)

###  SIII Symmetries on the cylinder

To implement charge-flux threading, we work in the real-space formalism with
periodic boundary conditions and vector potential modified according to Eq.
(24). We then get

$\displaystyle t_{(x,y+1)\leftarrow(x,y)}^{\phi}$
$\displaystyle=t\exp\left(i\pi x\right)\exp\left(i\phi/L_{y}\right)$ (70)
$\displaystyle t_{(x+1,y)\leftarrow(x,y)}^{\phi}$ $\displaystyle=t$ (71)
$\displaystyle t_{(x+1,y-1)\leftarrow(x,y)}^{\phi}$
$\displaystyle=t\exp\left(-i\pi(x+1/2)\right)\exp\left(-i\phi/L_{y}\right),$
(72)

so that we pick up an additional phase of $\phi/L_{y}$ when $y$ increments by
$1.$ One finds that the symmetries of the system are sensitive not only to the
parity of $L_{y}$, but also to the threaded charge and spin flux.

####  SIII.1 Magnetic translations

Figure S2: Density-density connected correlations for each of the four studied
circumferences $L_{y}=3$ to $6$, with the Hubbard interaction tuned to the
values at which the correlation length is maximal at the largest available
bond dimension, namely $U(L_{y}=3)=11.125$, $U(L_{y}=4)=11.0$,
$U(L_{y}=5)=11.6$, and $U(L_{y}=6)=11.4$, respectively. The even-circumference
cylinders are threaded with $\Phi_{\text{ext}}=\pi/2$, as in the main text.

When $L_{y}$ is even, the Hamiltonian can be restored by composing the bare
$T_{x}$ with a large gauge transformation. In contrast, by Fig. S1(b), we know
that the effect of translations on the fluxes through loops cannot be gauged
away when $L_{y}$ is odd. This manifests in the fact that $T_{x}^{\text{mag}}$
[see Eq. (48)] is ill-defined, which we see by imposing periodic boundary
conditions and comparing $\bm{r}+L_{y}\bm{a}_{2}$ with $\bm{r}$; note that

$\displaystyle\exp\left(\frac{i}{2}\bm{b}_{2}\cdot(\bm{r}+L_{y}\bm{a}_{2})\right)=\exp\left(i\pi
L_{y}\right)\exp\left(\frac{i}{2}\bm{b}_{2}\cdot\bm{r}\right)\neq\exp\left(\frac{i}{2}\bm{b}_{2}\cdot\bm{r}\right).$
(73)

####  SIII.2 Particle-hole

Now, let’s fix $\Phi_{\text{par}}=2\Phi_{\triangle}=+\pi$ and interrogate
whether placing the system on a YC cylinder changes this conclusion. Focusing
on a particular ring $r$ around the cylinder with flux $\Phi_{r},$ we note
that

$\displaystyle\tilde{\mathcal{\mathcal{P}}}:\Phi_{r}\mapsto\begin{cases}-\Phi_{r}&L_{y}\text{
even},\\\ -\Phi_{r}+\pi&L_{y}\text{ odd}.\end{cases}$ (74)

Therefore, $\tilde{\mathcal{\mathcal{P}}}$ is a symmetry (up to a gauge
transformation) when

$\displaystyle\Phi_{r}\in\begin{cases}\pi\mathbb{Z}&L_{y}\text{ even},\\\
\frac{\pi}{2}+\pi\mathbb{Z}&L_{y}\text{ odd},\end{cases}$ (75)

for all rings $r.$

When $L_{y}$ is even, Gauss’s law dictates that the flux $\Phi_{r_{1,2}}$
through two adjancent rings $r_{1,2}$ must be equal — note that this allows
$T_{x}$ to be a symmetry (up to a gauge transformation). Therefore, possibly
after threading $\phi$ external flux through each ring of the cylinder, it is
possible to tune all the $\Phi_{r}$ simultaneously (to $0$ or $\pi$) so that
$\tilde{\mathcal{\mathcal{P}}}$ is respected. Only in those cases is
$T_{x}\tilde{\mathcal{\mathcal{P}}}$ also a symmetry (up to gauge).

On the other hand, when $L_{y}$ is odd, Gauss’s law instead dictates that

$\displaystyle\Phi_{r_{2}}-\Phi_{r_{1}}=\pi\qquad\text{mod }2\pi.$ (76)

This always rules out $T_{x}$ as a symmetry. However, so long as
$\Phi_{r_{1}}$ is $\pi/2$ or $-\pi/2,$ both Eqs. (75) and (76) can be
respected, so that $\tilde{\mathcal{\mathcal{P}}}$ is a symmetry (up to
gauge). However, odd $L_{y}$ is special in that if we tune $\Phi_{r_{1}}$ to 0
or $\pi,$ then $T_{x}\tilde{\mathcal{\mathcal{P}}}$ is a symmetry (up to
gauge) even though neither $T_{x}$ nor $\tilde{\mathcal{\mathcal{P}}}$ are
themselves symmetries.

##  Appendix D Observable and Correlation Functions

###  SI Heisenberg Operator in terms of Fermions

If the sites are spinless FermionSites and up and down spins are encoded in
adjacent FermionSites, we can compute spin correlations indirectly using the
fermion creation and annihilation operators. Since the spin correlation
operator $\vec{S}_{i}\cdot\vec{S}_{j}$ involves the product of two spin
operators, we need to express it in terms of the fermion operators. In a
system with spinless FermionSites, we can define the spin operators as:

$\displaystyle S_{i}^{+}=c_{i\uparrow}^{\dagger}c_{i\downarrow},\quad
S_{i}^{-}=c_{i\downarrow}^{\dagger}c_{i\uparrow},\quad
S_{i}^{z}=\frac{1}{2}(c_{i\uparrow}^{\dagger}c_{i\uparrow}-c_{i\downarrow}^{\dagger}c_{i\downarrow}),$
(77)

where $c_{i\uparrow}^{\dagger}$ creates an up-spin fermion on site $i$ and
$c_{i\downarrow}^{\dagger}$ creates a down-spin fermion on site $i$. The spin
correlation operator $\vec{S}_{i}\cdot\vec{S}_{j}$ can be written in terms of
these fermion operators as:

$\displaystyle\vec{S}_{i}\cdot\vec{S}_{j}=\frac{1}{2}\left(c_{i\uparrow}^{\dagger}c_{i\downarrow}c_{j\downarrow}^{\dagger}c_{j\uparrow}+c_{i\downarrow}^{\dagger}c_{i\uparrow}c_{j\uparrow}^{\dagger}c_{j\downarrow}\right)+S_{i}^{z}S_{j}^{z}.$
(78)

###  SII Staggered Heisenberg Order Parameter

Here we express the staggered-Heisenberg order parameter in the hybrid
$(x,k_{y})$ operator representation, where $x$ indexes cylinder rings. Because
of translation invariance around the cylinder, it is useful to define the
$y$-averaged inter-ring NN Heisenberg operator

$\displaystyle\mathcal{O}_{\text{H}}(x)=\frac{1}{L_{y}}\sum_{y}\sum_{a=x,y,z}S^{a}(x,y)S^{a}(x+1,y),$
(79)

where

$\displaystyle
S^{a}(x,y)=c^{\dagger}(x,y)\frac{\sigma^{a}}{2}c(x,y)=\frac{1}{L_{y}}\sum_{k,k^{\prime}}e^{i(k-k^{\prime})y}c^{\dagger}(x,k)\frac{\sigma^{a}}{2}c(x,k^{\prime}).$
(80)

Since both the IQH and CSL ground states exhibit a two-ring translation
invariance (i.e., are invariant under $(T_{x})^{2}$ in addition to $T_{y}$)
then we can measure the local order parameter

$\displaystyle\mathcal{O}^{\delta\text{Heis}}_{0}=\mathcal{O}_{\text{H}}(1)-\mathcal{O}_{\text{H}}(0).$
(81)

In general, the hybrid representation reads

$\displaystyle\mathcal{O}_{\text{H}}(x)$
$\displaystyle=\frac{1}{L_{y}}\sum_{y}\sum_{a=x,y,z}S^{a}(x,y)S^{a}(x+1,y)$
$\displaystyle=\frac{1}{L_{y}^{2}}\sum_{a=x,y,z}\sum_{k,k^{\prime}}\sum_{p,p^{\prime}}\left(\frac{1}{L_{y}}\sum_{y}e^{i(k-k^{\prime}+p-p^{\prime})y}\right)c^{\dagger}(x,k)\frac{\sigma^{a}}{2}c(x,k^{\prime})\cdot
c^{\dagger}(x+1,p)\frac{\sigma^{a}}{2}c(x+1,p^{\prime})$
$\displaystyle=\frac{1}{L_{y}^{2}}\sum_{a=x,y,z}\sum_{k,k^{\prime}}\sum_{p,p^{\prime}}c^{\dagger}(x,k)\frac{\sigma^{a}}{2}c(x,k^{\prime})\cdot
c^{\dagger}(x+1,p)\frac{\sigma^{a}}{2}c(x+1,k-k^{\prime}+p).$

##  Appendix E Correlation Lengths

Figure S3: Correlation length data for even-circumference geometries. Left
panel: Illustration of the external magnetic flux penetrating each loop of the
infinite cylinder, namely no flux, $\Phi=0$. Center panels: Maximum
correlation length at $L_{y}=4$ in the $Q=0e$ charge sector [blue] for a range
of bond dimensions increasing with increasing darkness; and in the $Q=1e$
charge sector [green]. Right panel: The same for $L_{y}=6$. Figure S4:
Correlation length data for even-circumference geometries. Left panel:
Illustration of the external magnetic flux penetrating each loop of the
infinite cylinder, which alternates between $\pm\pi/2$, corresponding to
threaded flux $\Phi_{\text{ext}}=\pi/2$ in our conventions. Center panels:
Maximum correlation length at $L_{y}=3$ in the $Q=0e$ charge sector [blue] for
a range of bond dimensions increasing with increasing darkness; and in the
$Q=1e$ charge sector [green]. Right panel: The same for $L_{y}=5$.

Here we briefly comment on additional data concerning the correlation lengths
for geometries not displayed in the text. In particular, Fig. S3 shows
correlation length data for the $L_{y}=4$ and $6$ cylinders which are threaded
by no external flux. In particular, $L_{y}=6$ reproduces the behavior
presented in Kuhlenkamp et al., [11]. We remark that the correlation lengths
are quantitatively smaller than those at $\Phi_{\text{ext}}$ external flux,
which are shown in the main text. We similarly show data for the $L_{y}=3$ and
$5$ cylinders threaded by $\Phi_{\text{ext}}=\pi/2$ external flux. Unlike at
$\Phi_{\text{ext}}=0$ in the main text, the flux breaks the glide-PH symmetry,
and thus we see no diverging or quantitatively large correlation length (as is
consistent with no spontaneous symmetry breaking).

##  Appendix F Correlation Length Extrapolation

Figure S5: (a) Extrapolated inverse correlation length in the charge-neutral
sector at $L_{y}=4$ and $\Phi_{\text{ext}}=\pi/2$ obtained following the
method in Ref. [52], as detailed in Sec. F, for a range of Hubbard strengths
[different colors] and for a sequence of increasing bond dimensions [same-
colored points moving sequentially to the left]. The extrapolated inverse
correlation length saturates at a minimum value exceeding $0.15$; the
correlation length is equivalently bounded above by $6.6$ ring spacings. (b)
The same for $L_{y}=5$ and $\Phi_{\text{ext}}=0$. In this case, the
extrapolated inverse correlation length reaches and descends beyond 0,
signaling the need for larger bond dimensions to accurately estimate large
correlation lengths.

To obtain the extrapolated correlation lengths shown in the main text, we
employ a recently-developed method for extrapolating correlation lengths in
DMRG [52, 59]. Namely, we compute several of the largest eigenvalues of the
transfer matrix $TX_{i}=\lambda_{i}X_{i}$ in the electric charge $Q_{E}=1$
sector for each $k_{y}$, and select those from the same “excitation branch”
[52], namely carrying a consistent complex phase. These are related to the
correlation lengths as

$\displaystyle\epsilon_{i}=1/\xi_{i}=-\log\left|\lambda_{i}\right|$ (82)

with $\epsilon_{i}\leq\epsilon_{i+1}$. We report $\xi$ in units of cylinder
rings, or equivalently $u=a\sqrt{3}/2$ where $a$ is the lattice spacing of the
underlying triangular lattice. Correspondingly, $\epsilon$ is reported in
units of $u^{-1}$. For a system that is gapped in a charge sector
$q=(Q,S_{z},k_{y})$, both $\epsilon_{1}^{q},\epsilon_{2}^{q}$ converge to the
same infinite value as $\chi\to\infty$, namely the finite inverse correlation
length. As a result, the quantity

$\delta^{q}=\epsilon_{2}^{q}-\epsilon_{1}^{q}$ (83)

vanishes at $\chi\to\infty$, serving as a good “scaling variable” that
measures the deviation from convergence. To estimate $\xi(\chi\to\infty)$, we
therefore compute $\epsilon(\delta)$ at each available
$\epsilon(\chi),\delta(\chi)$ and extrapolate to $\delta=0$ with a polynomial
fit, namely

$\displaystyle\epsilon(\delta)=a+b\delta^{c}.$ (84)

This extrapolation is shown in Fig. S5 for both $L_{y}=4$ at
$\Phi_{\mathrm{ext}}=\pi/2$ and $L_{y}=5$ at $\Phi_{\mathrm{ext}}=0$, yielding
the extrapolated values shown in the main text. In both cases, the correlation
length is computed in the $(Q,S_{z},k_{y})=\bm{0}$, which hosts largest
correlation length.

##  Appendix G Density-density correlations at all circumferences

Figure S6: Density-density connected correlations for each of the four studied
circumferences $L_{y}=3$ to $6$, with the Hubbard interaction tuned to the
values at which the correlation length is maximal at the largest available
bond dimension, namely $U(L_{y}=3)=11.125$, $U(L_{y}=4)=11.0$,
$U(L_{y}=5)=11.6$, and $U(L_{y}=6)=11.4$, respectively. The even-circumference
cylinders are threaded with $\Phi_{\text{ext}}=\pi/2$, as in the main text.

In Fig. S6 we show the ring-averaged density-density connected correlations,
defined as $\langle n_{\text{av}}(x)n_{\text{av}}(0)\rangle,$ where

$\displaystyle n_{\text{av}}(x)$
$\displaystyle=\frac{1}{L_{y}}\sum_{y}c^{\dagger}(x,y)c(x,y)=\frac{1}{L_{y}}\sum_{k}c^{\dagger}(x,k)c(x,k).$
(85)

Panels (a,c), which correspond to the odd circumferences $L_{y}=3,5$
respectively, exhibit apparently algebraic correlations at long distances. On
the other hand, panels (b,d) exhibit exponentially-decaying correlations that
nonetheless feature a near-algebraic plateau, over a range of $x$ less than
their respective correlation lengths.