# Strong-coupling expansion of multi-band interacting models: mapping onto the
transverse-field $J_{1}$-$J_{2}$ Ising model

Xiaoyu Wang  National High Magnetic Field Laboratory, Tallahassee, FL 32310,
USA  Morten H. Christensen  Niels Bohr Institute, University of Copenhagen,
Denmark  Erez Berg  Department of Condensed Matter Physics, Weizmann Institute
of Science, Rehovot 76100, Israel  Rafael M. Fernandes  School of Physics and
Astronomy, University of Minnesota, Minneapolis, MN 55455, USA

###### Abstract

We investigate a class of two-dimensional two-band microscopic models in which
the inter-band repulsive interactions play the dominant role. We first
demonstrate three different schemes of constraining the ratios between the
three types of inter-band interactions – density-density, spin exchange, and
pair-hopping – that render the model free of the fermionic sign-problem for
any filling and, consequently, amenable to efficient Quantum Monte Carlo
simulations. We then study the behavior of these sign-problem-free models in
the strong-coupling regime. In the cases where spin-rotational invariance is
preserved or lowered to a planar symmetry, the strong-coupling ground state is
a quantum paramagnet. However, in the case where there is only a residual
Ising symmetry, the strong-coupling expansion maps onto the transverse-field
$J_{1}$-$J_{2}$ Ising model, whose pseudospins are associated with local
inter-band magnetic order. We show that by varying the band structure
parameters within a reasonable range of values, a variety of ground states and
quantum critical points can be accessed in the strong-coupling regime, some of
which are not realized in the weak-coupling regime. We compare these results
with the case of the single-band Hubbard model, where only intra-band
repulsion is present, and whose strong-coupling behavior is captured by a
simple Heisenberg model.

##  I Introduction

In systems of interacting electrons, a strong Coulomb repulsion can give rise
to correlated insulating states in which charge carriers become localized. The
low-energy properties are then usually determined by emergent charge-neutral
excitations. A prime example is the Mott-insulating phase observed in several
transition metal oxides, including the parent compounds of the high-
temperature cuprate superconductors [1, 2, 3]. The low energy excitations are
typically magnetic in nature, since the local spins usually order at low
enough temperatures inside the Mott state. More exotic correlated phenomena
may arise in systems where the electrons have additional degrees of freedom
besides spin. Examples include the orbital-selective Mott transition [4, 5, 6]
and the Hund’s metallic state [7, 8] in multi-orbital systems, as well as
ferromagnetic and Chern insulating phases in graphene-based systems with
valley degrees of freedom [9, 10, 11, 12].

From a theoretical perspective, while these materials have important
structural and chemical differences, it is useful to consider simple models
that may capture universal emergent behaviors associated with these correlated
phases [13]. In this regard, the Hubbard model [14, 15, 16] is certainly among
the most studied models in condensed matter physics, consisting of a kinetic
hopping term (with coefficient $t$) and an onsite Hubbard repulsion term (with
coefficient $U$) involving electrons on a single orbital. Upon increasing $U$,
one generally expects a metal to Mott-insulator transition [17]. However, this
is not the full story. As Phil Anderson showed in Ref. 18, a perturbative
calculation in the strong-coupling regime ($U\gg t$) reveals that the spins of
the localized charge carriers experience a superexchange interaction promoted
by virtual hopping processes. As a result, at half-filling, the insulating
state is expected to display long-range magnetic Néel order – unless
frustration is present, in which case spin liquid phases may appear [1].
Moving away from half-filling, one obtains the rich $t$-$J$ model [3].
Interestingly, in the weak-coupling regime ($U\ll t$), perturbative
calculations generically find a Néel state at half-filling [19, 20], which can
be a metal or a Slater insulator depending on additional hopping parameters.
As a result, the magnetic order in the weak-coupling and strong-coupling
regimes are the same.

For intermediate coupling strengths and away from half-filling, unconventional
superconductivity is generally expected from both strong-coupling [1, 21] and
weak-coupling perspectives [22, 23, 24]. Assessing this regime of moderate
correlations, however, is theoretically challenging due to its non-
perturbative nature. Numerical methods have played an important role in
bridging the gap between the strong- and weak-coupling regimes of the Hubbard
model [25]. These include density-matrix renormalization group methods [26,
27, 28], dynamical mean-field theory [17, 29, 30, 31], and Quantum Monte Carlo
(QMC) simulations [32, 33, 34]. While the latter is a powerful method, due to
its exact and unbiased nature, it suffers from the infamous fermionic sign
problem, which, in the case of the Hubbard model, can only be avoided exactly
at half filling and for bipartite lattices [35].

One of the main motivations to study the Hubbard model has undoubtedly been
the cuprate high-temperature superconductors. While questions remain about
whether this single-band model can capture their rich phenomenology [36, 37],
the rise of iron-based superconductors, ruthenates, and other multi-band
systems has revived the interest in multi-orbital generalizations of the
Hubbard model, the so-called Hubbard-Kanamori models (see, for example, Ref.
38). Several works have unearthed the unique properties of these models,
including the importance of the Hund’s rule coupling in promoting strong-
coupling behavior [7, 17] and the rich interplay between nesting-driven spin-
density wave and unconventional superconductivity [39]. Strong-coupling
expansions involving both spin and orbital degrees of freedom have also been
widely employed [40], resulting in complex Kugel-Khomskii effective models
[41]. From a numerical perspective, the fact that the Hubbard model suffers
from the fermionic sign-problem may discourage the use of QMC methods to
investigate the more complicated Hubbard-Kanamori models. However, as realized
in Ref. 34, by extending the number of bands of certain low-energy models that
describe electrons interacting with bosonic excitations, it is possible to
completely avoid the sign problem due to the emergence of an anti-unitary
symmetry [42]. A similar reasoning was put forward previously in Ref. 43 in
the context of the Hubbard-Kanamori model. Therefore, multi-band interacting
models may provide a unique window into the regime of moderate correlations
that is usually difficult to access in single-band models.

Figure 1: Phase diagram of the two-band electronic model with inter-band only
repulsion, see Eq. (4). The dashed black line denotes a metal-to-insulator
transition/crossover, whereas the green lines denote the superconducting (SC)
dome. The dashed red line marks a first-order antiferromagnetic (AFM) phase
transition. The AFM phase has a dome-like structure bounded by two putative
quantum phase transitions. The inset depicts the Fermi surface of the non-
interacting Hamiltonian. The color code refers to the charge compressibility
$\chi_{c}$. Figure reproduced from Ref. 44. Copyright 2020 by the American
Physical Society.

In a previous work [44], together with Y. Schattner, we showed that a
particular realization of the Hubbard-Kanamori model, formulated in band space
rather than in orbital space and with spin-anisotropic interactions, is
amenable to be simulated with a sign-problem-free QMC method. The key point is
to set the intra-band repulsion to zero and consider only inter-band repulsion
terms. While the opposite limit of large intra-band repulsion and vanishing
inter-band terms should map directly onto the usual Hubbard model, the inter-
band dominated regime that we considered in Ref. 44 has been little explored.
As such, it has the potential to provide important insights onto the
properties of multi-band interacting systems. Importantly, such a limit is not
as artificial as it may first look: a weak-coupling renormalization-group (RG)
analysis of this model shows that the inter-band terms grow much faster than
the intra-band ones under the RG flow, in the case of a nearly-nested Fermi
surface [39]. Moreover, the model remains sign-problem free even away from
half-filling and for longer-range hopping parameters.

In Fig. 1, we reproduce the temperature-interaction strength phase diagram
obtained by us in Ref. 44, via QMC simulations for a model with nearly nested
bands (see the inset). Despite the absence of intra-band repulsion, a metal-
to-insulator crossover takes place for intermediate coupling strengths (dashed
black line). Superconductivity (SC, green line) is also found in the metallic
state near the onset of Néel antiferromagnetic (AFM) order. The main feature
of this phase diagram, however, is the emergence of an AFM dome (red line).
Starting from the weak-coupling limit, it is not surprising that AFM order is
only seen after the interaction strength overcomes a threshold value, since
the nesting between the two bands is not perfect. What is more surprising is
the apparent lack of AFM order in the strong-coupling limit, since Néel order
is a hallmark of the Mott insulating state of the standard Hubbard model. In
Ref. [44], we performed a strong-coupling expansion of this inter-band
interacting model, and found the ground state to be in the quantum
paramagnetic phase of an effective transverse-field Ising model, in agreement
with the QMC results.

In this paper, we perform a strong-coupling expansion of the inter-band
interacting model for an arbitrary band dispersion and for different types of
spin-anisotropic interactions, thereby extending our previous results. In the
case of inter-band interactions that preserve the SU$(2)$ spin rotational
symmetry or lower it to a planar symmetry, we find a non-degenerate ground
state for the single site problem, indicating a “trivial” quantum paramagnetic
ground state in the strong-coupling regime. On the other hand, for the case of
inter-band interactions that lower the SU$(2)$ symmetry to an Ising symmetry,
the ground state of the single-site problem is doubly degenerate. This gives
rise to a pseudospin that corresponds to the two polarizations of the inter-
band Ising-like magnetization. In terms of this pseudospin, the strong-
coupling expansion of our microscopic interacting model maps onto a
transverse-field Ising model with extended exchange interactions, i.e.
nearest-neighbor interactions, next-nearest-neighbor interactions, etc. In
particular, we show that, by changing the band dispersion parameters across a
reasonable range of values, one can in principle access the entire phase
diagram of the transverse-field $J_{1}$-$J_{2}$ Ising model. This includes the
regime where the critical behavior is that of the simple transverse-field
Ising model – as was the case for the particular band parameters we considered
in Ref. 44 – or that of the transverse field Ashkin-Teller (four-state clock)
model. We discuss how the latter can shed new light on the emergence of
vestigial nematicity in systems that display stripe-type magnetic order, with
possible implications for the coupled magnetic-nematic transitions of iron-
based superconductors [45]. Overall, our work reveals the richness of the
strong-coupling regime of microscopic models in which inter-band interactions
dominate, which can be quite different from the strong-coupling behavior of
the standard intra-band dominated Hubbard model.

Our paper is organized as follows: In Section II we introduce the microscopic
two-band model, and show how restrictions placed on the interaction parameters
lead to the emergence of an anti-unitary symmetry, which allows the model to
be simulated using sign-problem free QMC. In Section III we perform a strong
coupling expansion of the model with an inter-band spin-spin interaction of
the Ising-type, and show that it is mapped onto the transverse-field
$J_{1}$-$J_{2}$ Ising model. In Section IV, we discuss the rich phase diagram
of the transverse-field $J_{1}$-$J_{2}$ Ising model, as well as the choice of
microscopic tight-binding parameters to achieve each ground state. Our
conclusions are presented in Sec. V.

##  II Multi-band interacting model

We start from a two-band electronic Hamiltonian with onsite interactions only:

$\begin{split}H&=H_{2}+H_{4},\\\
H_{2}&=\sum_{ij,\mu}\left[(t^{c}_{ij}-\mu^{c}\delta_{ij})c^{\dagger}_{i\mu}c_{j\mu}+(t^{d}_{ij}-\mu^{d}\delta_{ij})d^{\dagger}_{i\mu}d_{j\mu}\right],\\\
H_{4}&=\sum_{i,\mu\nu}\left[U_{1}c^{\dagger}_{i\mu}c_{i\mu}d^{\dagger}_{i\nu}d_{i\nu}-U_{2}c^{\dagger}_{i\mu}c_{i\nu}d^{\dagger}_{i\nu}d_{i\mu}\right.\\\
&+\frac{U_{3}}{2}\left(c^{\dagger}_{i\mu}c^{\dagger}_{i\nu}d_{i\nu}d_{i\mu}+h.c.\right)\\\
&\left.+U_{4}c^{\dagger}_{i\mu}c_{i\mu}c^{\dagger}_{i\nu}c_{i\nu}+U_{5}d^{\dagger}_{i\mu}d_{i\mu}d^{\dagger}_{i\nu}d_{i\nu}\right].\end{split}$
(1)

Here, $c^{\dagger}_{i,\mu}$ ($d^{\dagger}_{i,\mu}$) creates an electron in
band $c$ ($d$) at site $i$ of a square lattice with spin $\mu$. The non-
interacting electronic dispersions $\varepsilon^{c}(\mathbf{k})$ and
$\varepsilon^{d}(\mathbf{k})$ are obtained from the corresponding hopping
parameters $t_{ij}^{c,d}$. The terms $\mu^{c}$ and $\mu^{d}$ contain
implicitly both the chemical potential, given by $(\mu^{c}+\mu^{d})/2$, and
the onsite energies $\pm(\mu^{c}-\mu^{d})/2$ associated with each band. There
are five types of onsite electronic interactions, corresponding to intra-band
density-density repulsion ($U_{4}$ and $U_{5}$), inter-band density-density
repulsion $(U_{1})$, spin exchange $(U_{2})$ and pair hopping $(U_{3})$. The
model can been viewed as a projection of the usual two-orbital Hubbard-
Kanamori model on the band basis [39], with the assumption that the angle
dependence of the projected interaction parameters can be neglected.

Depending on the choice of band and interaction parameters, various low-energy
phenomena can be explored. For example, if the non-interacting band
dispersions are nearly nested (i.e. one hole-like Fermi pocket and another
electron-like Fermi pocket of similar size and shape), density-wave order in
different channels (i.e. spin, charge, loop-current) can be promoted at weak
coupling depending on the values of the inter-band interactions $U_{1}$,
$U_{2}$, and $U_{3}$ [39]. Unconventional superconductivity, characterized by
gaps of opposite signs on the two bands, is a close competitor of the density-
wave order. On the other hand, if the intra-band interactions $U_{4}$ and
$U_{5}$ are dominant, the model reduces to two nearly-independent copies of
the single-band Hubbard model. Alternatively, by constraining the inter-band
interactions $U_{1}$, $U_{2}$, and $U_{3}$ to be twice the strength of the
intra-band interactions, the model maps onto a two-layer Hubbard-model with
nearest-neighbor hopping between the layers, which has been employed to study
Cooper pairing due to an incipient band [46].

As discussed in the introduction, our interest here is on the strong-coupling
limit in the regime of dominant inter-band repulsion, as a counterpoint of the
more widely studied regime of dominant intra-band repulsion. Therefore,
hereafter we set $U_{4}=U_{5}=0$. An appealing feature of the model with
inter-band interactions only is that, by properly setting the ratios between
the three inter-band interactions, $U_{1}$, $U_{2}$, and $U_{3}$, the
Hamiltonian can be efficiently solved numerically via sign-problem-free QMC
simulations. This is because the interaction term can then be written as an
effective attractive term in the inter-band spin-channel. Introducing the
notation
$\psi_{i}=(c_{i\uparrow},c_{i\downarrow},d_{i\uparrow},d_{i\downarrow})^{T}$,
we define the inter-band spin order parameter as:
$M^{a}_{i}\equiv\sum_{\alpha\beta}\psi^{\dagger}_{i\alpha}(\sigma^{a}\rho^{x})_{\alpha\beta}\psi_{i\beta}$,
where $\sigma$ and $\rho$ are Pauli matrices acting in spin and band space
respectively, and $(\alpha,\beta)$ are indices of the vector space spanned by
$\psi_{i}$.

There are three different possibilities to do such a rewriting of the
interaction term. The first one corresponds to choosing $U_{1}=4U$,
$U_{2}=2U$, and $U_{3}=6U$, which leads to:

$H_{4,\text{Heisenberg}}=-U\sum_{a=x,y,z}\sum_{i}M_{i}^{a}M_{i}^{a}\,.$ (2)

Here, the subscript “Heisenberg” is used to emphasize the fact that the
interaction term is invariant under spin SU(2) rotational symmetry,
reminiscent of the Heisenberg model. Other ratios between $U_{1}$, $U_{2}$,
and $U_{3}$ allow for similar types of rewriting in terms of “XY” and “Ising”
inter-band spins. Of course, in these cases the interactions must break spin-
rotational symmetry. This is not an unreasonable assumption, since spin-orbit
coupling naturally breaks spin-rotational symmetry in actual materials. In our
treatment, such effects are treated on a phenomenological level only. In
particular, setting $U_{1}=4U(1-\delta_{\mu\nu})$, $U_{2}=0$, and $U_{3}=4U$
we obtain

$H_{4,\text{XY}}=-U\sum_{a=x,y}\sum_{i}M_{i}^{a}M_{i}^{a}\,.$ (3)

Note that the interaction term possesses a residual $U(1)$ (or planar)
symmetry, similar to the XY model. Finally, in the case where
$U_{1}=4U\delta_{\mu\nu}$, $U_{2}=2U$, and $U_{3}=2U$, $H_{4}$ can be
rewritten as

$H_{4,\text{Ising}}=-U\sum_{i}M_{i}^{z}M_{i}^{z}\,,$ (4)

where only a $\mathbb{Z}_{2}$ symmetry is preserved.

The key motivation to introduce these restrictions on the inter-band
repulsions is that the resulting Hamiltonian can be simulated with QMC without
the sign-problem. This can be seen by decoupling the quartic terms via a
Hubbard-Stratonovich field $\\{\phi_{i}^{a}(\tau)\\}$, where $\tau$ denotes
imaginary time. As a result, the partition function can be represented as

$\mathcal{Z}=\int
D[\phi]\det\\{\widehat{G}^{-1}(\phi)\\}\exp(-\mathcal{S}_{\phi})\,.$ (5)

Here
$\mathcal{S}_{\phi}=\int_{0}^{\beta}\mathrm{d}\tau\frac{1}{4U}[\phi_{i}^{a}(\tau)]^{2}$
and $\widehat{G}\equiv(\partial_{\tau}+H_{2}+H^{\phi}_{2})^{-1}$ is the
fermionic Green’s function, with
$H_{2}^{\phi}=\sum_{i}\phi_{i}^{a}(\tau)M^{a}_{i}$. In the single-particle
Hilbert space, the quadratic Hamiltonian ($H_{2}+H_{2}^{\phi}$) has an anti-
unitary symmetry $\widehat{U}=i\sigma^{y}\rho^{z}K$, where $K$ denotes complex
conjugation. Since $\widehat{U}^{2}=-1$, the eigenstates of the single
particle spectrum are always doubly degenerate (an analogue of a Kramer’s
doublet), which guarantees a positive fermionic determinant [42]. In Ref.
[44], we used determinantal QMC to study the Ising case in Eq. (4), resulting
in the phase diagram of Fig. 1 discussed in the Introduction.

It is interesting to note that the restriction on the ratios between $U_{1}$,
$U_{2}$, and $U_{3}$ that is needed to “eliminate” the fermionic sign-problem
is a much milder constraint than the corresponding restriction on the single-
band Hubbard model. In the latter case, one needs to impose half-filling and
hopping parameters that preserve the bipartite nature of the square lattice.
In the present case, on the other hand, there are no restrictions on the
filling and on the type of hopping parameters. The reason is because the
avoidance of the fermionic sign-problem arises from an anti-unitary symmetry
related to the intrinsic two-band nature of the problem, in the same spirit as
in Ref. 42.

##  III Strong-coupling expansion

While in our previous work [44] we performed a strong-coupling expansion of
the Ising case [Eq. (4)] for a specific tight-binding parameters set, here we
explore the strong-coupling regime of the three cases presented in Eqs.
(2)–(4) for an arbitrary tight-binding parametrization. The procedure consists
of first neglecting the kinetic terms and then solving the interacting
Hamiltonian exactly on a single site. Next, we treat the kinetic terms
perturbatively, and discuss the strong-coupling physics using an effective
Hamiltonian in terms of the available degrees of freedom [47]. As we will
show, the Ising case provides the only non-trivial ground state at strong-
coupling, mapping onto the rich transverse-field $J_{1}$-$J_{2}$ Ising model.

###  III.1 Single-site exact solutions

The energy spectra of the interacting Hamiltonians in Eqs. (2)–(4) on a single
site are illustrated in Figure 2. The local Fock space can be written as
$\ket{\eta_{c};\eta_{d}}$ where
$\eta_{c},\eta_{d}\in\\{0,\uparrow,\downarrow,\uparrow\downarrow\\}$. This is
a 16-dimensional vector space. Due to number conservation, the Hamiltonians
can be analyzed within any given integer electron filling $n=0,\dots,4$. The
linear dimension for each filling factor is given by the binomial coefficient
$d_{n}=C_{4}^{n}$. The $n=0$ and $n=4$ states have the highest energy, $E=0$.
All eight $n=1$ and $n=3$ states have equal energy, $E=-3U$ for the Heisenberg
case, $E=-2U$ in the XY case, and $E=-U$ in the Ising case. In all three
cases, the $n=2$ sector contains the lowest energy states. In total, there are
six states in the $n=2$ sector, given by:

$\Phi=\begin{pmatrix}\ket{0;\uparrow\downarrow}&\ket{\uparrow;\uparrow}&\ket{\uparrow;\downarrow}&\ket{\downarrow;\uparrow}&\ket{\downarrow;\downarrow}&\ket{\uparrow\downarrow;0}\end{pmatrix}^{T}\,,$
(6)

The matrix elements of $H_{4}$ in this sector are

Figure 2: Energy spectrum and level degeneracy of the single-site interacting
Hamiltonian $H_{4}$ for the Heisenberg [Eq. (2)], XY [Eq. (3)], and Ising [Eq.
(4)] cases. States in the $n=2$ filling sector are denoted in red.

$\mathcal{H}_{4,\text{Heisenberg}}=-U\begin{pmatrix}6&0&0&0&0&-6\\\
0&4&0&0&0&0\\\ 0&0&2&2&0&0\\\ 0&0&2&2&0&0\\\ 0&0&0&0&4&0\\\
-6&0&0&0&0&6\end{pmatrix},$ (7)
$\mathcal{H}_{4,\text{XY}}=-U\begin{pmatrix}4&0&0&0&0&-4\\\ 0&4&0&0&0&0\\\
0&0&0&0&0&0\\\ 0&0&0&0&0&0\\\ 0&0&0&0&4&0\\\ -4&0&0&0&0&4\end{pmatrix},$ (8)
$\mathcal{H}_{4,\text{Ising}}=-U\begin{pmatrix}2&0&0&0&0&-2\\\ 0&0&0&0&0&0\\\
0&0&2&2&0&0\\\ 0&0&2&2&0&0\\\ 0&0&0&0&0&0\\\ -2&0&0&0&0&2\end{pmatrix}.$ (9)

The single-site energy spectrum of each case is shown in Fig. 2. The states
for the filling $n=2$ are shown in red, whereas the states for the other
filling sectors are shown in blue. Both the Heisenberg and XY Hamiltonians
have a unique ground state given by
$\ket{a_{1}}\equiv\frac{1}{\sqrt{2}}\left(\ket{\uparrow\downarrow;0}-\ket{0;\uparrow\downarrow}\right)$,
with energies $-12U$ and $-8U$ respectively. For the Ising Hamiltonian, on the
other hand, the lowest-energy state $-4U$ is two-fold degenerate, consisting
of both $\ket{a_{1}}$ and
$\ket{a_{2}}\equiv\frac{1}{\sqrt{2}}\left(\ket{\uparrow;\downarrow}+\ket{\downarrow;\uparrow}\right)$.
A general ground state can therefore be written as:

$\ket{\varphi}=\alpha\ket{a_{1}}+\beta\ket{a_{2}}\,,$ (10)

where $|\alpha|^{2}+|\beta|^{2}=1$. This degeneracy is a result of the
commutation relation $[H_{4,\text{Ising}},M^{z}_{i}]=0$, which does not hold
in the Heisenberg or XY cases. Because $M^{z}_{i}\ket{a_{1}}=-2\ket{a_{2}}$
and $M^{z}_{i}\ket{a_{2}}=-2\ket{a_{1}}$, the combinations
$\frac{1}{\sqrt{2}}(\ket{a_{1}}-\ket{a_{2}})$ and
$\frac{1}{\sqrt{2}}(\ket{a_{1}}+\ket{a_{2}})$ are eigenstates of $M_{i}^{z}$
with eigenvalues $\pm 2$. These combinations can thus be interpreted as the
two possible polarizations of the inter-band Ising magnetization. Therefore,
in the two-dimensional space spanned by the spinor in Eq. (10), the inter-band
magnetization is represented by the $\tau^{x}$ pseudospin.

This analysis shows that, in the strong-coupling limit, the Heisenberg and XY
cases display only a featureless quantum paramagnetic state. On the other
hand, the Ising case has a residual SU(2) degeneracy related to the inter-band
magnetization degree of freedom. Perturbative interactions between the local
inter-band magnetizations, which we will study in the next subsection, may
result in non-trivial ground states. Therefore, within our model, the Ising
case is the closest analogue of the single-band Hubbard model, whose strong-
coupling limit is characterized by perturbative interactions between local
(intra-band) spins.

###  III.2 Effective Hamiltonian of the Ising case

We proceed to discuss the effects of the kinetic terms given by $H_{2}$. For
the Heisenberg and XY cases, the many-body ground state is unique and gapped,
given by $\ket{\Psi_{0}}=\Pi_{i=1}^{L^{2}}\ket{a_{1}}$. Note that this is a
quantum paramagnetic state with
$\langle\vec{M}_{i}e^{i\mathbf{Q}\cdot\mathbf{r}_{i}}\rangle=0$ for arbitrary
wavevector $\mathbf{Q}$. In the Ising case, however, the lowest energy state
on each site is a spinor, analogous to the physical electron spin in the one-
band Hubbard model. As a result, the ground state has an
$\text{SU(2)}^{L^{2}}$-degeneracy. Perturbations due to the kinetic terms can
lift this degeneracy, leading to non-trivial correlations in the Hilbert space
spanned by the local spinors.

The energetics of low-energy excitations can be studied using an effective
Hamiltonian approach [47] in terms of the pseudospins $\tau^{\mu}$ previously
introduced. Recall that, in the basis of Eq. (10), inter-band magnetic order
corresponds to a finite expectation value for $\tau^{x}$. For the Ising
Hamiltonian we find, up to second order in peturbation theory,

$\begin{split}H_{\text{eff},ss^{\prime}}&\approx-4UL^{2}\delta_{ss^{\prime}}+\bra{\Psi_{0,s}}H_{2}\ket{\Psi_{0,s^{\prime}}}\\\
&+\sum_{n\neq
0,t}\frac{\bra{\Psi_{0,s}}H_{2}\ket{\Psi_{n,t}}\bra{\Psi_{n,t}}H_{2}\ket{\Psi_{0,s^{\prime}}}}{E_{0}-E_{n}},\end{split}$
(11)

where $\ket{\Psi_{0,s}}=\Pi_{i=1}^{L^{2}}\ket{\varphi_{i,s}}$, and
$\ket{\varphi_{i,s}}=\alpha_{i,s}\ket{a_{1}}+\beta_{i,s}\ket{a_{2}}$ is a
configuration of the local spinor. $\ket{\Psi_{n,t}}$ denotes an excited
state, having e.g. 1 electron on site $i_{1}$ and 3 electrons on site $i_{2}$.

The contribution from the single-particle onsite $\mu^{c,d}$ terms to the
effective Hamiltonian is given by

$\begin{split}H_{\text{eff}}^{(\mu)}&\approx-\sum_{i}\left[(\mu^{c}+\mu^{d})+\frac{1}{8U}(\mu^{c}-\mu^{d})^{2}\right]\tau_{i}^{0}\\\
&-\sum_{i}\frac{1}{8U}(\mu^{c}-\mu^{d})^{2}\tau^{z}_{i},\end{split}$ (12)

Note that only the difference in the onsite energies between the two bands,
but not the chemical potential, gives non-trivial energetics. It corresponds
to a transverse field, since magnetic order is given by $\tau^{x}$. This
difference between onsite energies is a tuning parameter absent from the
strong-coupling limit of one-band Hamiltonians, such as the Hubbard model.

The hopping parameter $t_{ij}$ generates spinor correlations between sites
$\\{i,j\\}$. To second order, this leads to the superexchange interactions:

$H_{\text{eff}}^{(t)}\approx\sum_{:ij:}\left(-\frac{(t_{ij}^{c})^{2}+(t_{ij}^{d})^{2}}{6U}\tau_{i}^{0}\tau^{0}_{j}+\frac{t_{ij}^{c}t_{ij}^{d}}{3U}\tau_{i}^{x}\tau_{j}^{x}\right),$
(13)

where $:ij:$ denotes an ordered pair of sites, i.e., permuting the two indices
does not lead to a new term in the Hamiltonian. We note that the superexchange
interaction is of an Ising-type, as expected from the fact that the model
lacks spin-rotational symmetry. Furthermore, the signs of the exchange
interactions depend on the relative signs between the hopping parameters of
the two bands. Hence, equal (opposite) signs favor antiferromagnetic
(ferromagnetic) alignment of the inter-band magnetization. To summarize, in
the strong coupling limit, the two-band electronic model with Ising
interaction maps onto the generalized transverse-field Ising model containing
longer-range exchange interactions:

$H_{\text{eff}}\approx\sum_{:ij:}J_{ij}\tau_{i}^{x}\tau^{x}_{j}-h\sum_{i}\tau_{i}^{z}\,,$
(14)

where $h_{i}\equiv(\mu_{i}^{c}-\mu_{i}^{d})^{2}/8U$ and $J_{ij}\equiv
t_{ij}^{c}t_{ij}^{d}/3U$. Note that we have omitted an overall shift of the
ground state energy.

##  IV The transverse-field $J_{1}$-$J_{2}$ Ising model

Figure 3: (a) Schematic phase diagram at zero transverse field ($h=0$), and
(b) Schematic phase diagram at zero-temperature ($T=0$) of the transverse-
field $J_{1}$-$J_{2}$ Ising model for antiferromagnetic exchange interactions.
Solid (dashed) lines correspond to continuous (first order) phase boundaries.
PM denotes the paramagnetic phase and VBS denotes a possible string valence-
bond solid.

Let us focus on the case where $H_{2}$ has hopping parameters defined up to
next-nearest-neighbor bonds, yielding the band dispersions
$\varepsilon^{i}(\mathbf{k})=-\mu^{i}+2t^{i}_{1}(\cos k_{x}+\cos
k_{y})+4t^{i}_{2}\cos k_{x}\cos k_{y}$. According to Eq. (14), this leads to
the square-lattice transverse-field $J_{1}$-$J_{2}$ Ising model, with

$\begin{split}J_{1}&=\frac{t^{c}_{1}t^{d}_{1}}{3U}\,,\\\
J_{2}&=\frac{t^{c}_{2}t^{d}_{2}}{3U}\,,\\\
h&=\frac{(\mu^{c}-\mu^{d})^{2}}{8U}\,.\end{split}$ (15)

This model has been studied extensively in the recent literature [48, 49, 50,
51, 52, 53], and here we give a brief overview of the proposed phase diagrams
of the frustrated case, where $J_{2}>0$. Because the model is invariant upon
changing the signs of $J_{1}$ or $h$, hereafter we focus on the case of
$J_{1},h>0$. For $h=0$, the classical phase diagram is rather well-
established, and is schematically shown in Fig. 3(a) based on the Monte Carlo
results of Ref. 48. For $J_{2}<J_{1}/2$, the ground state is a Néel
antiferromagnet described by the order parameter
$\Delta_{\text{N}}=L^{-2}\sum_{i}\langle\tau^{x}_{i}e^{i\mathbf{Q}_{\text{N}}\cdot\mathbf{r}_{i}}\rangle$,
where $\mathbf{Q}_{\text{N}}\equiv(\pi,\pi)$ is the ordering wave-vector (note
that the ground state would be a ferromagnet if $J_{1}<0$). On the other hand,
for $J_{2}>J_{1}/2$, the ground state is a striped antiferromagnet (regardless
of the sign of $J_{1}$), with an order parameter
$\Delta_{\text{S}_{n}}=L^{-2}\sum_{i}\langle\tau^{x}_{i}e^{i\mathbf{Q}_{\text{S}_{n}}\cdot\mathbf{r}_{i}}\rangle$,
where $\mathbf{Q}_{\text{S}_{1}}=(\pi,0)$ and
$\mathbf{Q}_{\text{S}_{2}}=(0,\pi)$. The transition between the Néel and the
stripe phases is first-order, as expected from the fact that they break
different symmetries.

Upon increasing temperature, the Néel to paramagnetic (PM) transition is
second-order and belongs to the 2D-Ising universality class – except possibly
in a narrow region near $J_{2}=J_{1}/2$, where the transition may become first
order [48]. As for the stripe ground state, it is important to note that it is
four-fold degenerate, since there are two stripe ordering wave-vectors and two
Ising spin polarizations. In the range $1/2<J_{2}/J_{1}\lessapprox 0.67$, the
stripe-PM transition is first-order. However, for $J_{2}/J_{1}\gtrapprox
0.67$, the stripe-PM transition is second-order and described by the 2D four-
state clock model [48]. A special property of this model is that it only has
weak universality, in the sense that only the anomalous critical exponent
$\eta$ (and consequently the $\delta$ exponent) is universal, while the other
critical exponents are non-universal [54]. In the particular case of the
classical $J_{1}$-$J_{2}$ Ising model, it was shown in Ref. 48 that the non-
universal critical exponents as a function of the ratio $J_{2}/J_{1}$ map onto
the critical exponents of another model that belongs to the four-state clock
weak universality: the Ashkin-Teller model. In particular, for
$J_{2}/J_{1}\approx 0.67$, the critical exponents are those of the four-state
Potts model, whereas for $J_{2}/J_{1}\rightarrow\infty$, they are those of the
2D Ising model.

The quantum phase diagram ($T=0$) of the model remains widely debated,
although some properties seem to be consistent across different methods [49,
50, 51, 52, 53]. In Fig. 3(b), we show a schematic candidate phase diagram
based on these works. As in the classical case, the Néel state is realized for
$J_{2}/J_{1}<1/2$ and the stripe state is realized for $J_{2}/J_{1}>1/2$. For
zero transverse field, the transition at $J_{2}=J_{1}/2$ is first-order, like
in the classical phase diagram. As $h$ increases, however, the situation is
less clear. For instance, Ref. 50 reported the onset of a string valence-bond
solid for finite transverse fields. The quantum phase transition from the Néel
to the PM state is believed to be second-order and in the 3D Ising
universality class [53] – although some methods report a regime of first-order
transition near $J_{2}/J_{1}=1/2$ [51]. The stripe-PM quantum transition seems
to display a tricritical point separating a first-order transition line from a
second-order transition line – similarly to the classical case [52].
Interestingly, the position of the quantum tricritical point seems to be
closer to the degeneracy point than the classical tricritical point [52, 53],
i.e. $(J_{2}/J_{1})_{\mathrm{q}}<(J_{2}/J_{1})_{\mathrm{cl}}\approx 0.67$, as
illustrated schematically in Fig. 3(a) and (b) by the red dots. The nature of
the second-order quantum stripe-PM transition remains unclear. Because the
classical transition is described by the four-state clock model, it is natural
to expect that the quantum transition should be described by the quantum
version of the same model. Based on the recent results of Ref. 55 on the
quantum $q$-state clock model, this transition is expected to belong to the 3D
XY universality class. However, Ref. 53 found that the quantum stripe-PM
transition has non-universal critical exponents.

This rich landscape of possible ground states of the two-band model with
dominant inter-band interactions in the strong-coupling regime contrasts with
the simple Néel state obtained for the single-band Hubbard model. We now show
that the $(h/J_{1},\,J_{2}/J_{1})$ phase diagram of Fig. 3(b) can in principle
be traversed with reasonable band structure parameters, contrasting the
strong-coupling and weak-coupling behaviors of the model.

Figure 4: (a-d) Two types of non-interacting Fermi surfaces featuring one hole-like pocket centered at $\Gamma$ and one electron-like pocket centered at either (a,b) $\mathbf{Q}_{\text{N}}=(\pi,\pi)$ or (c,d) $\mathbf{Q}_{\text{S}}=(\pi,0)/(0,\pi)$. (e-f) Two identical copies of Fermi surfaces with a larger hopping parameter for (e) the nearest-neighbor hopping and (f) the next-nearest-neighbor hopping, respectively. The band parameters are given in Table 1. | $\mu^{c}$ | $t^{c}_{1}$ | $t^{c}_{2}$ | $\mu^{d}$ | $t^{d}_{1}$ | $t^{d}_{2}$ | $J_{2}/J_{1}$ | $h/J_{1}$ | G.S. ($U\gg t$)
---|---|---|---|---|---|---|---|---|---
4(a) | $t$ | $t$ | $0$ | $-t$ | $t$ | $0$ | $0$ | $3/2$ | Néel
4(b) | $2t$ | $t$ | $0$ | $-2t$ | $t$ | $0$ | $0$ | $6$ | PM
4(c) | $t$ | $0.6t$ | $0.8t$ | $-t$ | $0.2t$ | $t$ | $20/3$ | $25/6$ | Stripe
4(d) | $2t$ | $0.6t$ | $0.8t$ | $-2t$ | $0.2t$ | $t$ | $20/3$ | $50/3$ | PM
4(e) | $t$ | $t$ | $0$ | $t$ | $t$ | $0$ | $0$ | $0$ | Néel
4(f) | $t$ | $0.8t$ | $t$ | $t$ | $0.8t$ | $t$ | $25/16$ | $0$ | Stripe
Table 1: Parameters of the dispersions shown in Fig. 4 and discussed in the
text, and their respective ground states (G.S.) at strong coupling.

A commonly studied situation is when the two bands give rise to a hole-like
and an electron-like Fermi pocket, as illustrated in Fig. 4 (a)-(d). In (a),
the $c$-band creates a hole-like Fermi pocket centered at $\Gamma$ in the
Brillouin zone, whereas the $d$-band gives rise to an electron-like Fermi
pocket of identical size centered at $\mathbf{Q}_{\text{N}}$. This is achieved
by a dominant nearest-neighbor hopping, as well as opposite onsite energies,
$\mu^{c}=-\mu^{d}$, of magnitudes comparable to the bandwidth. As shown in
Table 1, the tight-binding parametrization of Fig. 4(a) favors a Néel ground
state in the strong-coupling regime. Upon increasing $(\mu^{c}-\mu^{d})$,
which corresponds to shrinking the sizes of the hole-like and electron-like
pockets, the value of the effective random field $h/J_{1}$ increases and moves
the system towards the quantum paramagnetic ground state. The corresponding
Fermi surface is shown in Fig. 4 (b). Therefore, without changing the
electronic occupation, it is possible to induce a quantum phase transition
from the Néel phase to the paramagnetic phase. Physically, changing the band
offset may be achieved via pressure.

It is also interesting to compare the strong-coupling ground state with the
weak-coupling one. The parametrization in Fig. 4 (a) satisfies the perfect
nesting condition
($\varepsilon^{c}_{\mathbf{k}}=-\varepsilon^{d}_{\mathbf{k}+\mathbf{Q}_{\text{N}}}$),
which leads to a spin-density wave order, even for infinitesimal small $U$,
described by the same Néel order parameter. Varying $(\mu^{c}-\mu^{d})$ does
not spoil the perfect nesting condition, and therefore is not expected to
drive a transition to a paramagnetic phase, in contrast to the strong-coupling
limit.

By including sizable next-nearest-neighbor hopping, i.e. increasing
$t_{2}/t_{1}$, the $d$-band electron-like Fermi pockets become centered at
$\mathbf{Q}_{\text{S}_{1}}=(\pi,0)$ and $\mathbf{Q}_{\text{S}_{2}}=(0,\pi)$,
as shown in Fig. 4(c). Here, a stripe state is preferred at strong-coupling,
as shown in Table 1. Similarly to the previous case, by increasing the onsite
energy difference – see Fig. 4(d) – the system moves towards a quantum
paramagnetic state due to the increase in the transverse field value. In the
weak-coupling regime, the stripe magnetic transition temperature is suppressed
upon making the nesting conditions poorer. Moreover, as discussed in Ref. 56,
in the weak-coupling regime, strong deviations from perfect nesting can change
the magnetic ground state from the stripe phase to a charge-spin density-wave
– i.e. a collinear double-$\mathbf{Q}$ state consisting of a linear
combination of the order parameters $\Delta_{\mathrm{S_{1}}}$ and
$\Delta_{\mathrm{S_{2}}}$. Therefore, the magnetic ground states in the weak-
and strong-coupling regimes may be different.

There is an important difference between the Néel state and the stripe state.
The fourfold degeneracy of the latter corresponds to two distinct Ising
symmetries: one related to the polarization of the Ising-magnetic order and
one related to the tetragonal symmetry of the lattice – since there are two
possible stripe directions that lower the tetragonal symmetry to orthorhombic
in different ways. The latter is thus associated with nematic order [45, 57],
described in terms of the composite order parameter
$\Delta_{\mathrm{S_{1}}}^{2}-\Delta_{\mathrm{S_{2}}}^{2}$ [58]. Such a nematic
phase is called a vestigial order of the underlying stripe phase [59, 60]. In
the weak-coupling regime, the quantum nematic phase transition is generally
expected to be first-order and simultaneous to the stripe one [61]. However,
in the strong-coupling regime of the model, both order parameters can onset
simultaneously at a _single_ quantum critical point, which likely belongs to
the 3D XY universality class. This result has important implications for the
possibility of nematic and magnetic quantum criticality being realized in
iron-based superconductors [62, 63, 64, 45], whose band structure contains
hole pockets and electron pockets separated by the wave-vectors
$\mathbf{Q}_{\text{S}_{n}}$.

The above examples demonstrate that the magnetic ground states at weak and at
strong coupling can break the same symmetry, therefore allowing for a smooth
connection at moderate coupling strengths. However, this is by no means
necessary. To illustrate this point, we consider a simple example of identical
tight-binding parameters, $\mu^{c}=\mu^{d}$ and $t^{c}_{1(2)}=t^{d}_{1(2)}$,
as illustrated in Figs. 4(e) and (f) . In this case, the transverse field
vanishes, and the ground state in the strong-coupling regime is either a Néel
state or a stripe state, depending on the ratio $t_{2}/t_{1}$ (see Table 1).
This is in stark contrast to what one would expect from a weak-coupling
approach, since this case features two identical Fermi surfaces, which are far
from satisfying any nesting condition.

##  V Conclusions

In this work, we generalized the strong-coupling expansion of the single-band
Hubbard model, as pioneered by Phil Anderson [18], to a two-band electronic
model with dominant inter-band repulsion. While the former maps onto the
Heisenberg model, the latter maps onto the transverse-field Ising model with
extended exchange interactions. In particular, the onsite energy difference
between the two electronic bands gives rise to a transverse field, while the
Ising superexchange interactions arise from virtual hopping processes
involving nearest-neighbors and next-nearest-neighbors. Importantly, by fixing
the ratio between the inter-band repulsive interactions, this electronic model
can be simulated using sign-problem free QMC for arbitrary electronic filling
and hopping parameters. In contrast, the Hubbard model, where intra-band
repulsion dominates, is only sign-problem-free at half-filling and for
bipartite kinetic Hamiltonians.

We also showed how the rich $(h/J_{1},\,J_{2}/J_{1})$ phase diagram of the
transverse-field $J_{1}$-$J_{2}$ model can be probed by appropriately tuning
the microscopic band parameters over a reasonable range of values. This opens
an interesting avenue for future sign-problem-free QMC studies to explore the
impact of different types of strong-coupling ground states on the emergence of
superconductivity and other emergent phenomena at intermediate interaction
strengths.

###### Acknowledgements.

We thank Andrey Chubukov, Anders Sandvik, Yoni Schattner, Jun Takahashi, and
Oskar Vafek for fruitful discussions. XW acknowledges financial support from
National MagLab, which is funded by the National Science Foundation
(DMR-1644779) and the state of Florida. MHC acknowledges financial support
from the Villum foundation. EB was supported by the European Research Council
(ERC) under grant HQMAT (grant no. 817799), the US-Israel Binational Science
Foundation (BSF), and the Minerva foundation. RMF is supported by the U.S.
Department of Energy, Office of Science, Basic Energy Sciences, Materials
Science and Engineering Division, under Award No. DE-SC0020045. RMF also
acknowledges partial support from the Research Corporation for Science
Advancement via the Cottrell Scholar Award.

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