# Spin susceptibility anomaly in cluster Mott insulators on a partially-filled anisotropic Kagome lattice: applications to LiZnMoO

###### Abstract

Motivated by recent experiments on the quantum-spin-liquid candidate material LiZnMoO, we study a single-band extended Hubbard model on an anisotropic Kagome lattice with the 1/6 electron filling. Due to the partial filling of the lattice, the inter-site repulsive interaction is necessary to generate Mott insulators, where electrons are localized in clusters, rather than at lattice sites. We provide examples of such cluster Mott insulators and study the phase transitions between metallic states and cluster Mott insulators on an anisotropic Kagome lattice. It is shown that these cluster Mott insulators are generally U(1) quantum spin liquids with spinon Fermi surfaces. However, the nature of charge excitations in different cluster Mott insulators could be quite different and we show that there exists a novel cluster Mott insulator where charge fluctuations around the hexagonal cluster induce a plaquette charge order (PCO). The spinon excitation spectrum in this spin-liquid cluster Mott insulator is reconstructed due to the PCO so that only 1/3 of the total spinon excitations are magnetically active. The strong coupling limit of the same model is also analyzed via a Kugel-Khomskii-like model. Based on these results, we propose that the anomalous behavior of the finite-temperature spin-susceptibility in LiZnMoO may be explained by finite-temperature properties of the cluster Mott insulator with the PCO as well as fractionalized spinon excitations. Existing and possible future experiments on LiZnMoO, and other Mo-based cluster magnets are discussed in light of these theoretical predictions.

###### pacs:

75.10.Kt, 75.10.Jm## I Introduction

If there is no spontaneous symmetry breaking, the ground state of a Mott insulator with
odd number of electrons per unit cell may be a quantum spin liquid (QSL)Hastings (2004).
The QSL is an exotic quantum phase of matter with a long-range quantum entanglementWen (2007),
which is characterized by fractionalized spin excitations
and an emergent gauge structures at low energiesBalents (2010).
It is now clear that some frustrated Mott insulating systems which are proximate to
Mott transitions may provide physical realizations of QSL
phases.Lee and Lee (2005); Motrunich (2005, 2006); Podolsky *et al.* (2009); Chen and Kim (2013)
These U(1) QSLs arise from strong charge fluctuations in the weak Mott regime,
which can generate sizable long range spin exchanges or spin ring exchanges and
suppress possible magnetic orderings.Motrunich (2005, 2006)
Several QSL candidate materials, such as the 2D
triangular lattice organic materials -(ET)Cu(CN)
and EtMeSb[Pd(dmit)],
and a 3D hyperkagome system NaIrOHiroi *et al.* (2001); Shimizu *et al.* (2003); Okamoto *et al.* (2007),
are expected to be in this weak Mott regime.
These weak Mott-insulator U(1) QSLs are obtained as a deconfined phase of an
emergent U(1) lattice gauge theoryFlorens and Georges (2004); Lee and Lee (2005), where
the electron is fractionalized into spin-carrying spinons and charged bosons.
The charge excitations are gapped and
the low-energy physics of the QSLs is described by a spinon Fermi surface
coupled to the emergent U(1) gauge field.

In this work, motivated by the recent experiments on a new QSL candidate material
LiZnMoOSheckelton *et al.* (2012, 2014); Mourigal *et al.* (2014),
we consider a 1/6-filled extended Hubbard model with nearest-neighbor repulsions
and propose a U(1) QSL with spinon Fermi surfaces and a plaquette
charge order (PCO) as a possible ground state.
The Mott insulators in partially filled systems arise due to
large nearest-neighbor repulsions and localization of the charge
degrees of freedom in certain cluster units. Hence such Mott
insulators may be called “cluster Mott insulators” (CMIs).

In LiZnMoO, as described in
Ref. Sheckelton *et al.*, 2012,
each MoO triangular cluster hosts one unpaired localized electron
with moment. These MoO clusters
are organized into
a triangular lattice structure (see Fig. 1).Sheckelton *et al.* (2012, 2014); Mourigal *et al.* (2014)
No magnetic ordering is detected in neutron scattering, in
NMR and SR measurement down to K.Sheckelton *et al.* (2012, 2014); Mourigal *et al.* (2014)
In particular, the spin susceptibility shows a very puzzling anomaly:
below about 100K the spin susceptibility is governed by a different
Curie-Weiss law with a much smaller
Curie-Weiss temperature (K)
from the high temperature one (K) and
a much reduced Curie constant which is 1/3 of the high temperature one.

In a recent theoretical workFlint and Lee (2013), Flint and Lee considered the possibility of an emergent honeycomb lattice with weakly coupled dangling spins in the centers of the hexagons. In their description, the emergent honeycomb system may form a gapped QSL phase while the remaining weakly-coupled dangling spin moments comprise 1/3 of the total magnetic moments and dominate the low-temperature magnetic properies which then explains the “1/3 anomaly” in the spin susceptibility. Their theory invokes the lattice degrees of freedom to work in a way to generate the emergent honeycomb lattice for the spin system. Such a scenario might be plausible but needs to be confirmed by further experiments. In this paper, however, we explore an alternative explanation for the experiments that is based on electronic degrees of freedom and their interactions.

Instead of working with the exchange model between the local spin moments, we consider a single-band extended Hubbard model for the unpaired Mo electrons of the MoO clusters. The Hubbard model is the parent model of the spin exchange interactions and may contain the crucial physics that is not described by the spin exchange model. Moreover, the Mo electrons are in electron orbital states, and electron systems are often not in the strongly localized regime due to the substantial spatial extension of the orbital wavefunction. Therefore, we think it is more appropriate to model the system by a Hubbard-like model. Our single-band extended Hubbard model is defined on the anisotropic Kagome lattice that is formed by the Mo sites (see Fig. 1a) and is given by

(1) | |||||

where the spin index is implicitly summed, () creates (annihilates) an electron with spin at lattice site , and and are the nearest-neighbor electron hopping and interaction in the up-pointing triangles (denoted as ‘u’) and the down-pointing triangles (denoted as ‘d’) (see Fig. 1b), respectively. is the electron occupation number at site . Since there exists only one electron in each Kagome lattice unit cell, the electron filling for this Hubbard model is . In Sec. II, we shall provide a motivation to consider this single-band Hubbard model using a quantum chemistry analysis.

We include the on-site Hubbard- interaction as well as two inter-site repulsions and in the extended Hubbard model. Although the down-triangles are larger in size than the up-triangles in LiZnMoO, because of the large spatial extension of the Mo electron orbitals we think it is necessary to include the inter-site repulsion for the down-triangles. Moreover, for LiZnMoO we expect and . Keeping the Hubbard- interaction as the largest energy scale in the model, we study the phase diagram in terms of .

In the case of the fractional filling, the Mott localization is driven by the inter-site repulsions () rather than the on-site Hubbard interaction and the electrons are localized in the (elementary) triangles of the Kagome lattice instead of the lattice sites. Because of the asymmetry between the up-triangles and down-triangles of the Kagome lattice, the Mott localization in the up-triangles and down-triangles does not need to occur simultaneously. Therefore, two types of CMIs are expected.

Refs. Sheckelton *et al.*, 2012; Flint and Lee, 2013 assume that LiZnMoO
is in the type-I CMI phase, where the electrons are only localized in the up-triangles while
the electron number in the down-triangles remains strongly fluctuating.
In our work,
we propose that the system may be more likely to be in the type-II CMI,
where the inter-site repulsions ()
are strong enough to localize the electrons in both up-triangles and down-triangles
even though .
The electron number in every triangle is then fixed to be one.
Although a single electron cannot hop from one triangle to another,
a (collective) ring hopping of 3 electrons on the perimeter
of an elementary hexagon on the Kagome lattice is allowed
and gives rise to a long-range
plaquette charge order (PCO) in the type-II CMI (see Fig. 1c).
The emergence of PCO in the type-II CMI is a quantum effect and cannot
be obtained from the classical treatment of the electron interaction.

With the PCO, 1/3 of the elementary hexagons become resonating. As shown in Fig. 1c, these resonating hexagons form an emergent triangular lattice (ETL). The PCO triples the original unit cell of the Kagome lattice, and the localized electron number in the enlarged unit cell now becomes 3, which is still odd. Therefore, the type-II CMI with the PCO is not connected to a trivial band insulator and the QSL is still expected. In the resulting U(1) QSL, we obtain 9 mean-field spinon bands for the type-II CMI with the PCO, compared to the 3 spinon bands in the U(1) QSL for the type-I CMI without the PCO. A direct band gap separates the lowest spinon band from other spinon bands in the presence of PCO. The lowest spinon band is completely filled by 2/3 of the spinons, leaving the remaining 1/3 of the spinons to partially fill the second and third lowest spinon bands. Because of the band gap, the only active degrees of freedom at low energies are the spinons in the partially filled spinon bands, and the fully-filled lowest spinon band is inert to external magnetic field at low temperatures as long as the PCO persists. Therefore, only 1/3 of the magnetic degrees of freedom are active at low temperatures. If one then considers the local moment formation starting from the band filling picture of the spinons (just like electrons occupying the same band structure) only the 1/3 of the spinons from the partially filled upper bands would participate in the local moment formation. This means the type-II CMI phase with the PCO would be continuously connected to the Curie-Weiss regime with the 1/3 Curie constant (compared to the case when all spinons can participate in the local moment formation) at high temperature. This would explain the “1/3 anomaly” in the spin susceptibility data of LiZnMoO.

Alternatively, we could consider the strong coupling regime where the PCO is very strong. Here the 3 electrons are strongly localized in each resonating hexagon and an effective local moment model appears. The three electrons in one individual resonating hexagon are then locally entangled which leads to 4-fold degenerate ground states. This 4-fold degeneracy is characterized by one time-reversal-odd spin-1/2 and one time-reversal-even pseudospin-1/2 degrees of freedom. It is then shown that the local quantum entanglement of the three electrons in each resonating hexagon only gives rise to one magnetically active spin-1/2 moment. The spins and pseudospins are weakly coupled and are described by a Kugel-Khomskii modelKugel and Khomskii (1982) on the emergent triangular lattice (see Fig. 1c). We show that this strong coupling result is also consistent with the “1/3 anomaly” in the spin susceptibility data of LiZnMoO.

The PCO in the type-II CMI breaks the discrete lattice symmetries of
the Kagome system. At the finite-temperature transition,
the PCO is destroyed and the lattice symmetries are restored.
This transition should occur at a temperature that is on the
order of the electron ring-hopping energy scale.
Moreover, this transition is found to be strongly first order in a clean system
but would be smeared out in a disordered LiZnMoO sample.
As shown in Fig. 2, there exists another crossover temperature
above which the electron localization
in the down-triangles is thermally violated.
Between and ,
the electron localization with one electron in each triangle is still obeyed
but the PCO is destroyed.
The electron occupation configuration in this intermediate regime
is extensively degenerate just like the spin configuration in
a classical Kagome spin iceWills *et al.* (2002), so we name this intermediate temperature phase
as “Kagome charge ice” (KCI).
Above the crossover temperature , the system
can be thought as the higher temperature regime of
the type-I CMI, where the charge localization occurs only in the up-triangles.
Besides the distinct finite-temperature charge behaviors, we also expect
thermal crossovers in the spin susceptibility (see Fig. 2).
Above , each electron would contribute
a local spin-1/2 moment, and hence,
the anomalous low-temperature spin susceptibility
changes into regular Curie-Weiss behavior whose
Curie constant, ,
is 3 times the low-temperature Curie constant,
.

The rest of the paper is structured as follows. In Sec. II, we start from the molecular orbitals of the MoO clusters, introduce an appropriate atomic state representation, and provide a microscopic justification for the single-band Hubbard model in Eq. (1). In Sec. III.1, we formulate the charge sector of the type-II CMI as a compact U(1) gauge theory on the dual honeycomb lattice (DHL). Then in Sec. III.2, we introduce a new slave-particle construction for the electron to obtain the mean-field phase diagram that includes the type-I CMI, type-II CMI and a Fermi-liquid metal (FL-metal). In Sec. III.3, we focus on the type-II CMI phase. We obtain the PCO in the charge sector by mapping the low-energy charge sector Hamiltonian into a quantum dimer model on the DHL. We generalize the Levin-Wen string mean-field theory to study the reconstruction of the spinon band structure by the PCO. We explain the consequence of this reconstructed spinon band structure and discuss the low-temperature magnetic susceptibility. In Sec. IV, we consider the strong coupling regime of the type-II CMI with the PCO and identify the structure of the local moments formed by the 3 electrons in an individual resonating hexagon. The interaction between these local moments is described by a Kugel-Khomskii model on the ETL. In Sec. V.1 and Sec. V.2, we connect our theory to the experiments on LiZnMoO and suggest possible future experiments. Finally in Sec. V.3, we discuss other Mo based cluster magnets. Some details of the computations are included in the Appendices.

## Ii Molecular orbitals and the Hubbard model

As suggested by Refs. Cotton, 1964; Sheckelton *et al.*, 2012, the Mo electrons in an
isolated MoO cluster form molecular orbitals
because of the strong Mo-Mo bonding. Among the 7 valence electrons in
the cluster, 6 of them fill the lowest three molecular orbitals {A,
E, E} in pairs, and the seventh electron remains
unpaired in a totally symmetric A
molecular orbital with equal contributions from all three Mo atoms (see Fig. 3).

We first consider the molecular orbital states in the group {A, E, E}. This group can be described by a linear combination of an atomic state at each Mo site (which is in turn a linear combination of five atomic orbitals).

(2) | |||||

(3) | |||||

(4) |

where () labels the three Mo sites in the cluster and the atomic state is the contribution from the Mo atom at . The atomic states at different Mo sites are related by the 3-fold rotation about the center of the cluster. Likewise, the fully-filled {A, E, E} and other unfilled molecular orbitals at higher energies are constructed from the atomic state and other atomic states (), respectively. Here, the atomic states () represent a distinct orthonormal basis from the five atomic orbitals that are the eigenstates of the local Hamiltonian of the MoO octahedron.

We group the molecular orbitals based on the atomic state from which they are constructed. In this classification, for example, {A, E, E} fall into one group while {A, E, E} fall into another group as they are constructed from two different atomic states.

In LiZnMoO, the different molecular orbitals of the neighboring clusters MoO overlap and form molecular bands. To understand how the molecular orbitals overlap with each other, we consider the wavefunction overlap of different atomic states . Since the down-triangle has the same point group symmetry as the up-triangle in LiZnMoO, the wavefunction overlap of the atomic states in the down-triangles should approximately resemble the one in the up-triangles. More precisely, the wavefunction of the atomic state (e.g. ) has similar lobe orientations both inward into and outward from the MoO cluster, with different spatial extensions due to the asymmetry between up-triangles and down-triangles. Consequently, the orbital overlap between the molecular orbitals from the same group is much larger than the one between the molecular orbitals from the different groups. Therefore, each molecular band cannot be formed by one single molecular orbital but is always a strong mixture of the three molecular orbitals in the same group.

We now single out the three molecular bands that are primarily formed by
the group of {A, E, E} molecular orbitals.
There are four energy scales associated with these three
molecular orbitals and bands:

(1) the energy separation
between the {A, E, E} group
and other groups of orbitals (both filled and unfilled),

(2) the total bandwidth of the three molecular bands
formed by the {A, E, E} molecular orbitals,

(3) the intra-group interaction between two electrons on
any one or two orbitals of the {A, E, E} group,not

(4) the inter-group interaction between the electron
on an orbital of the {A, E, E} group
and the other electron on an orbital of a different group.
It is expected, from the previous wavefunction overlap argument, that the inter-group
interaction is much weaker than the intra-group interaction
and thus can be neglected at the first level of approximationNot .

In this paper, we assume that the energy separation
is larger than the total bandwidth and the intra-group interaction.
In this regime,
the large separates these three molecular bands
from other molecular bands (both filled and unfilled) so that
the fully filled {A, E, E} orbitals remain
fully-filled and the unfilled molecular orbitals remain unfilled even after they
form bands.
Moreover, the large also prevents a band-filling reconstruction
due to the interaction (in principle, the system can gain interaction energy by
distributing the electrons evenly among different groups of orbitals).
Therefore, we can ignore both the fully-filled and unfilled molecular bands
and just focus on the three partially filled bands.
It also means one will have to consider three-band model
with all of {A, E, E} orbitals
on the triangular lattice formed by the MoO clusters.
In this case, alternatively one could simply consider
atomic states as the starting point. Then the relevant model
would be a single-band Hubbard model based on the atomic state
at each Mo site of the anisotropic Kagome lattice.
We take the latter approach in this paper.
Finally, since only one atomic state is involved at
each Mo site, the orbital angular momentum of the electrons are trivially quenched
so that we can neglect the atomic spin-orbit coupling
at the leading orderWitczak-Krempa *et al.* (2014).

The corresponding single-band Hubbard model is given by Eq. (1), where we include the on-site and nearest-neighbor electron interactions. Now it is clear that the physical meaning of the electron operator () in Eq. (1) is to create (annihilate) an electron on the state with spin at the Kagome lattice site .

## Iii Generic phase diagram

As explained in Sec. I, the extended Hubbard model in Eq. (1) can support two types of CMIs with distinct electron localization patterns. Besides the insulating phases, the model includes a FL-metal when the interaction is weak. To describe different phases and study the Mott transitions in this model, we first employ the standard slave-rotor representation for the electron operator,Florens and Georges (2004); Lee and Lee (2005) , where the bosonic rotor () carries the electron charge and the fermionic spinon () carries the spin quantum number. To constrain the enlarged Hilbert space, we introduce an angular momentum variable , , where is conjugate to the rotor variable with . Moreover, since the on-site interaction is assumed to be the biggest energy scale, in the large limit the double electron occupation is always suppressed. Hence, the angular variable primarily takes () for a singly-occupied (empty) site.

Via a decoupling of the electron hopping term into the spinon and rotor sectors, we obtain the following two coupled Hamiltonians for the spin and charge sectors, respectively,

(5) | |||||

(6) | |||||

where and (), () for the bond on the up-triangles (down-triangles). is a Lagrange multiplier that imposes the Hilbert space constraint. Here, we have chosen the couplings to respect the symmetries of the Kagome lattice. The Hamiltonians and are invariant under an internal U(1) gauge transformation, , and . This internal U(1) gauge structure is then referred as the U(1) gauge field in the following.

Since the electron is not localized on a lattice site in the CMIs, it will be shown that the rotor variable is insufficient to describe all the phases in a generic phase diagram, except for certain special limits for the type-I CMI which we analyze in Appendix. A. To remedy this issue, we will extend the slave-rotor representation to a new parton construction for the electron operator in the following sections and then generate the phase diagram.

### iii.1 Charge sector of type-II CMI as a compact U(1) gauge theory

To introduce a new parton construction, we need to first understand the low-energy physics of the charge sector, especially in the type-II CMI. We will show the charge localization pattern in the type-II CMI leads to an emergent compact U(1) lattice gauge theory description for the charge-sector quantum fluctuations. In the slave-rotor formalism, the charge-sector Hamiltonian is given by

(7) | |||||

where we have dropped the interaction term because in the large limit. This charge sector Hamiltonian can be thought as a Kagome lattice spin-1/2 XXZ model in the presence of an external magnetic field upon identifying the rotor operators as the spin ladder operators, where

(8) |

Thus the corresponding effective spin- model reads

(9) | |||||

in which we have made a uniform mean-field approximation such that . The 1/6 electron filling is mapped to the total “magnetization” condition , where is the total number of Kagome lattice sites.

The type-II CMI appears when the interactions are dominant over the hoppings . In terms of the effective spin , the electron charge localization condition in the type-II CMI is

(10) |

In the type-II CMI, the allowed effective spin configuration is “2-down 1-up” in every triangle. These allowed classical spin configuration are extensively degenerate. The presence of the transverse effective spin exchanges lifts the classical ground state degeneracy and the effective interaction can be obtained from a third-order degenerate perturbation theory. The resulting effective ring exchange Hamiltonian is given as

(11) |

where “” refers to the elementary hexagon of the Kagome lattice, 5). and “1,2,3,4,5,6” are the 6 vertices on the perimeter of the elementary hexagon on the Kagome lattice (see Fig.

We now map the effective Hamiltonian into
a compact U(1) lattice gauge theory on the DHL.
We introduce the lattice U(1) gauge fields () by
definingHermele *et al.* (2004)

(12) | |||||

(13) |

where , , and . The centers (labelled as ) of the triangles form a dual honeycomb lattice (see Fig. 4). The fields and are identified as the electric field and the vector gauge field of the compact U(1) lattice gauge theory and With this identification, the local “2-down 1-up” charge localization condition in Eq. (10) is interpreted as the “Gauss’ law” for the emergent U(1) lattice gauge theory. The effective ring exchange Hamiltonian reduces to a gauge “magnetic” field term on the DHL,

(14) |

where is a lattice curl defined on the ‘’ that refers to the elementary hexagon on the honeycomb lattice. As this internal gauge structure emerges at low energies in the charge sector, in the following we will refer this gauge field as the U(1) gauge field.

### iii.2 Slave-particle construction and mean-field theory

Since the gauge theory in the charge sector is a compact U(1) gauge theory defined on a 2D lattice, it would be confining due to the well-known non-perturbative instanton effect if all the elementary excitations (except for “photon”) is gapped. However, in our case, the spinon excitations are gapless and possess spinon Fermi surfaces. While these spinons do not directly couple to U(1) gauge field, they would interact with charge excitations in terms of U(1) gauge field and then can indirectly couple to U(1) gauge field via the charge excitations. Thus, a deconfined phase of the U(1) gauge field may still be allowed if spinon Fermi surface fluctuations can suppress instanton events, which would then support fractionalized charge excitations inside the type-II CMI. Resolving this issue requires non-perturbative computations and is left for a future work. In the following, we introduce a new parton formulation and obtain the mean-field phase diagram for the extended Hubbard model. In this subsection, we shall first ignore the instanton effect (and the related charge sector symmetry breaking) which will become important in the type-II CMI when we consider quantum fluctuations beyond the mean-field theory.

#### iii.2.1 Generalized parton construction

Before introducing the new parton formalism, we would like to explain the connection and
the difference between the current problem and the fractional charge liquid (FCL) Mott insulating phase
in our previous work for a
3D pyrochlore lattice Hubbard model with a 1/4 or 1/8 electron filling.Chen *et al.* (2014)
Similar to the type-II CMI in the Kagome lattice case discussed here,
the low-energy physics of the charge sector in the FCL
is described by a compact lattice U(1) (or U(1))
gauge theory on a 3D diamond lattice.
Because it is defined in 3D, the U(1) gauge field for the
pyrochlore lattice case can easily be deconfined in the Mott insulating phase,
which supports the charge quantum number fractionalization
in the FCL. Therefore, in the absence of instanton effect, we can use the same
construction here and represent the electron creation operator as

(15) |

where , is
the same fermionic spinon creation operator in the slave-rotor representation,
()
is the creation (annihilation) operator for the bosonic charge excitation
in the triangle that is centered at , and
is an open string operator
of the U(1) gauge field that connects the two charge
excitations in the neighboring triangles at and
. In the following, we use
the string or the U(1) field interchangeably.
This parton representation for the electron operator is connected
to the slave-rotor representation by identifyingLee *et al.* (2012); Savary and Balents (2012)
for .
The original Hilbert space constraint in the slave-rotor representation is also needed here.
To match with the underlying lattice U(1)
gauge theory description, we define the following
operator, Lee *et al.* (2012); Savary and Balents (2012)

(16) |

that measures the local U(1) (electric) gauge charge.
Here, () for ()
and .
We further supplement this definition with a Hilbert space constraintLee *et al.* (2012); Savary and Balents (2012)

(17) |

such that the new representation of the electron operator is restricted to the physical Hilbert space. For the ground state, we have in every triangle. An effective spin-flip operator or the local charge excitation (with ) creates two neighboring defect triangles (at and ) that violate the “2-down 1-up” charge localization condition of the type-II CMI. Thus, these two defect triangles carry the U(1) gauge charges and , which are assigned to the operators and , respectively. Under an internal U(1) gauge transformation, , and .

#### iii.2.2 Type-I & Type-II CMIs and mean-field phase diagram

Using the new parton construction, the microscopic Hubbard model becomes

(18) | |||||

which is supplemented with the Hilbert space constraint. In a mean-field theory treatment, we decouple the electron kinetic terms and obtain four mean-field Hamiltonians for each sector,

(19) | |||||

(20) | |||||

(21) | |||||

(22) | |||||

where

Here we explain a few things related to the above mean-field equations. First, in the above choices of mean-field couplings, we have assumed that these couplings respect all the symmetries of the original Hubbard model on the Kagome lattice. Second, the Lagrange multipliers, which are used to fix the Hilbert space constraints, are expected to vanish from the same argument as noted in Appendix. A, so we do not explicitly write them out in the mean-field Hamiltonians. Third, the electron hoppings on the bonds of down-triangles (up-triangles) mediate the tunnelling of the charge bosons in the up-triangles (down-triangles). Therefore, the charge bosons on the up-triangles and down-triangles do not mix, and hence, we have two separate charge boson mean-field Hamiltonians and that are defined in the up-triangle and down-triangle subsystems, respectively.

Type-II CMI | for . |
---|---|

Type-I CMI | for , for . |

Type-I CMI | for , for . |

FL metal | for , for . |

The string (or U(1) gauge-field) sector mean-field Hamiltonian is a simple
effective spin-1/2 ferromagnetic XY model on the Kagome lattice, which is solved classically.
The low-energy charge sector ring hopping Hamiltonian in Eq. (14)
favors a zero net U(1) gauge flux, which we choose
for the U(1) gauge flux seen by the charge bosons in the mean-field theory.
Therefore, the mean-field U(1) gauge flux is obtained
based on a non-mean-field perturbative argument.
This treatment is used
to obtain the phase of the quantum spin ice in the 3D pyrochlore lattice
in Ref. Lee *et al.*, 2012.
Of course, this treatment misses the phases in other flux sectors which
may potentially be stabilized by other larger couplings.
To resolve this issue, one needs much more
exhaustive analysis of the energetics of different flux sectors
which is beyond the scope of this work. For the phases in which
we are interested, we expect such a treatment is sufficient.
In the following, we fix the gauge for the mean-field value of
the string operator by requiring
where is a real parameter. In the semi-classical approximation of the string
mean-field Hamiltonian , we can set .

We solve the mean-field Hamiltonians self-consistently (see Appendix. B.2). The mean-field phase diagrams for different choices of the couplings are depicted in Fig. 6. The four different phases correspond to different behaviors of the charge bosons that we list in Tab. 1. When the charge bosons from both up-triangles and down-triangle subsystems are condensed, the FL-metal phase is realized. When they are both gapped and uncondensed, we have the type-II CMI. When the charge bosons from one triangle subsystem are condensed and the other is uncondensed, we have the type-I CMI. Here we introduce a subindex ‘u’ or ‘d’ to the type-I CMI to indicate which triangles the electrons are localized in. All the transitions are continuous in the mean-field theory, except the transition between the type-I and the type-I CMIs which is strongly first order. Beyond the mean-field theory, the transition between the FL-metal and the type-I CMIs will probably remain continuous and quantum XY typeSenthil (2008a, b) while the transition into the type-II CMI may depend on the detailed charge structure inside the type-II CMI.

Now we explain the phase boundaries. We begin with the phase boundary between the type-I CMI and the FL-metal. As we increase , the effective electron hopping on the up-triangle bonds gets suppressed which effectively enhances the kinetic energy gain through the down-triangle bonds. As a result, a larger is required to drive a Mott transition. A similar argument applies to the phase boundary between the type-I and the type-II CMIs. A larger is needed to compete with the kinetic energy gain on the upper triangle bonds for a larger in the type-I CMI and to drive a transition to the type-II CMI. For , electrons are more likely to be localized in the up-triangles to gain the intra-cluster kinetic energy. Thus, a smaller is needed to drive a Mott transition and a larger is needed to drive the system from the type-I to the type-II CMIs.

In the type-I CMI, the spin sector forms a U(1) QSL with a spinon Fermi surface, which is the same as the U(1) QSL in Appendix. A with . The U(1) gauge field also behaves similarly. To be concrete, we first discuss the type-I CMI. The U(1) gauge field acquires a mass when the condensation of the charge bosons in down-triangles occurs. The two fractionally-charged charge bosons then are combined back into the original unit-charged charge rotor . While the charge fractionalization does not exist in the type-I CMI, the spin-charge separation still survives. The condensation of the charge bosons from the down-triangles leads to the local “metallic” clusters in the up-triangles such that the localized electron can move more or less freely within each up-triangle. Because of the local “metallic” clusters, only the U(1) gauge field living on the down-triangle bonds that connect the up-triangles remains active and continues to fluctuate at the low energies. The low-energy physics is described by the spinon Fermi surface coupled with a fluctuating U(1) gauge field, leading to a U(1) QSL in the triangular lattice formed by the up-triangles.

In the type-II CMI, the charges remain strongly fluctuating which is described by the compact U(1) gauge theory in Sec. III.1. The mean-field theory, however, does not capture this charge quantum fluctuation inside the type-II CMI and thus cannot not give a reliable prediction for the behaviors of charge excitations and spinons. We will discuss this issue in Sec. III.3. Nevertheless, the mean-field theory does obtain qualitatively correct phase boundaries.

### iii.3 Type-II CMI with the plaquette charge order

In this subsection, we focus on the type-II CMI and discuss the PCO due to the leading quantum fluctuation in the charge sector. Then we explain how the spin physics is influenced by the charge sector in the type-II CMI.

#### iii.3.1 The plaquette charge order via quantum dimer model

As we point out in Sec. III.1, the extensive degeneracy of the classical charge ground state of the type-II CMI is lifted by the ring hopping Hamiltonian in Eq. (11), which arises from the collective tunnelling of the 3 electrons on the perimeter of the elementary hexagons on the Kagome lattice. This ring hopping Hamiltonian is the dominant interaction in the charge sector of the type-II CMI and should be treated first. We now map into a quantum dimer model on the DHL. As depicted in Fig. 7, a dimer is placed on the corresponding link of the DHL if the center of the link (or the Kagome lattice site) is occupied by an electron charge. The rotor operator simply adds or removes the dimer that is centered at the Kagome lattice site . So is mapped into the quantum dimer model with only a resonant term,

(23) |

where and
refer to the two dimer covering configurations in the elementary hexagon
of the DHL as shown in Fig. 7.
In Ref. Moessner *et al.*, 2001, Moessner, Sondhi and Chandra have studied
the phase diagram of the quantum dimer model on the honeycomb lattice quite extensively.
In the case with only the resonant term, they found a
translational symmetry breaking phase with a plaquette dimer order,
in which the system preferentially gains dimer resonating (or kinetic) energy
through the resonating hexagons on the DHL (see Fig. 8).
The plaquette dimer order of the quantum dimer model is then mapped back to
the plaquette charge order (PCO) on the Kagome lattice (see Fig. 8).
This is a quantum mechanical effect and cannot be obtained from treating the
inter-site electron interactions in a classical fashion.

With the PCO, the electrons are preferentially hopping around the perimeters of the resonating hexagons on the Kagome lattice. These resonating hexagons are periodically arranged, forming an emergent triangular lattice (ETL). Due to the translational symmetry breaking, this ETL has a larger unit cell than the original Kagome lattice.

We note this PCO has already been obtained in the same Hubbard model
in an isotropic Kagome lattice with 1/6, 1/3 and 2/3 electron filling
in certain parameter regimes in previous works.Pollmann *et al.* (2008); Rüegg and Fiete (2011); Polmann *et al.* (2014); Ferhat and Ralko (2014)
The result was obtained either through perturbatively mapping to the
quantum dimer model or by a Hatree-Fock mean-field calculation.
In particular, Ref. Banerjee *et al.*, 2008 applied the quantum Monte Carlo technique
to simulate a hardcore boson Hubbard model on an isotropic Kagome lattice and
discovered a direct weakly-first-order Mott transition from the superfluid phase
to the type-II CMI with the PCO for 1/3 and 2/3 boson fillings.

#### iii.3.2 Effective low energy theory of the type-II CMI

We now turn to the spin sector physics of the type-II CMI.
It has been shown by exact diagonalization in Ref. Pollmann *et al.*, 2008
that the 1/6-filled extended Hubbard model in
an isotropic Kagome lattice supports a ferromagnetic state for the
type-II CMI in the limit and
.
The underlying physical mechanism for such a ferromagnetic state is
similar to the Nagaoka’s ferromagnetism.
This ferromagnetic state, however, is very unstable to the introduction
of the antiferromagnetic (AFM)
spin interaction between the electron spins.
In the actual material, AFM spin (exchange) interaction is always
present, and moreover, the experiments did not find any evidence of ferromagnetic ordering.
So we do not consider the possibility of the ferromagnetic ordering
in the extreme limit that is considered in Ref. Pollmann *et al.*, 2008.

To understand how the PCO in the charge sector influences the spin sector. we first consider the low-energy effective ring hopping model in the type-II CMI that is expressed in terms of the original electron operators,

(24) | |||||

where and are readily obtained from the same third-order degenerate perturbation calculation as in Sec. III.1. Here, , and “1,2,3,4,5,6” are the 6 vertices in the elementary hexagon of the Kagome lattice and should not be confused with the sublattice labellings in Fig. 9 for the ETL. Using the slave-rotor representation for the electron operator , the ring hopping model can be decoupled as

(25) |

where we have singled out the charge sector and treated the spinon sector in a mean-field fashion and where the lattice sites are arranged either clockwise or anti-clockwise. Here is evaluated in the spinon mean-field ground state, which we explain below. For a time-reversal invariant system, we expect . If we further assume the translational invariance for the spinon sector, the resulting charge sector model is equivalent to in Eq. (11) and also to the quantum dimer model in Eq. (23) except for the different couplings. Since the PCO breaks the translational symmetry of the Kagome lattice, the spinon sector should be influenced by this symmetry breaking in the charge sector. In the following, we want to understand how the spinon band structure is affected by the underlying PCO in the charge sector and how the modified spinon band structure feeds back into the charge sector. To this end, we take the enlarged unit cell of the ETL and introduce the following spinon mean-field (hopping) Hamiltonian (see Fig. 9),

(26) | |||||

where labels the unit cell of the ETL and “1,2,3,4,5,6,7,8,9” label the 9 sublattices of the ETL. Moreover, in Eq. (26), the spinon hoppings are given by

(27) | |||||

(28) | |||||

(29) | |||||

(30) |

For the type-I CMI, we should have and due to the 3-fold rotational symmetry around the center of each triangle on the Kagome lattice. For the type-II CMI, however, we expect and due to the presence of the PCO. The PCO enhances the hoppings of the charge rotors and spinons in the resonating hexagons and weakens the ones in the non-resonating hexagons. The enhanced spinon hopping in the resonating hexagons further strengthen the couplings of the in the resonating hexagons through . Thus the PCO would become more stable if the coupling between spinon and charge excitations is switched on.

#### iii.3.3 Levin-Wen’s string mean-field theory and the spinon band structure

We consider the combination of the spinon hopping model with the ring model . We extend Levin-Wen’s string mean-field theoryLevin and Wen (2007) for the quantum dimer (or string) model to solve the coupled charge and spinon problem. In Levin and Wen’s original work, this mean-field theory is designed for the quantum dimer model. As discussed previously in Sec. IV.1, our ring hopping model is a quantum dimer model if the charge occupation is mapped to the dimer covering on the DHL. The new ingredients here are the presence of the spinon degrees of freedom, and the coupling and the mutual feedback between the spinons and dimers (or charge fluctuations).

We describe the variational wavefunction that is used to optimize the Hamiltonian in Eq. (25) for the dimer sector. Following Levin and Wen, the wavefunction is parametrized by a set of variational parameters where is defined on each bond of the DHL. These variational parameters are also termed as string fugacity by Levin and Wen.Levin and Wen (2007) Notice that the bonds on the DHL are also parametrized by the Kagome lattice