Exact hole-induced SU ( N ) flavor-singlets in certain U = ∞ SU ( N ) Hubbard models

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I. INTRODUCTION
The SU (2) Hubbard model in the presence of a hole doping has been extensively studied as it is expected to capture essential features of the high-temperature superconductivity in cuprate superconductors [1][2][3][4].Despite its deceptively simple form, the model presents significant challenges and complexity due to competing tendencies to develop various types of distinct ordered phases [5].Even in the strong coupling (U = ∞) limit, an analytical solution on a bipartite lattice (e.g., square lattice) exists only for single-hole doping on a finite-sized system-the celebrated "Nagaoka theorem" states that such a system leads to a fully polarized ferromagnet [6][7][8][9][10][11].In a physical context, the Nagaoka theorem implies the formation of the ferromagnetic Nagaoka polaron, which has been observed in numerics [12,13] and in cold-atom experiments [14].
On the other hand, it is known that the hole motion in the U = ∞ SU (2) Hubbard model on a non-bipartite lattice (e.g., triangular lattice) induces antiferromagnetic correlations around it [15][16][17][18].However, for such a nonbipartite lattice, even the single-hole problem is poorly understood due to the frustration inherent in antiferromagnetism.The problem has been recently solved in a frustration-free version of a non-bipartite lattice, which unambiguously demonstrated that a hole is surrounded by resonating valence bond (RVB)-like correlations [19].Such a result suggests the formation of an RVB polaron on a more realistic non-bipartite lattice.
For systems with an emergent (or exact) SU (N ) symmetry with N > 2 [20][21][22][23][24][25], e.g., systems with degenerate multiple valleys or flavors [26][27][28][29][30], their physics may be characterized by the SU (N ) Hubbard model or its generalizations under suitable circumstances.If so, the magnetism at the 1  N th filling (one fermion per site) in the strong coupling regime, U ≫ t, is captured by the SU (N ) Heisenberg model with exchange interactions J = 4t 2 /U .However, when t ≫ J (U → ∞ limit), it is the motion of a hole that is responsible for the magnetism upon hole doping of such a Mott insulator.Therefore, SU (N ) generalizations of the Nagaoka and counter-Nagaoka theorems are needed.In Refs.[31,32], it is shown that, with the "unusual" sign of the hopping matrix element, t < 0, a single hole motion in such a U = ∞ SU (N ) Hubbard model leads to a fully flavor-polarized ground state.However, less is understood for the same problem with the "usual" sign of hopping t > 0, again due to the frustration inherent in antiferromagnetism.
In this paper, we study the dynamics of a single hole doped at the 1  N th filling of the U = ∞ SU (N ) Hubbard and t-J models on certain solvable graphs.We first consider such a problem on an (N + 1)-site graph that satisfies the connectivity condition (as defined later), and show that the ground state is in the SU (N ) flavor-singlet sector (Sec.II).Any other flavor configurations frustrate the hole motion.From such an (N + 1)-site subgraph, we construct a subgraph tree, on which the single hole prob- lem in the SU (N ) t-J model is exactly solvable (Secs.III-IV).The ground state is a positive superposition of SU (N ) flavor-singlet covering states.In Sec.V, we speculate on the possibility of exotic phases of matter in the presence of a dilute but finite hole concentration.
We note that the exact solvability of the single hole problem in a subgraph tree is due to the existence of an extensive number of local SU (N ) symmetries-in some sense, this is Hilbert space fragmentation [33][34][35] from restricted hole motion.
We start by solving a single hole problem in the U = ∞ SU (N ) Hubbard model (N ≥ 2) on an (N +1)-site graph that satisfies the connectivity condition (to be defined below).We assume that the hopping matrix elements are positive but otherwise arbitrary t ij > 0: Here, a = 1, 2, ..., N is a flavor index of a fermion in the fundamental representation, i = 0, 1, 2, ..., N is a site index, and ⟨i, j⟩ is an edge of the graph.V ({n i }) describes arbitrary on-site terms and density-density interactions The last U = ∞ term forbids any double occupancy.Lemma: The ground state of the Hamiltonian (1) on an (N +1)-site graph that satisfies the connectivity condition in the single hole sector is a unique SU (N ) flavor-singlet state.
In order to prove the Lemma, it is convenient to work in a particular many-body basis in a single hole sector.In doing so, we restrict ourselves to a flavor-balanced subspace, where each flavor a = 1, 2, ..., N appears exactly once [36].For example, is a flavor-balanced state, where |∅⟩ is the vacuum state with no fermions and 0 in the third expression denotes that the site i = 0 is unoccupied.This can be reexpressed as the final expression by creating a ghost fermion with flavor a = 0 at the hole site and annihilating it.This is a useful notation that will be used throughout the paper.From this state, we form a complete orthonormal basis in a flavor-balanced subspace by applying a permutation of (N + 1) objects (a hole and N fermions), σ ∈ S N +1 , where S N +1 is the symmetric group of (N + 1) objects: where we assumed that the ith site is occupied by a hole, i.e., σ(i) = 0 and again we introduced a ghost fermion with flavor a = 0 in the last expression for convenience.Among the states in the flavor-balanced subspace are the completely flavor-antisymmetric, SU (N ) flavorsinglet (FS) states with the hole at site i, , ..., σ(N )⟩ . ( Connectivity condition: An (N + 1)-site graph is said to satisfy the connectivity condition if all the basis states in Eq. ( 4) can be reached from one another by repeated applications of hopping operators in Eq. ( 1), are examples of graphs that satisfy the connectivity condition.In particular, in Fig. 1 (c), starting from the state |0, 1, 2, ..., N ⟩, moving a hole around the triangular loop induces a transposition (1 2) and moving it around the largest, length (N + 1), loop induces the N -cycle (1 2 ... N ).These two permutations, together with hopping operations, generate S N +1 .More generally, Theorem 2 of Ref. [32] provides the sufficient condition for the connectivity condition.
Proof of the Lemma: Any two basis states in the flavor-balanced subspace, |σ⟩ and |τ ⟩, have a nonzero hopping matrix element only when they differ by one transposition involving a hole: σ −1 (0) = τ −1 (a) and τ −1 (0) = σ −1 (a) for some flavor a, and Also, the interaction term V ({n i }) only contributes to diagonal matrix elements.Therefore, the Perron-Frobenius theorem ensures that there exists a unique ground state |Ψ 0 ⟩ which is a positive superposition of all the basis states (A σ > 0): Since this state has a nonzero overlap with a flavor-singlet state |i, FS⟩, it must be a flavor-singlet state [if it were instead a superposition of multiple irreps of SU (N ), then it is possible to construct degenerate ground states, in contradiction to the uniqueness of the ground state].There- fore, it is possible to rewrite Eq. ( 7) as a positive superposition (A(i) > 0) of |i, FS⟩: See Fig. 1 (d) for the illustration of such a state.□

III. SU (N ) FLAVOR-SINGLETS IN A SUBGRAPH TREE
It is now straightforward to generalize the previous result to a "subgraph tree" constructed as follows.Starting from an (N + 1)-site subgraph that satisfies the connectivity condition, we attach other (N + 1)-site subgraphs to some (or all) of the vertices of the initial subgraph, in such a way that it does not create any other cycles (loops of length l ≥ 3 in which only the first and the last vertices are equal) than those contained within each subgraph.This generates depth 1 tree of (N + 1)-site subgraph.Continuing this n times will generate a depth n subgraph tree, which has the property that all the cycles of the graph are contained within each subgraph.Let N SG be the number of subgraphs constituting such a subgraph tree.The number of sites in such a graph is N N SG + 1. Figures 2 (a-b) are examples of such graphs.We will consider the Hamiltonian (1) on such a graph in the presence of a single hole.
The advantage of such a subgraph tree is that there is an SU (N ) symmetry associated with each subgraph as can be seen as follows [37].First, a many-body basis can be constructed by locating the site of the hole i, and then specifying the flavor configuration on the rest of the sites.Once the hole location is specified, it is easy to see that there is a unique N -mer covering of the lattice [see Fig. 2 (c) for the illustration of such a covering].In any step in which the hole hops to a neighboring site, one N -mer is moved, but in such a way that it remains inside the initial (N + 1)-site subgraph in which it was contained.Thus, we can label the N -mers uniquely by a subgraph index s = 1, ..., N SG , and the total flavor SU (N ) symmetry is preserved for N fermions contained in each s N -mer.That is, there exists SU (N Thanks to such SU (N ) NSG symmetry, it is enough to consider a subspace that is flavor-balanced in each s Nmer.Any other states in the Hilbert space can be reached by repeated applications of raising and lowering operators on each s N -mer.(See Appendix B for the expression of those raising/lowering operators.)We now construct a many-body basis restricted in such a flavor-balanced subspace analogously to Eqs. ( 3) and ( 4).We first occupy a hole at a particular location (call it i = 0), which will define a unique N -mer covering as discussed above.For each s N -mer, we label the sites contained in it by i where again, the ghost flavor index a = 0 is introduced for convenience in c 0,0 .Using the fact that fermions in different N -mers do not exchange one another due to the restricted dynamics of a hole, we might as well treat them as distinguishable and rename a flavor index a in s Nmer to be (s − 1)N + a.Hence, the basis state (9) can be denoted by From this state, any other basis state that is flavorbalanced for each s N -mer can be reached by repeated applications of hopping operators Tij .There are (N N SG + 1)(N !) NSG different such (orthonormal) basis states.Each such basis state has a permutation operator σ ∈ S N NSG+1 associated with it defined as a relative flavor configuration from the initial one in Eq. ( 10): if site i is occupied by the flavor a, then σ(i) ≡ a. (We emphasize that flavor indices are renamed to have values a = 0, 1, ..., N N SG ).Let P be the collection of all such permutations σ.We define the basis states {|σ⟩ : σ ∈ P } with a particular sign structure analogous to Eq. ( 4): where we again assumed that the ith site is occupied by a hole, i.e., σ(i) = 0.The sign structure again allows us to write the SU (N ) flavor-singlet covering (FSC) state, the state with an SU (N ) flavor-singlet on every N -mer, as a uniform superposition of the basis states that have their hole at site i: The following Theorem is the main result of our paper.Theorem: The ground state of the Hamiltonian (1) on a "subgraph tree" in the single hole sector is unique and is a positive (A(i) > 0) superposition of the SU (N ) flavor-singlet covering (FSC) states [38]: (See Fig. 2 (c) for an illustration of this state) Proof of Theorem: It is straightforward to show that any nonzero off-diagonal element of the Hamiltonian matrix is negative, ⟨σ| Tij |τ ⟩ = −t ij < 0, as in Eq. ( 6).Also, since any basis state |σ⟩ can be reached from one another by repeated applications of Tij , one concludes from the Perron-Frobenius theorem that the ground state is unique and is a positive (A σ > 0) superposition of all the basis states: This has a positive overlap with a flavor-singlet covering state |i, FSC⟩, and hence N fermions in every N -mer must be a flavor-singlet.Hence, |Ψ 0 ⟩ can be rewritten as a superposition of flavor-singlet covering states as in Eq. ( 13).□ In Fig. 3 of Appendix A, we show the result of the finite size exact diagonalization study on the SU (3) U = ∞ Hubbard model on the four-tetrahedron geometry.The result agrees with the Theorem.

IV. SU (N ) t-J MODEL
Now we generalize the previous results to the SU (N ) t-J model.In the presence of a finite but large U (≫ t) term, U 2 i ni (n i − 1), one can obtain the SU (N ) t-J model from the SU (N ) Hubbard model by projecting out the states with multiply occupied sites [39][40][41]: Here Ĥ is the Hamiltonian for the U = ∞ Hubbard model ( 1), J ij = 4t 2 ij /U and K ijk = 2t ij t jk /U , ⟨i, j, k⟩ denotes the triplet of sites such that j is a nearest neighbor to i and k, and ∆ab ) is the annihilation operator of a flavor-antisymmetric state on a bond ⟨i, j⟩.λi = ( λ(1) i , ..., λ(N 2 −1)

i
) denotes (N 2 −1) generators of the SU (N ) group at site i with the normalization Tr(λ [42].The Heisenberg operator can be rewritten in terms of a flavor-permutation operator Pij as J ij ( λi The following two corollaries generalize the Lemma and the Theorem to certain t-J models. Corollary 1: Let us define Ĥt-J on an (N + 1)-site graph that satisfies the connectivity condition.J ij ≥ 0 and K ijk ≥ 0 do not have to be related to one another and can be arbitrary independent parameters.Then, the ground state of Ĥt-J in the presence of a single hole is unique and is a positive superposition of flavor-singlet states (8) as in the Lemma.
Corollary 2: For Ĥt-J defined on a subgraph tree, let J ij = J s ≥ 0 be uniform within each subgraph and connect any two sites within it.Also, let K ijk ≥ 0 terms act only on three sites ⟨i, j, k⟩ fully contained within a subgraph.Again, J ij and K ijk can be independent parameters, except for the above constraints.Then, the ground state of Ĥt-J in the presence of a single hole is unique and is a positive superposition of flavor-singlet covering states (13) as in the Theorem [43].
Proof of Corollary 1: For a single hole problem in an (N + 1)-site graph, the total SU (N ) symmetry is intact even in the presence of Ĵ and K terms, and one can work in the flavor-balanced basis (4).Again, it is then sufficient to show that all the nonzero off-diagonal matrix elements are negative.In particular, for σ ̸ = τ , ⟨σ| Ĵij |τ ⟩ is nonzero only when σ and τ differ by one transposition between occupied sites: σ(i) = τ (j) ̸ = 0, σ(j) = τ (i) ̸ = 0, and σ(k) = τ (k) for k ̸ = i, j.In such a case, one obtains Similarly, any nonzero off-diagonal element of Kijk is negative This completes the proof.□ Proof of Corollary 2: For a subgraph tree, consider first the case when Ĵ = 0.When K ijk are nonzero only for triplets of sites ⟨i, j, k⟩ fully contained in a subgraph, SU (N ) NSG symmetry is intact.Thus, one can still work in the flavor-balanced basis (11) and the same proof as in the Theorem can be applied.
When Ĵ ̸ = 0, the SU (N ) NSG symmetry is lost.However, for the special case where J ij = J s is uniform within each subgraph and connects any two sites within it, one can rewrite the Heisenberg term as [density-density interactions in Ĵ can be absorbed in V term in Eq. ( 1)]: ⟨i,j⟩ Here (s, i) denotes the site i = 1, 2, ..., N +1 in a subgraph s.This Heisenberg operator takes the lowest possible eigenvalue for the flavor-singlet covering states (12): This means that the ground state of Ĥt-J is still in the flavor-singlet covering subspace spanned by states (12) and is of the form Eq. ( 13) with positive A(i) > 0. □

V. DISCUSSION
Our result demonstrates the fundamental importance of the sign of the hopping matrix elements t ij on a kinetic magnetism of the Hubbard model, which in turn, manifests as a particle-hole asymmetry in the magnetic phase diagram.More precisely, in the usual SU (2) Hubbard model, the particle-hole transformation c i,σ → c † i,σ , with σ =↑, ↓, maps the single doublon problem to the single hole problem with the opposite sign of t ij [4].This implies that for a bipartite lattice-where the sign of t ij can be changed by a gauge transformation-the phase diagram is particle-hole symmetric around half-filling.On the other hand, for a non-bipartite lattice the phase diagram exhibits a particle-hole asymmetry.For example, the single hole dynamics in the triangular lattice U = ∞ Hubbard model leads to a 120 • antiferromagnetic ordering [16,17] whereas the single doublon problem satisfies the Nagaoka theorem and leads to a fully polarized ferromagnet (except for one singlet for a doublon).Performing such a particle-hole transformation to the SU (N ) Hubbard model, one maps a single hole problem at 1/N filling to a single N -on (N fermions at a site) problem at (N − 1)/N filling with the opposite sign of t ij .Since (N − 1) fermions at the same site must be completely flavor-antisymmetric, such N − 1 electrons form a complex conjugate representation N of the fundamental representation.Hence we see that with the usual sign of the hopping t ij > 0, while the Nagaoka ferromagnetic state appears for a single fermion doping of the (N − 1)/N filled Mott insulator, a single hole doping at 1/N filling generically induces antiferromagnetism.
We note that in a more realistic non-bipartite lattice (e.g. a triangular or pyrochlore lattice), it is likely that the hopping operators T and exchange interactions Ĵ (or singlet hopping terms K) favor different local magnetic correlations.In such a case, the hole can only delocalize in a finite number of sites, leading to the formation of an RVB polaron.
Going from such a single RVB/RFS polaron problem to a multi-polaron (or multi-hole) problem requires yet another technical development, but we can speculate on possible outcomes (apart from a trivial phase separation scenario).First, it is possible to have a broken-SU (N )-symmetry phase with a long-range flavorantiferromagnetic order when flavor-singlets are supported over a sufficiently long distance [16][17][18]44].When SU (N ) flavor-singlets are supported on sufficiently short distances, one can have a flavor-disordered phase with topological order [45][46][47][48][49].The flavor-disordered state with a broken translation symmetry corresponds to various topologically ordered crystalline phases [50,51].Finally, it is possible to have various exotic liquid phases with a topological character such as a Z N topologically ordered Fermi liquid (FL* phase) [52] or hightemperature superconductivity.
ACKNOWLEDGEMENT K-S.K. would like to acknowledge the hospitality of the Massachusetts Institute of Technology, where the majority of this work was done.K-S.K. appreciates Samuel Alipour-fard, Zhaoyu Han, and Pavel Nosov for helpful discussions.K-S.K. also thanks Aidan Reddy for valuable discussions on the exact diagonalization method.H.K. would like to thank Tadahiro Miyao for useful comments on the manuscript.K-S.  when the number of fermions is N + 1, every site of the subgraph is occupied, so λi

FIG. 1 .
FIG. 1. (a),(b) Examples of a complete graph with fully connected edges.(c) An example of a non-complete graph which nevertheless satisfies the connectivity condition.(d) The ground state of the U = ∞ SU (3) Hubbard model in the presence of a single hole on a tetrahedron with uniform tij = t and V = 0 in Eq. (1).Magenta trimers denote SU (3) flavor-singlets, i.e., three fermions with complete flavor-antisymmetry, and circles denote the location of the hole.The signs associated with the many-body states appearing in |Ψ0⟩ are defined implicitly in Eq. (5).

FIG. 2 .
FIG. 2. (a),(b) Examples of subgraph tree.In (a), sites are numbered in the way specified in the main text above Eq.(9).(c) The ground state in the single hole sector of the U = ∞ SU (3) Hubbard [Theorem] and certain t-J models [Corollary 2] is a positive (A(i) > 0) superposition of the SU (3) flavorsinglet covering states.

Fig. 2 .
(a) for the illustration of such a site numbering scheme along with subgraph indices s) and occupy it with fermions with flavors a = 1, • • • , N, respectively.This defines one basis state |0, (1, ..., N ) , (1, ..., N ) , ..., (1, ..., N )⟩ K. was supported by the Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering, under contract DE-AC02-76SF00515.H.K. was supported by JSPS KAKENHI Grants No. JP18K03445, No. JP23H01093, No. 23H01086, and MEXT KAKENHI Grant-in-Aid for Transformative Research Areas A "Extreme Universe" (KAKENHI Grant No. JP21H05191).the SU (3) U = ∞ Hubbard model in the single-hole sector on a four-tetrahedron geometry as shown in Fig. 3 (13-sites; shown in the inset).Hopping matrix elements are set to t ij = 1 for all bonds.The ground state is the positive superposition of SU (3) flavor-singlet covering states, consistent with the result of the Theorem.

FIG. 3 .
FIG. 3. Ground state and several first excited states of SU (3) U = ∞ Hubbard model on the four-tetrahedron geometry.The horizontal axis denotes the Young diagram corresponding to the irrep of the SU (3) group of the trimer (shown in gray) in the specified subgraph s.The three first excited states shown are the ones that have adjoint representation in s = 2, 3, 4 trimer, respectively; trimers in the subgraphs that are not specified are SU (3) singlets.FSC is the sector which has SU (3) singlet on every trimer.The ground state is of the form Eq. (13) and has the same energy as the noninteracting problem with the same hopping matrix elements, EGS = −4.1326383.