Symmetry-protected Bose-Einstein condensation of interacting hardcore Bosons

We introduce a mechanism stabilizing a one-dimensional quantum many-body phase, characterized by a certain wave vector $k_0$, from a $k_0$-modulated coupling to a center site, via the protection of an emergent $\mathbb Z_2$ symmetry. We illustrate this mechanism by constructing the solution of the full quantum many-body problem of hardcore bosons on a wheel geometry, which are known to form a Bose-Einstein condensate. The robustness of the condensate is shown numerically by adding nearest-neighbor interactions to the wheel Hamiltonian. We identify the energy scale that controls the protection of the emergent $\mathbb Z_2$ symmetry. We discuss further applications such as geometrically inducing finite-momentum condensates. Since our solution strategy is based on a generic mapping from a wheel geometry to a projected ladder, our analysis can be applied to various related problems with extensively scaling coordination numbers.

Cold atom experiments have become a versatile platform to realize various exotic quantum phases of matter [1][2][3][4][5][6][7][8].Available experimental setups nowadays allow for the control of both geometry and interactions of simulated model systems.It is thus crucial to theoretically identify physical mechanisms that improve the stability and scaling properties of exotic quantum phases, which then might be realized and tested in experiments.In that context, remarkable progress in understanding the stability of Bose-Einstein condensate (BEC) has been made by analyzing spectral properties of a wheel of hardcore bosons (HCB) [9][10][11][12] as depicted in Fig. 1a.This model features an energy scale ∼ √ L that is generated by the extensively scaling coordination number of a center site.While large coordination numbers appear in several theoretical approaches [13][14][15][16][17][18], the exact solution as well as the stability against perturbations remained an open question.Besides others, the problem of finding exact expressions for ground states of long-range coupled HCB Hamiltonians is a major obstacle.Here, arbitrarily longranged interactions appear when expressing the HCB degrees of freedom in terms of spinless fermions (SF) via a Jordan-Wigner transformation (JWT).
In this letter, we present a mapping that allows us to construct the full solution of a family of quantum manybody problems with arbitrary k 0 -modulated ring-to-center hoppings s j = se ik0j , and to analyze the formation of a BEC phase with momentum k 0 .In the context of central spin models [19][20][21][22][23][24] a solution strategy to a similar problem is based on the Bethe ansatz and has been applied to describe for instance Rydberg impurities in ultracold atomic quantum gases [25].In contrast, we derive the solution by introducing a mapping to a ladder system of SF, yielding closed analytical expressions.We emphasize that this mapping can be applied in various other setups to analytically tackle problems with an extensively scaling coordination number.In the context of hardcore bosons, our approach reveals that the sta-  FIG. 1.The main plot illustrates the single-particle dispersion relation (middle, red curve) of the wheel geometry (a), emerging from projecting down the dispersion from the ladder geometry (b) (upper orange and lower blue curve).Note the appearance of two single-particle states at k0 = 0 (red crosses).This is because the Hilbert space of the wheel is obtained by projecting out all modes on the inner ring, except for the zero momentum states |N ,k=0 .Momentum conservation then couples this central mode to the particular mode on the outer ring respecting the k0-modulated ring-tocenter hopping, which generates an extensively scaling level splitting (red circle and crosses).
bilizing mechanism for the BEC is the extensively scaling coordination number of the center site, introducing a robust discrete Z 2 symmetry that protects the ordered quantum many-body phase against local perturbations on the outer ring.Furthermore, we trace back the protection to odd-parity k = k 0 single-particle states that are gapped out ∼ s √ L ≡ s.This property allows us to show that in the thermodynamic limit the system immediately transitions into a BEC, as long as there is a finite ring-tocenter hopping rate s > 0, which remarkably also holds when adding local interactions to the outer ring.We demonstrate, beyond previous work, the robust protection of the BEC numerically, using density-matrix renormalization group (DMRG) [26,27] simulations to calculate the k 0 -condensate fraction when adding nearestneighbor (NN) interactions, for a wide parameter range and various particle densities.As a consequence, the Z 2 symmetry in principle allows to experimentally tune the transition temperature of a gas of interacting HCB by modifying the wheel's coordination number.Here, we show that the central quantity is the ratio V s between the interaction strength V and the renormalized ring-to-center hopping, which we demonstrate by further numerical results.Finally, our analysis implies that the emergent Z 2 symmetry is generically induced by the model's geometry.Therefore, general k 0 -modulated hoppings give rise to corresponding protected k 0 -modes and the respective single-particle states are gapped out ∼ s √ L. This paves the way to a generic mechanism that can be exploited in various contexts, for instance, to stabilize exotic quantum many-body phases such as k 0 = 0 BEC [28][29][30][31].
Analytical solution via wheel-to-ladder mapping.-Weconsider HCB on a L-sited ring with an additional center site [10,12] (see Fig. 1a).The model exhibits k 0modulated ring-to-center hopping s j = se ik0j while the homogeneous hopping on the ring is tuned by a parameter t.The corresponding Hamiltonian reads where ĥ( †) j is the HCB ladder operator on the j-th site of the ring and ĥ( †) on the center site, spanning the overall Hilbert space H ⊗L+1

2
. In the limit s t → 0 (ring geometry) the model exhibits a quasi BEC, i.e., the ground state is a bosonic condensate, whose occupation scales as √ N [32,33], where N denotes the number of HCB.The opposite limit, s t → ∞ (star geometry), has been shown recently to feature a real BEC where the occupation in the ground state scales as Lρ (1 − ρ + 1/L) with ρ = N /L [11].
In order to construct a full analytical solution, we introduce a mapping from the wheel Eq. ( 1) to a ladder geometry of HCB with periodic boundary conditions (see Fig. 1b).The overall solution strategy is sketched schematically in Fig. 2. The crucial step is to identify the central Hilbert space of the HCB wheel with the subspace of the single-particle momentum states |N ,k=0 on the inner ring of the ladder (enforcing occupations N ,k=0 ≡ N ≤ 1).The projector Π to this subspace allows us to construct a solution on the expanded Hilbert space of the ladder geometry and eventually project down.Thereby, the long-range coupled wheel Hamiltonian can be mapped to an only next-nearest-neighbor (NNN) coupled ladder Hamiltonian Ĥlad :  1).The HCB wheel is transformed to a ladder , which is then mapped to a ladder of spinless SF via a JWT.From the ladder of SF, the single-particle spectrum |k, ± and therefrom projected Slater determinants |n k 0 , FSN−n k 0 are constructed, utilizing the projector to the N ≤ 1 subspace, Π .The constructed many-particle Slater determinants finally allow for the analytic solution of the full HCB wheel diagonalizing a 4 × 4 matrix M. Note that no closed solution of the SF ladder Hamiltonian is required (only its projected counterpart).
While the full details of the mapping can be found in [34], the most important observation is that a JWT of Ĥlad introduces only local parity operators e iπ n ,j : ,j ) denotes the fermionic ladder operator on the j-th site of the outer (inner) ring and the single-site number operator on the inner ring is given by n ,j = ĉ † ,j ĉ ,j .We emphasize the appearance of a rescaled ring-to-center hopping amplitude s = s √ L, that allows to connect to the known solutions when taking the thermodynamic limit L → ∞.In fact, in the thermodynamic limit, the wheel immediately collapses to the star geometry whenever there is a fixed, finite ratio s t , and the ground state is a true BEC.However, the question remains what happens for fixed ratios s t .This matters for finite system sizes, as is the case for mesoscopic systems, in ultracold atomic gas experiments or Rydberg atoms [35].In particular, we are interested in the impact of the extensive energy scale set by s on the formation and stability of the BEC, which requires a more in-depth analysis of the ground state of Eq. (3).Note that for now and in the following, we refer to the scaling of the ringto-center hopping s = s √ L as extensive in the system size.
As shown in Fig. 1, the corresponding single-particle spectrum is identical to that of a tight-binding chain (i.e., ε k = 2t cos k) except for the k = k 0 states whose single-particle energies are characterized by the splitting These k = k 0 single-particle eigenstates Eq. ( 4) separate ∝ |s| ∝ √ L from the remaining spectrum giving rise to a single-particle gap.Referring to Eq. (3), in the limit s t → ∞, the hopping on the outer ring can be neglected, and the same holds for the impact of the JWT on the overall eigenstate.Consequently, the single-particle gap can be expected to control the many-body spectrum, in this limit.Additionally, from ∆ ± s/t→∞ −→ ±1 we find that the corresponding wavefunction is characterized by a maximally mixing of the distinguished mode |k 0 on the outer ring with the state |N = 1 on the inner ring.This highly non-local wavefunction, generated from the extensive scaling of Eq. ( 5), already suggests the stability of the BEC under local perturbations on the outer ring.
In order to further elaborate on the extensive scaling property, we now return to the full solution of Eq. ( 1) with the complete derivation detailed in [34].Here, the key observation is that Slater determinants constructed from single-particle states Eq. ( 4) with k = k 0 are also eigenstates of Ĥ = Π Ĥlad Π : where |FS N = |k 1 , . . ., k N denotes a Slater determinant labeled by a set of N single-particle eigenstates with k l = k 0 .This observation can be understood by noting that the projected parity operator in Eq. ( 3) can be written in terms of the zero momentum density N on the inner ring and thus Π e iπ n ,j Π |FS N = |FS N .Particle-number conservation of the wheel Hamiltonian then motivates to construct an ansatz for the N -particle eigenstates, superimposing all possible occupations of the k 0 mode that belong to the same overall particle number sector with complex coefficients α 0,1±,2 .These states describe a superposition of either empty (∝ α 0 ) or doubly occupied (∝ α 2 ) k 0 states and highly non-local states ∝ α 1± in Clustering of the many-particle eigenstates for a wheel composed of 10 lattice sites in the N = 4 particle number sector as a function of the ring-to-center hopping s.Different colors correspond to the clustered energies generated from the different eigenvalues of Eq. ( 9).Indicated are also the two gaps defining the two critical ring-to-center hoppings sc,1, sc,2.
which the k 0 mode on the outer ring is coupled to the |N = 1 mode on the inner ring.
Using the orthogonality of different Slater determinants, it is a straightforward calculation to find that the general solution of the eigenvalue problem Π Ĥlad Π |FS N = E Π |FS N reduces to the diagonalization of a 4 × 4 matrix.Fixing a Slater determinant |FS N and two modes k , k = k 0 so that ĉk |FS N = |FS N −1 as well as ĉk ĉk |FS N = |FS N −2 , and labeling the 4 basis states by their possible occupations of the k = k 0 mode n k0 = 0, 1 ± , 2, the resulting eigenvalue problem is of the form We emphasize the existance of a hidden Z 2 symmetry of the many-body eigenstates.This symmetry is an immediate consequence of the modulation of the hopping to the center site, i.e., it characterizes the k 0 -occupation.Furthermore, condensation requires a breaking of particle number conservation on the outer ring, which is possible only in the n k0 = 1 ± subspace.Thus, an odd Z 2 symmetry of the ground state signals the formation of a BEC.
Upon solving Eq. ( 9), a special structure of the manybody spectrum appears that is characterized by a clustering of eigenstates belonging to the same k 0 -parity sector, which is exemplified in Fig. 3. Therefrom, for a given filling fraction ρ = N/L we can extract the scaling of two critical parameters separating the low-lying odd-parity cluster (blue in Fig. 3), which hosts the BEC ground state, from the remaining eigenstates.In what follows we set t ≡ 1 as unit of energy.The first critical hopping sc,1 and gap ∆ 1 arise once the clustered odd-parity eigenstates constitute the overall ground state, indicating the condensation of bosons into the k 0 mode (abbreviating The second critical hopping is defined by the complete separation of the odd-parity cluster from the even-parity many-particle eigenstates: Note that s > sc,2 implies that scattering between states with even and odd k 0 parity, caused by external perturbations, can only occur if the energy barrier ∆ 2 can be overcome.
Interactions on the outer ring.-Theanalytical solution and, in particular, the property of BEC ground states exhibiting odd k = k 0 parity allows to draw some striking conclusions on the stability of the BEC in the presence of local perturbations on the outer ring.Adding interactions acting on a finite subset of outer ring sites only, we note that in general the single-particle description breaks down in favor of a Luttinger liquid [36][37][38][39].Therefore, interactions generically couple the two parity sectors in the k 0 subspace and one might expect a breaking of the Z 2 symmetry.However, mixing of the k 0 parity sectors caused by local interactions connecting d neighboring sites on the outer ring is of the order of ∼ V d s where V is the largest interaction strength.The consequence is that increasing the number of lattice sites, i.e., the center site's coordination number, Z 2 symmetry of the k 0 modes is approximately restored, stabilizing BEC in the presence of interactions on the outer ring.
We numerically checked the robustness of the BEC in the thermodynamic limit for k 0 = 0 and finite values of the ring-to-center hopping.To this end, we calculated the ground-state occupation n k0 (s, L) of the k 0 = 0 mode [1,40] using DMRG.Normalizing with respect to the upper bound on the condensate occupation n max (L) = Lρ (1 − ρ + 1/L) [11], we extrapolated the condensate fraction into the thermodynamic limit n k0 (s)/n max = lim L→∞ n k 0 (s,L) nmax(L) (see [34] for the details).
FIG. 4. (a) Ground-state BEC condensate fraction n(s) normalized to the maximally possible value nmax [11] and extrapolated to the thermodynamic limit.(b) Asymptotically the normalized condensate fraction for a fixed number of lattice sites is a function of the ratio V /s, only.Results are shown for different NN interaction strengths V and densities on the ring ρ.Note that for very small fillings ρ = 1 /16 (inset of (b)), there are significant deviations of the observed connection between the condensate fraction and the ratio V /s.This originitates from the flat single-particle dispersion around k = π (see Fig. 1).Thereby, the complete separation of the odd-parity states (controlled by ∆2) occurs already for small ring-tocenter hoppings, mainly independent on the number of lattice sites.
The resulting extrapolations are shown in Fig. 4a for interaction strengths between V = 0 and V = 1 and particle densities between ρ = 1 /16 and 1 /2.Note that even though there is a renormalization of the overall condensate fraction, we always observe a finite condensate density in the thermodynamic limit, even for strong interactions and high particle densities.To further demonstrate the asymptotic robustness of the Z 2 symmetry protection, in Fig. 4b the dependency of the condensate fraction at finite system sizes and as a function of V /s is shown.We also observe the behavior expected from our previous analysis, namely that the condensate occupation dominantly depends on the system size and the ratio between the interaction and the extensively scaling renormalized ring-to-center hopping s = s √ L. In accordance to the scaling of ∆ 2 , the maximally possible condensation is reached if s ∼ L. We emphasize that these relations can be translated to experimental realizations to determine the necessary coordination number, i.e., number of sites on the ring, to detect BEC in the presence of interactions.
Conclusion.-We introduced a solution strategy for models on a wheel with k 0 modulated ring-to-center hopping s j = se ik0j , which we applied to a system of HCB [9][10][11][12].Our central finding is the protection of a BEC by a Z 2 symmetry emerging from the model-specific modulation of the hopping to the center site.We traced back this remarkable feature to an extensively scaling separation ∝ s √ L of the k = k 0 single-particle modes, generated from the extensive coordination of the center with the ring sites.This scaling renders the BEC robust against local perturbations on the ring.We demonstrated this feature numerically by calculating the HCB k 0 -condensate fraction of the ground state in the presence of NN interactions and for various particle number densities.Our calculations clearly show the protection of BEC where the condensate fraction is controlled by the ratio V s √ L and approaches the maximally possible value [11], even in the presence of strong interactions.
Our findings imply important consequences for both experimental and theoretical realizations of wheel geometries in general.First of all, a particular single-particle mode can be gapped out by a proper modulation of the ring-to-center hopping, allowing the general protection of ordered phases that are characterized by a certain wave vector.We believe that such a modulation of the ring-to-center hopping provides an experimentally feasible approach to realize exotic, finite-momentum BEC in the framework of ultracold or Rydberg atoms [28-31, 35, 41, 42].Second, there is a many-body gap separating the BEC-carrying states from the remaining spectrum ∼ s √ L, i.e., large gaps can be realized by increasing the coordination number of the center site.The resulting robustness against interactions on the ring can be exploited to increase critical temperatures for phase transitions into otherwise highly fragile quantum phases.Possible applications are mesoscopic setups where a conducting center site may be contacted to one-dimensional ring geometries via tunnel contacts, allowing the stabilization of ordered states on the ring against perturbations.Such a scaling could also be exploited to increase the stability of superconducting qubits by means of an all-to-all-coupling of a set of noisy stabilizer qubits to a central qubit [43].Moreover, we believe that the wheelto-ladder mapping could prove useful in the analysis of hidden fermions [44].Further interesting questions are the incorporation of disorder on both the ring and center site, as well as the effect of (artificial) gauge fields and a rescaled ring-to-center hopping s → s √ L with regard to the crossover from one to an infinite number of dimensions.

WHEEL-TO-LADDER MAPPING
Here, we outline the mapping from the wheel geometry to the projected ladder in detail.The wheel Hamiltonian is given by Ĥ = −t L−1 j=0 ĥ † j ĥj+1 + ĥ † j+1 ĥj − L−1 j=0 s j ĥ † j ĥ + h.c., (S1) with ĥ( †) j ĥ( †) describing hardcore bosonic degrees of freedom located on the outer ring (center site) and s j = e ik0j with k 0 = 2π L n (n ∈ Z) a reciprocal lattice vector.We consider periodic boundary conditions on the outer ring, i.e., ĥ( †) L = ĥ( †) 0 .The model is defined on the tensor product Hilbert space H L+1 = H ⊗L 2 ⊗ H , where H 2 (H ) is the single-particle Hilbert space of a hardcore boson on the outer ring (center site).We define an enlarged Hilbert space of two concentric rings, where the Hilbert space of the inner ring is a copy of the Hilbert space of the outer ring.
To describe hopping to the inner ring, we introduce ĥ( †) ,j as the corresponding ladder operators acting on the sites of the inner ring.Note that sites with the same index on the inner and outer ring can be aligned vertically, yielding a ladder geometry.We introduce the second quantization basis for the inner ring |n ,1 , n ,2 , . . ., n ,L ∈ H ,L as well as the vacuum state |∅ ∈ H ,L .
In order to represent the same physical situation as the wheel system, the total occupation of all sites on the inner ring must be either zero or one.Furthermore, we enforce the allowed states in H ,L to transform under rotations of the inner ring, which is an a priori constraint so far, but will turn out to be very useful in the forthcoming discussion.We introduce H by the set of states |ω that meet these constraints H = span |ω ∈ H ,L .
On this subspace of H ,L , we then must have where n = 0, 1, . . ., L − 1 and R applies a rotation to the inner ring: R : and addition in the site indices is performed modulo L. The allowed states satisfying the above constraints are given by for any j ∈ {1, . . ., L}.
Let us from now on identify |0 |1 with the empty (occupied) inner site of the wheel.Having introduced the allowed states, we construct a Hamiltonian exhibiting the same matrix elements in P = H L ⊗ H as Eq.(S1) in s j ĥ † j ω + h.c. .
Here, we defined operators ω = |0 1| and ω † = |1 0| via their action on |1 and |0 , respectively.In order to obtain a representation of L in P we write where ĥ( †) ,l acts on the l-th site of the inner ring, as well as a projector that projects to F ,1 ⊂ H ,L , i.e., the Fock space spanned by empty and singly-occupied states on the inner ring: with ne ,j = 1 ,j − n ,j and n ,j = ĥ † ,j ĥ ,j .Since P † = P and P 2 = P , P indeed is a projector.Rewriting ω and projecting down to F ,1 , we obtain Ĥproj = P L P = P L ,t P + s √ L j,l P e ik0j e −iql ĥ † j ĥ ,l + h.c.
which now acts on H L ⊗ F ,1 .We introduced |k , the eigenstates of R , labeled by the respective rotation angle k = 2π L n with n ∈ {0, . . ., L − 1} so that R |k = e ik |k .In the single-particle subspace on the inner ring, these states are given by In order to obtain a representation that is block-diagonal in the |k -basis we choose q = 0: Ĥproj = P Ĥlad P , Ĥlad = −t j ĥ † j ĥj+1 + h.c.− s j e ik0j ĥ † j ĥ ,j + h.c. .
We note that Ĥproj has long-ranged hoppings while Ĥlad does not.Importantly, the eigenstates of R are eigenstates of Ĥproj due to [ Ĥproj , R ] = 0, while this is not the case for Ĥlad , since [ R , Ĥlad ] = 0. Since [ N , R ] = 0 holds, with N being the total particle number on the inner ring, we may set up the simultaneous eigenstates and group them by their corresponding eigenvalues of R : is uniquely specified in the subspace with N = 0, (S9) They span the Fock space F ,1 .The allowed states Eq. ( S3) are obtained by taking only the states with k = 0 in Eq. (S7).For brevity, we define |∅ be the projector into the k = 0 subspace on the inner ring.We can thus write the initial Hamiltonian Eq. (S1), mapped to the ladder geometry as This establishes the mapping between the eigenstates of Π Ĥlad Π , which will be computed explicitly later on, and the desired eigenstates of Eq. (S1), i.e., we need to find the many-particle eigenstates of Eq. (S6) projected into the k = 0 sector.For that purpose, we employ a Jordan-Wigner transformation expressing the hardcore bosonic ladder operators in terms of fermionic ones.Implementing the ladder geometry via the mappings ĥ( †) j → ĥ( †) 2j and ĥ( †) ,j → ĥ( †) 2j+1 yields the fermionic ladder operators ĉ( †) j = k<j e iπ nk ĥ( †) j , with j = 0, . . ., 2L − 1. (S12) After transforming the Hamiltonian, we reintroduce fermionic operators on the inner and outer ring via ,j and obtain

SOLUTION OF THE SINGLE-PARTICLE PROBLEM
In the single-particle subspace, the Jordan-Wigner transformed ladder Hamiltonian becomes It is instructive to explicitly construct the single-particle eigenstates subject to the projection into the q = 0 subspace, where for brevity we introduce the decomposition Ĥlad = Ĥlad,t + Ĥlad,s .We begin by considering the action of the where k = 2t cos(k) is the single-particle dispersion relation of non-interacting, spinless fermions.For k = k 0 , Π |ψ k indeed satisfies the eigenvalue equation.In order to determine the k = k 0 single-particle eigenstates we make the ansatz Π Setting c k = δ k ,k0 c k0 and choosing c k0 = 1, we can solve for ∆ Thus, for k = k 0 , there are two orthogonal single-particle eigenstates In this representation, projecting down the eigenstates into the zero-momentum sector on the inner ring is particularly easy: Using Eq. (S11), the single-particle problem of the wheel Hamiltonian Eq. (S1) is thus solved by expanding the ladder problem in terms of the |ψ k,µ and projecting into the q = 0 subspace where ε ± = 1 2 ε 0 ± sgn(s) ε 2 0 + 4s 2 .For later convenience, we also introduce ladder operators ψ( †) k,µ annihilating (creating) single-particle modes |ψ k,µ .From Eq. (S22) it can be readily checked that they obey fermionic anticommutation relations

SOLUTION OF THE MANY-PARTICLE PROBLEM
The solution of the full many-particle problem Eq. (S1) is done with the help of Slater determinants constructed from the single-particle eigenstates Eq. (S23).Here, the important point is that all many-particle Slater determinants with either empty or doubly occupied k = k 0 modes are already eigenstates of the projected ladder Eq. (S11) and thereby of the wheel Hamiltonian.In order to prove this observation, we define for a given set of N modes k N = (k 1 , . . ., k N ) with k l = 2π L n l = k 0 and 0 ≤ n l < L projected Slater determinants of single-particle eigenstates of Ĥlad wherein we fixed the global phase by normal ordering the modes: Within this ordering, Slater determinants with modes |ψ k=k0,± occupied are always moved to the left and denoted by: In order to proceed, in the first sum we expand the operators acting on the inner ring in terms of rotations, i.e., perform a Fourier transformation (using k n = 2π L n) and apply the projection to the q = 0 subspace ĉ † In a similar manner, we expand the second sum in terms of rotations of both the outer and inner ring yielding Having found Eq. (S31) the solution strategy for the many-particle problem is straightforward.Employing total particle-number conservation, we decompose the many-particle Hilbert space into orthogonal subspaces H N with fixed total particle number N .Each N -particle subspace is then stratified into 4-dimensional subspaces H k ,k ,k N −2 that are parametrized by a set of N different modes k N = k N −2 ∪{k , k } with k l , k , k = k 0 , and we specified two distinct modes k , k .A subspace H k ,k ,k N −2 is given by the linear hull of states with complex coefficients α 0,1+,1−,2 ∈ C. Using orthogonality of the Slater determinants, it can be checked that this stratification yields a complete decomposition of the N -particle Hilbert space by counting the dimensionalities.
Varying the set of modes k N −2 ∪ {k , k }, the number of orthogonal basis states is obtained from the right hand side of Eq. (S32) via dim The last equality shows that indeed, the choosen parametrization generates a complete basis set for H N .It is also easy to see that Π Ĥlad Π does not mix states belonging to different subspaces (S37) These calculations imply a clustering of the many-particle eigenvalues as shown in Fig. 3 in the main text.For the set of all allowed modes k N −2 ∪ {k , k } excluding k = k 0 , each of the 4 energies consitutes a bulk of many-particle eigenvalues to the total spectrum with a bandwidth ∼ E(k N ).Furthermore, for large system sizes L 1, the level spacing in the bulk of the clustered many-particle eigenvalues scales as ∼ 2π L .Importantly, the clustered eigenvalues behave differently, depending on the occupations of the k = k 0 modes.For the case of empty or doubly occupied k = k 0 modes, up to corrections ∼ 1 L the spectrum is given by the summed single-particle energies (E(k N ) and (E(k N −2 ) + 2)).If, however the k = k 0 mode is singly occupied, we obtain a separation of the clustered manyparticle eigenvalues ∼ ±|s|.We want to point out that in the latter case, the overall dependency of the many-particle eigenvalues on the ring-to-center coupling closely resembles that of single-particle dispersion relation of the underlying fermionic ladder Hamiltonian upon replacing k → E(k N −1 ).This is not by accident but a result of the extensively scaling coordination number of the central site, restoring the single-particle character.
The previous discussion suggest the definition two energy gaps ∆ 1 , ∆ 2 characterizing the transition to the BEC phase as indicated in Fig. 3 in the main text.Note that as soon as ∆ 1 > 0 the ground state is characterized by singly occupied k = k 0 modes, i.e., an odd k = k 0 parity that is separated from the clustered many-body states with

1 |n k 0 ,FIG. 2 .
FIG.2.Solution strategy for Eq.(1).The HCB wheel is transformed to a ladder , which is then mapped to a ladder of spinless SF via a JWT.From the ladder of SF, the single-particle spectrum |k, ± and therefrom projected Slater determinants |n k 0 , FSN−n k 0 are constructed, utilizing the projector to the N ≤ 1 subspace, Π .The constructed many-particle Slater determinants finally allow for the analytic solution of the full HCB wheel diagonalizing a 4 × 4 matrix M. Note that no closed solution of the SF ladder Hamiltonian is required (only its projected counterpart).