Symmetry-protected Bose-Einstein condensation of interacting hardcore bosons

Abstract

The large practical potential of exotic quantum states is often precluded by their notorious fragility against external perturbations or temperature. Here, we introduce a mechanism stabilizing a one-dimensional quantum many-body phase exploiting an emergent ${{\mathbb{Z}}}_{2}$-symmetry based on a simple geometrical modification, i.e. a site that couples to all lattice sites. 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 Bose-Einstein condensates. The robustness of the condensate against interactions is shown numerically by adding nearest-neighbor interactions, which typically destroy Bose-Einstein condensates. We discuss further applications such as geometrically inducing finite-momentum condensates. Since our solution strategy is based on a generic mapping, our findings are applicable in a broader context, in which a particular state should be protected, by introducing an additional center site.

Introduction

The ability to control and manipulate quantum systems has seen a remarkable development in the past two decades. For instance, 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 condensates (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 $\sim \sqrt{L}$ that is generated by the extensively scaling coordination number of a center site, which in this geometry is given by the number of ring sites L. While large coordination numbers appeared in several theoretical approaches^{13,14,15,16,17,18}, the underlying mechanism as well as its stability against perturbations remained an open question.

Fig. 1: Wheel geometry, ladder geometry and the single-particle dispersion.

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 ${\big\vert {N}_{\odot ,k = 0}\big\rangle }_{\odot }$. Momentum conservation then couples this central mode to the particular mode on the outer ring respecting the k₀-modulated ring-to-center hopping, which generates an extensively scaling level splitting (red circle and crosses).

Here we show that the observed stability of the BEC phase on the wheel lattice is no physical oddity, but a consequence of the geometry of the wheel lattice itself, hosting a hidden ${{\mathbb{Z}}}_{2}$-symmetry. For that purpose, we present a mapping that allows us to construct the solution of a family of quantum many-body problems with arbitrary k₀-modulated ring-to-center hoppings ${s}_{j}=s{{{{{{{{\rm{e}}}}}}}}}^{{{{{{{{\rm{i}}}}}}}}{k}_{0}j}$, and to analyze the formation of a BEC phase with momentum k₀. 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 gases²⁵. In contrast, we derive an analytical expression by introducing a mapping to a ladder system of spinless fermions (SF) which becomes exact both in the thermodynamic limit and at low densities. 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 stabilizing mechanism for the BEC is the extensively scaling coordination number of the center site, introducing a robust discrete ${{\mathbb{Z}}}_{2}$-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 = k₀ single-particle states that are gapped out $\sim s\sqrt{L}\equiv \tilde{s}$. This property implies that in the thermodynamic limit the system immediately transitions into a BEC, as long as there is a finite ring-to-center 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)^26,27,28,29 simulations to calculate the k₀-condensate fraction when adding nearest-neighbor interactions, for a wide parameter range and various particle number densities. As a consequence, the ${{\mathbb{Z}}}_{2}$-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 $\frac{V}{\tilde{s}}$ between the interaction strength V and the renormalized ring-to-center hopping, as we demonstrate by further numerical results. Finally, our analysis implies that the emergent ${{\mathbb{Z}}}_{2}$-symmetry is generically induced by the model’s geometry. Therefore, general k₀-modulated hoppings give rise to corresponding protected k₀ modes and the respective single-particle states are gapped out $\sim s\sqrt{L}$. 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 k₀ ≠ 0, i.e. finite momentum BEC^30,31,32,33.

Results

In order to demonstrate our mapping and its implications, we consider HCB on a L-sited ring with an additional center site^10,12 (see Fig. 1a). The model exhibits k₀-modulated ring-to-center hopping ${s}_{j}=s{{{{{{{{\rm{e}}}}}}}}}^{{{{{{{{\rm{i}}}}}}}}{k}_{0}j}$ while the homogeneous hopping on the ring is tuned by a parameter t. The corresponding Hamiltonian reads

$$\hat{H}\equiv -t\mathop{\sum }\limits_{j=0}^{L-1}\left({\hat{h}}_{j}^{{{{\dagger}}} }{\hat{h}}_{j+1}+{{\mbox{h.c.}}}\right)-\mathop{\sum }\limits_{j=0}^{L-1}\left({s}_{j}{\hat{h}}_{j}^{{{{\dagger}}} }{\hat{h}}_{\odot }+{{\mbox{h.c.}}}\right),$$

(1)

where ${\hat{h}}_{j}^{({{{\dagger}}} )}$ is the HCB ladder operator on the j-th site of the ring and ${\hat{h}}_{\odot }^{({{{\dagger}}} )}$ on the center site, spanning the overall Hilbert space ${{{{{{{{\mathcal{H}}}}}}}}}_{2}^{\otimes L+1}$. In the limit $\frac{s}{t}\to 0$ (ring geometry) the model exhibits off-diagonal quasi long-range order, indicated by the algebraic decay of the single-particle density matrix (SPDM) ${\rho }_{ij} \sim | {x}_{i}-{x}_{j}{| }^{-\frac{1}{2}}$^34,35. This leads to the formation of a quasi-BEC, i.e. the ground state is a bosonic condensate whose occupation scales as $\sqrt{N}$^36,37, where N denotes the number of HCB.

The opposite limit, $\frac{s}{t}\to \infty$ (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/L¹¹.

Analytical solution via wheel-to-ladder mapping

The property of Eq. (1) to interpolate between two superficially disconnected physical situations, one exhibiting no stable long-range order ($\frac{s}{t}=0$) and one featuring true long-range order ($\frac{s}{t}\to \infty$), calls for a deeper understanding of the underlying physics. For that purpose, we demonstrate how to construct an analytical solution of Eq. (1) and, thereby, learn about the origin of the emergence of true long-range order. From a theoretical point of view, finding the eigenstates is hindered by the fact that under a Fourier transformation, HCB loose their hardcore property. Typically, this is accounted for by mapping the HCB to spinless fermions via a Jordan-Wigner transformation (JWT) which, however, requires to introduce a normal ordering, for instance via a chain mapping. Here, the additional center site complicates the situation, since it couples to any other lattice site in the chain mapping, generating arbitrarily long-ranged interactions. We introduce a mapping from the wheel Eq. (1) to a ladder geometry of HCB with periodic boundary conditions (see Fig. 1b) to resolve this issue. Physically, the coupling to an inner ring allows to disentangle the complications arising from arbitrary long-ranged hoppings of particles on the outer ring that are generated by second order processes. 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 ${\big\vert {N}_{\odot ,k = 0}\big\rangle }_{\odot }$ on the inner ring of the ladder (enforcing occupations N_⊙,k=0 ≡ N_⊙ ≤ 1). The projector ${\hat{\Pi }}_{\odot }$ to this subspace allows us to construct a solution on the expanded Hilbert space of the ladder geometry and eventually project down. In Supplementary Note 3, we observe that the JWT of the projector becomes asymptotically diagonal in both the thermodynamic limit and at low densities. Therefore, the long-range coupled wheel Hamiltonian can be mapped to an only next-nearest-neighbor (NNN) coupled ladder Hamiltonian ${\hat{H}}_{{{{{{{{\rm{lad}}}}}}}}}$:

$$\hat{H}={\hat{\Pi }}_{\odot }{\hat{H}}_{{{{{{{{\rm{lad}}}}}}}}}{\hat{\Pi }}_{\odot }.$$

(2)

While the full details of the mapping can be found in Supplementary Note 1, the most important observation is that a JWT of ${\hat{H}}_{{{{{{{{\rm{lad}}}}}}}}}$ introduces only local parity operators ${{{{{{{{\rm{e}}}}}}}}}^{{{{{{{{\rm{i}}}}}}}}\pi {\hat{n}}_{\odot ,j}}$:

$${\hat{\Pi }}_{\odot }{\hat{H}}_{{{{{{{{\rm{lad}}}}}}}}}{\hat{\Pi }}_{\odot }= \, t\mathop{\sum}\limits_{j}{\hat{\Pi }}_{\odot }\left({\hat{c}}_{j}^{{{{\dagger}}} }{{{{{{{{\rm{e}}}}}}}}}^{{{{{{{{\rm{i}}}}}}}}\pi {\hat{n}}_{\odot ,j}}{\hat{c}}_{j+1}+\,{{\mbox{h.c.}}}\,\right){\hat{\Pi }}_{\odot }\\ -\tilde{s}\mathop{\sum}\limits_{j}{\hat{\Pi }}_{\odot }\left({{{{{{{{\rm{e}}}}}}}}}^{{{{{{{{\rm{i}}}}}}}}{k}_{0}j}{\hat{c}}_{j}^{{{{\dagger}}} }{\hat{c}}_{\odot ,j}+\,{{\mbox{h.c.}}}\,\right){\hat{\Pi }}_{\odot },$$

(3)

wherein ${\hat{c}}_{j}^{({{{\dagger}}} )}\,$$({\hat{c}}_{\odot ,j}^{({{{\dagger}}} )})$ 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 ${\hat{n}}_{\odot ,j}={\hat{c}}_{\odot ,j}^{{{{\dagger}}} }{\hat{c}}_{\odot ,j}$. We emphasize the appearance of a rescaled ring-to-center hopping amplitude $\tilde{s}=s\sqrt{L}$, which 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 $\frac{s}{t}$, and the ground state is a true BEC. However, the question remains what happens for fixed ratios $\frac{\tilde{s}}{t}$. This matters for finite system sizes, as for example in mesoscopic systems, and ultracold atomic gas experiments, in particular Rydberg atoms. In particular, we are interested in the impact of the extensive energy scale set by $\tilde{s}$ 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 ring-to-center hopping $\tilde{s}=s\sqrt{L}$ as extensive in the system size.

Fig. 2: Solution strategy for Eq. (1).

The hardcore boson (HCB) wheel ⊙ is transformed to a ladder ⊚ , which is then mapped to a ladder of spinless fermions (SF) via a Jordan-Wigner transform (JWT). From the ladder of SF, the single-particle spectrum $\left\vert k,\pm \right\rangle$ and therefrom projected Slater determinants $\big\vert {n}_{{k}_{0}},{{{{{{{{\rm{FS}}}}}}}}}_{N-{n}_{{k}_{0}}}big\rangle$ are constructed, utilizing the projector to the N_⊙ ≤ 1 subspace, ${\hat{\Pi }}_{\odot }$. The constructed many-particle Slater determinants finally allow for the analytic solution of the HCB wheel diagonalizing a 4 × 4 matrix M. Note that no closed solution of the SF ladder Hamiltonian is required (only its projected counterpart).

As we show in Supplementary Note 2, it is instructive to first solve Eq. (3) for the single-particle eigenstates $\left\vert k,\pm \right\rangle$, fulfilling $\langle {{{{{{{{\rm{e}}}}}}}}}^{{{{{{{{\rm{i}}}}}}}}\pi {\hat{n}}_{\odot ,j}}\rangle \equiv 1$:

$$\left\vert k,\pm \right\rangle =\left\{\begin{array}{ll}{c}_{k}^{{{{\dagger}}} }\left\vert \varnothing \right\rangle \hfill\quad &{{{{{{{\rm{if}}}}}}}}\,k\ne {k}_{0},\\ {\psi }_{\pm }\left({\hat{c}}_{k}^{{{{\dagger}}} }+{\Delta }_{\pm }{\hat{c}}_{\odot ,k = 0}^{{{{\dagger}}} }\right)\left\vert \varnothing \right\rangle \quad &{{{{{{{\rm{if}}}}}}}}\,k={k}_{0}.\end{array}\right.$$

(4)

Here, ${\hat{c}}_{(\odot ),k}^{{{{\dagger}}} }=\frac{1}{\sqrt{L}}\mathop{\sum }\nolimits_{j = 0}^{L-1}{{{{{{{{\rm{e}}}}}}}}}^{-{{{{{{{\rm{i}}}}}}}}kj}{\hat{c}}_{(\odot ),j}^{{{{\dagger}}} }$ with ψ_(k),± being a normalization constant and $\left\vert \varnothing \right\rangle$ denotes the vacuum state. As shown in Fig. 1, the corresponding single-particle spectrum is identical to that of a tight-binding chain (i.e., ${\varepsilon }_{k}=2t\cos k$) except for the k = k₀ states whose single-particle energies are characterized by the splitting ${\Delta }_{\pm }=\frac{{\varepsilon }_{0}}{2\tilde{s}}\pm \frac{\sqrt{{\varepsilon }_{0}^{2}+4{\tilde{s}}^{2}}}{| 2\tilde{s}| }$:

$${\varepsilon }_{\pm }=\frac{1}{2}\left({\varepsilon }_{0}\pm {{{{{{{\rm{sgn}}}}}}}}(\tilde{s})\sqrt{{\varepsilon }_{0}^{2}+4{\tilde{s}}^{2}}\right)=\tilde{s}{\Delta }_{\pm }.$$

(5)

These k = k₀ single-particle eigenstates Eq. (4) separate $\propto | \tilde{s}| \propto \sqrt{L}$ from the remaining spectrum giving rise to a single-particle gap. Referring to Eq. (3), in the limit $\frac{\tilde{s}}{t}\to \infty$, 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 ${\Delta }_{\pm }\mathop{\longrightarrow }\limits^{\tilde{s}/t\to \infty }\pm 1$ we find that the corresponding wavefunction is characterized by maximally mixing the distinguished mode $\left\vert {k}_{0}\right\rangle$ on the outer ring with the state ${\left\vert {N}_{\odot } = 1\right\rangle }_{\odot }$ 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 solution of Eq. (1) with the complete derivation detailed in the Supplementary Note 4. Here, the key observation is that Slater determinants $\left\vert {{{{{{{{\rm{FS}}}}}}}}}_{N}\right\rangle =\left\vert {k}_{1},\ldots ,{k}_{N}\right\rangle$ constructed from a set of N single-particle states Eq. (4) with k ≠ k₀ are also eigenstates of $\hat{H}={\hat{\Pi }}_{\odot }{\hat{H}}_{{{{{{{{\rm{lad}}}}}}}}}{\hat{\Pi }}_{\odot }$:

$${\hat{\Pi }}_{\odot }{\hat{H}}_{{{{{{{{\rm{lad}}}}}}}}}{\hat{\Pi }}_{\odot }\left\vert {{{{{{{{\rm{FS}}}}}}}}}_{N}\right\rangle =\left(\mathop{\sum }\limits_{l=1}^{N}{\varepsilon }_{{k}_{l}}\right){\hat{\Pi }}_{\odot }\left\vert {{{{{{{{\rm{FS}}}}}}}}}_{N}\right\rangle .$$

(6)

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

$${\hat{\Pi }}_{\odot }{e}^{{{{{{{{\rm{i}}}}}}}}\pi {\hat{n}}_{\odot ,j}}{\hat{\Pi }}_{\odot }={\hat{1}}_{\odot ,j}-\frac{2}{L}{\hat{c}}_{\odot ,k = 0}^{{{{\dagger}}} }{\hat{c}}_{\odot ,k = 0},$$

(7)

and thus ${\hat{\Pi }}_{\odot }{e}^{{{{{{{{\rm{i}}}}}}}}\pi {\hat{n}}_{\odot ,j}}{\hat{\Pi }}_{\odot }\left\vert {{{{{{{{\rm{FS}}}}}}}}}_{N}\right\rangle =\left\vert {{{{{{{{\rm{FS}}}}}}}}}_{N}\right\rangle$. Particle-number conservation of the wheel Hamiltonian then motivates to construct an ansatz for the N-particle eigenstates, superimposing all possible occupations of the k₀ mode that belong to the same overall particle number sector

$$\left\vert {\psi }_{N}\right\rangle = \, {\alpha }_{0}\left\vert {{{{{{{{\rm{FS}}}}}}}}}_{N}\right\rangle +\left({\alpha }_{{1}_{+}}{\hat{\psi }}_{+}^{{{{\dagger}}} }+{\alpha }_{{1}_{-}}{\hat{\psi }}_{-}^{{{{\dagger}}} }\right)\left\vert {{{{{{{{\rm{FS}}}}}}}}}_{N-1}\right\rangle \\ +{\alpha }_{2}{\hat{\psi }}_{+}^{{{{\dagger}}} }{\hat{\psi }}_{-}^{{{{\dagger}}} }\left\vert {{{{{{{{\rm{FS}}}}}}}}}_{N-2}\right\rangle ,$$

(8)

with complex coefficients ${\alpha }_{0,{1}_{\pm },2}$. These states describe a superposition of either empty ( ∝ α₀) or doubly occupied ( ∝ α₂) k₀ states and highly non-local states $\propto {\alpha }_{{1}_{\pm }}$ in which the k₀ mode on the outer ring is coupled to the ${\left\vert {N}_{\odot } = 1\right\rangle }_{\odot }$ 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 ${\hat{\Pi }}_{\odot }{\hat{H}}_{{{{{{{{\rm{lad}}}}}}}}}{\hat{\Pi }}_{\odot }\left\vert {{{{{{{{\rm{FS}}}}}}}}}_{N}\right\rangle =E{\hat{\Pi }}_{\odot }\left\vert {{{{{{{{\rm{FS}}}}}}}}}_{N}\right\rangle$ is reduced to the diagonalization of a 4 × 4 matrix. Fixing a Slater determinant $\left\vert {{{{{{{{\rm{FS}}}}}}}}}_{N}\right\rangle$ and two modes ${k}^{{\prime} },{k}^{{\prime\prime} }\ne {k}_{0}$ so that ${\hat{c}}_{{k}^{{\prime} }}\left\vert {{{{{{{{\rm{FS}}}}}}}}}_{N}\right\rangle =\left\vert {{{{{{{{\rm{FS}}}}}}}}}_{N-1}\right\rangle$ as well as ${\hat{c}}_{{k}^{{\prime\prime} }}{\hat{c}}_{{k}^{{\prime} }}\left\vert {{{{{{{{\rm{FS}}}}}}}}}_{N}\right\rangle =\left\vert {{{{{{{{\rm{FS}}}}}}}}}_{N-2}\right\rangle$, and labeling the 4 basis states by their possible occupations of the k = k₀ mode ${n}_{{k}_{0}}=0,{1}_{\pm },2$, the resulting eigenvalue problem is of the form

(9)

with ${h}_{0}=\langle 0| {\hat{H}}_{{{{{{{{\rm{lad}}}}}}}}}| 0\rangle$, ${h}_{1}=\langle {1}_{\mu }| {\hat{H}}_{{{{{{{{\rm{lad}}}}}}}}}| {1}_{{\mu }^{{\prime} }}\rangle$ for $\mu ,{\mu }^{{\prime} }\in \{+,-\}$, and ${h}_{2}=\langle 2| {\hat{H}}_{{{{{{{{\rm{lad}}}}}}}}}| 2\rangle$. Note the block-diagonal structure that reflects the different k = k₀ parities, i.e.

$${{{{{{{{\rm{e}}}}}}}}}^{{{{{{{{\rm{i}}}}}}}}\pi {\hat{n}}_{{k}_{0}}}\left\vert {n}_{{k}_{0}}\right\rangle =\left\{\begin{array}{ll}\left\vert {n}_{{k}_{0}}\right\rangle ,\quad &{{{{{{{\rm{if}}}}}}}}\,{n}_{{k}_{0}}=0,2,\\ -\left\vert {n}_{{k}_{0}}\right\rangle ,\quad &{{{{{{{\rm{if}}}}}}}}\,{n}_{{k}_{0}}={1}_{+},{1}_{-}.\end{array}\right.$$

(10)

We emphasize the existence of a hidden ${{\mathbb{Z}}}_{2}$-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 k₀-occupation. Furthermore, condensation requires a breaking of particle number conservation on the outer ring, which is possible only in the ${n}_{{k}_{0}}={1}_{\pm }$ subspace. Thus, an odd ${{\mathbb{Z}}}_{2}$-symmetry of the ground state signals the formation of a BEC.

Upon solving Eq. (9), a special structure of the many-body spectrum appears that is characterized by a clustering of eigenstates belonging to the same k₀-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 ${\tilde{s}}_{{{{{{{{\rm{c}}}}}}}},1}$ and gap Δ₁ arise once the clustered odd-parity eigenstates constitute the overall ground state, indicating the condensation of bosons into the k₀ mode (abbreviating ${X}_{\rho }=\frac{\sin (\pi \rho )}{\pi }$):

$${\tilde{s}}_{{{{{{{{\rm{c}}}}}}}},1}=\sqrt{8{X}_{\rho }},$$

(11)

$${\Delta }_{1}=-\left(1+2{X}_{\rho }\right)+\sqrt{{\left(1-2{X}_{\rho }\right)}^{2}+{\tilde{s}}^{2}}.$$

(12)

The second critical hopping is defined by the complete separation of the odd-parity cluster from the even-parity many-particle eigenstates:

$${\tilde{s}}_{{{{{{{{\rm{c}}}}}}}},2}=4L\sqrt{{X}_{\rho }^{2}+{{{{{{{\mathcal{O}}}}}}}}({L}^{-1})},$$

(13)

$${\Delta }_{2}=-4L{X}_{\rho }-\left(1-2{X}_{\rho }\right)+\sqrt{{\left(1+2{X}_{\rho }\right)}^{2}+{\tilde{s}}^{2}}.$$

(14)

Note that $\tilde{s} > {\tilde{s}}_{{{{{{{{\rm{c}}}}}}}},2}$ implies that scattering between states with even and odd k₀ parity, caused by external perturbations, can only occur if the energy barrier Δ₂ can be overcome.

Fig. 3: 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 eigenenergies E₀, E_1,±, E₂ of ${\hat{H}}_{{{{{{{{\rm{lad}}}}}}}}}$ in the k₀-subspace Eq. (9), generating a clustering in the many-body spectrum. For each energy, we show all possible k ≠ k₀ Slater determinants. Indicated are also the gaps defining the two critical ring-to-center hoppings, s_c,1 and s_c,2, describing the starting and complete separation of even and odd parity sectors, respectively.

Discussion

The analytical solution and, in particular, the property of BEC ground states exhibiting odd k = k₀ 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 more complicated many-body state. Therefore, interactions generically couple the two parity sectors in the k₀ subspace and one might expect a breaking of the ${{\mathbb{Z}}}_{2}$-symmetry. However, mixing of the k₀ parity sectors caused by local interactions connecting d neighboring sites on the outer ring is of the order of $\sim \frac{Vd}{\tilde{s}}$ 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, the ${{\mathbb{Z}}}_{2}$ symmetry of the k₀ modes is approximately restored, thus stabilizing BEC in the presence of interactions on the outer ring.

We numerically checked the robustness of the BEC in the thermodynamic limit for k₀ = 0 and finite values of the ring-to-center hopping. To this end, we calculated the ground-state occupation ${n}_{{k}_{0}}(s,L)$ of the k₀ = 0 mode^1,38 using DMRG. Normalizing with respect to the upper bound on the condensate occupation ${n}_{\max }(L)=L\rho (1-\rho +1/L)$¹¹, we extrapolated the condensate fraction into the thermodynamic limit ${n}_{{k}_{0}}(s)/{n}_{\max }={\lim }_{L\to \infty }\frac{{n}_{{k}_{0}}(s,L)}{{n}_{\max }(L)}$, see Supplementary Note 5. 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 ${{\mathbb{Z}}}_{2}$-symmetry protection, in Fig. 4b the dependency of the condensate fraction at finite system sizes and as a function of $V/\tilde{s}$ 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 $\tilde{s}=s\sqrt{L}$. In accordance to the scaling of Δ₂, the maximally possible condensation is reached if $\tilde{s} \sim L$. 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.

Fig. 4: Numerical analysis of the robustness of the Bose-Einstein condensate (BEC).

a Ground-state condensate fraction ${n}_{{k}_{0}}$ normalized to the maximally possible value ${n}_{\max }$¹¹ and extrapolated to the thermodynamic limit. b Asymptotically the normalized condensate fraction for a fixed number of lattice sites L is a function of the ratio of nearest-neighbor interaction strength and rescaled hopping amplitude $V/\tilde{s}$, only. Results are shown for different V and densities on the ring ρ. Note that for very small fillings ρ = 1/16 (inset), there are significant deviations of the observed connection between the condensate fraction and the ratio $V/\tilde{s}$. This originates from the flat single-particle dispersion around k = π (see Fig. 1). Thereby, the complete separation of the odd-parity states (controlled by the energy gap Δ₂) occurs already for small ring-to-center hoppings, mainly independent on the number of lattice sites.

Conclusion

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^{30,31,32,33,39,40,41}. Second, there is a many-body gap separating the BEC-carrying states from the remaining spectrum $\sim s\sqrt{L}$, 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 achieve exceptionally stable logical qubits by coupling a set of noisy qubits to a central qubit. Experimental platforms providing the required all-to-one couplings are for example NV centers^42,43 or superconducting qubits in circuit QED settings⁴⁴. Moreover, we believe that the wheel-to-ladder mapping could prove useful in the analysis of hidden fermions⁴⁵. 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 $s\to \frac{s}{\sqrt{L}}$ with regard to the crossover from one to an infinite number of dimensions.

Methods

Wheel-to-ladder mapping

One of our methodical key findings is a mapping that allows to utilize a JWT to construct the solution to the many-body problem. Closed analytic expressions for models exhibiting a true BEC are rare, in particular for the case of HCBs, which can be obtained, for instance, in the Bose-Hubbard model in the limit of infinitely strong on-site repulsion. We therefore elaborate on the mapping in more detail.

To solve the many-particle problem of the wheel Hamiltonian $\hat{H}$, we introduce a mapping between the long-range coupled wheel Hamiltonian and the NNN coupled ladder Hamiltonian ${\hat{H}}_{{{{{{{{\rm{lad}}}}}}}}}$ with periodic boundary conditions depicted in Fig. 1. Since they exhibit the same matrix elements, we identify the subspace of the center site with the single-particle momentum states ${\left\vert k = 0,{N}_{\odot } = 0,1\right\rangle }_{\odot }$ in the inner ring of the ladder, where N_⊙ is the occupation of the inner ring of the periodic ladder. Utilizing the projector to this subspace ${\hat{\Pi }}_{\odot }$, we can relate both Hamiltonians:

$$\hat{H}={\hat{\Pi }}_{\odot }{\hat{H}}_{{{{{{{{\rm{lad}}}}}}}}}{\hat{\Pi }}_{\odot }.$$

(15)

A JWT allows to express the hardcore bosonic ladder operators in terms of fermionic ones

$${\hat{c}}_{j}^{({{{\dagger}}} )}=\mathop{\prod}\limits_{k < j}{{{{{{{{\rm{e}}}}}}}}}^{{{{{{{{\rm{i}}}}}}}}\pi {\hat{n}}_{k}}{\hat{h}}_{j}^{({{{\dagger}}} )},{{{{{{{\rm{with}}}}}}}}\,j=0,\ldots ,2L-1.$$

(16)

When transforming the hardcore bosonic Hamiltonian, the JWT leads to cancellations of arbitrarily long-ranged interactions and only a single phase-factor ${{{{{{{{\rm{e}}}}}}}}}^{{{{{{{{\rm{i}}}}}}}}\pi {\hat{n}}_{\odot ,j}}$ remains in the hopping terms on the outer ring ${\hat{\Pi }}_{\odot }{\hat{h}}_{j}^{{{{\dagger}}} }{\hat{h}}_{j+1}{\hat{\Pi }}_{\odot }={\hat{\Pi }}_{\odot }{\hat{c}}_{j}^{{{{\dagger}}} }{{{{{{{{\rm{e}}}}}}}}}^{{{{{{{{\rm{i}}}}}}}}\pi {\hat{n}}_{\odot ,j}}{\hat{c}}_{j+1}^{{{{\dagger}}} }{\hat{\Pi }}_{\odot }$. At this point, the projected ladder Hamiltonian can be solved analytically.

Numerical methods

All numerical results were obtained using DMRG in its matrix-product state representation²⁷ implemented in the SymMPS toolkit⁴⁶. More precisely, the calculations were performed with a bond dimension up to 1200, which allowed the discarded weight to always stay below 2 ⋅ 10⁻⁸ and usually below 10⁻¹⁰. Since DMRG works best in one-dimensional systems, the wheel is projected onto a chain in a way that reduces the long-range interaction to a (rather large) minimum, see Fig. 5.

Fig. 5: Chosen projection of the wheel geometry (top) onto a chain (bottom).

This reduces the long-range interaction from the first to the last site that is replaced by multiple NNN interactions compared to a straight forward periodic boundary implementation. The center site (star) of the wheel is placed in the middle of this chain—again in order to reduce the long-range interaction to a minimum.

Observables

The observable of interest, as shown in Fig. 4, is the normalized condensate fraction of the distinguished k₀ mode extrapolated to the thermodynamic limit. Note that in our calculations we chose k₀ = 0. In order to obtain this quantity, we need to get the SPDM of the ground state,

$${\rho }_{j,{j}^{{\prime} }}=\langle {\hat{c}}_{j}^{{{{\dagger}}} }{\hat{c}}_{{j}^{{\prime} }}\rangle$$

(17)

for multiple system sizes. The condensate fraction is then obtained by Fourier transforming the SPDM:

$${n}_{k}=\frac{1}{L}\mathop{\sum}\limits_{j,{j}^{{\prime} }}{{{{{{{{\rm{e}}}}}}}}}^{-2ik(j-{j}^{{\prime} })/L}{\rho }_{j,{j}^{{\prime} }}.$$

(18)

In order to be able to compare the condensate fractions for different system sizes, it is necessary to normalize them w.r.t. the maximally possible value. This is given by¹¹

$${n}_{\max }(L)=(L-N+1)\cdot N/L.$$

(19)

We chose four different sizes of the outer ring (32, 64, 128, 256) and extrapolated these normalized results via a 1/L fit.