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 onedimensional quantum manybody 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 manybody problem of hardcore bosons on a wheel geometry, which are known to form BoseEinstein condensates. The robustness of the condensate against interactions is shown numerically by adding nearestneighbor interactions, which typically destroy BoseEinstein condensates. We discuss further applications such as geometrically inducing finitemomentum 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.
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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 BoseEinstein 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.
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 manybody problems with arbitrary k_{0}modulated ringtocenter hoppings \({s}_{j}=s{{{{{{{{\rm{e}}}}}}}}}^{{{{{{{{\rm{i}}}}}}}}{k}_{0}j}\), 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 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 manybody phase against local perturbations on the outer ring. Furthermore, we trace back the protection to oddparity k = k_{0} singleparticle 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 ringtocenter 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 densitymatrix renormalization group (DMRG)^{26,27,28,29} simulations to calculate the k_{0}condensate fraction when adding nearestneighbor 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 ringtocenter 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 k_{0}modulated hoppings give rise to corresponding protected k_{0} modes and the respective singleparticle 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 manybody phases such as k_{0} ≠ 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 Lsited ring with an additional center site^{10,12} (see Fig. 1a). The model exhibits k_{0}modulated ringtocenter 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
where \({\hat{h}}_{j}^{({{{\dagger}}} )}\) is the HCB ladder operator on the jth 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 offdiagonal quasi longrange order, indicated by the algebraic decay of the singleparticle density matrix (SPDM) \({\rho }_{ij} \sim  {x}_{i}{x}_{j}{ }^{\frac{1}{2}}\)^{34,35}. This leads to the formation of a quasiBEC, 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/L^{11}.
Analytical solution via wheeltoladder mapping
The property of Eq. (1) to interpolate between two superficially disconnected physical situations, one exhibiting no stable longrange order (\(\frac{s}{t}=0\)) and one featuring true longrange 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 longrange 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 JordanWigner 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 longranged 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 longranged 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 singleparticle 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 longrange coupled wheel Hamiltonian can be mapped to an only nextnearestneighbor (NNN) coupled ladder Hamiltonian \({\hat{H}}_{{{{{{{{\rm{lad}}}}}}}}}\):
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 \({{{{{{{{\rm{e}}}}}}}}}^{{{{{{{{\rm{i}}}}}}}}\pi {\hat{n}}_{\odot ,j}}\):
wherein \({\hat{c}}_{j}^{({{{\dagger}}} )}\,\)\(({\hat{c}}_{\odot ,j}^{({{{\dagger}}} )})\) denotes the fermionic ladder operator on the jth site of the outer (inner) ring and the singlesite 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 ringtocenter 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 indepth analysis of the ground state of Eq. (3). Note that for now and in the following, we refer to the scaling of the ringtocenter hopping \(\tilde{s}=s\sqrt{L}\) as extensive in the system size.
As we show in Supplementary Note 2, it is instructive to first solve Eq. (3) for the singleparticle eigenstates \(\left\vert k,\pm \right\rangle\), fulfilling \(\langle {{{{{{{{\rm{e}}}}}}}}}^{{{{{{{{\rm{i}}}}}}}}\pi {\hat{n}}_{\odot ,j}}\rangle \equiv 1\):
Here, \({\hat{c}}_{(\odot ),k}^{{{{\dagger}}} }=\frac{1}{\sqrt{L}}\mathop{\sum }\nolimits_{j = 0}^{L1}{{{{{{{{\rm{e}}}}}}}}}^{{{{{{{{\rm{i}}}}}}}}kj}{\hat{c}}_{(\odot ),j}^{{{{\dagger}}} }\) with ψ_{(k),±} being a normalization constant and \(\left\vert \varnothing \right\rangle\) denotes the vacuum state. As shown in Fig. 1, the corresponding singleparticle spectrum is identical to that of a tightbinding chain (i.e., \({\varepsilon }_{k}=2t\cos k\)) except for the k = k_{0} states whose singleparticle energies are characterized by the splitting \({\Delta }_{\pm }=\frac{{\varepsilon }_{0}}{2\tilde{s}}\pm \frac{\sqrt{{\varepsilon }_{0}^{2}+4{\tilde{s}}^{2}}}{ 2\tilde{s} }\):
These k = k_{0} singleparticle eigenstates Eq. (4) separate \(\propto  \tilde{s} \propto \sqrt{L}\) from the remaining spectrum giving rise to a singleparticle 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 singleparticle gap can be expected to control the manybody 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 nonlocal 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 singleparticle states Eq. (4) with k ≠ k_{0} are also eigenstates of \(\hat{H}={\hat{\Pi }}_{\odot }{\hat{H}}_{{{{{{{{\rm{lad}}}}}}}}}{\hat{\Pi }}_{\odot }\):
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 \({\hat{\Pi }}_{\odot }{e}^{{{{{{{{\rm{i}}}}}}}}\pi {\hat{n}}_{\odot ,j}}{\hat{\Pi }}_{\odot }\left\vert {{{{{{{{\rm{FS}}}}}}}}}_{N}\right\rangle =\left\vert {{{{{{{{\rm{FS}}}}}}}}}_{N}\right\rangle\). Particlenumber conservation of the wheel Hamiltonian then motivates to construct an ansatz for the Nparticle eigenstates, superimposing all possible occupations of the k_{0} mode that belong to the same overall particle number sector
with complex coefficients \({\alpha }_{0,{1}_{\pm },2}\). These states describe a superposition of either empty ( ∝ α_{0}) or doubly occupied ( ∝ α_{2}) k_{0} states and highly nonlocal states \(\propto {\alpha }_{{1}_{\pm }}\) in which the k_{0} 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 \({\hat{c}}_{{k}^{{\prime} }}\left\vert {{{{{{{{\rm{FS}}}}}}}}}_{N}\right\rangle =\left\vert {{{{{{{{\rm{FS}}}}}}}}}_{N1}\right\rangle\) as well as \({\hat{c}}_{{k}^{{\prime\prime} }}{\hat{c}}_{{k}^{{\prime} }}\left\vert {{{{{{{{\rm{FS}}}}}}}}}_{N}\right\rangle =\left\vert {{{{{{{{\rm{FS}}}}}}}}}_{N2}\right\rangle\), and labeling the 4 basis states by their possible occupations of the k = k_{0} mode \({n}_{{k}_{0}}=0,{1}_{\pm },2\), the resulting eigenvalue problem is of the form
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 blockdiagonal structure that reflects the different k = k_{0} parities, i.e.
We emphasize the existence of a hidden \({{\mathbb{Z}}}_{2}\)symmetry of the manybody 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 \({n}_{{k}_{0}}={1}_{\pm }\) 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 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 lowlying oddparity 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 Δ_{1} arise once the clustered oddparity eigenstates constitute the overall ground state, indicating the condensation of bosons into the k_{0} mode (abbreviating \({X}_{\rho }=\frac{\sin (\pi \rho )}{\pi }\)):
The second critical hopping is defined by the complete separation of the oddparity cluster from the evenparity manyparticle eigenstates:
Note that \(\tilde{s} > {\tilde{s}}_{{{{{{{{\rm{c}}}}}}}},2}\) implies that scattering between states with even and odd k_{0} parity, caused by external perturbations, can only occur if the energy barrier Δ_{2} can be overcome.
Discussion
The analytical 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 singleparticle description breaks down in favor of a more complicated manybody state. Therefore, interactions generically couple the two parity sectors in the k_{0} subspace and one might expect a breaking of the \({{\mathbb{Z}}}_{2}\)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 \(\sim \frac{Vd}{\tilde{s}}\) 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 k_{0} 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 k_{0} = 0 and finite values of the ringtocenter hopping. To this end, we calculated the groundstate occupation \({n}_{{k}_{0}}(s,L)\) of the k_{0} = 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 ringtocenter hopping \(\tilde{s}=s\sqrt{L}\). In accordance to the scaling of Δ_{2}, 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.
Conclusion
Our findings imply important consequences for both experimental and theoretical realizations of wheel geometries in general. First of all, a particular singleparticle mode can be gapped out by a proper modulation of the ringtocenter hopping, allowing the general protection of ordered phases that are characterized by a certain wave vector. We believe that such a modulation of the ringtocenter hopping provides an experimentally feasible approach to realize exotic, finitemomentum BEC in the framework of ultracold or Rydberg atoms^{30,31,32,33,39,40,41}. Second, there is a manybody gap separating the BECcarrying 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 onedimensional 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 alltoone couplings are for example NV centers^{42,43} or superconducting qubits in circuit QED settings^{44}. Moreover, we believe that the wheeltoladder mapping could prove useful in the analysis of hidden fermions^{45}. 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 ringtocenter hopping \(s\to \frac{s}{\sqrt{L}}\) with regard to the crossover from one to an infinite number of dimensions.
Methods
Wheeltoladder mapping
One of our methodical key findings is a mapping that allows to utilize a JWT to construct the solution to the manybody 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 BoseHubbard model in the limit of infinitely strong onsite repulsion. We therefore elaborate on the mapping in more detail.
To solve the manyparticle problem of the wheel Hamiltonian \(\hat{H}\), we introduce a mapping between the longrange 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 singleparticle 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:
A JWT allows to express the hardcore bosonic ladder operators in terms of fermionic ones
When transforming the hardcore bosonic Hamiltonian, the JWT leads to cancellations of arbitrarily longranged interactions and only a single phasefactor \({{{{{{{{\rm{e}}}}}}}}}^{{{{{{{{\rm{i}}}}}}}}\pi {\hat{n}}_{\odot ,j}}\) 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 matrixproduct state representation^{27} implemented in the SymMPS toolkit^{46}. 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 onedimensional systems, the wheel is projected onto a chain in a way that reduces the longrange interaction to a (rather large) minimum, see Fig. 5.
Observables
The observable of interest, as shown in Fig. 4, is the normalized condensate fraction of the distinguished k_{0} mode extrapolated to the thermodynamic limit. Note that in our calculations we chose k_{0} = 0. In order to obtain this quantity, we need to get the SPDM of the ground state,
for multiple system sizes. The condensate fraction is then obtained by Fourier transforming the SPDM:
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 by^{11}
We chose four different sizes of the outer ring (32, 64, 128, 256) and extrapolated these normalized results via a 1/L fit.
Data availability
The data that support the findings of the current study are available from the authors upon reasonable request.
Code availability
The software package SymMPS^{46} is available via www.symmps.eu. The source code is available from the authors upon reasonable request.
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Acknowledgements
We thank F. Grusdt and U. Schollwöck for carefully reading the manuscript. Furthermore, we thank M. Bramberger and M. Grundner for very fruitful discussions. T.K. acknowledges financial support by the ERC Starting Grant from the European Union’s Horizon 2020 research and innovation program under grant agreement number 758935. F.A.P. acknowledges funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) via Research Unit FOR 2414 under project number 277974659. R.H.W., F.A.P. and S.P. acknowledge support from the Munich Center for Quantum Science and Technology. The authors gratefully acknowledge the funding of this project by computing time provided by the Paderborn Center for Parallel Computing (PC^{2}).
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S.P., T.K. and R.H.W. developed the idea of the wheeltoladder mapping. S.P., R.H.W. and F.A.P. worked out the theoretical analysis. T.K. performed all DMRG simulations. All authors contributed to the writing of the manuscript.
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Wilke, R.H., Köhler, T., Palm, F.A. et al. Symmetryprotected BoseEinstein condensation of interacting hardcore bosons. Commun Phys 6, 182 (2023). https://doi.org/10.1038/s4200502301303z
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DOI: https://doi.org/10.1038/s4200502301303z
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