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# Long-lived phantom helix states in Heisenberg quantum magnets

## Abstract

Exact solutions for quantum many-body systems are rare but provide valuable insights for the description of universal phenomena such as the non-equilibrium dynamics of strongly interacting systems and the characterization of new forms of quantum matter. Recently, specific solutions of the Bethe ansatz equations for integrable spin models were found. They are dubbed phantom Bethe states and can carry macroscopic momentum yet no energy. Here, we show experimentally that there exist special helical spin patterns in anisotropic Heisenberg chains which are long-lived, relaxing only very slowly in dynamics, as a consequence of such states. We use these phantom spin-helix states to directly measure the interaction anisotropy, which has a major contribution from short-range off-site interactions. We also generalize the theoretical description to higher dimensions and other non-integrable systems and find analogous stable spin helices, which should show non-thermalizing dynamics associated with so-called quantum many-body scars. These results have implications for the quantum simulation of spin physics, as well as many-body dynamics.

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## Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## Code availability

The code that supports the findings of this study is available from the corresponding author upon reasonable request.

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## Acknowledgements

We thank J. Rodriguez-Nieva for helpful discussions and J. Amato-Grill for important advice. We thank J. Xiang for experimental assistance, J. de Hond for comments on the manuscript and M. Zwierlein for sharing equipment. We acknowledge support from the NSF through the Center for Ultracold Atoms and grant no. 1506369, the Vannevar-Bush Faculty Fellowship and DARPA (grant no. W911NF2010090). W.W.H. is supported in part by the Stanford Institute of Theoretical Physics. Y.K.E.L. is supported in part by the National Science Foundation Graduate Research Fellowship under grant no. 1745302.

## Author information

Authors

### Contributions

P.N.J., Y.K.E.L., H.L., W.W.H. and W.K. conceived the experiment. P.N.J., Y.K.E.L., H.L., I.D. and Y.M. performed the experiment. P.N.J., Y.K.E.L. and H.L. analysed the data. W.W.H. generalized the findings to higher dimensions. All authors discussed the results and contributed to the writing of the manuscript.

### Corresponding author

Correspondence to Yoo Kyung ‘Eunice’ Lee.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

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## Extended data

### Extended Data Fig. 1 Calibration of the wavevector Q.

Starting with all spin aligned, under a constant magnetic field gradient $${B}^{\prime}$$, the wavevector Q grows linearly starting from 0 as a function of winding time twind until at time twind = T = 15.608(4) ms we reach a wavevector Q = 2π/a and all spins are aligned again. All observed wavevectors fall on a line which determines the magnetic field gradient as $$\mu {B}^{\prime}/\hslash =(2\uppi /a)/T=2\uppi \times 64.068(15)\,{{{\rm{Hz}}}}/a$$ at a bias field of B = 882.612 G. The error bar of T implies an uncertainty in Q of at most ± 0.001 × 2π/a.

### Extended Data Fig. 2 Dephasing at the chain ends and holes.

If a spin has only one nearest neighbour, it experiences a torque and dephases rapidly at a rate ~Jxy (as opposed to a spin with two nearest neighbours, whose interactions cancel exactly in a phantom helix). This is the case for the spins at the two ends of the spin chain, as well as next to a hole. Additionally, the same spins experience an effective magnetic field $${h}_{z}=4{\tilde{t}}^{2}/{U}_{\uparrow \uparrow }-4{\tilde{t}}^{2}/{U}_{\downarrow \downarrow }$$ (created by superexchange), which is reduced by a factor of two, compared to the bulk. This modifies the boundary condition at the ends, and can lead to dephasing.

### Extended Data Fig. 3 Numerically calculated contrast decay of finite, open chains.

The simulation is performed using spin chains with 8 sites (blue) and 15 sites (red), for Δ = 0 (left) and 0.5 (right). The decay rate γ(Q) is determined using a short-time quadratic expansion as derived in ref. 17. The open chain boundary condition introduces a finite contrast decay even when the phantom condition is met (dashed line), but does not shift the Q-value at which the minimum decay rate occurs.

### Extended Data Fig. 4 Phantom spin helices for triangular and kagome lattices.

As in the case of the square lattice (see Fig. 4), we label each vertex with its azimuthal angle in the transverse plane. However, in the case of a triangular lattice (top) or kagome lattice (bottom), only ϕ = Qp = ±2π/3 and Δ = −1/2 defines a valid phantom spin helix. Thus, the spin can only point in one of three directions in the Sx-Sy plane (denoted by three different colours), with relative angle 2π/3 between them. Note that the colinear neighbours of each red spin are blue and green spins, and the interactions of the blue and green spins with the red spin cancel for each line. In this way, one can understand the 2D phantom helix states as arising from simple ‘stacking’ together of phantom helices of 1D chains.

### Extended Data Fig. 5 Level spacing statistics for the extended spin-1/2 XXZ model.

The model is defined by equation (33) with J1 = 1, Jn > 2 = 0, $${\varDelta }_{n}=\cos (n{Q}_{p})$$, and variable J2. We numerically simulate a system of N = 22 spins on a ring, and choose Qp = 2πm/N where m = 3, which gives Δ1 ≈ 0.654 and Δ2 ≈ − 0.142. The spin helix with wavevector Qp can be proven to be an exact quantum many-body eigenstate for any J2. We resolve global symmetries and compute eigenvalues in the sector with momentum k = 0, spin-flip G = 1, reflection parity R = 1, and magnetization M = 0. As can be seen, at J2 = 0 the average r-parameter 〈r〉 ≈ 0.39 (bottom dashed line), consistent with Poissonian statistics indicating an integrable system. When J2 ≠ 0, 〈r〉 → 0.53 (top dashed line) for large enough J2, consistent with Wigner-Dyson statistics in the Gaussian Orthogonal Ensemble (GOE), indicating that integrability has been broken. We expect the behaviour of 〈r〉 at J2 = 0 to be singular in the limit of very large system sizes; that is, any infinitesimal perturbation J2 in the thermodynamic limit will be sufficient to render the model non-integrable.

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Jepsen, P.N., Lee, Y.K.‘., Lin, H. et al. Long-lived phantom helix states in Heisenberg quantum magnets. Nat. Phys. 18, 899–904 (2022). https://doi.org/10.1038/s41567-022-01651-7

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• DOI: https://doi.org/10.1038/s41567-022-01651-7

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