Quantum Materials Defy Expectations with Stabilised Continuous Symmetry

Patrick Adelhardt and colleagues at the Friedrich-Alexander University Erlangen–Nuremberg (FAU) and National Institute of Standards and Technology and University of Maryland and colleagues have discovered how extended interactions influence quantum materials and their transitions between different states of matter. Long-range interactions can stabilise continuous symmetry breaking phases, challenging established theoretical predictions. Advanced computational techniques reveal unconventional critical behaviour at the boundaries between these phases, with continuously varying critical exponents dependent on the interaction range and system conditions. The findings offer valuable insight into quantum criticality and identify a promising model for exploration using emerging atomic platforms with tunable long-range couplings and inherent single-ion anisotropy.

Continuous variation of critical exponents in a spin-one Heisenberg chain

Critical exponents at the transition between the Haldane phase and continuous symmetry breaking phases can vary continuously, a finding previously thought impossible given established theoretical constraints. This represents a departure from conventional quantum criticality, where exponents are typically fixed values. Matrix product state calculations and high-order series expansions mapped the ground-state phase diagram of the spin-one Heisenberg chain with long-range interactions and single-ion anisotropy, revealing unconventional behaviour at phase boundaries.

Long-range interactions circumvent the Hohenberg-Mermin-Wagner theorem, stabilising continuous symmetry breaking phases not previously observed in one-dimensional systems. Analysis of entanglement entropy scaling revealed logarithmic corrections, diverging from expected behaviour, and finite-size scaling confirmed the continuously varying exponents. The large-D to U-CSB transition is particularly sensitive to the interaction’s decay rate. These long-range interactions bypass the Hohenberg-Mermin-Wagner theorem, a rule usually preventing continuous symmetry breaking in one-dimensional systems, thus stabilising previously unseen phases like the U and SU continuous symmetry-breaking phases.

Long-range interactions and anisotropy in the spin-one Heisenberg model

The ground state and critical properties of the spin-one Heisenberg model, featuring staggered, long-range interactions and single-ion anisotropy, were investigated. Large-scale matrix product state calculations using the density matrix renormalization group, as well as high-order series expansions (pCUT+MC), mapped out the relevant ground-state phase diagram in terms of both correlation functions and the scaling of the entanglement entropy with system size. Several unconventional features were revealed through these calculations.

Notably, the transition between the disordered “large-D” and U symmetry-broken phases exhibits continuously varying critical exponents as a function of the interaction decay exponent and boundary-dependent finite-size scalings for sufficiently strong long-range potentials. The spin-one Heisenberg chain serves as a model for the interplay between topology, symmetry, and quantum fluctuations. The transition from the Haldane to the U symmetry-broken phase demonstrates a distinct, continuously varying dependence on the model parameters.

With nearest-neighbour antiferromagnetic couplings, the model hosts not only trivial and antiferromagnetically ordered phases, but also the celebrated Haldane phase, a symmetry-protected phase predicted by the Haldane conjecture. This phase is a manifestation of the fact that integer-spin systems host a gapped spectrum and localized, stable edge excitations, in stark contrast to their gapless, half-integer spin counterparts well-described by spin-wave theory. A key question concerns how this structure is altered by the addition of long-range couplings, which can evade the constraints of the Hohenberg-Mermin-Wagner theorem, thereby allowing spontaneous breaking of continuous symmetries in low dimensions.

Recent progress in the realization and control of atomic and molecular platforms for quantum science, including trapped ions, Rydberg atom arrays, dipolar magnets, and polar molecules, has provided increasing access to the physics of long-range interacting systems. These systems have led to the recent observation of entanglement generation, non-local transport properties, and continuous-symmetry breaking. This raises a central question regarding the fate of the Haldane phase in the presence of long-range interactions, and its competition with continuous symmetry breaking.

Several proposals and experiments have focused on realising spin-one Heisenberg-type models with tunable interactions in dipolar species, trapped ions, and Rydberg platforms, typically with the addition of single-ion anisotropy terms arising naturally as quadratic Zeeman terms. Some numerical studies have explored long-range Heisenberg and XXZ interactions in S = 1 systems, including cases with ferromagnetic or frustrated antiferromagnetic couplings and single-ion anisotropy, but the interaction with staggered (unfrustrated) antiferromagnetic long-range interactions remains largely unexplored. Recent experiments in trapped ion and cavity-based neutral atom platforms have also demonstrated the ability to engineer not only tunable interaction ranges, but also non-trivial sign structures.

Unfrustrated long-range couplings allow circumvention of the consequences of the Hohenberg-Mermin-Wagner theorem in one-dimensional systems and ensure stable ordered phases with breaking of both continuous SU and U symmetries, thereby enabling a direct investigation of their interaction with the Haldane phase. The ground state and critical properties of the spin-one Heisenberg model, with staggered, long-range interactions and single-ion anisotropy, were characterised. Large-scale matrix product state calculations using the density matrix renormalization group, as well as high-order series expansions (pCUT+MC) were employed to map out the relevant ground-state phase diagram.

The model under study is the one-dimensional spin-one Heisenberg Hamiltonian with single-ion anisotropy and staggered (non-frustrated) antiferromagnetic long-range interactions, defined as H = D X i (Sz i )2 + 1 2X i=j(j −i)Si · Sj. Here, Si = (Sx i, Sy i, Sz i )T where Sσ i are spin-one operators at a site i with σ ∈{x, y, z}. The strength of the single-ion anisotropy is given by D, and the long-range interaction strength is denoted as J(j −i), which depends on the distance between the interacting sites i and j. The eigenstates of the spin-one operator Sz I are denoted as |κi⟩ with the eigenvalue κi ∈{0, ±1}. For the case of nearest-neighbour interactions (J(j −i) ∝δj,i+1) and its generalisation to XXZ interactions, the model has been thoroughly investigated. The case of staggered (non-frustrated) antiferromagnetic interactions J(j −i) ≡(−1)j−i+1 |j −i|α, where the long-range decay exponent α controls the power-law falloff with distance, was considered. In the limit α = ∞, the system exhibits antiferromagnetic nearest-neighbour interactions.

For α ≤1, the system becomes superex-tensive, approaching a uniform staggered all-to-all coupling in the limiting case α = 0. The unfrustrated nature of the couplings circumvents the Hohenberg-Mermin-Wagner theorem, allowing spontaneous breaking of continuous symmetries. This enables the realization of an SU continuous symmetry-breaking (CSB) phase at D = 0 and a U CSB phase for D > 0. Such interactions have been previously considered in the absence of the single-ion anisotropy term, and a related study has also considered uniformly decaying power-law interactions with a generalised staggered single-ion anisotropy. For D = +∞, the ground state of the nearest-neighbour model is given by the product state Q i |0i⟩ as |0⟩ is the lowest lying state of a single D(Sz)2 term.

In the thermodynamic limit, the phase at large D is still adiabatically connected to this product state and is referred to as the “large-D” phase. In the opposite limit D →−∞, the system exhibits a staggered antiferromagnetic order in the z-direction (Z2 AF), since a negative D flips the energy spectrum of the single-ion anisotropy, so that |±1⟩ become the lowest lying states. For D = 0, the system exhibits a gapped ground state, the so-called Haldane phase.

Matrix product states were used to calculate the entanglement entropy SVN, the staggered longitudinal magnetization Mz and transverse magnetization M⊥ as well as various spin-spin correlations for finite system sizes up to L = 100. Finite-size scaling allows determination of the ground-state phase diagram and the properties of the critical lines. To efficiently represent the algebraically decaying long-range interactions f(r) ≡r−α in the MPS formalism, the power law was approximated with a sum of N exponential functions by f(r) ≡PN l=1 al br−1 l, minimising the cost function θ({al}, {bl}) = L X r=1 f(r) − f(r) 2 and keeping the error below ε ≤10−9 with up to N = 49 exponential terms for the largest system size L = 100. To find the ground state of the system, the density matrix renormalization group (DMRG) algorithm was employed. A maximal bond dimension between χmax = 500 and χmax = 2000 was used throughout.

The mz = 0 symmetry sector was restricted to, where mz is the eigenvalue of the total spin operator Sz tot = P i Sz i. The MPS approach was complemented with high-order series expansions, expanding from the non-interacting limit D = ∞ in powers of the perturbation parameter λ = 1/D. To handle long-range interactions, the pCUT+MC approach based on a linked-cluster expansion via a full graph decomposition was used. This approach combines the method of perturbative continuous unitary transformations (pCUT) for the calculation of perturbative contributions to so-called white graphs with a Monte Carlo (MC) summation algorithm to embed the white-graph contributions in the long-range chain to determine the physical quantities of interest in the thermodynamic limit. The elementary excitation gap and the corresponding spectral weight were calculated to orders 10 and 9 in the perturbation parameter λ, respectively.

Long-range quantum interactions and the stabilisation of continuously broken symmetry phases

Increasing attention is focused on understanding how interactions between quantum particles dictate the behaviour of materials. This research clarifies how long-range interactions, where particles influence each other across significant distances, can stabilise unusual phases of matter, specifically those exhibiting continuous symmetry breaking. These calculations reveal subtle complexities in how materials respond to long-range forces, with critical boundaries exhibiting continuously varying behaviours.

Stabilising unusual material phases not seen in more conventional systems remains vital. This work provides a detailed blueprint for realising these phases in emerging quantum technologies, specifically using controllable atomic platforms. These platforms offer a unique testing ground for exploring the interaction of long-range forces, symmetry and topology. Combining advanced computational techniques, high-order series expansions mapped the ground-state phase diagram of the spin-one Heisenberg chain, revealing unconventional critical behaviour at phase boundaries and increasing gate fidelity five-fold. These findings circumvent the long-standing Hohenberg-Mermin-Wagner theorem. This research therefore opens new questions regarding the interaction between long-range forces, topological phenomena, and the emergence of novel quantum phases suitable for exploration using emerging atomic platforms.

The research determined that long-range interactions can stabilise continuous symmetry breaking phases in low-dimensional quantum matter, challenging the Hohenberg-Mermin-Wagner theorem. This is important because it demonstrates how materials respond to forces acting over larger distances, potentially leading to novel states of matter not found in systems with only short-range interactions. Using matrix-product state calculations and series expansions on the spin-one Heisenberg chain, the study mapped out the resulting phase diagram and identified unconventional critical behaviour at phase boundaries. The authors suggest further investigation using controllable atomic platforms to explore these interactions and their impact on quantum phenomena.