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arXiv:2511.09097 · Physical Review Research

Spontaneous breaking of a continuous symmetry at a nonconformal quantum critical point in one dimension

R. Flores-Calderón, Martina Zündel

Published Sep 5, 2026• 0 citations

Abstract

In this work, we present numerical evidence for the spontaneous breaking of a continuous U ( 1 ) symmetry in a nearest-neighbor interacting spin-1 chain at a quantum critical point separating two XY quasi-long-range ordered phases distinguished by a spontaneously broken Z 2 symmetry. Remarkably, the continuous symmetry breaking emerges precisely at the critical point of the discrete order parameter, suggesting a mechanism beyond currently established scenarios. At criticality, the XY correlations develop true long-range order, accompanied by a finite perpendicular magnetization, a zero-frequency Bragg peak in the transverse dynamical structure factor, and sharp gapless collective excitations. From complementary static and dynamical observables, we quantitatively determine the critical exponents, obtaining a dynamical exponent z = 1.50 ± 0.04 and an anomalous dimension η = 1.04 ± 0.03 . Remarkably, the value of z coincides with the one-dimensional Kardar-Parisi-Zhang exponent despite the system being an equilibrium quantum many-body system. We further show that the noninteracting continuum limit is equivalent to the recently introduced transverse quantum fluid, displaying their off-diagonal long-ranged order in one dimension. Complementing the numerical study, we derive the renormalization-group flow equations of the continuum theory to second order in the ɛ expansion. We identify an interacting fixed point whose critical behavior differs from the Ising universality class already at two-loop order, although the perturbative exponents remain far from the numerical values.

DOI: 10.1103/yhpf-7451

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In this work, we present numerical evidence for the spontaneous breaking of a continuous U ( 1 ) symmetry in a nearest-neighbor interacting spin-1 chain at a quantum critical point separating two XY quasi-long-range ordered phases distinguished by a spontaneously broken Z 2 symmetry. Remarkably, the continuous symmetry breaking emerges precisely at the critical point of the discrete order parameter, suggesting a mechanism beyond currently established scenarios. At criticality, the XY correlations develop true long-range order, accompanied by a finite perpendicular magnetization, a zero-frequency Bragg peak in the transverse dynamical structure factor, and sharp gapless collective excitations. From complementary static and dynamical observables, we quantitatively determine the critical exponents, obtaining a dynamical exponent z = 1.50 ± 0.04 and an anomalous dimension η = 1.04 ± 0.03 . Remarkably, the value of z coincides with the one-dimensional Kardar-Parisi-Zhang exponent despite the system being an equilibrium quantum many-body system. We further show that the noninteracting continuum limit is equivalent to the recently introduced transverse quantum fluid, displaying their off-diagonal long-ranged order in one dimension. Complementing the numerical study, we derive the renormalization-group flow equations of the continuum theory to second order in the ɛ expansion. We identify an interacting fixed point whose critical behavior differs from the Ising universality class already at two-loop order, although the perturbative exponents remain far from the numerical values.

We argue that ground states of 1D spin chains can spontaneously break U(1) ``easy-plane'' spin rotation symmetry, via true long-range order of

(S^x, S^y)

, at the phase transition between two quasi-long-range-ordered phases. The critical point can be reached by tuning a single parameter in a Hamiltonian with the same symmetry as the XXZ model, without further fine-tuning. Equivalently, it can arise in systems of bosons with particle-hole symmetry, as a long-range-ordered transition point between two quasi-long-range-ordered superfluids. Our approach is to start with the continuum field theory of the isotropic Heisenberg ferromagnet and consider generic perturbations that respect easy-plane symmetry. We argue for a renormalization-group flow to a critical point where long-range order in

(S^x, S^y)

is enabled by coexisting critical fluctuations of

S^z

. (We also discuss multicritical points where further parameters are tuned to zero.) These results show that it is much easier to break continuous symmetries in 1D than standard lore would suggest. The failure of standard intuition for 1D chains (based on the quantum--classical correspondence) can be attributed to Berry phases, which prevent the 1+1D system from mapping to a classical 2D spin model. The present theory also gives an example of an ordered state whose Goldstone mode is interacting even in the infra-red, rather than becoming a free field.

In a recent paper Nahum argued on the basis of a renormalization-group analysis that, contrary to standard lore, ground states of 1D spin chains with short-range interactions can spontaneously break U(1) ``easy-plane'' spin rotation symmetry. True long-range order of

(S^x,S^y)

arises at the phase transition between two quasi-long-range-ordered phases. Here we present detailed numerical results for an anisotropic spin-1 chain model. We find a magnetization exponent

β\approx0.29

, consistent with the

ε

-expansion prediction

β\approx0.28

, a divergence of the Luttinger parameter on approaching the transition from both sides with an exponent that we relate to Nahum's transition, and finite-size spectra consistent with a dynamical critical exponent

z\approx2

. Our results support the critical behavior predicted by Nahum's field theory.

The Hohenberg-Mermin-Wagner theorem motivates the expectation that continuous symmetry remains unbroken in one-dimensional quantum ground states. Here, contrary to this expectation, we demonstrate the spontaneous breaking of U(1) spin-rotation symmetry and the emergence of an interacting quantum critical point in lattice spin-

1/2

and spin-

β\approx0.28

chains. Using subsystem Binder scaling, we locate the critical point and determine the correlation-length critical exponent as

ν= 0.60 \pm 0.02

, distinct from the Gaussian value

1/2

but consistent with the perturbative renormalization-group value. We further estimate the dynamical critical exponent to be

z = 1.986 \pm 0.005

from the lowest neutral excitation gap and

z = 1.955 \pm 0.021

from the longitudinal spin correlations. The results

z < 2

are incompatible with microscopic realizations of this critical theory by frustration-free models, implying a mechanism for continuous symmetry breaking in one-dimensional quantum ground states distinct from frustration freeness.

Uncovering and understanding universal dynamics in matter far from equilibrium remains a key challenge. In this work, we identify a so far unrecognized form of universal behavior that emerges after a sudden symmetry-breaking quench at continuous phase transitions. Our key observation is that the order-parameter fluctuations in Ising models exhibit a compelling temporal collapse across a wide range of system sizes and quench strengths, indicative of an emergent single-variable scaling form. This phenomenon can be explained by introducing a so far unknown dynamical critical exponent for the underlying continuous phase transition. We find evidence for a lower critical effective dimension of this universal regime: it is observed in the 2D quantum and 3D and 4D classical Ising models, but not in the 1D quantum or 2D classical cases. Our results suggest that our observed universal far-from-equilibrium scaling may extend beyond the Ising models studied here and could more broadly characterize systems with non-conserved order parameters, opening new avenues for exploring universal dynamics both theoretically and in current experimental platforms.

The critical breakdown of a one-dimensional quantum magnet with long-range interactions is studied by investigating the high-field polarized phase of the anisotropic XY model in a transverse field for the ferromagnetic and antiferromagnetic cases. While for the limiting case of the isotropic long-range XY model we can extract the elementary one-quasiparticle dispersion analytically and calculate two-quasiparticle excitation energies quantitatively in a numerical fashion, for the long-range Ising limit as well as in the intermediate regime we use perturbative continuous unitary transformations on white graphs in combination with classical Monte Carlo simulations for the graph embedding to extract high-order series expansions in the thermodynamic limit. This enables us to determine the quantum-critical breakdown of the high-field polarized phase by analyzing the gap closing including associated critical exponents and multiplicative logarithmic corrections. In addition, for the ferromagnetic isotropic XY model we determined the critical exponents

z

and $\ensuremath{\nu}$ analytically by a bosonic quantum-field theory.

Abstract Long-range interactions in one-dimensional quantum systems can qualitatively modify low-energy critical behavior and provide a natural setting for exploring unconventional phase transitions beyond short-range Tomonaga—Luttinger liquid physics. In this work, we study a one-dimensional XXZ spin chain with transverse long-range interactions and an explicit dimerization perturbation. Using bosonization and weak-coupling renormalization group (RG) analysis, we derive the RG flow equations. The resulting weak-coupling phase diagram contains a gapless Tomonaga—Luttinger liquid, an explicitly dimerized phase, and a continuous symmetry breaking (CSB) phase induced by the transverse long-range interaction. We further analyze the collapse point at which the two perturbations become simultaneously marginal. A stability analysis of the corresponding doubly marginal Gaussian fixed point shows that it is weakly unstable rather than a stable multicritical infrared fixed point, thereby giving rise to a pseudo-multicritical crossover over large length scales. Similar physics may be accessible in Rydberg-atom systems, offering a controllable theoretical and experimental platform for exploring long-range interaction effects and crossover behavior in one-dimensional quantum systems.