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arXiv:2610.06682

Interacting quantum criticality with U(1) symmetry breaking in 1+1 dimensions

Mutsumi Shimomura, Kohei Kawabata

cond-mat.str-elcond-mat.stat-mechquant-phSubmitted Oct 5, 2026• Announced Oct 6, 2026• 0 citations• Revised Oct 20, 2026 (v2)
v1 · Oct 5, 2026v2 · Oct 20, 2026
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Abstract

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-$1$ 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.

Comments
22 pages, 7 figures, 4 tables

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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-1/21/21/2 and spin-111 chains. Using subsystem Binder scaling, we locate the critical point and determine the correlation-length critical exponent as ν=0.60±0.02ν= 0.60 \pm 0.02ν=0.60±0.02, distinct from the Gaussian value 1/21/21/2 but consistent with the perturbative renormalization-group value. We further estimate the dynamical critical exponent to be z=1.986±0.005z = 1.986 \pm 0.005z=1.986±0.005 from the lowest neutral excitation gap and z=1.955±0.021z = 1.955 \pm 0.021z=1.955±0.021 from the longitudinal spin correlations. The results z<2z < 2z<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.
The Hohenberg-Mermin-Wagner theorem states that there is no spontaneous breaking of continuous internal symmetries in spatial dimensions d≤2 at finite temperature. At zero temperature, the quantum-to-classical mapping further implies the absence of such symmetry breaking in one dimension, which is also known as Coleman's theorem in the context of relativistic quantum field theories. One route to violate this "folklore" is requiring an order parameter to commute with a Hamiltonian, as in the classic example of the Heisenberg ferromagnet and its variations. However, a systematic understanding has been lacking. In this Letter, we propose a family of one-dimensional models that display spontaneous breaking of a U(1) symmetry at zero temperature, where the order parameter does not commute with the Hamiltonian. While our models can be deformed continuously within the same phase, there exist symmetry-preserving perturbations that render the observed symmetry breaking fragile. We argue that a more general condition for this behavior is that the Hamiltonian is frustration-free.
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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 (Sx,Sy)(S^x,S^y)(Sx,Sy) 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≈0.29β\approx0.29β≈0.29, consistent with the εεε-expansion prediction β≈0.28β\approx0.28β≈0.28, 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≈2z\approx2z≈2. Our results support the critical behavior predicted by Nahum's field theory.
Abstract The Mermin-Wagner theorem states that spontaneous continuous symmetry breaking is prohibited in systems with short-range interactions at spatial dimension D ≤ 2. For long-range interactions with a power-law form (1/ r α ), the theorem further forbids ferromagnetic or antiferromagnetic order at finite temperature when α ≥ 2 D . However, the situation for α ∈ (2, 4) at D = 2 is not covered by the theorem. To address this, we conduct large-scale quantum Monte Carlo simulations and field theoretical analysis. Our findings show spontaneous breaking of S U (2) symmetry in the ferromagnetic Heisenberg model with 1/ r α -form long-range interactions at D = 2. We determine critical exponents through finite-size analysis for α < 3 (above the upper critical dimension with Gaussian fixed point) and 3 ≤ α < 4 (below the upper critical dimension with non-Gaussian fixed point). These results reveal new critical behaviors in 2D long-range Heisenberg models, encouraging further experimental studies of quantum materials with long-range interactions beyond the Mermin-Wagner theorem’s scope.