Can U(1) symmetry break spontaneously in one-dimensional quantum ground states?

Why one-dimensional ground states were expected to resist breaking

The Hohenberg-Mermin-Wagner theorem rules out spontaneous breaking of a continuous symmetry at nonzero temperature in one- and two-dimensional classical equilibrium systems with sufficiently short-range interactions. Through the quantum-classical correspondence, which maps a d-dimensional quantum system onto a classical system in d+1 dimensions, this result motivates the expectation that continuous symmetry stays unbroken in one-dimensional quantum ground states. The Tomonaga-Luttinger liquid is the standard illustration: logarithmic phase fluctuations replace true long-range order with algebraically decaying quasi-long-range order.

The paper stresses that this correspondence does not automatically carry over the assumptions of the classical theorem. In one-dimensional quantum lattice systems, rigorous restrictions on continuous symmetry breaking are derived only under additional assumptions. The isotropic ferromagnetic Heisenberg chain is the familiar counterexample, where fully polarized ground states break SU(2) spin-rotation symmetry even in one dimension, but there the order parameter commutes with the Hamiltonian and the ground-state manifold contains polarized product states. More recent frustration-free constructions break continuous symmetry without a conserved order parameter, so the general mechanism remained open.

The Berry phase ingredient that classical theories miss

The paper's field-theoretic starting point is the spin coherent-state path integral. Overlaps of coherent states at neighboring imaginary times generate a Berry-phase term fixed geometrically by the solid angle traced by the spin orientation. This term is purely imaginary in the Euclidean action and has no direct counterpart in the Boltzmann weight of classical statistical mechanics.

In ferromagnets, the Berry phase makes the azimuthal phase and the longitudinal magnetization canonically conjugate, changing long-distance fluctuations away from those of a relativistic theory. Building on Nahum's proposal, the authors consider reducing SU(2) spin-rotation symmetry to U(1) while keeping magnetization-reversal and spatial symmetries. That reduction permits a local quartic interaction of the longitudinal magnetization, which is relevant in one spatial dimension and destabilizes the Gaussian fixed point. Tuning the quadratic density term at fixed positive quartic interaction is then supposed to give access to an interacting fixed point where phase fluctuations and their conjugate magnetization remain coupled even at long distances.

Subsystem Binder scaling locates the transition

The spin-1 model conserves the total longitudinal magnetization, so eigenstates can be chosen with definite M_z, and a nondegenerate finite-chain ground state must sit in the M_z = 0 sector. In that sector both the second and fourth moments of the magnetization density vanish, making the conventional Binder ratio undefined. The authors instead use the magnetization of half the chain, which can fluctuate even when the total magnetization is fixed, and define R_half as the ratio of its fourth moment to the square of its second moment.

DMRG ground states in the M_z = 0 sector give R_half as a function of the anisotropy parameter g_1 for several system sizes. The curves cross in a narrow region near g_1 approximately -0.826, and the crossing points g_x(L, L+2) between sizes L and L+2 converge as L grows. Fitting R_half = F_p(x) + L^{-omega}(b_0 + b_1 x) with x = (g_1 - g_1c) L^{1/nu}, polynomial degree p = 3, 4, 5, and a fitting window |g_1 + 0.8258| <= 0.01 yields g_1c = -0.8258 +/- 0.0002 and nu = 0.60 +/- 0.02. The quoted uncertainties are conservative systematic ranges reflecting sensitivity to fitting choices, not statistical errors of individual fits.

nu = 0.60 signals interacting rather than Gaussian criticality

The estimated nu = 0.60 +/- 0.02 is distinct from the Gaussian value nu_G = 1/2. Equivalently, the RG eigenvalue 1/nu of the relevant perturbation differs from its Gaussian value 2, which the authors present as numerical evidence for interacting critical behavior. The value also lies between the perturbative RG results nu approximately 0.5833 at order epsilon and nu approximately 0.6258 at order epsilon squared.

The spin-1/2 chain, where tuning g_1 changes the coefficients of both the quadratic and gradient terms in the coherent-state Lagrangian, shows the same qualitative behavior. Its half-chain Binder ratio crosses in a narrow coupling interval, and the same scaling form with L_max = 68 gives g_1c = 0.3826 +/- 0.0003 and nu = 0.61 +/- 0.02. That nu is consistent with the spin-1 result and again larger than the Gaussian value, so the interacting criticality is not tied to a single spin length.

Positive transverse magnetization in the thermodynamic limit

Because the ground state preserves U(1) symmetry, the expectation values of M_x and M_y vanish individually. The authors therefore track the squared transverse magnetization density m_perp^2(L) = <M_x^2 + M_y^2>_0 / L^2, evaluated in the lowest-energy state of the M_z = 0 sector. A positive limit m_perp^2 = lim_{L to infinity} m_perp^2(L) > 0 is sufficient for spontaneous U(1) symmetry breaking, since a nonzero transverse magnetization persists when an infinitesimal transverse field is removed after the thermodynamic limit.

At the critical coupling g_1 = -0.8258, DMRG on periodic spin-1 chains with a quadratic fit m_perp^2(L) = m_perp^2 + a_1/L + a_2/L^2 over 20 <= L <= 72 gives m_perp^2 approximately 0.9473. Varying L_min from 20 to 40 at fixed L_max = 72 keeps estimates between 0.9469 and 0.9473, and comparing linear, quadratic, and cubic fits at L_min = 20 gives values between 0.94687 and 0.94889. The adopted estimate is m_perp^2 = 0.947 +/- 0.003, clearly positive. For comparison, the ferromagnetic Heisenberg chain has the exact finite-size result m_perp^2(L) = 1 + 1/L in the ground state with S_tot = L and M_z = 0.

Finite-entanglement scaling confirms order on the infinite chain

To study the infinite chain directly, the authors use VUMPS with a U(1)-symmetric matrix product state ansatz. Here the cutoff is not system size but bond dimension chi, which limits the entanglement the MPS can represent, so observables must be extrapolated through finite-entanglement scaling. Because S_i^+ carries U(1) charge q = 1, the transverse correlations probe the q = 1 transfer-matrix sector, and the longest correlation length in that sector, xi_1(chi), serves as the scaling variable.

The authors first remove finite-entanglement effects at fixed block length and then extrapolate the block length to infinity, taking the chi to infinity limit before probing long-range order. The resulting estimate is m_perp^2 approximately 0.949, consistent with the finite-size extrapolation. For the spin-1/2 chain, the finite-size scaling of m_perp^2(L) at g_1 = 0.3826 gives a thermodynamic-limit estimate of 0.212, with a conservative interval 0.209 <= m_perp^2 <= 0.215, compared with the exact Heisenberg result m_perp^2(L) = 1/4 + 1/(2L).

z below 2 rules out frustration-free realizations

The authors analyze the finite-size scaling of the lowest neutral excitation gap and obtain the dynamical critical exponent z = 1.986 +/- 0.005, which characterizes the low-energy dispersion omega proportional to |k|^z. This is a small but significant deviation from the Gaussian value z_G = 2. The paper also checks the longitudinal spin correlations with VUMPS: comparing trial scaling dimensions, the value Delta_psi = 0.507 inferred from z reduces the relative root-mean-square spread of the rescaled correlation curves by about 29.5 percent relative to the Gaussian Delta_psi = 0.5. The authors caution that this improved collapse is not a precise independent determination of Delta_psi, since minimizing the spread gives an effective dimension drifting from about 0.520 to 0.531 as chi_min increases from 512 to 1536 at fixed chi_max = 3072.

The key implication the authors draw is that z < 2 is incompatible with microscopic realizations of this critical theory by frustration-free models, since frustration-free Hamiltonians require z >= 2. Continuous symmetry breaking in one dimension is therefore not exhausted by frustration-free constructions, implying a mechanism distinct from frustration freeness. The paper also notes that the same classical field theory has a microscopic realization in the nonequilibrium Ising lattice gas, whose Monte Carlo simulations give nu = 0.62 +/- 0.03 and nu = 0.61 +/- 0.03, consistent with the DMRG value, while the corresponding effective dynamical exponents 1.94 +/- 0.02 and 1.965 +/- 0.005 deviate from the reported z, a discrepancy the authors flag as meriting further investigation.

What to compare before trusting the breaking

The most decision-relevant comparison in this paper is not whether some one-dimensional chain breaks a continuous symmetry, but which diagnostic establishes it and which exponent distinguishes the mechanism. The breaking claim rests on the extrapolated squared transverse magnetization density, m_perp^2 = 0.947 +/- 0.003 for the spin-1 chain and 0.212 with interval 0.209 to 0.215 for the spin-1/2 chain, obtained in symmetry-preserving states rather than by adding a symmetry-breaking field. The interacting character rests on nu = 0.60 +/- 0.02 and nu = 0.61 +/- 0.02, both above the Gaussian 1/2 and bracketed by the perturbative RG values. The mechanism claim rests on z = 1.986 +/- 0.005 falling below 2, the threshold that frustration-free Hamiltonians require.

Readers weighing this result should keep three boundaries in view. The exponents are quoted as conservative systematic ranges tied to fitting-window and polynomial-degree choices, not statistical confidence intervals. The critical coupling for the spin-1 chain, g_1c = -0.8258 +/- 0.0002, agrees with the earlier g_c = -0.826 +/- 0.005, but the reported z differs substantially from the earlier z = 1.50 +/- 0.04, and combining that earlier beta = 0.30 +/- 0.01 with its z through the scaling relation gives nu = 0.40 +/- 0.02, incompatible with the value here. The Delta_psi collapse improvement is suggestive rather than a precise determination. The paper's own priority is therefore the z < 2 threshold: it is the single number that separates this critical theory from frustration-free realizations and makes the case that one-dimensional continuous symmetry breaking has more than one route.

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