Evidence for spontaneous breaking of a continuous symmetry at a non-conformal quantum critical point in one dimension

TL;DR

This work demonstrates spontaneous breaking of a continuous U(1) symmetry at a non-conformal quantum critical point in a one-dimensional spin-1 chain, enabled by a Berry-phase term in the action. Using matrix product state methods, the authors observe XY quasi-long-range order with a finite perpendicular magnetization at criticality, a Bragg peak in the dynamical structure factor, and gapless modes, yielding a dynamical exponent z ≈ 3/2 and anomalous dimension η ≈ 1, consistent with a KPZ-like scaling in 1D. A perturbative RG analysis about d_c=2 reveals an interacting fixed point with exponents differing from Ising at two-loop order, but the small corrections hint at a non-perturbative fixed point, motivating a non-perturbative starting point for quantitative predictions. The results challenge conventional expectations for 1D continuous-symmetry breaking at equilibrium and invite experimental exploration in platforms such as Rydberg arrays and quantum gas microscopes to probe the non-conformal critical point and its unusual dynamical scaling.

Abstract

In this work, we present evidence for the spontaneous breaking of a continuous symmetry in a nearest-neighbour interacting spin-1 chain tuned to a quantum critical point at $T=0$ between two XY quasi-long-range order phases differing by the spontaneous breaking of a $\mathbb{Z}_2$ symmetry. Despite the one-dimensional nature of the system, which typically prevents such a continuous symmetry breaking due to the Hohenberg-Mermin-Wagner theorem, the presence of a Berry phase term in the quantum model allows us to observe, using matrix product state methods, a finite perpendicular magnetization. Moreover, the quasi-long-range decay of the correlation function becomes truly long-range order, and the dynamical structure factor displays a characteristic Bragg peak together with sharp gapless modes. Our results imply the quantum phase transition has an anomalous dimension of $η\simeq 1$ together with the dynamical critical exponent $z\simeq 3/2$, known from the Kardar-Parisi-Zhang universality class in one dimension. We perform a perturbative renormalization group calculation about the upper critical dimension $d_c=2$ that we could close at second loop order. We find an interacting fixed point with critical exponents distinct from the Ising ones. Together, our findings suggest the nature of the fixed point to be non-perturbative. We propose a field-theory that we believe to improve the quantitative results.

Figures (7)

• Figure 1: a) Order parameter $m_z^2$ vs coupling $g_1$ calculated in the ground state of Eq. \ref{['Ham']} using iDMRG. We vary the bond dimension $\chi$ and perform a power-law fit at $\chi=100$, finding a critical exponent of $\beta = 0.30\pm 0.01$ and the fixed point value $g_c=-0.826\pm0.005$. b) The perpendicular magnetization $m_\perp$ shows a decreasing behaviour with increasing bond dimension, consistent with a QLRO of the XY type. Nevertheless, at the critical point of a) the magnetization stays approximately constant. It becomes a local maximum as a function of $g_1$. c) Log-log plot of the equal-time connected correlation function for the $S^z_i$ component of the spin. We plot the correlations for different values of $\chi$, finding a linear regime, where we observe a collapse for different bond dimensions. Performing a fit at $\chi=100$, we find the relation of Eq. \ref{['corr_z']}. d) Plot of the XY correlation function as a function of position for $g_1=g_c$. We observe that close to the critical point and at large distances, the correlator behaves like a constant, i.e. the XY component is LRO.
• Figure 2: Dynamical spin structure factor as a function of frequency and momentum for the a) perpendicular Eq. \ref{['DSF_perp']} and b) parallel spin components Eq. \ref{['DSF_z']}. For the simulations we use an MPS of bond dimension $\chi=100$ and extent $L=200$, the time evolution is done for $\Delta t =0.1$ and 600 time steps. A Gaussian window function with $\sigma=0.2$ is applied to the time data. We extract the maxima of $\mathcal{S}_{zz}$ and perform a log-log fit of the data as a function of frequency and momentum for long-wavelengths, shown in the inset of b). We find the relation $\omega\propto k^z$ with $z=1.51\pm0.03$.
• Figure S3: Entanglement entropy as a function of time and space for the time evolution of the wave function used to calculate the main text dynamical structure factors a ) $\mathcal{S}_{\perp}(k,\omega)$ and b) $\mathcal{S}_{zz}(k,\omega)$. We use the critical coupling $g_1=g_c,\ g_2=J=1$ and $\Delta t =0.1,\chi=100,L=200$ and $T_{\text{max}}=60$.
• Figure S4: iDMRG results for different bond dimension $\chi$ of the model in the main text with $g_2=1,J=1$. We plot in a) the the XY correlation function as a function of the number of sites for $g_1$ close to the critical point. b) Entanglement entropy as a function of coupling $g_1$. c) Correlation length as a function of coupling $g_1$.
• Figure S5: Dynamical structure factor cuts for the main text plot showcasing a) Bragg peak or condensed fraction shown in $\mathcal{S}_\perp(k=0,\omega)$. b) Longitudinal dynamical structure factor as a function of momentum $\mathcal{S}_\perp(k,\omega=0)$, fit to a power-law $\eta=1.01\pm 0.05$ consistent with the static results of the main text. c) Power-law fit with slope $z=1.51\pm0.03$. Here we use the centers of a family of Gaussians fitted to each cut of the longitudinal spin structure factor at fixed momentum ranging from zero to one.