Continuous Symmetry Breaking Research Articles

The interplay between liquid–liquid and ferroelectric phase transitions in supercooled water

The distinctive characteristics of water, evident in its thermodynamic anomalies, have implications across disciplines from biology to geophysics. Considered a valid hypothesis to rationalize its unique properties, a liquid-liquid phase transition in water below the freezing point, in the so-called supercooled regime, has nowadays been observed in several molecular dynamics simulations and is being actively researched experimentally. The hypothesis of ferroelectric phase transition in supercooled water can be traced back to 1977, due to Stillinger. In this work, we highlight intriguing and far-reaching implications of water: The ferroelectric and liquid-liquid phase transitions can be designed as two facets of the same underlying phenomenon. Our results are based on the analysis of extensive molecular dynamics simulations and are explained in the context of a classical density functional theory in mean-field approximation valid for a polar liquid, where dipolar interaction is treated perturbatively. The theory underpins the potential role of ferroelectricity in promoting the liquid-liquid phase transition, being the density-polarization coupling inherent in the dipolar interaction potential. The existence of ferroelectric order in supercooled low-density liquid water is confirmed by the observation in molecular dynamics simulations of collective modes in space-time polarization correlation functions, traceable to spontaneous breaking of continuous rotational symmetry. Our work sheds light on water's supercooled behavior and opens the door to experimental investigations of the static and dynamic behavior of water's polarization.

Critical Spontaneous Breaking of U(1) Symmetry at Zero Temperature in One Dimension.

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.

Nano and micro-structural complexity of nematic liquid crystal configurations

Of our interest are frustration-driven pattern generating mechanisms in systems which in bulk equilibrium display spatially homogeneous long-range orientational order in absence of perturbations. As testbed material, we select thermotropic nematic liquid crystals. In bulk, they exhibit weakly discontinuous order–disorder phase transformation on varying temperature where the ordered nematic phase features spatially uniform axial order along an arbitrary symmetry breaking direction. However, due to continuous symmetry breaking (CSB) the established order is extremely susceptible to various perturbations which are in real systems in general always present. We theoretically illustrate how diverse complex patterns could be excited. Particularly intriguing configurations could appear if topological defects are present that could be generated via CSB. Our analysis is based on a relatively simple Lebwohl-Lasher-type model in which we could get analytical insight into phenomena of our interest. Using it we illustrate history dependent early stage isotropic-nematic phase evolution and final patterns in presence of “impurities” (e.g., nanoparticles). We show how characteristic effective interaction characteristics predict qualitatively different emerging patterns. Our analysis is based on CSB which is ubiquitous in nature. Consequently, demonstrated mechanisms are expected to manifest also in other condensed matter systems whose ordered phase is formed via CSB. We illustrate how kinetics and impurities could impact key structural properties of the systems of our interest.

Continuous symmetry breaking of low-dimensional systems driven by inhomogeneous oscillatory driving forces.

The driving forces of chiral active particles and deformations of cells are often modeled by spatially inhomogeneous but temporally periodic driving forces. Such inhomogeneous oscillatory driving forces have only recently been proposed in the context of active matter, and their effects on the systems are not yet fully understood. In this work, we theoretically study the impact of spatially inhomogeneous oscillatory driving forces on continuous symmetry breaking. We first analyze the linear model for the soft modes in the ordered phase to derive the lower critical dimension of the model, and then analyze the spherical model to investigate more detailed phase behaviors. Interestingly, our analysis reveals that symmetry breaking occurs even in one and two dimensions, where the Hohenberg-Mermin-Wagner theorem prohibits continuous symmetry breaking in equilibrium. Furthermore, fluctuations of conserved quantities, such as density, are anomalously suppressed in the long-wavelength, i.e., show hyperuniformity.

Observation of a phase transition from a continuous to a discrete time crystal

Discrete (DTCs) and continuous time crystals (CTCs) are novel dynamical many-body states, that are characterized by robust self-sustained oscillations, emerging via spontaneous breaking of discrete or continuous time translation symmetry. DTCs are periodically driven systems that oscillate with a subharmonic of the external drive, while CTCs are continuously driven and oscillate with a frequency intrinsic to the system. Here, we explore a phase transition from a continuous time crystal to a discrete time crystal. A CTC with a characteristic oscillation frequency ωCTC is prepared in a continuously pumped atom-cavity system. Modulating the pump intensity of the CTC with a frequency ωdr close to 2ωCTC leads to robust locking of ωCTC to ωdr/2 , and hence a DTC arises. This phase transition in a quantum many-body system is related to subharmonic injection locking of non-linear mechanical and electronic oscillators or lasers.

The simplified energy landscape of the φ 4 model and the phase transition

The on-lattice φ 4 model is a paradigmatic example of a continuous real-variable model undergoing a continuous symmetry-breaking phase transition (SBPT). Here, we study the Z2 -symmetric mean-field case without the quadratic term in the local potential. We show that the Z2 -SBPT is not affected by the quadratic term and that the potential energy landscape is greatly simplified from a geometric–topological viewpoint. In particular, only three critical points exist to confront, with a number growing as eN (N is the number of degrees of freedom) of the model with a negative quadratic term. We focus on the properties of the equipotential surfaces with the aim to deepen the link between SBPTs and the essential properties of a potential that is capable of entailing them. The results are interpreted in view of of some recent achievements regarding rigorous necessary and sufficient conditions for a Z2 -SBPT.

Continuous symmetry breaking in adaptive quantum dynamics

Continuous symmetry breaking in adaptive quantum dynamics

Constants of motion characterizing continuous symmetry-broken phases.

We present a theory characterizing the phases emerging as a consequence of continuous symmetry-breaking in quantum and classical systems. In symmetry-breaking phases, dynamics is restricted due to the existence of a set of conserved charges derived from the order parameter of the phase transition. Their expectation values are determined by the privileged direction appearing in the ordered phase as a consequence of symmetry breaking, and thus they can be used to determine whether this direction is well defined or has quantum fluctuations. Our theory is numerically exemplified via the two-dimensional limit of the vibron model, a fully connected system invariant under a rotation operator which generates the continuous symmetry-breaking.

When the moduli space is an orbifold: spontaneous breaking of continuous non-invertible symmetries

We investigate theories of Nambu-Goldstone bosons where the spontaneously broken continuous symmetry is non-invertible. In such theories, the vacua generically parameterize an orbifold. We study in detail the simplest example of a single free scalar with shift symmetry, modded by reflection symmetry. At singular points of the vacuum manifold, we show that the spectrum of NG excitations is reduced, in particular there are no single-particle states. At the smooth points, on the other hand, single NG modes are present. We show that this is a consequence of the fact that at those points one can construct invertible operators implementing the continuous symmetry on the Hilbert space.

A symmetry principle for gauge theories with fractons

Fractonic phases are new phases of matter that host excitations with restricted mobility. We show that a certain class of gapless fractonic phases are realized as a result of spontaneous breaking of continuous higher-form symmetries whose conserved charges do not commute with spatial translations. We refer to such symmetries as nonuniform higher-form symmetries. These symmetries fall within the standard definition of higher-form symmetries in quantum field theory, and the corresponding symmetry generators are topological. Worldlines of particles are regarded as the charged objects of 1-form symmetries, and mobility restrictions can be implemented by introducing additional 1-form symmetries whose generators do not commute with spatial translations. These features are realized by effective field theories associated with spontaneously broken nonuniform 1-form symmetries. At low energies, the theories reduce to known higher-rank gauge theories such as scalar/vector charge gauge theories, and the gapless excitations in these theories are interpreted as Nambu-Goldstone modes for higher-form symmetries. Due to the nonuniformity of the symmetry, some of the modes acquire a gap, which is the higher-form analogue of the inverse Higgs mechanism of spacetime symmetries. The gauge theories have emergent nonuniform magnetic symmetries, and some of the magnetic monopoles become fractonic. We identify the ’t Hooft anomalies of the nonuniform higher-form symmetries and the corresponding bulk symmetry-protected topological phases. By this method, the mobility restrictions are fully determined by the choice of the commutation relations of charges with translations. This approach allows us to view existing (gapless) fracton models such as the scalar/vector charge gauge theories and their variants from a unified perspective and enables us to engineer theories with desired mobility restrictions.

Non-equilibrium entanglement asymmetry for discrete groups: the example of the XY spin chain

Entanglement asymmetry is a novel quantity that, using entanglement methods, measures how much a symmetry is broken in a part of an extended quantum system. So far, it has only been used to characterise the breaking of continuous Abelian symmetries. In this paper, we extend the concept to cyclic ZN groups. As an application, we consider the XY spin chain, in which the ground state spontaneously breaks the Z2 spin parity symmetry in the ferromagnetic phase. We thoroughly investigate the non-equilibrium dynamics of this symmetry after a global quantum quench, generalising known results for the standard order parameter.

Electrons interacting with Goldstone modes and the rotating frame

We consider electronic systems with a spontaneously broken continuous symmetry. The scattering vertex between electrons and Goldstone modes is calculated over the entire Brillouin zone using the random phase approximation. This calculation reveals two things: (1) electrons always couple to both ϕ and ∂tϕ, where ϕ is the Goldstone field, and (2) quasiparticles in a state with continuous symmetry breaking have to be defined in a rotating frame, which locally follows the fluctuations of the order parameter. The implications of these findings for electron spectral functions in both symmetry-broken and thermally disordered systems are discussed, and the examples of antiferromagnetism in the Hubbard model and spin spiral order in the three-band model are worked out in detail. Published by the American Physical Society 2024

Animating hydrogel knotbots with topology-invoked self-regulation

Steering soft robots in a self-regulated manner remains a grand challenge, which often requires continuous symmetry breaking and recovery steps for persistent motion. Although structural morphology is found significant for robotic functions, geometric topology has rarely been considered and appreciated. Here we demonstrate a series of knotbots, namely hydrogel-based robots with knotted structures, capable of autonomous rolling and spinning/rotating motions. With symmetry broken by external stimuli and restored by self-regulation, the coupling between self-constraint-induced prestress and photothermal strain animates the knotbots continuously. Experiments and simulations reveal that nonequilibrium processes are regulated dynamically and cooperatively by self-constraints, active deformations, and self-shadowing effect of the photo-responsive gel. The active motions enable the knotbots to execute tasks including gear rotation and rod climbing. This work paves the way to devise advanced soft robots with self-regulated sustainable motions by harnessing the topology.

Correlated noise and critical dimensions.

In equilibrium, the Mermin-Wagner theorem prohibits the continuous symmetry breaking for all dimensions d≤2. In this work, we discuss that this limitation can be circumvented in nonequilibrium systems driven by the spatiotemporally long-range anticorrelated noise. We first compute the lower and upper critical dimensions of the O(n) model driven by the spatiotemporally correlated noise by means of the dimensional analysis. Next we consider the spherical model, which corresponds to the large-n limit of the O(n) model and allows us to compute the critical dimensions and critical exponents, analytically. Both results suggest that the critical dimensions increase when the noise is positively correlated in space and time and decrease when anticorrelated. We also report that the spherical model with the correlated noise shows the hyperuniformity and giant number fluctuation even well above the critical point.

Continuous symmetry breaking in a trapped-ion spin chain.

One-dimensional systems exhibiting a continuous symmetry can host quantum phases of matter with true long-range order only in the presence of sufficiently long-range interactions1. In most physical systems, however, the interactions are short-ranged, hindering the emergence of such phases in one dimension. Here we use a one-dimensional trapped-ion quantum simulator to prepare states with long-range spin order that extends over the system size of up to 23 spins and is characteristic of the continuous symmetry-breaking phase of matter2,3. Our preparation relies on simultaneous control over an array of tightly focused individual addressing laser beams, generating long-range spin-spin interactions. We also observe a disordered phase with frustrated correlations. We further study the phases at different ranges of interaction and the out-of-equilibrium response to symmetry-breaking perturbations. This work opens an avenue to study new quantum phases and out-of-equilibrium dynamics in low-dimensional systems.

Finite-temperature critical behaviors in 2D long-range quantum Heisenberg model

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/rα), the theorem further forbids ferromagnetic or antiferromagnetic order at finite temperature when α ≥ 2D. 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 SU(2) symmetry in the ferromagnetic Heisenberg model with 1/rα-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.

Nonequilibrium Phase Transition to Temporal Oscillations in Mean-Field Spin Models.

We propose a mean-field theory to describe the nonequilibrium phase transition to a spontaneously oscillating state in spin models. A nonequilibrium generalization of the Landau free energy is obtained from the joint distribution of the magnetization and its smoothed stochastic time derivative. The order parameter of the transition is a Hamiltonian, whose nonzero value signals the onset of oscillations. The Hamiltonian and the nonequilibrium Landau free energy are determined explicitly from the stochastic spin dynamics. The oscillating phase is also characterized by a nontrivial overlap distribution reminiscent of a continuous replica symmetry breaking, in spite of the absence of disorder. An illustration is given on an explicit kinetic mean-field spin model.

Continuous symmetry breaking in a two-dimensional Rydberg array.

Spontaneous symmetry breaking underlies much of our classification of phases of matter and their associated transitions1-3. The nature of the underlying symmetry being broken determines many of the qualitative properties of the phase; this is illustrated by the case of discrete versus continuous symmetry breaking. Indeed, in contrast to the discrete case, the breaking of a continuous symmetry leads to the emergence of gapless Goldstone modes controlling, for instance, the thermodynamic stability of the ordered phase4,5. Here, we realize a two-dimensional dipolar XY model that shows a continuous spin-rotational symmetry using a programmable Rydberg quantum simulator. We demonstrate the adiabatic preparation of correlated low-temperature states of both the XY ferromagnet and the XY antiferromagnet. In the ferromagnetic case, we characterize the presence of a long-range XY order, a feature prohibited in the absence of long-range dipolar interaction. Our exploration of the many-body physics of XY interactions complements recent works using the Rydberg-blockade mechanism to realize Ising-type interactions showing discrete spin rotation symmetry6-9.

Quantum local-equilibrium approach to dissipative hydrodynamics.

The macroscopic hydrodynamic equationsare derived for many-body systems in the local-equilibrium approach, using the Schrödinger picture of quantum mechanics. In this approach, statistical operators are defined in terms of microscopic densities associated with the fundamentally conserved quantities and other slow modes possibly emerging from continuous symmetry breaking, as well as macrofields conjugated to these densities. Functional identities can be deduced, allowing us to identify the reversible and dissipative parts of the mean current densities, to obtain general equationsfor the time evolution of the conjugate macrofields, and to establish the relationship to projection-operator methods. The entropy production is shown to be nonnegative by applying the Peierls-Bogoliubov inequality to a quantum integral fluctuation theorem. Using the expansion in the gradients of the conjugate macrofields, the transport coefficients are given by Green-Kubo formulas and the entropy production rate can be expressed in terms of quantum Einstein-Helfand formulas, implying its nonnegativity in agreement with the second law of thermodynamics. The results apply to multicomponent fluids and can be extended to condensed matter phases with broken continuous symmetries.

Kibble-Zurek Mechanism for Dynamical Ordering in a Driven Vortex System.

The Kibble-Zurek mechanism describes the formation of topological defects in systems crossing a continuous symmetry-breaking phase transition at a finite quench rate. While this mechanism has been extensively studied for equilibrium transitions, its applicability to nonequilibrium transitions has not yet been fully examined. Recent simulation has shown the applicability of the Kibble-Zurek mechanism to dynamical ordering transitions in particlelike assemblies, including superconducting vortices, driven over random disorder. Here, we experimentally study the configurational order of vortices in the course of dynamical ordering with various quench rates. We verify a power-law scaling of the defect density with the quench rate and an impulse-adiabatic crossover on the ordered side of the transition, which are key predictions of the Kibble-Zurek mechanism. Our results suggest the applicability of the Kibble-Zurek mechanism to other nonequilibrium phase transitions.