Discrete Symmetry Breaking Research Articles

Long-lived topological time-crystalline order on a quantum processor

Topologically ordered phases of matter elude Landau’s symmetry-breaking theory, featuring a variety of intriguing properties such as long-range entanglement and intrinsic robustness against local perturbations. Their extension to periodically driven systems gives rise to exotic new phenomena that are forbidden in thermal equilibrium. Here, we report the observation of signatures of such a phenomenon—a prethermal topologically ordered time crystal—with programmable superconducting qubits arranged on a square lattice. By periodically driving the superconducting qubits with a surface code Hamiltonian, we observe discrete time-translation symmetry breaking dynamics that is only manifested in the subharmonic temporal response of nonlocal logical operators. We further connect the observed dynamics to the underlying topological order by measuring a nonzero topological entanglement entropy and studying its subsequent dynamics. Our results demonstrate the potential to explore exotic topologically ordered nonequilibrium phases of matter with noisy intermediate-scale quantum processors.

Kitaev honeycomb antiferromagnet in a field: quantum phase diagram for general spin

We use tensor-network methods and high-order linked-cluster expansions to explore the quantum phase diagram of the antiferromagnetic Kitaev honeycomb model in a magnetic field for general spin S values. Tensor network calculations for the pure Kitaev model confirm the absence of fluxes and spin-spin correlations beyond nearest neighbors, while revealing discrete orientational symmetry breaking for S ∈ 1, 3/2, 2, consistent with the semiclassical limit. An intermediate region between Kitaev phases and the high-field polarized phase is identified for all considered spin values, showing a sequence of potential phases characterized by distinct local magnetization patterns while the total magnetization increases smoothly as a function of the field. Linked-cluster expansions for the high-field zero-momentum gap and spectral weight indicate a quantum critical breakdown of the polarized phase, suggesting exotic physics at intermediate Kitaev couplings.

Spin-polarized second-order nonlinear Hall effect in 8-Pmmn monolayer borophene

The second-order nonlinear Hall effect in 8-Pmmn monolayer borophene under the influence of an out-of-plane electric field and intrinsic spin–orbit interaction is reported. This unconventional response sensitive to the breaking of discrete and crystal symmetries can be tuned by the applied electric field, which can vary the bandgap induced by spin–orbit coupling. It is described by a Hall conductivity tensor that depends quadratically on the applied electric field. We find that the nonlinear Hall effect strongly depends on the spin polarization. In particular, it exhibits out of the phase character for spin-up and spin-down states. Remarkably, it undergoes a phase flip in the spin-up state at a large out-of-plane electric field that generates a staggered sublattice potential greater than the spin–orbit interaction strength. It is shown that the nonlinear Hall effect in the system originates from the broken inversion symmetry that plays an indispensable role in developing finite Berry curvature and its relevant dipole moment. It is found that at zero temperature, the nonlinear Hall response is maximal when the Fermi energy is twice the bandgap parameter and vanishes at large Fermi energies. Notably, the peak of nonlinear Hall response shifts to lower Fermi energies at finite temperature.

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.

Leptogenesis, dark matter and gravitational waves from discrete symmetry breaking

We analyse a model that connects the neutrino sector and the dark sector of the universe via a mediator Φ, stabilised by a discrete 𝒵 4 symmetry that breaks to a remnant 𝒵 2 upon Φ acquiring a non-zero vacuum expectation value (vϕ ). The model accounts for the observed baryon asymmetry of the universe via additional contributions to the canonical Type-I leptogenesis. The 𝒵 4 symmetry breaking scale (vϕ ) in the model not only establishes a connection between the neutrino sector and the dark sector, but could also lead to gravitational wave signals that are within the reach of current and future experimental sensitivities.

Entanglement-enabled symmetry-breaking orders

A spontaneous symmetry-breaking order is conventionally described by a tensor-product wavefunction of some few-body clusters; some standard examples include the simplest ferromagnets and valence bond solids. We discuss a type of symmetry-breaking orders, dubbed entanglement-enabled symmetry-breaking orders, which cannot be realized by any such tensor-product state. Given a symmetry breaking pattern, we propose a criterion to diagnose if the symmetry-breaking order is entanglement-enabled, by examining the compatibility between the symmetries and the tensor-product description. For concreteness, we present an infinite family of exactly solvable gapped models on one-dimensional lattices with nearest-neighbor interactions, whose ground states exhibit entanglement-enabled symmetry-breaking orders from a discrete symmetry breaking. In addition, these ground states have gapless edge modes protected by the unbroken symmetries. We also propose a construction to realize entanglement-enabled symmetry-breaking orders with spontaneously broken continuous symmetries. Under the unbroken symmetries, some of our examples can be viewed as symmetry-protected topological states that are beyond the conventional classifications.

Heavy axions from twin dark sectors with [formula omitted]-characterized mirror symmetry

The QCD Lagrangian contains a CP violating gluon density term with a physical coefficient θ¯. The upper bound on the electric dipole moment of neutron requires the value of θ¯ to be extremely small. The tiny θ¯ is commonly known as the strong CP problem. In order to solve this puzzle, we construct a θ¯-characterized mirror symmetry between a pair of twin dark sectors with respective discrete symmetries. By taking a proper phase rotation of dark fields, we can perfectly remove the parameter θ¯ from the full Lagrangian. In our scenario, the discrete symmetry breaking, which are responsible for the mass generation of dark colored fermions and dark matter fermions, can be allowed near the TeV scale. This means different phenomena from the popular axion models with high scale Peccei-Quinn global symmetry breaking.

Gravitational wave signature of generic disappearance of Z2-symmetry breaking domain walls

Breaking of discrete parity at high scale gives rise to Z2-domain walls (DW). The metastability of such walls can make them relatively long lived and contradict standard cosmology. We consider two classes of theories with similar underlying feature, the left right symmetric theories and two Higgs doublet models. Both of them possess some breaking of Z2 discrete symmetries. As a first step, domains form at a high energy scale during parity breaking. In the second step, these domains further decompose into subdomains due to Z2 symmetry breaking in two Higgs doublet models closer to the electroweak scale. We show that after this two step formation of domains and subdomains, a QCD instanton induced energy difference can remove the domain walls as well as the subdomain walls at around the same time successfully. The removal occurs purely as the result of a chance event taking place with probability very close to 0.25, and does not require one to introduce any non-renormalisable Z2-symmetry breaking term to the Lagrangian. We then investigate the gravitational waves arising from the collapse of such domain walls and show that the peak frequency of these waves lies in the 10−7–10−6 Hz band, corresponding to annihilation temperatures of 1–10 GeV. This frequency band is sensitive to pulsar timing array based experiments such as SKA and NANOGrav. The recent NANOGrav results rule out our DW collapse model for higher values of parity breaking scale above 107 GeV. Our DW collapse model with parity breaking scales below 107 GeV remains consistent with the current NANOGrav results and has a good chance of being seriously tested in future pulsar timing based experiments.

Ground state of Rydberg-dressed Bose gas confined in a toroidal trap

The experimental realization of Rydberg dressing technology in ultracold atomic systems provides another superior platform for studying novel states of matter and macroscopic quantum phenomena. In this work, based on the mean-field theory, we have investigated the ground-state phases of a two-component Bose–Einstein condensate with Rydberg interaction and confined in a toroidal trap. The effects of the Rydberg interaction and external potential, especially the Rydberg blockade radius, on the ground-state structure of such a system have been investigated in full parameter space. Our results show that the Rydberg blockade radius, which can be regarded as another controllable parameter, can be used to obtain a variety of ground-state phases. More interestingly, it is found that for weak Rydberg interactions, the Rydberg blockade radius breaks the spontaneous rotational symmetry of the system, leading to the formation of a discrete unit cell structure. For strongly interacting cases, it can be used to realize different orders of discrete rotational symmetry breaking.

QCD-collapsed domain walls: QCD phase transition and gravitational wave spectroscopy

For a discrete symmetry that is anomalous under QCD, the domain walls produced in the early universe from its spontaneous breaking can naturally annihilate due to QCD instanton effects. The gravitational waves generated from wall annihilation have their amplitude and frequency determined by both the discrete symmetry breaking scale and the QCD scale. The evidence of stochastic gravitational waves at nanohertz observed by pulsar timing array experiments suggests that the discrete-symmetry-breaking scale is around 100 TeV, assuming the domain-wall explanation. The annihilation temperature is about 100 MeV, which could naturally be below the QCD phase transition temperature. We point out that the QCD phase transition within some domains with an effective large QCD θ angle could be a first-order one. To derive the phase diagram in θ and temperature, we adopt a phenomenological linear sigma model with three quark flavors. The domain-wall explanation for the NANOGrav, EPTA, PPTA and CPTA results hints at a first-order QCD phase transition, which predicts additional gravitational waves at higher frequencies. If the initial formation of domain walls is also a first-order process, this class of domain-wall models predicts an interesting gravitational wave spectroscopy with frequencies spanning more than ten orders of magnitude, from nanohertz to 100 Hz.

Discrete time translation symmetry breaking in a Josephson junction laser

A Josephson junction laser is realized when a microwave cavity is driven by a voltage-biased Josephson junction. Through the ac Josephson effect, a dc voltage generates a periodic drive that acts on the cavity and generates interactions between its modes. A sufficiently strong drive enables processes that downconvert a drive resonant with a high harmonic into photons at the cavity fundamental frequency, breaking the discrete time translation symmetry set by the Josephson frequency. Using a classical model, we determine when and how this transition occurs as a function of the bias voltage and the number of cavity modes. We find that certain combinations of mode number and voltage tend to facilitate the transition which emerges via an instability within a subset of the modes. Despite the complexity of the system, there are cases in which the critical drive strength can be obtained analytically.

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.

Classification of Abelian domain walls

We discuss domain walls from spontaneous breaking of Abelian discrete symmetries $Z_N$. A series of different domain wall structures are predicted, depending on the symmetry and charge assignments of scalars leading to the spontaneous symmetry breaking (SSB). A widely-existing type of domain walls are those separating degenerate vacua which are adjacent in the field space. We denote these walls as adjacency walls. In the case that $Z_N$ terms are small compared with the $U(1)$ terms, the SSB of $U(1)$ generates strings first and then adjacency walls bounded by strings are generated after the SSB of $Z_N$. For symmetries larger than $Z_3$, non-adjacent vacua exist, we regard walls separating them as non-adjacency walls. These walls are unstable if $U(1)$ is a good approximation. If the discrete symmetry is broken via multiple steps, we arrive at a complex structure that one kind of walls wrapped by another type. On the other hand, if the symmetry is broken in different directions independently, walls generated from the different breaking chains are blind to each other.

Period Tripling due to Parametric Down-Conversion in Circuit QED.

Discrete time-translation symmetry breaking can be observed in periodically driven systems oscillating at a fraction of the frequency of the driving force. However, with the exception of the parametric instability in period doubling, multiperiodic driving does not lead to an instability threshold. In this Letter, we point out that quantum vacuum fluctuations can be generically employed to induce period multiplication. Inparticular, we discuss the period-tripled states in circuit QED and propose a microwave setup. We show that for weak dissipation or strong driving, the system exhibits a nonequilibrium phase transition in the sense that the timescale over which the period-tripled state is generated can be arbitrarily separated from the timescale of the subsequent dephasing.

Domain Formation Driven by the Entropy of Topological Edge Modes.

In this Letter we study interacting systems with spontaneous discrete symmetry breaking, where the degenerate symmetry-broken states are topologically distinct gapped phases. Edge modes appear at domain walls between the two topological phases. In the presence of a weak disorder field conjugate to the order parameter, we find that the entropy of the edge modes drives a thermal transition between a gapped uniform phase and a phase with a disorder-induced domain structure. We characterize this transition using a phenomenological Landau functional, and corroborate our conclusions with a concrete microscopic model. Finally, we discuss the possibilities of experimental signatures of this phase transition, and propose graphene-based moiré heterostructures as candidate materials in which such a phase transition can be detected.

Discrete symmetry breaking defines the Mott quartic fixed point

Because Fermi liquids are inherently non-interacting states of matter, all electronic levels below the chemical potential are doubly occupied. Consequently, the simplest way of breaking Fermi liquid theory is to engineer a model in which some of those states are singly occupied keeping time-reversal invariance intact. We show that breaking an overlooked local-in-momentum space $\mathbb Z_2$ symmetry of a Fermi liquid does precisely this. As a result, while the Mott transition from a Fermi liquid is correctly believed to obtain without the breaking of any continuous symmetry, a discrete symmetry is broken. This symmetry breaking serves as an organizing principle for Mott physics whether it arises from the tractable Hatsugai-Kohmoto (HK) model or the intractable Hubbard model. That both are controlled by the same fixed point we establish through a renormalization group analysis. An experimental manifestation of this fixed point is the onset of particle-hole asymmetry, a widely observed phenomenon in strongly correlated systems. Theoretically, the singly-occupied region of the spectrum gives rise to a surface of zeros of the single-particle Green function, denoted as the Luttinger surface. Using K-homology, we show that the Bott topological invariant guarantees the stability of this surface to local perturbations. Our proof demonstrates that the strongly coupled fixed point only corresponds to those Luttinger surfaces with co-dimension $p+1$ with $p$ odd. We conclude that the Hubbard and HK models both lie in the same high temperature universality class and are controlled by the broken $\mathbb Z_2$ symmetry quartic fixed point.

Quantum theory of the nonlinear Hall effect

The nonlinear Hall effect is an unconventional response, in which a voltage can be driven by two perpendicular currents in the Hall-bar measurement. Unprecedented in the family of the Hall effects, it can survive time-reversal symmetry but is sensitive to the breaking of discrete and crystal symmetries. It is a quantum transport phenomenon that has deep connection with the Berry curvature. However, a full quantum description is still absent. Here we construct a quantum theory of the nonlinear Hall effect by using the diagrammatic technique. Quite different from nonlinear optics, nearly all the diagrams account for the disorder effects, which play decisive role in the electronic transport. After including the disorder contributions in terms of the Feynman diagrams, the total nonlinear Hall conductivity is enhanced but its sign remains unchanged for the 2D tilted Dirac model, compared to the one with only the Berry curvature contribution. We discuss the symmetry of the nonlinear conductivity tensor and predict a pure disorder-induced nonlinear Hall effect for point groups C3, C3h, C3v, D3h, D3 in 2D, and T, Td, C3h, D3h in 3D. This work will be helpful for explorations of the topological physics beyond the linear regime.

Quasi-Locality Bounds for Quantum Lattice Systems. Part II. Perturbations of Frustration-Free Spin Models with Gapped Ground States

We study the stability with respect to a broad class of perturbations of gapped ground-state phases of quantum spin systems defined by frustration-free Hamiltonians. The core result of this work is a proof using the Bravyi–Hastings–Michalakis (BHM) strategy that under a condition of local topological quantum order (LTQO), the bulk gap is stable under perturbations that decay at long distances faster than a stretched exponential. Compared to previous work, we expand the class of frustration-free quantum spin models that can be handled to include models with more general boundary conditions, and models with discrete symmetry breaking. Detailed estimates allow us to formulate sufficient conditions for the validity of positive lower bounds for the gap that are uniform in the system size and that are explicit to some degree. We provide a survey of the BHM strategy following the approach of Michalakis and Zwolak, with alterations introduced to accommodate more general than just periodic boundary conditions and more general lattices. We express the fundamental condition known as LTQO by means of an indistinguishability radius, which we introduce. Using the uniform finite-volume results, we then proceed to study the thermodynamic limit. We first study the case of a unique limiting ground state and then also consider models with spontaneous breaking of a discrete symmetry. In the latter case, LTQO cannot hold for all local observables. However, for perturbations that preserve the symmetry, we show stability of the gap and the structure of the broken symmetry phases. We prove that the GNS Hamiltonian associated with each pure state has a non-zero spectral gap above the ground state.

Hierarchy of higher-order Floquet topological phases in three dimensions

Following a general protocol of periodically driving static first-order topological phases (supporting surface states) with suitable discrete symmetry breaking Wilson-Dirac masses, here we construct a hierarchy of higher-order Floquet topological phases in three dimensions. In particular, we demonstrate realizations of both second-order and third-order Floquet topological states, respectively supporting dynamic hinge and corner modes at zero quasienergy, by periodically driving their static first-order parent states with one and two discrete symmetry breaking Wilson-Dirac mass(es). While the static surface states are characterized by codimension ${d}_{c}=1$, the resulting dynamic hinge (corner) modes, protected by antiunitary spectral or particle-hole symmetries, live on the boundaries with ${d}_{c}=2\phantom{\rule{4pt}{0ex}}(3)$. We exemplify these outcomes for three-dimensional topological insulators and Dirac semimetals, with the latter ones following an arbitrary spin-$j$ representation.

Bistability and time crystals in long-ranged directed percolation

Stochastic processes govern the time evolution of a huge variety of realistic systems throughout the sciences. A minimal description of noisy many-particle systems within a Markovian picture and with a notion of spatial dimension is given by probabilistic cellular automata, which typically feature time-independent and short-ranged update rules. Here, we propose a simple cellular automaton with power-law interactions that gives rise to a bistable phase of long-ranged directed percolation whose long-time behaviour is not only dictated by the system dynamics, but also by the initial conditions. In the presence of a periodic modulation of the update rules, we find that the system responds with a period larger than that of the modulation for an exponentially (in system size) long time. This breaking of discrete time translation symmetry of the underlying dynamics is enabled by a self-correcting mechanism of the long-ranged interactions which compensates noise-induced imperfections. Our work thus provides a firm example of a classical discrete time crystal phase of matter and paves the way for the study of novel non-equilibrium phases in the unexplored field of driven probabilistic cellular automata.