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1. Pattern formation in a four-ring reaction-diffusion network with heterogeneity

   https://doi.org/10.1103/PhysRevE.105.024310  

   Hunter, Ian; Norton, Michael M.; Chen, Bolun; Simonetti, Chris; Moustaka, Maria Eleni; Touboul, Jonathan; Fraden, Seth (February 2022, Physical Review E)

   In networks of nonlinear oscillators, symmetries place hard constraints on the system that can be exploited to predict universal dynamical features and steady states, providing a rare generic organizing principle for far-from-equilibrium systems. However, the robustness of this class of theories to symmetry-disrupting imperfections is untested in free-running (i.e., non-computer-controlled) systems. Here, we develop a model experimental reaction-diffusion network of chemical oscillators to test applications of the theory of dynamical systems with symmeries in the context of self-organizing systems relevant to biology and soft robotics. The network is a ring of four microreactors containing the oscillatory Belousov-Zhabotinsky reaction coupled to nearest neighbors via diffusion. Assuming homogeneity across the oscillators, theory predicts four categories of stable spatiotemporal phase-locked periodic states and four categories of invariant manifolds that guide and structure transitions between phase-locked states. In our experiments, we observed that three of the four phase-locked states were displaced from their idealized positions and, in the ensemble of measurements, appeared as clusters of different shapes and sizes, and that one of the predicted states was absent. We also observed the predicted symmetry-derived synchronous clustered transients that occur when the dynamical trajectories coincide with invariant manifolds. Quantitative agreement between experiment and numerical simulations is found by accounting for the small amount of experimentally determined heterogeneity in intrinsic frequency. We further elucidate how different patterns of heterogeneity impact each attractor differently through a bifurcation analysis. We show that examining bifurcations along invariant manifolds provides a general framework for developing intuition about how chemical-specific dynamics interact with topology in the presence of heterogeneity that can be applied to other oscillators in other topologies. 

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2. Lattice Hamiltonian for adjoint QCD2

   https://doi.org/10.1007/JHEP08(2024)009  

   Dempsey, Ross; Klebanov, Igor R; Pufu, Silviu S; Søgaard, Benjamin T (August 2024, Journal of High Energy Physics)

   A<sc>bstract</sc> We introduce a Hamiltonian lattice model for the (1 + 1)-dimensional SU(N_c) gauge theory coupled to one adjoint Majorana fermion of massm. The discretization of the continuum theory uses staggered Majorana fermions. We analyze the symmetries of the lattice model and find lattice analogs of the anomalies of the corresponding continuum theory. An important role is played by the lattice translation by one lattice site, which in the continuum limit involves a discrete axial transformation. On a lattice with periodic boundary conditions, the Hilbert space breaks up into sectors labeled by theN_c-alityp= 0, …N_c− 1. Our symmetry analysis implies various exact degeneracies in the spectrum of the lattice model. In particular, it shows that, form= 0 and evenN_c, the sectorspandp′ are degenerate if |p−p′| =N_c/2. In theN_c= 2 case, we explicitly construct the action of the Hamiltonian on a basis of gauge-invariant states, and we perform both a strong coupling expansion and exact diagonalization for lattices of up to 12 lattice sites. Upon extrapolation of these results, we find good agreement with the spectrum computed previously using discretized light-cone quantization. One of our new results is the first numerical calculation of the fermion bilinear condensate. 

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3. Spontaneous breaking of U(1) symmetry in coupled complex SYK models

   https://doi.org/10.1007/JHEP11(2020)162  

   Klebanov, Igor R.; Milekhin, Alexey; Tarnopolsky, Grigory; Zhao, Wenli (November 2020, Journal of High Energy Physics)

   null (Ed.)

   A bstract As shown in [1], two copies of the large N Majorana SYK model can produce spontaneous breaking of a Z 2 symmetry when they are coupled by appropriate quartic terms. In this paper we similarly study two copies of the complex SYK model coupled by a quartic term preserving the U(1) × U(1) symmetry. We also present a tensor counterpart of this coupled model. When the coefficient α of the quartic term lies in a certain range, the coupled large N theory is nearly conformal. We calculate the scaling dimensions of fermion bilinear operators as functions of α . We show that the operator $$ {c}_{1i}^{\dagger }{c}_{2i} $$ c 1 i † c 2 i , which is charged under the axial U(1) symmetry, acquires a complex dimension outside of the line of fixed points. We derive the large N Dyson-Schwinger equations and show that, outside the fixed line, this U(1) symmetry is spontaneously broken at low temperatures because this operator acquires an expectation value. We support these findings by exact diagonalizations extrapolated to large N . 

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4. Quantized Axial Charge of Staggered Fermions and the Chiral Anomaly

   https://doi.org/10.1103/PhysRevLett.134.021601  

   Chatterjee, Arkya; Pace, Salvatore D; Shao, Shu-Heng (January 2025, Physical Review Letters)

   In the $1+1\mathrm{D}$ultralocal lattice Hamiltonian for staggered fermions with a finite-dimensional Hilbert space, there are two conserved, integer-valued charges that flow in the continuum limit to the vector and axial charges of a massless Dirac fermion with a perturbative anomaly. Each of the two lattice charges generates an ordinary U(1) global symmetry that acts locally on operators and can be gauged individually. Interestingly, they do not commute on a finite lattice and generate the Onsager algebra, but their commutator goes to zero in the continuum limit. The chiral anomaly is matched by this non-Abelian algebra, which is consistent with the Nielsen-Ninomiya theorem. We further prove that the presence of these two conserved lattice charges forces the low-energy phase to be gapless, reminiscent of the consequence from perturbative anomalies of continuous global symmetries in continuum field theory. Upon bosonization, these two charges lead to two exact U(1) symmetries in the XX model that flow to the momentum and winding symmetries in the free boson conformal field theory. Published by the American Physical Society2025 

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5. Gauging U(1) symmetry in (2+1)d topological phases

   https://doi.org/10.21468/SciPostPhys.12.6.202  

   Cheng, Meng; Jian, Chao-Ming (January 2022, SciPost Physics)

   We study the gauging of a global U(1) symmetry in a gapped system in(2+1)d. The gauging procedure has been well-understood for a finiteglobal symmetry group, which leads to a new gapped phase with emergentgauge structure and can be described algebraically using themathematical framework of modular tensor category (MTC). We develop acategorical description of U(1) gauging in a MTC, taking into accountthe dynamics of U(1) gauge field absent in the finite group case. Whenthe ungauged system has a non-zero Hall conductance, the gauged theoryremains gapped and we determine the complete set of anyon data for thegauged theory. On the other hand, when the Hall conductance vanishes, weargue that gauging has the same effect of condensing a special Abeliananyon nucleated by inserting 2\pi 2 π U(1) flux. We apply our procedure to theSU(2) _k k MTCs and derive the full MTC data for the \mathbb{Z}_k ℤ k parafermion MTCs. We also discuss a dual U(1) symmetry that emergesafter the original U(1) symmetry of an MTC is gauged. 

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