We study the near-equilibrium critical dynamics of the O(3) nonlinear sigma model describing isotropic antiferromagnets with a nonconserved order parameter reversibly coupled to the conserved total magnetization. To calculate response and correlation functions, we set up a description in terms of Langevin stochastic equations of motion, and their corresponding Janssen-De Dominicis response functional. We find that in equilibrium, the dynamics is well-separated from the statics, at least to one-loop order in a perturbative treatment with respect to the static and dynamical nonlinearities. Since the static nonlinear sigma model must be analyzed in a dimensional d=2+ɛ expansion about its lower critical dimension d_{lc}=2, whereas the dynamical mode-coupling terms are governed by the upper critical dimension d_{c}=4, a simultaneous perturbative dimensional expansion is not feasible, and the reversible critical dynamics for this model cannot be accessed at the static critical renormalization group fixed point. However, in the coexistence limit addressing the long-wavelength properties of the low-temperature ordered phase, we can perform an ε=4-d expansion near d_{c}. This yields anomalous scaling features induced by the massless Goldstone modes, namely subdiffusive relaxation for the conserved magnetization density with asymptotic scaling exponent z_{Γ}=d-2, which may be observable in neutron scattering experiments. Intriguingly, if initialized near the critical point, the renormalization group flow for the effective dynamical exponents recovers their universal critical values z_{c}=d/2 in an intermediate crossover region.We study the near-equilibrium critical dynamics of the O(3) nonlinear sigma model describing isotropic antiferromagnets with a nonconserved order parameter reversibly coupled to the conserved total magnetization. To calculate response and correlation functions, we set up a description in terms of Langevin stochastic equations of motion, and their corresponding Janssen-De Dominicis response functional. We find that in equilibrium, the dynamics is well-separated from the statics, at least to one-loop order in a perturbative treatment with respect to the static and dynamical nonlinearities. Since the static nonlinear sigma model must be analyzed in a dimensional d=2+ɛ expansion about its lower critical dimension d_{lc}=2, whereas the dynamical mode-coupling terms are governed by the upper critical dimension d_{c}=4, a simultaneous perturbative dimensional expansion is not feasible, and the reversible critical dynamics for this model cannot be accessed at the static critical renormalization group fixed point. However, in the coexistence limit addressing the long-wavelength properties of the low-temperature ordered phase, we can perform an ε=4-d expansion near d_{c}. This yields anomalous scaling features induced by the massless Goldstone modes, namely subdiffusive relaxation for the conserved magnetization density with asymptotic scaling exponent z_{Γ}=d-2, which may be observable in neutron scattering experiments. Intriguingly, if initialized near the critical point, the renormalization group flow for the effective dynamical exponents recovers their universal critical values z_{c}=d/2 in an intermediate crossover region.

arXiv:2204.11145Physical review. EJun 27, 20223 citations

Behaviour of systems near the polycritical point

A.Z. Patashinsky, B.I. Shumilo

Physics Letters AApr 1, 19770 citations

Light scattering and nonlinear optical response near a critical point

Robert M. McGraw

The connection between light scattering and nonlinear optical response is examined quantitatively for Kerr media and is generalized to include isotropic materials with nonlocal interactions near a critical point. A number of important quantities, including the nonlinear refractive index ( ${\mathit{n}}_{2}$ ) and the four-wave-mixing coefficient, are shown to acquire increasing grating wave-vector (q) dependence as the critical point is approached. This dependence is identical to the q dependence obtained for light scattering in the Ornstein-Zernike theory. Illustrative calculations are presented for near-critical xenon using the Redlich-Kwong equation of state and parameters obtained from independent measurements. These results indicate ${\mathit{n}}_{2}$ values of ${10}^{\mathrm{\ensuremath{-}}15}$--${10}^{\mathrm{\ensuremath{-}}14}$ ${\mathrm{m}}^{2}$ /W and response times in the 1--10-\ensuremath{\mu}sec range. Calculations are presented for the phase-conjugate reflectivity in degenerate four-wave mixing that include both nonsaturable background loss and light-scattering noise. Reflectivities in excess of unity are predicted for 1.06-\ensuremath{\mu}m wavelength and cw pump intensities in the 10-kW/ ${\mathrm{cm}}^{2}$ range.The connection between light scattering and nonlinear optical response is examined quantitatively for Kerr media and is generalized to include isotropic materials with nonlocal interactions near a critical point. A number of important quantities, including the nonlinear refractive index (${\mathit{n}}_{2}$) and the four-wave-mixing coefficient, are shown to acquire increasing grating wave-vector (q) dependence as the critical point is approached. This dependence is identical to the q dependence obtained for light scattering in the Ornstein-Zernike theory. Illustrative calculations are presented for near-critical xenon using the Redlich-Kwong equation of state and parameters obtained from independent measurements. These results indicate ${\mathit{n}}_{2}$ values of ${10}^{\mathrm{\ensuremath{-}}15}$--${10}^{\mathrm{\ensuremath{-}}14}$ ${\mathrm{m}}^{2}$/W and response times in the 1--10-\ensuremath{\mu}sec range. Calculations are presented for the phase-conjugate reflectivity in degenerate four-wave mixing that include both nonsaturable background loss and light-scattering noise. Reflectivities in excess of unity are predicted for 1.06-\ensuremath{\mu}m wavelength and cw pump intensities in the 10-kW/${\mathrm{cm}}^{2}$ range.

Physical Review AAug 1, 199014 citations

Dynamical aspects of critical phenomena

M. P. Kawatra, J. I. Budnick

Jan 1, 197240 citations

Tangent equations of motion for nonlinear response functions

Atsushi Ono

Nonlinear response functions, formulated as multipoint correlation functions or Volterra kernels, encode the dynamical and spectroscopic properties of physical systems and underpin a wide range of nonlinear transport and optical phenomena. However, their evaluation rapidly becomes prohibitive at high orders because of combinatorial (often factorial) scaling or severe numerical errors. Here, we establish a systematic and efficient framework to compute nonlinear response functions directly from real-time dynamics, without explicitly constructing multipoint correlators or relying on numerically unstable finite-difference methods for order-resolved extraction. Our approach is based on the Gateaux derivative with respect to the external field in function space, which yields a closed hierarchy of tangent equations of motion (TEOM). Propagating the TEOM alongside the original dynamics isolates each perturbative order with high accuracy, providing a term-by-term decomposition of physical contributions. The computational cost scales exponentially with response order in the fully general setting and reduces to polynomial complexity when all perturbation directions are identical; both regimes avoid the factorial scaling of explicit multipoint-correlator evaluations. We demonstrate the power of TEOM by computing frequency-resolved fifth-order response functions for a solid-state electron model and by obtaining nonlinear response functions up to the 49th order with controlled accuracy in a classical Duffing oscillator. We further show that our time-evolution formulation allows optical conductivities to be evaluated directly while remaining numerically stable even near zero frequency. TEOM can be incorporated seamlessly into existing real-time evolution methods, yielding a general framework for computing nonlinear response functions in quantum and classical dynamical systems.Nonlinear response functions, formulated as multipoint correlation functions or Volterra kernels, encode the dynamical and spectroscopic properties of physical systems and underpin a wide range of nonlinear transport and optical phenomena. However, their evaluation rapidly becomes prohibitive at high orders because of combinatorial (often factorial) scaling or severe numerical errors. Here, we establish a systematic and efficient framework to compute nonlinear response functions directly from real-time dynamics, without explicitly constructing multipoint correlators or relying on numerically unstable finite-difference methods for order-resolved extraction. Our approach is based on the Gateaux derivative with respect to the external field in function space, which yields a closed hierarchy of tangent equations of motion (TEOM). Propagating the TEOM alongside the original dynamics isolates each perturbative order with high accuracy, providing a term-by-term decomposition of physical contributions. The computational cost scales exponentially with response order in the fully general setting and reduces to polynomial complexity when all perturbation directions are identical; both regimes avoid the factorial scaling of explicit multipoint-correlator evaluations. We demonstrate the power of TEOM by computing frequency-resolved fifth-order response functions for a solid-state electron model and by obtaining nonlinear response functions up to the 49th order with controlled accuracy in a classical Duffing oscillator. We further show that our time-evolution formulation allows optical conductivities to be evaluated directly while remaining numerically stable even near zero frequency. TEOM can be incorporated seamlessly into existing real-time evolution methods, yielding a general framework for computing nonlinear response functions in quantum and classical dynamical systems.

cond-mat.str-elCross-listedarXiv:2603.22137Physical Review ResearchMar 24, 20260 citations

{\mathit{n}}_{2}

) and the four-wave-mixing coefficient, are shown to acquire increasing grating wave-vector (q) dependence as the critical point is approached. This dependence is identical to the q dependence obtained for light scattering in the Ornstein-Zernike theory. Illustrative calculations are presented for near-critical xenon using the Redlich-Kwong equation of state and parameters obtained from independent measurements. These results indicate

{\mathit{n}}_{2}

values of ${10}^{\mathrm{\ensuremath{-}}15}$--${10}^{\mathrm{\ensuremath{-}}14}$

{\mathrm{m}}^{2}

/W and response times in the 1--10-\ensuremath{\mu}sec range. Calculations are presented for the phase-conjugate reflectivity in degenerate four-wave mixing that include both nonsaturable background loss and light-scattering noise. Reflectivities in excess of unity are predicted for 1.06-\ensuremath{\mu}m wavelength and cw pump intensities in the 10-kW/

{\mathrm{cm}}^{2}

range.