We consider the chiral phase transition relevant for QCD matter at finite temperature but with vanishing baryon density. Presumably, the chiral phase transition is of second order for two-flavor QCD in the chiral limit [F. Cuteri et al., On the order of the QCD chiral phase transition for different numbers of quark flavours, <a href="http://dx.doi.org/10.1007/JHEP11(2021)141">J. High Energy Phys. 11 (2021) 141</a>.]. Near the transition temperature, we apply the Schwinger-Keldysh formalism and construct a low-energy effective field theory for the system, in which fluctuations and dissipations are systematically captured. The dynamical variables involve the chiral charge densities and order parameter (chiral condensate). Via the holographic Schwinger-Keldysh technique, the effective action is further confirmed within a modified anti–de Sitter QCD model. With higher-order terms suitably neglected, the stochastic equations derived from the effective field theory resemble those of model F in the Hohenberg-Halperin classification. Within the effective field theory, we briefly discuss the spontaneous breaking of chiral symmetry and Goldstone modes.

China

Yanyan Bu

Zexin Yang

Chiral phase transition: Effective field theory and holography

Chiral Anomalous Magnetohydrodynamics (CAMHD) provides a low-energy effective framework for describing chiral fluids in the presence of dynamical electromagnetic fields and axial anomaly. This theory finds applications across diverse physical systems, including heavy-ion collisions, the early universe, and Weyl/Dirac semimetals. Along with Schwinger-Keldysh (SK) effective theories, holographic models serve as a complementary tool to provide a systematic formulation of CAMHD that goes beyond the weak coupling regime. In this work, we explore holographic models with U(1) × U(1) symmetry, where the electromagnetic U(1) field is rendered dynamical through mixed boundary conditions applied to the bulk gauge field and the axial anomaly is introduced via a Chern-Simons bulk term. Through a detailed holographic SK analysis, we demonstrate that the low-energy effective action derived from this model aligns precisely with the SK field theory proposed by Landry and Liu and, in fact, it generalizes it to scenarios with finite background axial field. This alignment not only validates the holographic model but also paves the way for its use in exploring unresolved aspects of CAMHD, such as the recently proposed chiral magnetic electric separation wave and nonlinear chiral instabilities.

Matteo Baggioli

Xiyang Sun

Chiral anomalous magnetohydrodynamics in action: effective field theory and holography

We study a nearly critical superfluid system from two complementary approaches. Within the first approach, we formulate a Schwinger-Keldysh effective field theory (EFT) for the system when it is located slightly above the critical temperature. The dynamical variables in the EFT construction are two scalars: a neutral scalar associated with the conserved U(1) charge, and a complex scalar describing the order parameter. The set of symmetries, particularly the dynamical Kubo-Martin-Schwinger (KMS) symmetry and chemical shift symmetry, strictly constrains the form of EFT action. Within the second approach, using the holographic Schwinger-Keldysh technique, we derive the effective action for a “microscopic” holographic superfluid, confirming the EFT construction. A systematic inclusion of non-Gaussianity is one highlight of present study.

Hongfei Gao

Xin Gao

Zhiwei Li

Nearly critical superfluid: effective field theory and holography

We present a systematic treatment of non-Gaussianity in stochastic systems using the Schwinger-Keldysh effective field theory framework, in which the non-Gaussianity is realized as nonlinear terms in the fluctuation field. We establish two stochastic formulations of the Schwinger-Keldysh effective field theory, with those nonlinear terms manifested as multiple non-Gaussian noises in the Langevin equation and as higher order diffusive terms in the Fokker-Planck equation. The equivalence of the stochastic formulations with the original Schwinger-Keldysh effective field theory is demonstrated with nontrivial examples for arbitrary non-Gaussian parameters. The stochastic formulations will be more flexible and effective in studying nonequilibrium dynamics. We also reveal an ambiguity when coarse-graining timescale and non-Gaussian parameters vanish simultaneously, which may be responsible for the unphysical divergence found in perturbative analysis.

Shu Lin

Chang Lei

Non-Gaussianity from Schwinger-Keldysh effective field theory

We study the quasi-hydrodynamics of a system with a softly broken U(1) global symmetry using effective field theory (EFT) and holographic methods. In the gravity side, we consider a holographic Proca model in the limit of small bulk mass, which is responsible for a controllable explicit breaking of the U(1) global symmetry in the boundary field theory. We perform a holographic Schwinger-Keldysh analysis, which allows us to derive the form of the boundary effective action in presence of dissipation. We compare our results with the previously proposed EFT and hydrodynamic theories, and we confirm their validity by computing the low-energy quasi-normal modes spectrum analytically and numerically. Additionally, we derive the broken holographic Ward identity for the U(1) current, and discuss the recently proposed novel transport coefficients for systems with explicitly broken symmetries. The setup considered is expected to serve as a toy model for more realistic situations where quasi-hydrodynamics is at work, such as axial charge relaxation in QCD, spin relaxation in relativistic systems, electric field relaxation in magneto-hydrodynamics, or momentum relaxation in condensed matter systems.

France

Vaios Ziogas

U(1) quasi-hydrodynamics: Schwinger-Keldysh effective field theory and holography

An effective description is presented for a Brownian particle in a magnetized plasma. In order to systematically capture various corrections to linear Langevin equation, we construct effective action for the Brownian particle, to quartic order in its position. The effective action is first derived within nonequilibrium effective field theory formalism, and then confirmed via a microscopic holographic model consisting of an open string probing magnetic fifth-dimensional anti–de Sitter (<math><mrow><msub><mrow><mi>AdS</mi></mrow><mrow><mn>5</mn></mrow></msub></mrow></math>) black brane. For practical usage, the non-Gaussian effective action is converted into Fokker-Planck-type equation, which is an Euclidean analog of Schrödinger equation and describes time evolution of probability distribution for particle’s position and velocity.

Biye Zhang

Jingbo Zhang

Nonlinear effective dynamics of a Brownian particle in magnetized plasma

We construct effective field theory for SU(2) isospin charge diffusion, based on holographic Schwinger-Keldysh contour [1]. The holographic model consists of a probe SU(2) gauge field in a doubled Schwarzschild-AdS5 geometry. Accurate to first order in derivative expansion, we analytically compute the effective action up to quartic order in dynamical variables. The effective theory contains both non-Gaussianity for noises and nonlinear interactions between noises and dynamical variables. Moreover, the effective theory captures both thermal and quantum fluctuations, which perfectly satisfy dynamical Kubo-Martin-Schwinger (KMS) symmetry at quantum level. Interestingly, the dynamical KMS symmetry, which is crucial in formulating non-equilibrium effective field theory for a quantum many-body system, is found to have a nice holographic interpretation.

Holographic Schwinger-Keldysh field theory of SU(2) diffusion

We compute Schwinger-Keldysh effective action for a relativistic heavy quark (with constant background velocity) in strongly coupled <math><mrow><mi>N</mi><mo>=</mo><mn>4</mn></mrow></math> supersymmetric Yang-Mills plasma. The holographic dual description involves a noisy trailing string moving in Schwarzschild-<math><mrow><msub><mrow><mi>AdS</mi></mrow><mrow><mn>5</mn></mrow></msub></mrow></math> black brane. The noise is caused by Hawking radiation emitted from string world sheet horizon. Besides quadratic terms, the effective action contains cubic interactions, which are entirely induced by the constant background velocity. The nonlinear fluctuation-dissipation relations for three-point functions are discussed based on holographic results.

Schwinger-Keldysh effective action for a relativistic Brownian particle in the <math><mrow><mi>AdS</mi><mo>/</mo><mi>CFT</mi></mrow></math> correspondence

Under holographic prescription for Schwinger-Keldysh closed time contour for non-equilibrium system, we consider fluctuation effect of the order parameter in a holographic superconductor model. Near the critical point, we derive the time-dependent Ginzburg-Landau effective action governing dynamics of the fluctuating order parameter. In a semi-analytical approach, the time-dependent Ginzburg-Landau action is computed up to quartic order of the fluctuating order parameter, and first order in time derivative.

Mitsutoshi Fujita

Ginzburg-Landau effective action for a fluctuating holographic superconductor

An effective action for diffusion of a conserved U(1) charge is derived to all orders in the derivative expansion within a holographic model dual to the Schwinger-Keldysh closed time path. A systematic approach to solution of the 5D Maxwell equations in a doubled Schwarzschild-AdS5 black brane geometry is developed. Constitutive relation for the stochastic charge current is shown to have a term induced by thermal fluctuations (coloured noise). All transport coefficient functions parameterising the effective action and constitutive relations are computed analytically in the hydrodynamic expansion, and then numerically for finite momenta.