We study the additivity and Haag duality of the von Neumann algebra of a quantum field theory $${\mathcal{T}}_{\mathcal{F}}$$ with 0-form (and the dual ($\textit{d}$ − 2)-form) (non)-invertible global symmetry $$\mathcal{F}$$ . We analyze the symmetric (uncharged) sector von Neumann algebra of $${\mathcal{T}}_{\mathcal{F}}$$ with the inclusion of bi-local and bi-twist operators in it. We establish the connection between the existence of these non-local operators in $${\mathcal{T}}_{\mathcal{F}}$$ and certain properties of the Lagrangian algebra $$\mathcal{L}$$ of the extended operators in the corresponding symmetry topological field theory (SymTFT). We prove that additivity or Haag duality of the symmetric sector von Neumann algebra is violated when $$\mathcal{L}$$ satisfies specific criteria, thus generalizing the result of Shao, Sorce and Srivastava to arbitrary dimensions. We further demonstrate the SymTFT construction via concrete examples in two dimensions.

South Korea

Qiang Jia

China

Jiahua Tian

Symmetry, symmetry topological field theory and von Neumann algebra

In this work we study the quark-meson model within a real-time formulation of the functional renormalization group (FRG) on the Schwinger-Keldysh contour. First, we discuss in detail the symmetry of thermal equilibrium for the fermionic sector of the Keldysh action. We take into account dissipation for the bosonic degrees of freedom in the spirit of the Caldeira-Leggett model by coupling the system to an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>O</mi><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></math> invariant external heat bath. We study the effect of dissipation on static equilibrium properties, most prominently on the FRG flow of the effective potential and thus on the resulting phase diagram. We find that, unlike in classical systems, through the contributions from nonzero Matsubara modes the dissipative dynamics can in general have an effect on static observables. We investigate these effects within two phenomenological models for the temperature dependence of the pion damping to verify that they are quantitatively small. To estimate their largest possible influence, we consider limits where the damping constants approach infinity.

Germany

Johannes V. Roth

Yunxin Ye

Sören Schlichting

Lorenz von Smekal

Effects of dissipation on phase diagram and bosonic excitations in the quark-meson model

QCD at nonzero isospin chemical potentials has phenomenological relevance for a series of physical systems and provides an ideal test ground for the modeling of dense, strongly interacting matter. The two-flavor quark-meson model is known to effectively describe the condensation of charged pions in QCD that occurs in this setting. In this paper, we derive a renormalization group invariant mean-field formulation of the model and demonstrate that the resulting phase diagram and equation of state are in quantitative agreement with data from lattice QCD simulations at small and intermediate isospin chemical potentials. In particular, the speed of sound from the model shows an excess over the conformal bound, as previously seen in lattice computations, in agreement with chiral perturbation theory. We then consider the speed of sound in the limit of large isospin chemical potentials and see that it approaches the conformal limit from above, in qualitative agreement with recent lattice results and in quantitative agreement with perturbation theory in the presence of a Bardeen-Cooper-Schrieffer gap. Finally, we consider the phase diagram in the approach to the chiral limit. We find that within the model, the chiral phase transition connects to the pion condensation phase boundary in the chiral limit and we discuss the implications for the properties of the chiral transition point.

Bastian B. Brandt

Volodymyr Chelnokov

Hungary

Gergely Endrődi

Gergely Markó

Daniel Scheid

Renormalization group invariant mean-field model for QCD at finite isospin density

We use a novel real-time formulation of the functional renormalization group (FRG) for dynamical systems with reversible mode couplings to study model H, the conjectured dynamic universality class of the quantum chromodynamics (QCD) critical point. We emphasize the structural similarities with model G, conjectured to be the dynamic universality class of the chiral phase transition in the limit of two massless quark flavors. Importantly, our formulation of the real-time FRG preserves all relevant symmetries throughout the FRG flow which, e.g., guarantees the nonrenormalization of the reversible mode couplings, as but one exact result. We derive nonperturbative RG flow equations for the kinetic coefficients of both, model G and H, in parallel, and discuss commonalities and differences in the resulting fixed-point structure and dynamic critical exponents, such as weak-scaling relations which hold in either case versus the characteristic strong scaling of model G which is absent in model H.

Universal critical dynamics near the chiral phase transition and the QCD critical point

von Neumann algebras have been playing an increasingly important role in the context of gauge theories and gravity. The crossed product presents a natural method for implementing constraints through the commutation theorem, rendering it a useful tool for constructing gauge-invariant algebras. The crossed product of a Type III algebra with its modular automorphism group is semifinite, which means that the crossed product regulates divergences in local quantum field theories. In this article, we find a sufficient condition for the semifiniteness of the crossed product of a Type III algebra with any locally compact group containing the modular automorphism group. Our condition surprisingly implies the centrality of the modular flow in the symmetry group, and we provide evidence for the necessity of this condition. Under these conditions, we construct an associated trace that computes physical expectation values. We comment on the importance of this result and its implications for subregion physics in gauge theory and gravity.

United States

Shadi Ali Ahmad

Marc S. Klinger

Simon Lin

Semifinite von Neumann algebras in gauge theory and gravity

We study the critical dynamics of a scalar field theory with <math altimg='si1.svg'><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub></math> symmetry in the dynamic universality class of Model A in two and three spatial dimensions with classical-statistical lattice simulations. In particular, we measure the non-equilibrium behavior of the system under a quench protocol in which the symmetry-breaking external field is changed at a constant rate through the critical point. Using the well-established Kibble-Zurek scaling theory we compute non-equilibrium scaling functions of cumulants of the order parameter up to fourth order. Together with the static critical exponents and the dynamic critical exponent, these fully describe the universal non-equilibrium evolution of the system near the critical point. We further extend the analysis to include finite-size effects and observe good collapse of our data onto two-dimensional universal non-equilibrium and finite-size scaling functions.

Leon J. Sieke

Mattis Harhoff

Universal non-equilibrium scaling of cumulants across a critical point

It has been argued that, while the individual terms in the generalized entropy, <math><msub><mi>S</mi><mrow><mi>gen</mi></mrow></msub><mo>=</mo><mi>A</mi><mo>/</mo><mn>4</mn><msub><mi>G</mi><mi>N</mi></msub><mo>+</mo><msub><mi>S</mi><mrow><mi>ext</mi></mrow></msub></math>, are ill-defined in the semiclassical limit, their sum is well-defined if one takes into account perturbative quantum gravitational effects. The first term diverges as <math><msub><mi>G</mi><mi>N</mi></msub><mo>→</mo><mn>0</mn></math>, and the second diverges due to the infinite entanglement across the horizon which is characteristic of type III von Neumann algebras. It was recently shown that the von Neumann algebra of observables “gravitationally dressed” to the mass of a Schwarzschild-AdS black hole or the energy of an observer in de Sitter spacetime admit a well-defined trace. The algebras are type <math><msub><mrow><mi>II</mi></mrow><mi>∞</mi></msub></math> (which does not admit a maximum entropy state) and type <math><mrow><msub><mrow><mi>II</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></math> (which admits a maximum entropy state) respectively, and the von Neumann entropy of “semiclassical” states was found to be (up to an additive constant) the generalized entropy. However, these arguments rely on the existence of a stationary “equilibrium (KMS) state” and do not apply to, for example, black holes formed from gravitational collapse, Kerr black holes, or black holes in asymptotically de Sitter spacetime. These spacetimes are stationary but not in thermal equilibrium. In this paper, we present a general framework for obtaining the algebra of “gravitationally dressed” observables for a linear, Klein-Gordon field on any spacetime with a (bifurcate) Killing horizon. We prove, assuming the existence of a stationary state—which is not necessarily KMS—and suitable asymptotic decay of solutions, a “structure theorem” that the algebra of “gravitationally dressed” observables always contains a type II factor of observables “localized” on the horizon. These assumptions have been rigorously proven in most cases of interest in this paper. Applying our general framework to the algebra of observables in the exterior of an asymptotically flat Kerr black hole where the fields are dressed to the black hole mass and angular momentum we find that the algebra is the product of a type <math><msub><mrow><mi>II</mi></mrow><mi>∞</mi></msub></math> algebra on the horizon and a type <math><msub><mrow><mi>I</mi></mrow><mi>∞</mi></msub></math> algebra at past null infinity. The full algebra is type <math><msub><mrow><mi>II</mi></mrow><mi>∞</mi></msub></math>, and the von Neumann entropy of semiclassical states is the generalized entropy. In the case of Schwarzschild-de Sitter, despite the fact that we must introduce an observer, the algebra of observables dressed to the perturbed areas of the black hole and cosmological horizons is the product of type <math><msub><mrow><mi>II</mi></mrow><mi>∞</mi></msub></math> algebras on each horizon. The entropy of semiclassical states is given by the sum of the areas of the two horizons as well as the entropy of quantum fields in between the horizons. Our results suggest that in all cases where there exists another “boundary structure” (e.g., an asymptotic boundary or another Killing horizon) the algebra of observables is type <math><msub><mrow><mi>II</mi></mrow><mi>∞</mi></msub></math> and in the absence of such structures (e.g. de Sitter spacetime) the algebra is type <math><mrow><msub><mrow><mi>II</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></math>.

Jonah Kudler-Flam

Samuel Leutheusser

Gautam Satishchandran

Generalized black hole entropy is von Neumann entropy

In this work, we study the evolution of an evaporating black hole, described by the Kerr–Taub–NUT metric, which emits scalar particles. We found that allowing the black hole to radiate massless scalar particles increases the angular momentum loss rate while decreasing the loss rate of the NUT parameter and black hole mass. In fact, it means that angular momentum will disappear faster than the other black hole parameters (mass and NUT parameter) during the evaporation process. We also calculate the time evolution of the mass, angular momentum, and NUT parameter in order to get the evolution of the Von Neumann entropy of the black hole. We found that the entropy follows approximately the so-called Page curve, where the $$\beta $$ <math> <mi>β</mi> </math> parameter, which quantifies the amount of radiation, affects the evaporation process. Implying that high $$\beta $$ <math> <mi>β</mi> </math> values accelerate the evaporation process of a Kerr–Taub–NUT black hole.

Ecuador

Vicente Arévalo

David Andrade

Clara Rojas

Time evolution of the Von Neumann entropy for a Kerr–Taub–NUT black hole

The mixing between the chiral condensate and the density in hot and dense quantum chromodynamics (QCD) matter is familiar. We show that the mixing relevant for the ground state is considerably more extensive, and in particular also involves gluonic degrees of freedom. As a result, the Hessian of the QCD effective action is non-Hermitian, but retains a symmetry under combined charge and complex conjugation. This can lead to complex-conjugate pairs of eigenvalues of this Hessian, signaling regimes with spatially modulated correlations. Furthermore, based on the analytic structure of the quark determinant at a chiral critical point, we demonstrate that the corresponding massless critical mode is composed of the chiral condensate, the density and the Polyakov loops. Due to an avoided crossing, the critical mode turns out to be disconnected from the chiral condensate in vacuum. We present general arguments for all these features and illustrate them through explicit model calculations.

Maximilian Haensch

Fabian Rennecke

Medium induced mixing, spatial modulations, and critical modes in QCD

Euclidean path integrals for UV-completions of d-dimensional bulk quantum gravity were recently studied in [1] by assuming that they satisfy axioms of finiteness, reality, continuity, reflection-positivity, and factorization. Sectors <math> <msub> <mi>H</mi> <mi>B</mi> </msub> </math> $$ {\mathcal{H}}_{\mathcal{B}} $$ of the resulting Hilbert space were then defined for any (d − 2)-dimensional surface <math> <mi>B</mi> </math> $$ \mathcal{B} $$ , where <math> <mi>B</mi> </math> $$ \mathcal{B} $$ may be thought of as the boundary ∂Σ of a bulk Cauchy surface in a corresponding Lorentzian description, and where <math> <mi>B</mi> </math> $$ \mathcal{B} $$ includes the specification of appropriate boundary conditions for bulk fields. Cases where <math> <mi>B</mi> </math> $$ \mathcal{B} $$ was the disjoint union B ⊔ B of two identical (d − 2)-dimensional surfaces B were studied in detail and, after the inclusion of finite-dimensional ‘hidden sectors,’ were shown to provide a Hilbert space interpretation of the associated Ryu-Takayanagi entropy. The analysis was performed by constructing type-I von Neumann algebras <math> <msubsup> <mi>A</mi> <mi>L</mi> <mi>B</mi> </msubsup> </math> $$ {\mathcal{A}}_L^B $$ , <math> <msubsup> <mi>A</mi> <mi>R</mi> <mi>B</mi> </msubsup> </math> $$ {\mathcal{A}}_R^B $$ that act respectively at the left and right copy of B in B ⊔ B.

Donald Marolf

Daiming Zhang

When left and right disagree: entropy and von Neumann algebras in quantum gravity with general AlAdS boundary conditions