arXiv subjects

Samuel L. Braunstein

Publications and source records attributed to Samuel L. Braunstein.

At least 19 recordsLinked to original sources

arXiv2026

Extracting Chern numbers from the equilibrium transport noise of multiband bosons

We show that the equilibrium transport noise of a multiband bosonic system carries a measurable imprint of its Chern numbers, and we give the protocol that extracts them. For electrons the topological part of the Berry curvature is absent from the transport noise: over a filled band it yields the quantized anomalous Hall conductivity, yet at a Fermi surface the Chern number is invisible to the fluctuations of the Hall current. That cancellation is an accident of the Fermi surface: magnons and phonons populate every band thermally, each with its own Chern number, so the harmonic sector of the Hodge decomposition is not a single constant, and a variance does not annihilate it. What survives is the occupation-weighted dispersion of Chern numbers. That current noise sees band geometry is known, and at zero temperature the antisymmetric part of the noise sum rule already returns a Chern number; the new element is an object with no zero-temperature analogue, existing only when several bands are populated and weighted differently. We derive it from a fluctuating Boltzmann equation and resolve the energy-magnetization subtraction at the level of fluctuations: the subtracted term is a curl, contributing exactly zero to the current a transport measurement records, and no magnetization auto-correlation enters the antisymmetric cross-spectrum at any wavevector. Measurement uses an equilibrium sum rule on that cross-spectrum, with no drive and hence no suppression: its frequency-integrated form cannot in practice separate topological from geometric content, for a rank reason, whereas the frequency-resolved form does, at accessible resolution. The protocol is specified for Cu(1,3-bdc), with calibration and robustness characterized. A driven alternative lies twelve orders below the equilibrium floor. Two independent implementations agree on every convention-free quantity.

cond-mat.str-el↗

arXiv2026

The Hodge structure of Berry-phase transport: topology, geometry, and noise

We show that the Hodge-de Rham decomposition of the Berry curvature organises the transport of a Bloch band and its fluctuations within a single geometric structure. Splitting the curvature into $L^2$-orthogonal harmonic, exact, and co-exact sectors yields a dictionary for both moments of the current. In the mean response the harmonic sector carries the topological anomalous-Hall conductivity, the exact sector the Fermi-surface geometry and the antisymmetric Berry-curvature dipole of polar metals, and, in three dimensions, the co-exact sector the chiral anomaly, quantised by the Weyl-node charges. In the fluctuations, described by a particle-conserving stochastic Boltzmann equation constrained by the fluctuation-dissipation theorem (FDT), the harmonic sector is silent, so topological transport is noiseless, while the field-driven noise is sourced by the geometric sectors (solely the exact sector in two dimensions); current-noise spectroscopy therefore separates global band topology from local band geometry. We prove that the harmonic null-space protection is dimension-independent, and we settle the remaining sector: the co-exact (monopole) sector carries no conservation law and, under the thermal sampling measure, mixes with the exact sector at $\mathcal{O}(1)$, so it furnishes no clean noise observable. The separation the noise provides is therefore two-way, topology versus geometry, in both two and three dimensions.

cond-mat.mes-hall↗

arXiv2026

A sharp bound on spacetime distance from quantum entanglement

Ryu-Takayanagi established how boundary entanglement encodes bulk area. We provide the metric counterpart: boundary mutual information imposes a rigorous lower bound on bulk geodesic separation that diverges logarithmically as correlations vanish. A multiscale iteration promotes this local inequality to a global obstruction to bulk connectivity. For parallel strips in AdS$_5$/CFT$_4$, the bound necessitates a quantum resolution of the classical mutual-information transition and fixes the asymptotic growth of geodesic distance.

hep-th↗

arXiv2026

Boundary conditions select the viscous operator on Riemannian hypersurfaces: formal analysis and rigorous thin-shell limits

A viscous fluid confined to a thin layer around a curved surface is governed, as the layer thickness vanishes, by an effective viscous operator on the surface. We show that the wall conditions select this operator. The vorticity-free (free) and stress-free (Navier) conditions, which coincide on flat walls but differ on curved ones by the shape operator, yield respectively the Hodge Laplacian and the deformation Laplacian, and these differ universally, on any hypersurface, by twice the Ricci curvature; a one-parameter family of wall conditions joins them, with an effective operator that couples to the extrinsic geometry only in between. We prove this in two forms: formally, by matched asymptotics, on an arbitrary hypersurface, and rigorously, as Mosco convergence of the viscous energy forms, hence with resolvent, semigroup and spectral convergence, on surfaces of revolution. The stress-free limit on general surfaces is due to Miura and is recovered here; the rigorous vorticity-free limit beyond the sphere, via a uniform Gaffney inequality, together with the interpolating family and the spectral packaging, is new, and the analysis makes precise a conflation of the two conditions in the classical spherical treatment. The extension-dependence of the operator found on the ellipsoid is explained as a dependence on the wall condition.

math-ph↗

arXiv2026

Lubkin-Page typicality bounds for Type~II von~Neumann factors

Typicality arguments for emergent spacetime rely on the Lubkin-Page bounds, which show that generic quantum states have vanishing correlations between subsystems. These bounds assume a tensor-product Hilbert space (a Type~I von~Neumann algebra), but the observable algebras in quantum field theory and quantum gravity are generically Type~II or Type~III, raising the question of whether the bounds survive. We prove that they do for all Type~II von~Neumann factors. For the hyperfinite Type~II$_1$ factor with a tripartite decomposition $R \cong A \otimes B \otimes E$, the mutual information between subsystems $A$ and $B$ vanishes as $O((d_A d_B / d_E)^2)$ in finite-dimensional approximations, provided $d_A d_B \leq d_E$ (Theorem~1). For Type~II$_\infty$ factors, including the gravitational algebras constructed via the crossed-product method by Witten and by Chandrasekaran, Longo, Penington, and Witten, the bound acquires an additional exponential suppression controlled by the Bekenstein-Hawking entropy (Theorem~2). We identify the obstructions to extending the result to Type~III factors and discuss the open question of whether the commutant of the observable algebra can serve as a natural thermal bath that tightens the bound further.

hep-th↗

arXiv2026

Hodge Topology of Semiclassical Transport: A Coordinate-Free Geometric Framework for the Anomalous Hall Effect and Non-Linear Berry Dipole

We establish a coordinate-free differential geometric framework for anomalous transport in topological bands using the Hodge-de Rham decomposition of the Brillouin zone. Standard formulations face mathematical singularities (Dirac strings) when using the quantum Berry connection in bands with non-zero Chern numbers. Applying this decomposition to the Berry curvature 2-form isolates the quantized topological monopole flux from a globally smooth geometric 1-form proxy potential, $\mathcal{A}$. Substituting this regularized potential into semiclassical transport integrals yields distinct analytical advantages. For linear transverse transport, our cohomological decomposition enables an exact geometric derivation of Haldane's insight via the co-area formula, partitioning the response into a continuous Fermi sea topological background and a localized Fermi surface geometric line integral. For non-linear transport, this globally smooth proxy unifies the geometric description, reproducing the high numerical stability of scalar integration-by-parts techniques directly from its exact sector, accommodating arbitrary Chern numbers. By enforcing the continuous Coulomb-Hodge gauge ($δ\mathcal{A} = 0$) alongside vanishing harmonic holonomies over fundamental 1-cycles ($\oint_{γ_i} \mathcal{A} = 0$), we map the Hodge potential $\mathcal{A}$ to the Maximally Localized Wannier Function (MLWF) gauge in trivial bands, providing a non-singular computational proxy for topologically obstructed bands. Finally, we analytically demonstrate that solving the Hodge Laplacian for $\mathcal{A}$ zeroes the macroscopic Brillouin zone average (uniform $\mathbf{R}=0$ zero-mode) topological divergence, yielding a mathematically consistent covariant formulation that matches the algorithmic robustness of standard methods against discrete $\mathbf{k}$-grid noise.

cond-mat.mes-hall↗

arXiv2026

Quantum typicality survives non-Abelian gauge constraints: exact analytical prediction confirmed in $SU(2)$ lattice gauge theory

Arguments for emergent spacetime require that quantum typicality, the generic absence of inter-subsystem correlations, persists on the physical Hilbert space of a gauge theory, where non-Abelian constraints could in principle inject geometry-supporting entanglement. Using $SU(2)$ lattice gauge theory on two-dimensional tori ($d_{\mathrm{phys}}$ up to $4{,}193$), we show that it does: the typical mutual information between strictly disjoint links matches an exact parameter-free analytical prediction combining a microcanonical baseline with Haar-random fluctuations. The Kogut-Susskind Hamiltonian generates correlations from states of definite geometry (such as the electric vacuum), while generic states show only regression to the mean, establishing that the arrow of correlation growth requires a non-generic initial condition.

quant-ph↗

arXiv2026

Global exponential stability for the three-dimensional Navier-Stokes equations on hyperbolic space

We prove that the three-dimensional incompressible Navier-Stokes equations with the deformation Laplacian on hyperbolic 3-space $\HH^3$ admit a unique global mild solution for sufficiently small initial data in $L^3(\HH^3)$, and that this solution decays exponentially to zero. The exponential decay rate is $μλ_\Def^{(3)}$, where $μ$ is the kinematic viscosity and $λ_\Def^{(3)} = 26/9$ is the effective spectral gap of the deformation Laplacian in $L^3$. On flat $\R^3$, the corresponding Kato-type result gives only algebraic decay. The exponential stability is a macroscopic consequence of the spectral gap provided by negative curvature. We also show that the $L^2$ norm is supercritical on $\HH^3$ (as on $\R^3$), with the obstruction arising from the local ultraviolet scaling of the heat kernel, which is insensitive to global geometry. The boundary between what curvature can and cannot improve is located exactly: the Fujita-Kato integral has a scaling exponent $1/2 - 3/(2p)$ that depends only on the integrability of the initial data, not on the geometry of the manifold. For $p \geq 3$ (the Kato critical space), the integral is bounded and the spectral gap contributes exponential time decay. For $p < 3$, the integral diverges at $t = 0$ (and strictly diverges for all $t>0$ when $p \le 3/2$) regardless of the curvature.

math-ph↗

arXiv2026

Resolving the viscosity operator ambiguity on Riemannian manifolds via a kinematic selection principle

On a general Riemannian manifold the Navier-Stokes equations admit several inequivalent formulations, differing in the choice of viscous operator: the Hodge Laplacian, the Bochner Laplacian, or the deformation Laplacian. We show that a Lagrangian kinematic construction, in which the strain rate is built from the rate of change of inner products of Lie-dragged connecting vectors, uniquely selects the deformation Laplacian for fluids whose configuration space is intrinsically the manifold. The Hodge Laplacian is excluded at the kinematic step (before introducing constitutive assumptions) because the strain rate constructed from inner-product geometry is symmetric and has no antisymmetric part. We further show that when the fluid arises as a thin-shell limit of an ambient three-dimensional flow, the operator that emerges depends on the boundary condition imposed in the normal direction: stress-free (Navier slip) conditions recover the deformation Laplacian, while Hodge boundary conditions recover the Hodge Laplacian, via an explicit decomposition of the ambient Bochner Laplacian into intrinsic and extrinsic pieces. The intrinsic piece is the deformation Laplacian regardless of the boundary condition. As an analytical confirmation, we show that the kinematic selection is consistent with the known failure of the energy inequality for the Hodge Laplacian on the hyperbolic plane $\HH^2$: the deformation Laplacian is coercive on $\HH^2$ while the Hodge Laplacian is not, because the Ricci term has the opposite sign in the two operators. We further prove that on any complete two-dimensional manifold with Gaussian curvature bounded above by a negative constant, the incompressible Navier-Stokes equation with the deformation Laplacian admits a unique global weak solution with exponential energy decay, resolving the analytical obstruction preventing the corresponding result for the Hodge Laplacian.

math-ph↗

arXiv2026

Exponential stability for the three-dimensional Navier-Stokes equations on negatively curved manifolds

We extend the exponential stability theorem for the three-dimensional incompressible Navier-Stokes equations from hyperbolic 3-space $\HH^3$ (established in a companion paper) to complete simply connected Riemannian 3-manifolds $(M^3, g)$ with pinched negative sectional curvature $-b^2 \leq K \leq -a^2 < 0$ and bounded geometry (including a strictly positive injectivity radius). The deformation Laplacian $Δ_\Def = Δ_B + \Ric$ remains the viscous operator, selected by Lagrangian kinematics. We prove that the {exact} system admits a unique global mild solution for small $L^3$ data, with exponential decay at a rate determined by the spectral gap of the Stokes operator. The extension overcomes three obstacles absent on $\HH^3$: (i) the semigroup factorisation $e^{tΔ_\Def} = e^{-2t}e^{tΔ_B}$ fails because $\Ric$ is not a scalar multiple of the metric; (ii) the Leray projector no longer commutes with $Δ_\Def$; (iii) the exact spectral gap is unknown. We resolve (i) unconditionally, without any curvature restriction, by observing that the Ricci perturbation $V = \Ric + 2a^2 g$ is negative semi-definite and applying a Trotter product bound with the diamagnetic inequality. We resolve (ii) by an algebraic reduction of the commutator $[\PP, Δ_\Def]$ to the complementary projector $(I-\PP)$ applied to the shifted Ricci endomorphism, giving a clean zeroth-order bound proportional to the curvature variation $b^2 - a^2$. This is the sole source of a curvature pinching constraint. We resolve (iii) via McKean's theorem, the diamagnetic inequality, and the Weitzenböck identity. The Fujita-Kato temporal singularity exponent $1/2 - 3/(2p)$ is unchanged from the $\HH^3$ case, confirming that the ultraviolet scaling obstruction is local and geometry-independent, driven fundamentally by an unresolvable temporal scaling mismatch.

math.AP↗

arXiv2026

Non-Perturbative Closed Form for the Typical Bipartite Mutual Information of Haar-Random States

The average bipartite quantum mutual information $\langle I(A{:}B)\rangle$ of Haar-random pure states can be expressed exactly through Page's formula in terms of digamma functions. We show that this quantity admits a single non-perturbative closed form: $\langle I(A{:}B)\rangle = (d_A^2-1)(d_B^2-1)\,\mathcal{G}(d_A,d_B,d_E)$, where $\mathcal{G}$ is given by an explicit convergent integral over a Bose--Einstein kernel. The overall factor $(d_A^2-1)(d_B^2-1)=\dim[\mathfrak{su}(d_A)]\cdot\dim[\mathfrak{su}(d_B)]$ is exact, not merely asymptotic. The asymptotic expansion of $\mathcal{G}$ in $1/N$ yields a Bernoulli-factorised series whose coefficients involve $ζ(1{-}2k)$; this series diverges, and our integral is its exact Borel sum. The integral representation also makes $\langle I\rangle < (d_A^2{-}1)(d_B^2{-}1)/(2N)$ manifest via a scale-inversion symmetry of the kernel. Our derivation traces the mutual information's structure to an exact decomposition of Page's entropy into a diagonal (Dirichlet) contribution and a Schur-majorisation eigenvalue correction, whose assembly into the mutual information cleanly separates classical from quantum correlations.

quant-ph↗

arXiv2026

Exact Geometric Typicality and Bipartite Entanglement from the Projected Central Limit Theorem on Hyperspheres

Starting from the exact Projected Central Limit Theorem on hyperspheres, we rederive the Beta distribution for subsystem occupation probabilities and Lubkin's purity formula from elementary hyperspherical moments, quantifying the finite-size ``platykurtic'' suppression of tails relative to the Gaussian approximation used in standard eigenstate-thermalization and typicality treatments. Our main new result concerns the bipartite quantum mutual information $\langle I(A{:}B)\rangle$ for Haar-random pure states. We show that its full asymptotic expansion in $1/N$ admits a Bernoulli-factorized form in which every order $k \ge 1$ carries the symmetric factor $(d_A^{2k}-1)(d_B^{2k}-1)$ and all higher odd-order corrections vanish identically. Through an exact algebraic reorganization of Page's formula (conjectured in Ref.~\cite{Page1993} and subsequently proven~\cite{Foong1994, SanchezRuiz1995, Sen1996}), we establish that the leading finite-size correction separates into a dominant $\mathfrak{su}(d_A) \otimes \mathfrak{su}(d_B)$ bipartite quantum coherence contribution $(d_A^2 - 1)(d_B^2 - 1)/(2N)$ and a subtracted classical-probability (Cartan $\otimes$ Cartan) contribution $(d_A - 1)(d_B - 1)/(2N)$, and we trace this separation to the difference between diagonal and eigenvalue entropies via Schur's majorisation theorem, with the dimensional counts $(d-1)$ and $(d^2-1)$ acquiring meaning through the Cartan structure of the generalised Bloch decomposition. These results admit a single non-perturbative closed form: the exact typical mutual information factors as $\langle I(A{:}B)\rangle = (d_A^2-1)(d_B^2-1)\,\mathcal{G}(d_A,d_B,d_E)$, with $\mathcal{G}$ given by an explicit Bose--Einstein integral whose asymptotic expansion in $1/N$ reproduces the Bernoulli series.

quant-ph↗

arXiv2026

Logarithmic Sobolev inequality and hypercontractivity for the Navier-Stokes Fokker-Planck operator

The stochastic incompressible Navier-Stokes equations on $\TT^3$, completed by the fluctuation-dissipation noise, have a Fokker-Planck generator that decomposes into a self-adjoint Ornstein-Uhlenbeck (dissipative) part and an antisymmetric (convective) part. We prove two results about this generator. First, the logarithmic Sobolev inequality holds with the same optimal constant as the pure Ornstein-Uhlenbeck operator, $c_\mathrm{LSI} = νλ_1$ (where $ν$ is the viscosity and $λ_1$ is the smallest nonzero eigenvalue of the Laplacian on $\TT^3$), independent of the number of retained Fourier modes. Second, the full semigroup is hypercontractive with the same rate as the Ornstein-Uhlenbeck semigroup. Both results follow from a single structural property: the convective generator is antisymmetric in $L^2(P_\mathrm{eq})$ (where $P_\mathrm{eq}$ is the Gibbs measure), and therefore contributes nothing to the Dirichlet form or the $L^q$ norm evolution. The antisymmetry is a consequence of two properties of the incompressible Navier-Stokes nonlinearity: energy conservation and phase-space volume preservation (the Liouville property). These are the same properties that underpin the fluctuation-dissipation theorem for the nonlinear Navier-Stokes equations.

math-ph↗

arXiv2026

Physical completion of the Navier-Stokes equations

The incompressible Navier-Stokes equations contain viscous dissipation but no thermal noise. I show, using a topological argument based on Poincaré's lemma, that the fluctuation-dissipation relation for the full nonlinear dynamics can be derived without the linearisation or structural assumptions that all previous derivations require. The nonlinear convective term is Hamiltonian (energy-preserving and phase-space-volume-preserving) and drops out of the Fokker-Planck equilibrium condition exactly, so the noise derived from linearised fluctuations near equilibrium is in fact exact for the full nonlinear system. This result proves, rather than assumes, the reversible/irreversible decomposition that the GENERIC framework postulates, provided Poincaré's lemma holds on the phase space. The resulting stochastic system, with a physical molecular-scale spectral cutoff, is trivially globally well-posed: a finite-dimensional stochastic differential equation with non-degenerate noise and a confining Lyapunov function. It has a unique Gibbs equilibrium and converges to it exponentially. The difficulty of the Clay Millennium Prize Problem arises entirely from two idealisations, zero temperature and infinite spectral resolution, neither of which is satisfied by any physical fluid.

cond-mat.stat-mech↗

arXiv2026

Exponential thermalisation of viscous fluids on negatively curved manifolds

The deterministic incompressible Navier-Stokes equations are physically incomplete: any viscous fluid at finite temperature must exhibit thermal fluctuations whose form is dictated by the fluctuation-dissipation relation. We formulate the stochastic Navier-Stokes equations with the kinematically selected deformation Laplacian on compact Riemannian manifolds with strictly negative Ricci curvature. The fluctuation-dissipation relation, derived from a topological (Poincaré lemma) argument, uniquely determines the noise from the viscous operator. For the spectrally truncated system, we prove that the unique stationary distribution is the Gibbs measure (Gaussian in the mode amplitudes, because the nonlinear convective terms preserve energy), and that convergence to equilibrium is exponentially fast with rate at least $2νλ_\Def$, where $ν$ is the kinematic viscosity and $λ_\Def$ is the spectral gap of the deformation Laplacian. The spectral gap satisfies $λ_\Def \geq κ^2$ when $\Ric \leq -κ^2 g$, and is independent of the volume of the domain. On flat space, the analogous thermalisation rate vanishes in the infinite-volume limit. The equilibrium velocity-velocity correlation function decays exponentially in geodesic distance, in contrast to the algebraic decay on flat space. These results provide a rigorous statistical-mechanical foundation for viscous fluids on negatively curved manifolds and illustrate how the geometry of the domain controls not only the deterministic dynamics but also the approach to thermal equilibrium.

math-ph↗

arXiv2026

Kinematic selection of the viscous stress in relativistic dissipative hydrodynamics

All standard formulations of relativistic dissipative hydrodynamics, from Eckart through Israel-Stewart to the recent BDNK framework, assume that the viscous stress depends on the shear tensor $σ_{αβ}$ and the expansion scalar $θ$ but not on the vorticity $ω_{αβ}$ or the acceleration $a_α$. We derive this structure from a Lagrangian kinematic construction on Lorentzian spacetimes, extending a recent result on Riemannian manifolds. The spatial strain rate, constructed from the rate of change of spatial inner products of Lie-dragged connecting vectors, is the spatially projected Lie derivative of the projected metric $h_{αβ} = g_{αβ} + u_αu_β$. The acceleration terms drop out exactly under spatial projection, and the vorticity cancels by symmetry. We show that material frame-indifference fails for generic Killing perturbations by an amount $δ\mathfrak{h}_{αβ} = +ε(ξ_αa_β+ ξ_βa_α)$ proportional to the acceleration, and is restored only for flow-preserving isometries. We prove that the non-relativistic limit of the BDNK equations gives the deformation Laplacian universally in the viscous sector, with the BDNK parameter dependence identified by Hegade K R, Ripley, and Yunes arising entirely from the thermal (heat-flux) sector. As an application, we derive the Weinberg gravitational-wave damping formula directly from the kinematic strain rate in a perturbed FRW spacetime.

gr-qc↗

arXiv2025

Equality of ADM mass and generalized Komar energy in asymptotically-flat dynamical spacetimes

We find a relation between the ADM mass and a generalized Komar energy in asymptotically-flat spacetime. We do not need to assume the existence of either a Killing or even asymptotically-Killing vector field. Instead, our generalized Komar energy is constructed from the normal evolution vector (the lapse function times the future-directed unit normal to the spacelike hypersurfaces on which the ADM mass is measured). We find equality between the ADM mass and this generalized Komar energy even for dynamical asymptotically-flat spacetimes provided the 3-dimensional Einstein tensor drops off quickly enough at spatial infinity, in particular, whenever ${}^{(3)}G_{ij}=o(r^{-3})$. No additional assumptions are required for equality. As this generalized energy is fully covariant, it may provide a powerful tool for analyzing energy content in dynamical spacetimes containing compact objects.

gr-qc↗

arXiv2025

Black holes, fast scrambling and the breakdown of the equivalence principle

Under reasonable assumptions, black holes have been argued to form firewalls, burning up anything crossing their horizons. This argument finds that a firewall would appear very late in a black hole's lifetime, when Hawking radiation has caused the horizon to shrink to one-half its original area. For stellar-mass black holes, this process surpasses the universe's current age and so no such black hole would currently possess a firewall. However, black holes have recently been conjectured to scramble their interior degrees-of-freedom, with a scrambling time scale comparable to the time it takes light to travel a Schwartzschild radius' distance. We prove that local observers will already experience a firewall from the scrambling time onwards after the black hole's formation. Here `local' means that the observer couples to fewer than one-half the black hole's total interior `qubits.' Indeed, for observers to fail to be local in this manner, it would mean that they couple to more `qubits' within such black holes than exist in all the stars of the observable universe. Therefore we find that if black holes are indeed fast scramblers, then every astrophysical black hole in the universe will already have a fully developed firewall for any local physical process.

gr-qc↗