Tame the Umklapp Processes in Real-Time Lattice Simulation for Hydrodynamics: An Ising Field Theory Study

Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: We're looking at the paper: "Tame the Umklapp Processes in Real-Time Lattice Simulation for Hydrodynamics".

Mira: Here's the gist of the paper. The gist The study calculates the real-time symmetric correlation function of the stress-energy tensor for a nonintegrable Ising field theory using lattice Hamiltonian simulation, finding that Umklapp processes are suppressed in the scaling region and emergent sound modes of relativistic hydrodynamics appear at late time and long wavelength.

Kai: Lattice Formulation and Theory The paper investigates the 1+1-dimensional (1 + 1D) Ising field theory, which is obtained by deforming a 1 + 1D Ising conformal field theory that describes a free massless Majorana fermion The action of the theory can be written as SIFT = SICFT − τ Z d 2x σ(x), (1) The Ising field theory is characterized by the dimensionless quantity ξ = τ h 8/15.

Mira: (2) This family of Ising field theories can be formulated on a spatial lattice in the scaling region of the Ising model with both transverse and longitudinal fields H = L X−1 i=0 Jσz i σz i+1 + hxσ x i + hzσ z i (3) The scaling region is reached by taking J = −1, hz → 0 − and hz 8/15L → ∞ with hx = −1 − ξlathz 8/15 (4) For finite ξlat ≠ 0, the theory is non-integrable and the number of stable particles in the continuum theory drops from three to one as ξlat increases Stress-Energy Tensor Correlation Function The core observable computed is the real-time symmetric correlation function of stress energy tensors Gµν s (t, x) = ⟨Tµν(t, x), T µν(0, 0)⟩T = Tr

Tµν(t, x), T µν(0, 0)ρT: , (5) To compute this on the lattice, the energy density is given by T 00(0, i) → Hi = J squared σz(i−1)σz i + σz i σz(i+1) + hxσ x i + hzσ z i, (6) The total momentum operator Jtot = L−1 i=0 Ji is not conserved explicitly but its 2-norm can be bounded as

H, Jtot: 2 ≤ 4JhxhzL.

Kai: (12) In the continuum limit, this norm becomes zero, meaning the total momentum becomes conserved Suppression of Umklapp Processes The paper explicitly demonstrates the suppression of Umklapp processes in the scaling region by computing the symmetric correlator in Fourier space Gµν s (ω, k) = X p X En(p) X Em(p+k) 2πL ∆ωZ ⟨En(p)TµνEm(p + k)⟩2 × h e −βEn(p) + e −βEm(p+k)< ∆ω 2 ≈ X p X En(p) 2πL Z f 2µν(En, p, ω, k) e −βEn(p) 1 + e −βω (28) Outside the scaling region, the symmetric correlator shows a diffusive peak with a form bω+4Dε sin 2(k/2) in the range ω ∈

0, 0.13: Inside the scaling region hz = −0.3, the diffusive transport peak disappears This shows the suppression of the Umklapp processes in the scaling region of the lattice theory Emergent Hydrodynamic Sound Modes In Fig.

Title and authors: Kai: So we're looking at this paper, "Tame the Umklapp Processes in Real-Time Lattice Simulation for Hydrodynamics: An Ising Field Theory Study". It sounds like they're tackling how to get realistic fluid behavior out of these lattice simulations.

Mira: Yeah, and the authors are using a one plus1D Ising field theory that comes from deforming a one plus1D Ising conformal field theory describing a free massless Majorana fermion <ref:2606.09984#pg1>. That's where things get interesting because it connects quantum field theory to something that looks like hydrodynamics.

Lev: And they use lattice Hamiltonian simulation with classical exact diagonalization and matrix product state tensor network methods to compute the stress-energy tensor correlation function. That tells us how the system evolves in time on a discrete grid, which is pretty fundamental for testing any hydrodynamic idea.

Kai: The real point here seems to be that they're calculating the real-time symmetric correlation function of the stress-energy tensor, G mu nu s(t, x), to see how that system behaves dynamically.

Mira: And what they found is actually quite specific: in the scaling region of this lattice theory, they managed to suppress Umklapp processes while emergent sound modes of relativistic hydrodynamics appear at long wavelength and late time.

Lev: That suppression of Umklapp processes is key because those processes usually mess up the continuum limit when you're trying to study things like fluid dynamics.

The paper's summary: Kai: To unpack that, the paper sets up this one plus1D Ising field theory on a spatial lattice with fields H = L-one sum i=zero L-one J sigma z i sigma z i+one + hx sigma x i + hz sigma z i <ref:2606.09984#pg2,1+1D Ising field theory>. They then define the scaling region by taking J to be-one hz to zero minus, and hz eight/15L to infinity with a specific relationship for hx <ref:2606.09984#pg2>.

Mira: And they characterize this family of Ising field theories with a dimensionless quantity xi = tau h eight/fifteen which tells us how the theory behaves depending on those parameters <ref:2606.09984#pg2,this family of Ising field theories>. The paper says that if you have a finite xi value not equal to zero, then the theory becomes nonintegrable.

Lev: That nonintegrability is important because it means that the number of stable particles in the continuum theory drops from three down to just one as xi increases, so we have fewer degrees of freedom to work with when we try to take that limit.

Kai: Then they focus on the total momentum operator, J tot = L-one sum i=zero L-one J i, and they show that while it's not conserved explicitly, its two-norm is bounded by

H, J tot: squared 4JhxhzL <ref:2606.09984#pg2>.

Mira: And they point out that in the continuum limit, this norm goes to zero. That means the total momentum becomes conserved in the continuum limit, which is crucial for studying hydrodynamics on a lattice.

The paper's improvements: Kai: The main contribution they highlight is demonstrating that Umklapp processes are suppressed inside this scaling region, which they show by looking at the Fourier space correlator G mu nu s(omega, k). They find that outside the scaling region, you see a diffusive peak with a form b omega + 4D epsilon two(k/two) in the range omega in zero zero point one three.

Mira: But inside this scaling region, specifically when hz = -zero point three, that diffusive transport peak disappears entirely. This disappearance is what signals the suppression of Umklapp processes in the scaling region of this lattice theory.

Lev: So they're showing that you can isolate a regime where those tricky non-integrable effects are suppressed, which is a big deal because it't hard to get clean continuum physics when those things are present.

Kai: And then they look at the sound modes emerging in Fig four inside the scaling region when hz = -zero point one five. They find two peaks described by a Lorentzian shape, b zero(omega-b one) 2+b two squared, which suggests they might be sound modes if you could push hz even closer to zero while staying in the scaling region <ref:2606.09984#pg2>.

Mira: But the fitting results for those parameters, b one and b two don't follow the expected scaling laws that we'd see for a sound mode, like b one proportional to two (k/two) and b two proportional to four two(k/two) <ref:2606.09984#pg2>.

Conclusion: Kai: So to wrap up this paper, the study uses real-time lattice Hamiltonian simulation to calculate transport coefficients nonperturbatively for an Ising field theory with xi lat = zero point zero one. They extrapolate the speed of sound to be c s/c = zero point seven five nine plus or minus zero point zero one nine and the ratio of bulk viscosity to entropy density is zeta/s = fourteen point one nine two plus or minus zero point eight nine nine at a temperature T/m one about seven point one four <ref:2606.09984#pg1,ratio of bulk viscosity to entropy density is>.

Mira: That ratio of bulk viscosity to entropy density, zeta/s = fourteen point one nine plus or minus zero point nine zero, is what they extracted by fitting the late-time behavior of G 10s(t, k) to the damped oscillation form b 0e-gamma zeta k 2t squared (c skt + phi).

Lev: So that means if you look at that zeta/s ratio, it's about fourteen point two compared to the entropy density in the continuum limit <ref:2606.09984#pg2>. That's a concrete number for a transport coefficient derived from first principles, which is hard to get otherwise.

Kai: It shows how useful real-time lattice Hamiltonian simulation is for describing hydrodynamization and calculating these coefficients nonperturbatively, especially when dealing with systems that are nonintegrable.

Mira: Overall, this work on "Tame the Umklapp Processes in Real-Time Lattice Simulation for Hydrodynamics: An Ising Field Theory Study" gives us concrete values for c s/c and zeta/s based on first principles calculations for this field theory.

Lev: It's a solid piece of work that shows how these simulation methods can give you those transport coefficients directly, without relying on some sort of approximation that might break down at higher densities or temperatures.