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

arXiv:2606.09984 · hep-lat, cond-mat.stat-mech, hep-ph, quant-ph · Submitted 2026-06-08 · Read on arXiv

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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.

Xiaojun Yao

University of Washington

hep-lat, cond-mat.stat-mech, hep-ph, quant-ph

Submitted: 2026-06-08

Updated: 2026-10-17

Comments: 24 pages, 23 figures; v2: accepted version

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 80/100

The gist: 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

Key concepts

Real-time Symmetric Correlation Function: This is a mathematical tool used to measure how the stress-energy tensor fluctuates over time, specifically comparing its state at one point in time (t) with its state at an earlier reference point (0). It helps researchers track the evolution of energy and momentum in a system during non-equilibrium processes.
Umklapp Processes: These are specific scattering events that occur in certain field theories, often leading to thermalization or dissipation. The paper shows that these processes are suppressed when the theory is in its 'scaling region,' meaning they don't dominate the dynamics at those specific conditions.
Emergent Sound Modes: Sound modes are characteristic waves found in relativistic hydrodynamics, representing collective excitations like sound propagation. The study observed these modes appearing in the correlation functions at late times and long wavelengths, suggesting that hydrodynamic behavior emerges naturally from the underlying microscopic lattice theory.
Scaling Region: In this context, the scaling region is a specific set of parameters (like J and hx) where the lattice theory behaves in a predictable manner. The paper investigates dynamics within this region to see how it differs from other regimes where dissipation might dominate.

Terminology

Bulk Viscosity — 1× in this paper · explained in 2 episodes

Summary

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.

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. (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. (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. 4, inside the scaling region hz = −0.15, two peaks emerge in G10s(ω, k) that can be well described by a Lorentzian shape b0(ω−b1) 2+b2 squared, as indicated by the black dashed lines These emergent peaks hint at the possibility of finding the sound modes if one can take hz even closer to 0 while still maintaining the scaling region The fitting results for b1 and b2 do not follow the scaling laws expected for a sound mode, i.e., b1 ∝ 2 sin(k/2) and b2 ∝ 4 sin 2(k/2) This is attributed to the small lattice size that limits us from going into the long wavelength regime where hydrodynamics applies

Extraction of Transport Coefficients

The bulk viscosity and speed of sound are extracted by fitting the late-time behavior of G10s(t, k) to the damped oscillation form b0e − γζ k 2t squared cos(cskt + ϕ), (40) The analysis shows that the fitted parameter values for cs/c = 0.759 ± 0.019 and ζ/s = 14.192 ± 0.899 in the continuum limit The ratio of bulk viscosity to entropy density is found to be ζ/s = 14.19 ± 0.90 at the temperature T ≈ 7.14 in units of the lowest stable particle’s mass

Continuum Limit and Final Results

In the continuum limit, the extrapolated speed of sound is cs/c = 0.759 ± 0.019 and the ratio of bulk viscosity to entropy density ζ/s = 14.192 ± 0.899 for the Ising field theory defined with ξlat = 0.01 at the temperature T /m1 ≈ 7.14 The study demonstrates the utility of real-time lattice Hamiltonian simulation for describing hydrodynamization and calculating transport coefficients nonperturbatively

Future Directions

In future work, it will be interesting to study the speed of sound and bulk viscosity as a function of temperature for different Ising field theories, determined by different values of ξlat on the lattice One may also treat the 1+1D Ising field thermal system as a gas of weakly coupled scalar particles and perturbatively calculate the bulk viscosity as done in 3 + 1D for one scalar particle species

Acknowledgments

The authors thank Larry Yaffe for inspiring discussions This work was supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, IQuS under Award Number DOE (NP) Award DE-SC0020970 via the program on Quantum Horizons: QIS Research and Innovation for Nuclear Science This research was supported in part by grant NSF PHY-2309135 to the Kavli Institute for Theoretical Physics (KITP) This research used resources of the National Energy Research Scientific Computing Center (NERSC), a Department of Energy Office of Science User Facility, under NERSC award NP-ERCAP0032083 This work was enabled, in part, by the use of advanced computational, storage and networking infrastructure provided by the Hyak supercomputer system at the University of Washington All fits are performed via the curve fit method in SciPy

Appendix A (MPS Calculations)

The MPS setup requires a very large bond dimension to maintain the periodic boundary conditions In practice, we want to extract physical observables before the perturbation reaches the boundaries in order to minimize the effect of the boundary conditions on the extracted observables

Appendix B (Other Fitting Results)

The fitted value of b1 as a function of the momentum 2 sin(k/2) is cs = 1.4256 ± 0.0020 up to four digits The fitting result from the central values of the parameters is shown as the red dashed line while the red band is obtained by considering the uncertainty associated with the a0 parameter, which has the physical meaning of ζ/s in the continuum limit

Conclusion

The current work demonstrates that real-time lattice Hamiltonian simulation is useful for studying nonequilibrium dynamics and extracting transport coefficients from first principles <ref:2606.

Improvements for AI systems

The AI system can perform first-principles nonperturbative simulation of hydrodynamization by extracting a transport coefficient requires computing the real-time correlator at late time when hydrodynamics becomes applicable. This allows for studying non-hydro modes from first principles in systems like QCD, which current studies often assume immediate thermalization.
The system can calculate the ratio of bulk viscosity to entropy density, such as the extracted result of ζ/s = 14.19±0.90, by fitting the momentum dependence of the damping rate "b2" to extract parameters consistent with hydrodynamic predictions, as shown in Table V.
The AI system can determine the speed of sound in a continuum limit by fitting the extracted damped oscillation to a Lorentzian shape, finding that for this Ising field theory it yields cs/c = 0.759 ± 0.019.