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arXiv:2606.09984

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

Xiaojun Yao

hep-latcond-mat.stat-mechhep-phquant-phSubmitted Jun 8, 2026• Announced Jun 10, 2026• 0 citations• Revised Oct 17, 2026 (v2)
v1 · Jun 8, 2026v2 · Oct 17, 2026
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Abstract

We calculate the real-time symmetric correlation function of the stress-energy tensor for a near-integrable Ising field theory consisting of three stable scalar particles via lattice Hamiltonian simulation. Using classical exact diagonalization and the matrix product state tensor network methods, we find that in the scaling region of the lattice theory, Umklapp processes that break continuum momentum conservation are suppressed and the sound modes of relativistic hydrodynamics emerge at long wavelength and late time in non-equilibrium dynamics. The extracted ratio of bulk viscosity to entropy density is $ζ/s=3.06\pm 0.68$ and the speed of sound is $c_s/c=0.76 \pm 0.02$ at the temperature $T\approx 7.14$ in units of the lowest stable particle's mass. Our study demonstrates the utility of real-time lattice Hamiltonian simulation for describing hydrodynamization and calculating transport coefficients nonperturbatively.

Comments
24 pages, 23 figures; v2: accepted version

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We calculate the real-time symmetric correlation function of the stress-energy tensor for a near-integrable Ising field theory consisting of three stable scalar particles via lattice Hamiltonian simulation. Using classical exact diagonalization and the matrix product state tensor network methods, we find that in the scaling region of the lattice theory, Umklapp processes that break continuum momentum conservation are suppressed and the sound modes of relativistic hydrodynamics emerge at long wavelength and late time in non-equilibrium dynamics. The extracted ratio of bulk viscosity to entropy density is ζ/s=3.06±0.68ζ/s=3.06\pm 0.68ζ/s=3.06±0.68 and the speed of sound is cs/c=0.76±0.02c_s/c=0.76 \pm 0.02cs​/c=0.76±0.02 at the temperature T≈7.14T\approx 7.14T≈7.14 in units of the lowest stable particle's mass. Our study demonstrates the utility of real-time lattice Hamiltonian simulation for describing hydrodynamization and calculating transport coefficients nonperturbatively.
We search for emergent hydrodynamic modes in real-time Hamiltonian dynamics of 2 + 1 -dimensional SU(2) lattice gauge theory on a quasi-one-dimensional plaquette chain, by numerically computing symmetric correlation functions of energy densities on lattice sizes of about 20 with the local Hilbert space truncated at j max = 1 2 . Because of the Umklapp processes, we only find a mode for energy diffusion. The symmetric correlator exhibits transport peak near zero frequency with a width approximately proportional to momentum squared at small momentum, when the system is fully quantum ergodic, as indicated by the eigenenergy level statistics. This transport peak leads to a power-law t − 1 2 decay of the symmetric correlator at late time, also known as the long-time tail, as well as diffusionlike spreading in position space. We also introduce a quantum algorithm for computing the symmetric correlator on a quantum computer and find it gives results consistent with exact diagonalization when tested on the IBM emulator. Finally we discuss the future prospect of searching for the sound modes.
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