Hamiltonian truncation approach to quenches in the Ising field theory

TL;DR

The paper develops and validates a non-perturbative truncated Hilbert space method (TFSA) to study real-time quantum quenches in the 1+1D scaling Ising field theory. It benchmarks integrable quenches against known results for energy, work statistics, Loschmidt echo, and order-parameter dynamics, then explores integrability-breaking quenches with a longitudinal field, uncovering confinement-driven meson spectra in the ferromagnetic phase and perturbative magnon dynamics in the paramagnetic phase. The authors demonstrate that a few low-lying states dominate the post-quench dynamics, enabling quench spectroscopy of the post-quench spectrum and form factors, and they validate TFSA against iTEBD lattice simulations near criticality. The work establishes TFSA as a versatile tool for continuum quantum field theories, with potential extensions to other models, and highlights the universal, lattice-continuum connection in non-equilibrium settings.

Abstract

In contrast to lattice systems where powerful numerical techniques such as matrix product state based methods are available to study the non-equilibrium dynamics, the non-equilibrium behaviour of continuum systems is much harder to simulate. We demonstrate here that Hamiltonian truncation methods can be efficiently applied to this problem, by studying the quantum quench dynamics of the 1+1 dimensional Ising field theory using a truncated free fermionic space approach. After benchmarking the method with integrable quenches corresponding to changing the mass in a free Majorana fermion field theory, we study the effect of an integrability breaking perturbation by the longitudinal magnetic field. In both the ferromagnetic and paramagnetic phases of the model we find persistent oscillations with frequencies set by the low-lying particle excitations not only for small, but even for moderate size quenches. In the ferromagnetic phase these particles are the various non-perturbative confined bound states of the domain wall excitations, while in the paramagnetic phase the single magnon excitation governs the dynamics, allowing us to capture the time evolution of the magnetisation using a combination of known results from perturbation theory and form factor based methods. We point out that the dominance of low lying excitations allows for the numerical or experimental determination of the mass spectra through the study of the quench dynamics.

Figures (19)

• Figure 2.1: Schematic illustration of the quenches in the parameter 2D space spanned by the transverse and longitudinal couplings. To describe the quench from the Hamiltonian $H(M_{0},0)$ (blue dot) to the Hamiltonian $H(M,h)$ (green dot), one can use either the basis defined by the eigenvectors of pre-quench Hamiltonian $H(M_0,0)$ (blue dot) or the "intermediate" Hamiltonian $H(M,0)$ (yellow dot), both of which correspond to free fermions.
• Figure 4.1: Time-dependent expectation value of the energy operator $\varepsilon=i\!:\!\bar{\psi}\psi\!:$ after integrable mass quenches in the ferromagnetic phase at system size $ML=40.$ Left panel: $M_0=1.5M$ with cut-offs (from bottom to top) $\Lambda=6M,10M,14M.$ The TFSA results are shown in blue while the analytical result \ref{['eq:epsilontheory']} is plotted in red. Right panel: $M_0=3M$ with cut-off $\Lambda=14M.$
• Figure 4.2: The statistics of work $P(W)$ for integrable mass quenches within the ferromagnetic phase. Here, for convenience, the work is defined relative to the post-quench ground state as opposed to Eq. \ref{['eq:PW']}, which amounts to a horizontal shift by $E_0-\mathcal{E}_0.$ The dimensionless length of the system is $ML=40$. The blue stars show the result of the TFSA with cut-off $\Lambda=14M$ while the red open circles denote the theoretical result corresponding to Eq. \ref{['eq:initialStateExpansion']}.
• Figure 4.3: The Loschmidt echo $\mathcal{L}(t)$ for integrable mass quenches in the ferromagnetic phase for system size $ML=40$. The dashed lines are raw TFSA data, the solid red line is the cut-off extrapolated TFSA result, while the exact analytical result \ref{['eq:Lecho_prediction']} is shown in green dots. The thin dashed line shows the TFSA result obtained by using the exact overlaps. The small deviations towards the end of the curve are artifacts of the smoothening procedure.
• Figure 4.4: Time-dependent expectation value of the order parameter $\langle\sigma(t)\rangle$ after integrable mass quenches in the ferromagnetic phase for system size $ML=40$. The finite cut-off energy results are shown in dashed lines, the cut-off extrapolated value (solid red line) is compared to the theoretical result \ref{['eq:sigma_theor']} for the large time asymptotics (green dots). The TFSA results using the exact overlaps are shown in thin dashed line.