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arXiv:2609.21846

Sine-Gordon Model with Bosonic Tensor Networks: Continuum Matching and Soliton Scattering

Florian Hechenberger, Tommaso Rainaldi, Felix Ringer

quant-phhep-phhep-thnucl-thSubmitted Sep 18, 2026• Announced Sep 21, 2026• 0 citations
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Abstract

We use bosonic tensor networks to connect the lattice sine-Gordon model in the Hamiltonian formulation quantitatively to its continuum theory. Matching the lattice vertex operator to its conformal normalization at the free-boson ultraviolet fixed point yields the exact relation between the bare lattice coupling and the renormalized continuum mass parameter. The soliton mass then approaches Zamolodchikov's exact continuum prediction throughout the studied parameter range, without adjustable parameters. Using uniform matrix product states and a quasiparticle ansatz, we also recover the relativistic soliton dispersion and the two lightest breather masses at the percent level. We simulate real-time collisions of Gaussian soliton-antisoliton wave packets near a reflectionless point and extract the Wigner spatial displacement. We compare this displacement with the exact continuum prediction obtained from the momentum derivative of the transmission phase, recovering its characteristic rapidity dependence. Our bosonic simulations provide a foundation for nonintegrable extensions, offer lessons for renormalization in other Hamiltonian lattice field theories, including gauge theories, and provide benchmarks for continuous-variable quantum simulations.

Comments
13 pages, 5 figures

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The Exact Quantum Sine-Gordon Field Equation and Other Non-Perturbative Results

H. Babujian, M. Karowski, Technische Univ. Berlin (Germany). Sonderforschungsbereich 288 -Differentialgeometrie und Quantenphysik

Using the methods of the "form factor program" exact expressions of all matrix elements are obtained for several operators of the quantum sine Gordon model: all powers of the fundamental bose field, general exponentials of it, the energy momentum tensor and all higher currents. It is found that the quantum sineGordon field equation holds with an exact relation between the "bare" mass and the renormalized mass. Also a relation for the trace of the energy momentum is obtained. The eigenvalue problem for all higher conserved charges is solved. All results are compared with Feynman graph expansions and full agreement is found. PACS: 11.10.-z; 11.10.Kk; 11.55.Ds Keywords: Integrable quantum field theory, Form factors The classical sine-Gordon model is given by the wave equation 2'(t; x) + ff fi sin fi'(t; x) = 0: Since Coleman [1] found the wonderful duality between the quantum sine-Gordon and the massive Thirring model a lot of effort has been made to understand this quantum field theo...

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We use bosonic tensor networks to connect the lattice sine-Gordon model in the Hamiltonian formulation quantitatively to its continuum theory. Matching the lattice vertex operator to its conformal normalization at the free-boson ultraviolet fixed point yields the exact relation between the bare lattice coupling and the renormalized continuum mass parameter. The soliton mass then approaches Zamolodchikov's exact continuum prediction throughout the studied parameter range, without adjustable parameters. Using uniform matrix product states and a quasiparticle ansatz, we also recover the relativistic soliton dispersion and the two lightest breather masses at the percent level. We simulate real-time collisions of Gaussian soliton-antisoliton wave packets near a reflectionless point and extract the Wigner spatial displacement. We compare this displacement with the exact continuum prediction obtained from the momentum derivative of the transmission phase, recovering its characteristic rapidity dependence. Our bosonic simulations provide a foundation for nonintegrable extensions, offer lessons for renormalization in other Hamiltonian lattice field theories, including gauge theories, and provide benchmarks for continuous-variable quantum simulations.
In the quantum inverse scattering method, the sine-Gordon model is solved on a lattice, and its continuum and infinite-volume limits become uniform only after the length of the box is rescaled. We show that this rescaling, supplemented by an on-shell condition, defines a renormalization scheme. The whole dependence on the cut-off is absorbed into a factor ZLZ_LZL​ multiplying the length of the box, with an exponent fixed by the scaling dimension of the vertex operator; neither the field nor the coupling βββ is renormalized, no subtraction scale is introduced, and the mass parameter receives a finite renormalization, fixed by requiring that the lightest breather has the mass of the boson in the linearized theory. The soliton and breather masses follow from the eigenvalues of the monodromy operator as functions of parameters that do not run, and reduce to the classical and semiclassical results as β→0β\rightarrow0β→0. In particular, the mass ratios are produced directly from the regularized construction, while the relation between the lattice parameters and the physical scale is absorbed into ZLZ_LZL​ and never needs to be computed. The scheme is a reparameterization of the theory which makes explicit the interplay between integrability and renormalization. We compare it with Coleman's normal ordering and with conformal perturbation theory, and we argue that it extends to ultralocal integrable theories with a massive spectrum, possibly including asymptotically free models that admit an ultralocal lattice regularization.
Simulating the real-time dynamics of quantum field theories (QFTs) is one of the most promising applications of quantum simulators. Regularizing a bosonic QFT for quantum simulation purposes typically involves a truncation in Hilbert space in addition to a discretization of space. Here, we discuss how to perform such a regularization of scalar QFTs by explicitly constructing suitable many-body lattice Hamiltonians using multilevel or qudit systems and show that this enables quantitative predictions in the continuum limit by extrapolating results obtained for large-spin models. With extensive matrix-product-state simulations, we numerically demonstrate the sequence of extrapolations that leads to quantitative agreement of observables for the integrable sine-Gordon (sG) QFT. We further show how to prepare static and moving-soliton excitations and we analyze their scattering dynamics in the continuum limit, in agreement with a semiclassical model and with quantitative analytical predictions. Finally, we illustrate how a nonintegrable perturbation of the sG model gives rise to dynamics reminiscent of string breaking and plasma oscillations in gauge theories. Our methods are directly applicable in state-of-the-art analog quantum simulators, opening the door to quantitatively investigating a wide variety of scalar-field theories and tackling long-standing questions in nonequilibrium QFT such as the fate of the false vacuum.
The linear problem associated with the Lax operator of the classical sine-Gordon theory can be recast into the monodromy matrix form that can be extended to quantum theory as well. Product of the quantum monodromy matrices will have contributions from the singularities arising out of the operator product expansions of sine-Gordon field. This enables one to find the star-triangle relations. This is a generalization of the method used by Thacker for the non-relativistic nonlinear Schrödinger field theory. In the infinite volume limit, it leads to an unambiguous description of the algebra involving the scattering data operators. Starting from a vacuum the module of physical states are constructed by the application of chains of the scattering operators and they turn out to have definite eigenvalues of energy and momentum. The mass-spectra and two-particle S matrix elements can be calculated straightforwardly and they coincide with the well-known results of Zamolodchikov and Zamolodchikov. It is also shown that the particle states can be created by a generalized Jordon-Wigner transformation of the free fermionic creation operators — thus establishing a link between the S matrix element and certain co-cycles.
Using the results of previous investigations on sine-Gordon form factors, exact expressions of all breather matrix elements are obtained for several operators: all powers of the fundamental Bose field, general exponentials of it, the energy–momentum tensor and all higher currents. Formulae for the asymptotic behaviour of bosonic form factors are presented which are motivated by Weinberg's power counting theorem in perturbation theory. It is found that the quantum sine-Gordon field equation holds, and an exact relation between the 'bare' mass and the renormalized mass is obtained. Also a quantum version of a classical relation for the trace of the energy–momentum is proved. The eigenvalue problem for all higher conserved charges is solved. All results are compared with perturbative Feynman graph expansions and full agreement is found.
We study Gaussian continuous tensor network states (GCTNS) - a finitely parameterized subclass of Gaussian states admitting an interpretation as continuum limits of discrete tensor network states. We show that, at short distance, GCTNS correspond to vacua of free Lifshitz theories with even dynamical exponent, establishing a connection between certain entanglement properties of the two. Two schemes for approximating ground states of free bosonic field theories using GCTNS are presented: rational approximants to the exact dispersion relation and Trotterized imaginary-time evolution. We apply them to Klein-Gordon theory and characterize the resulting GCTNS, identifying the scale at which deviations from its conformal short-distance behavior appear. In particular, we construct an optimal rational approximation for which, at fixed accuracy, the bond dimension grows only logarithmically with the UV cutoff. These results provide a simple and analytically controlled setting to assess the strengths and limitations of GCTNS as variational ansätze for relativistic quantum fields.
Using the methods of the "form factor program" exact expressions of all matrix elements are obtained for several operators of the quantum sine Gordon model: all powers of the fundamental bose field, general exponentials of it, the energy momentum tensor and all higher currents. It is found that the quantum sineGordon field equation holds with an exact relation between the "bare" mass and the renormalized mass. Also a relation for the trace of the energy momentum is obtained. The eigenvalue problem for all higher conserved charges is solved. All results are compared with Feynman graph expansions and full agreement is found. PACS: 11.10.-z; 11.10.Kk; 11.55.Ds Keywords: Integrable quantum field theory, Form factors The classical sine-Gordon model is given by the wave equation 2'(t; x) + ff fi sin fi'(t; x) = 0: Since Coleman [1] found the wonderful duality between the quantum sine-Gordon and the massive Thirring model a lot of effort has been made to understand this quantum field theo...