---
title: Fully Discrete Massive Thirring Model
url: https://www.emergentmind.com/topics/fully-discrete-massive-thirring-model
type: topic
---

# Fully Discrete Massive Thirring Model

Searching arXiv for recent papers on fully discrete Massive Thirring model and related discretizations.
The fully discrete Massive Thirring Model denotes lattice analogues of the classical massive Thirring system in which both independent variables are discretized, most commonly in light-cone coordinates, while the integrable structure is preserved through a discrete zero-curvature condition, Hirota bilinearization, or hierarchy reduction. In the recent literature, the subject is represented most explicitly by a bilinear/KP-Toda construction of a fully discrete light-cone model and by a Lax-pair construction that also produces an associated Yang–Baxter map [2507.09803], [2408.13913]. Closely related work on semi-discrete systems, Darboux transformations, and lattice Hamiltonian regularizations clarifies that the phrase “fully discrete” is used in more than one technical sense across integrable-systems and lattice-field-theory communities [1805.07854], [1710.09993].

## 1. Continuous origin and coordinate dependence

The continuous massive Thirring model appears in several equivalent normalizations and coordinate systems. In one light-cone normalization, the model is
\[
\mathrm{i}u_x+v+u|v|^2=0,\qquad \mathrm{i}v_t+u+v|u|^2=0,
\]
which is the continuous target of the fully discrete bilinear construction [2507.09803]. In another Lax-pair normalization, the light-cone system is written for fields \(q,r,u,v\) as
\[
iq_t+u-quu=0,\qquad ir_t-v-vqr=0,
\]
\[
iu_x-q+qru=0,\qquad iv_x+r-vqr=0,
\]
and, under the scalar reduction \(r=q^*\), \(v=u^*\), becomes
\[
iq_t+u-\frac{u}{2}|q|^2=0,\qquad iu_x-q+\frac{q}{2}|u|^2=0.
\]
This is the continuous system recovered from the fully discrete Lax construction [2408.13913].

The coordinate choice is not a superficial matter. A separate line of work emphasizes laboratory coordinates,
\[
\mathrm{i}u_t + \mathrm{i}u_x + v = |v|^2 u, \qquad \mathrm{i}v_t - \mathrm{i}v_x + u = |u|^2 v,
\]
and shows that previously known integrable discretizations were mainly in characteristic variables rather than in the coordinates natural for the Cauchy problem [1805.07854]. This establishes that “fully discrete Massive Thirring Model” does not refer to a unique canonical lattice equation, but to a family of integrable discretizations whose precise form depends on normalization, reduction, and coordinate system.

## 2. Bilinear and KP-Toda construction of a fully discrete light-cone model

A direct fully discrete light-cone model is given by the pair
\[
\frac{\rm i}{a}(u^l_{k+1}-u^l_k)+v^l_{k+1}+u^l_{k+1}v^l_{k+1}\tilde v^l_k=0,
\]
\[
\frac{\rm i}{b}(v^{l+1}_k-v^l_k)+u^{l+1}_k+v^{l+1}_k u^{l+1}_k\tilde u^l_k=0,
\]
where \(k\) is the discrete spatial index, \(l\) is the discrete temporal index, and \(a,b\) are the lattice spacings in the two directions [2507.09803]. The dependent variables satisfy
\[
u^l_k=\frac{g^l_k}{\tilde f^l_k},\qquad v^l_k=\frac{h^l_k}{f^l_k},\qquad \tilde u^l_k=\frac{\tilde g^l_k}{f^l_k},\qquad \tilde v^l_k=\frac{\tilde h^l_k}{\tilde f^l_k},
\]
and the field equations bilinearize into
\[
\frac{\rm i}{a}(g^l_{k+1}f^l_k-g^l_k f^l_{k+1})+h^l_{k+1}\tilde f^l_k=0,
\]
\[
\frac{\rm i}{a}\bigl(f^l_{k+1}\tilde f^l_k-f^l_k\tilde f^l_{k+1}\bigr)+h^l_{k+1}\tilde h^l_k=0,
\]
\[
\frac{\rm i}{b}\bigl(h^{l+1}_k\tilde f^l_k-h^l_k\tilde f^{l+1}_k\bigr)+g^{l+1}_k f^l_k=0,
\]
\[
\frac{\rm i}{b}\bigl(f^{l+1}_k\tilde f^l_k-f^l_k\tilde f^{l+1}_k\bigr)=g^{l+1}_k\tilde g^l_k.
\]

The derivation proceeds from determinant tau functions of the two-component discrete KP-Toda hierarchy, followed by a period-2 reduction
\[
q_i=\bar p_i,\qquad \bar q_i=p_i,\qquad a_1=-a_2=a,\qquad b_1=-b_2=-b,
\]
and then by a complex-conjugate reduction with
\[
\mu=-\mathrm{i},\qquad \nu=1,\qquad \bar p_i=p_i^*.
\]
The final identification of hierarchy indices,
\[
k_1-k_2=k,\qquad l_1-l_2=l,
\]
produces the fully discrete massive Thirring equations themselves [2507.09803].

This hierarchy-based route is conceptually distinct from the use of auxiliary discrete indices in continuous KP reductions. In the rogue-wave literature, the tau functions \(\tau_{n,k,l}\) and the shift \((n,k,l)\mapsto(n+1,k+1,l-1)\) appear only as internal reduction machinery, and no discrete or fully discrete MT equation is obtained [2208.03747]. The fully discrete model above is therefore a genuine lattice equation rather than a continuous MT system written with auxiliary hierarchy labels.

## 3. Lax-pair formulation, discrete zero curvature, and Yang–Baxter structure

A second explicit construction starts from a Lax-pair representation in light-cone coordinates and discretizes both directions through discrete spatial and temporal Lax matrices of the same structural type [2408.13913]. The discrete compatibility condition is
\[
L(q_n,r_n;\mu,\nu,\xi,\eta)\,L(u_n,v_n;\alpha,\beta,\gamma,\delta)
=
L(u_{n+1},v_{n+1};\alpha,\beta,\gamma,\delta)\,L(q_n,r_n;\mu,\nu,\xi,\eta),
\]
which is the fully discrete zero-curvature equation. From this identity, the paper derives four rational matrix equations, equations (2.29)–(2.32), for the discrete evolution of \(q_n,r_n,u_n,v_n\).

The model admits a Hermitian reduction
\[
r_n=q_n^\dagger,\qquad v_n=u_n^\dagger,
\]
with parameter constraints
\[
\gamma=\alpha^*,\qquad \delta=\beta^*,\qquad \xi=\mu^*,\qquad \eta=\nu^*.
\]
For the scalar case, the parameter choice
\[
\alpha=-\gamma=\frac{2i}{h},\qquad \beta=\delta=1,\qquad \mu=1,\qquad \xi=-\eta=\frac{2i}{\Delta}
\]
yields the fully discrete scalar MTM, written explicitly in the paper as the rational lattice equations (2.38)–(2.39). The same framework also supplies a binary Bäcklund–Darboux transformation and a one-soliton solution in the scalar reduction [2408.13913].

A distinctive feature of this formulation is the associated Yang–Baxter map. Using the factorization identities
\[
L(q_n,r_n;\mu,\nu,\xi,\eta)\,L(r_n,q_n;\xi,\eta,\mu,\nu)=\mu\xi I,
\]
and the analogous identity for the temporal variables, the discrete dynamics is recast as a parameter-dependent Yang–Baxter map with explicit rational formulas, equations (4.3)–(4.6). The map satisfies the set-theoretic Yang–Baxter relation and has a continuous limit, so the fully discrete MTM is tied not only to a lattice zero-curvature representation but also to a factorization-based Yang–Baxter structure [2408.13913].

## 4. Semi-discrete precursors and the role of coordinates

The fully discrete theory is closely connected to semi-discrete precursor models. In laboratory coordinates, an integrable spatial semi-discretization was obtained from a gauge-modified MTM Lax pair together with a Bäcklund–Darboux transformation for the Ablowitz–Ladik lattice, producing the system
\[
2\mathrm{i}\,\frac{dU_n}{dt} +Q_n+Q_{n+1} +\mathrm{i}(R_{n+1}-R_n) +U_n^2(R_n+R_{n+1}) -U_n\left(Q_{n+1}+Q_n\right) -|U_n|^2\left(Q_{n+1}-Q_n\right)=0,
\]
supplemented by two difference relations for \(Q_n\) and \(R_n\) [1805.07854]. That paper explicitly identifies the result as the first integrable semi-discretization of the MTM in laboratory coordinates and emphasizes its relevance to the Cauchy problem.

A related semi-discretization in non-characteristic coordinates is built from the Lax pair
\[
Y_{n+1}=L_n(\zeta)\,Y_n,\qquad \frac{d}{dt}Y_n=M_n(\zeta)\,Y_n,
\]
with compatibility
\[
L_{n,t}=M_{n+1}L_n-L_nM_n,
\]
and, after the reduction \(r_n=q_n^*\), \(U_n=u_n^*\), recovers
\[
i(q_t+a q_x)-q+|q|^2u=0,\qquad i u_x-q+|q|^2u=0
\]
in the continuum limit [2505.08027]. The appendix identifies the discrete spatial Lax problem with a binary Bäcklund–Darboux transformation, showing why the semi-discrete model is integrable.

The fully discrete bilinear model reduces to the semi-discrete one as
\[
b\to 0,
\]
and then to the continuous MT model as
\[
a\to 0,
\]
while the fully discrete Lax construction passes first to a semi-discrete MTM in one limit and then to the continuous light-cone system [2507.09803], [2408.13913]. At the same time, the laboratory-coordinate literature stresses that one cannot simply transfer a characteristic-coordinate discretization back to laboratory coordinates, because the coordinate rotation mixes positive and negative powers of the spectral parameter in the Lax operators [1805.07854]. This is a central reason why distinct discrete MTM families coexist.

## 5. Lattice Hamiltonian formulations and the broader use of “fully discrete”

In lattice field theory and quantum many-body work, the phrase “fully discrete” is used in a different but related sense. One study starts from the continuum \(1+1\)-dimensional Thirring action,
\[
S_{\mathrm{Th}}[\psi,\bar\psi]=\int d^2x\left[\bar\psi\, i\gamma^\mu\partial_\mu\psi-m_0\bar\psi\psi-\frac{g}{2}(\bar\psi\gamma_\mu\psi)^2\right],
\]
discretizes space with staggered fermions,
\[
H_{\mathrm{Th}}^{(\mathrm{latt.})}
=
-\frac{i}{2a}\sum_{n=0}^{N-2}(c_n^\dagger c_{n+1}-c_{n+1}^\dagger c_n)
+m_0\sum_{n=0}^{N-1}(-1)^n c_n^\dagger c_n
+\frac{2g}{a}\sum_{n=0}^{N/2-1} c_{2n}^\dagger c_{2n}\,c_{2n+1}^\dagger c_{2n+1},
\]
maps it to a spin-\(\tfrac12\) chain by Jordan–Wigner, and studies the resulting discrete Hamiltonian with MPS/DMRG in the zero-charge sector [1710.09993]. The paper explicitly states that the computational study is performed on a fully discrete lattice Hamiltonian formulation.

Two quantum-computation papers make the same distinction from a Hamiltonian perspective. One uses a periodic 1D spatial lattice with Wilson fermions, continuous time, Jordan–Wigner encoding onto \(2N\) qubits, and a three-site benchmark to compute the mass gap by a hybrid classical-quantum method [1912.07767]. Another uses staggered fermions, Jordan–Wigner mapping, and an explicit \(N=4\) Pauli Hamiltonian,
\[
aH_E = h_{1}+am h_{2}-\frac{1}{4} g^{2}h_{3}-\frac{1}{4} g^{2}h_{4}, \qquad
aH_M = h_{1}+am h_{2}-\frac{1}{4} g^{2}h_{3}+\frac{1}{4} g^{2}h_{4},
\]
to study finite-temperature chiral and topological transitions with QMETTS [2412.00803]. These works discretize the many-body Hilbert space and spatial coordinate explicitly, but retain continuous time; they are therefore lattice-Hamiltonian realizations of the massive Thirring model rather than fully discrete integrable evolution equations.

## 6. Solitons, reductions, and structural issues

The fully discrete MTM is primarily valued for preserving integrability under discretization. In the bilinear/KP-Toda construction, integrability is established through Hirota bilinear form, reduction from an integrable hierarchy, and Gram-determinant tau functions that give explicit multi-bright soliton solutions [2507.09803]. The one-soliton solution is written in terms of
\[
\xi_i=-k\ln(1-a p_i)-l\ln\!\left(1+\frac{b}{p_i}\right), \qquad
\tilde\xi_i=k\ln(1+a p_i^*)+l\ln\!\left(1-\frac{b}{p_i^*}\right),
\]
and the \(N\)-soliton sector is given by determinant formulas of the same type. The paper also emphasizes that its light-cone discretization is different from the older discretizations of Nijhoff et al. and has no square singularity [2507.09803].

A technically important point is that, in the discrete bright-soliton solutions, the auxiliary fields \(\tilde u_k^l,\tilde v_k^l\) are not automatically the complex conjugates of \(u_k^l,v_k^l\). The conjugate structure is recovered only in the continuum limit \(a,b\to 0\) [2507.09803]. This is one of the most common sources of confusion when discrete MTM formulas are compared directly with the continuous model.

The Lax-pair construction adds a different integrability package: discrete zero curvature, binary Bäcklund–Darboux transformation, one-soliton solutions, and a Yang–Baxter map with continuous limit [2408.13913]. In parallel, a continuous Darboux-matrix analysis of the classical MTM develops polynomial Darboux matrices constrained by a dihedral \(D_2\) reduction, with root quadruplets
\[
\xi_n,\quad \xi_n^*,\quad -\xi_n^*,\quad -\xi_n,
\]
and explicit lowest-degree dressing formulas [1411.7965]. That work does not derive a semi-discrete or fully discrete MTM, but it identifies the algebraic ingredients usually required for one: symmetry-preserving Darboux steps, automorphic determinant polynomials, and an iterative dressing mechanism. Within the integrable-systems literature, this suggests that fully discrete MTM equations are best understood not as arbitrary finite-difference replacements, but as reductions of larger discrete Lax or hierarchy structures constrained by the same reduction symmetries as the continuous theory.

Source: https://www.emergentmind.com/topics/fully-discrete-massive-thirring-model