Rigorous thermodynamic limit for the spectral equations of the modular sinh-Gordon model: the ground-state density equation

The quantum sinh-Gordon model realized on Faddeev's modular double $\mathcal{U}_{q,q^*}(\widehat{sl}_2)$ has finite-chain spectral equations of Bethe type: the eigenfunction of the modular pair of Baxter $TQ$ equations must be free of all but kinematic poles, which imposes pole-cancellation conditions at the zeros $x_\nu$ of a $q$-difference Wronskian. Sergeev (arXiv:2303.17725, 2023) took the $N\to\infty$ thermodynamic limit of these conditions under three explicitly conjectural assumptions — a continuous analytic distribution of the Wronskian zeros, analytic straightening of the auxiliary-zero contours together with convergence of the auxiliary zeros to the principal zeros, and dominance of the leading $t^2$-orders of the bootstrap expansion — and obtained a closed integral equation for the ground-state root density, $P(x+i\eta)+P(x-i\eta)=P_A(x)+P_B(x)$, solved by convolution with the hyperbolic-secant kernel $[4\eta\cosh(\pi x/2\eta)]^{-1}$. The convergence bound underpinning the second assumption is verified in that work only self-consistently, inside the conjectural limit itself. The problem is to justify this thermodynamic limit rigorously — by a proof, or by a derivation with explicit finite-$N$ error estimates — or to refute the conjectures within the model's stated parameter domain. No resolution of the problem is known in the literature to date; the author's own 2024 follow-up, on a real-$q$ relative model, invokes the same assumptions without proof.

Background

The sinh-Gordon model — a scalar relativistic field theory with a $\cosh$-interaction — is the simplest massive integrable quantum field theory, and it admits an exact lattice realization built on Faddeev's modular double $\mathcal{U}_{q,q^*}(\widehat{sl}_2)$: a non-compact (infinite-dimensional) quantum-group symmetry in which the usual deformation parameter $q=e^{i\pi b^2}$ with $b=e^{i\theta}$, $0<\theta<\pi/2$, is accompanied by a dual parameter $q^*=e^{-i\pi b^{-2}}$ acting on the same Hilbert space (Faddeev 1995). Throughout, $\eta=(b+b^{-1})/2$. In this realization (Sergeev 2023) the model is a chain of length $N$ whose transfer matrix $T(u)$, in the spectral variable $u=e^{2\pi bx}$, commutes with a family of integrals of motion; the inhomogeneities $\alpha_\nu,\beta_\nu$ (geometric means $e^{\pm 2\pi b\mu}$) play the role of external fields, and an auxiliary parameter $t$ with $|t^2|<1$ plays the role of $q^{S^z}$, i.e. of a temperature-like variable.

In Baxter's framework, the spectrum of a transfer matrix is found from a functional relation between $T(u)$ and an auxiliary Q-operator; here the relation takes the form of a modular pair of $q$-difference $TQ$ equations, one for each of the two conjugated Weyl pairs of the modular double. The coefficients involve the Faddeev quantum dilogarithm $\varphi(x)$, a standard special function of non-compact quantum group theory defined by $\log\varphi(x)=\int_{\mathbb R+i0}e^{-2ixy}\,dy/(4\sinh(by)\sinh(b^{-1}y)\,y)$ and satisfying $\log\varphi(x-i\eta)-\log\varphi(x+i\eta)=\log[(1-u)(1-u^*)]$ with $u=e^{2\pi bx}$. Two distinguished solutions $\chi_+(u)$ and $\chi_-(u)$ of the scalar $q$-difference equation — holomorphic at zero and at infinity respectively — form a $q$-difference Wronskian $W(u)$, which turns out to be a constant multiple of a finite product of Jacobi theta functions, $W(u)=\varrho\prod_{\nu=1}^{N}\vartheta_1(u/u_\nu)$ with $\sum_\nu x_\nu=0$. The zeros $u_\nu=e^{2\pi bx_\nu}$ (Bethe-type variables, by analogy with the Bethe roots of spin chains) are the unknowns of the spectral problem.

The quantization principle of this family of models (Sergeev 2023; Kashaev and Sergeev 2018 for the modular oscillator) requires that the eigenfunction $Q(x)=Q_1(x)-\xi Q_2(x)$ of the $TQ$ pair have only the kinematic poles explicitly present in its building blocks; the poles at the Wronskian zeros must therefore cancel, giving the finite-$N$ spectral equations

$$\lim_{x\to x_\nu}\frac{Q_1(x)}{Q_2(x)}=\xi,\qquad \nu=1,\dots,N,$$

where $Q_{1,2}$ are explicit functionals of $\{x_\nu\}$ through $\chi_\pm$. The dependence of $\chi_\pm$ on $\{x_\nu\}$ is, however, only recursively defined: expanding in powers of $t^2$ yields a linear bootstrap that determines polynomial corrections uniquely step by step, but no manageable closed form is known beyond $N=1$; and the zero decompositions of $\chi_\pm$ into products $\prod_m\prod_\nu(1-q^{2m}u/u_{\nu,m})$ and $\prod_m\prod_\nu(1-q^{2m}u'_{\nu,m}/u)$ introduce infinitely many auxiliary zeros $u_{\nu,m}$ and $u'_{\nu,m}$ with asymptotics $u_\nu/u_{\nu,n}=1+t^{2n}\Delta'_{\nu,n}+O(t^{2n+2})$ and $u'_{\nu,n}/u_\nu=1+t^{2n}\Delta_{\nu,n}+O(t^{2n+2})$.

For ordinary compact Bethe-ansatz models the thermodynamic limit $N\to\infty$ is classical: the roots condense into a continuous distribution whose density satisfies a linear integral equation (the archetype is the root-density equation for the one-dimensional spin-$1/2$ chain). For the present non-compact model, Sergeev (2023) assumed this structure — (i) the ground-state roots $x_\nu\in\mathbb R$ condense into an analytic density $P(x)$; (ii) the auxiliary roots $x_{\nu,m}$ condense along complex contours $C_m$ that can be analytically straightened to the real axis with the same density, so that $x_{\gamma,m}\to x_\gamma$ for all $m$ as $N\to\infty$, warranted by the bound $\frac{1}{N}\log|\Delta_{\gamma,n}|<0$; (iii) the leading $t^2$-orders dominate the bootstrap sums. Under these assumptions the spectral equations reduce in the limit to the ground-state density equation

$$P(x_0+i\eta)+P(x_0-i\eta)=P_A(x_0)+P_B(x_0),$$

where $P_A,P_B$ are the $N\to\infty$ densities of the two inhomogeneity families (means $\pm\mu$, second moments $S_A,S_B$), with normalization $\int P\,dx_0=1$, $\int P\,x_0\,dx_0=0$; the solution is the convolution $P=K*(P_A+P_B)$ with the hyperbolic-secant kernel $K(x)=[4\eta\cosh(\pi x/2\eta)]^{-1}$, and on it $\int P\,x_0^2\,dx_0=S_P=\eta^2+(S_A+S_B)/2$. The result passes two physics-level consistency checks: analytically continued to imaginary $x$ and $\mu$ it reproduces the exact free energy of the Faddeev-Volkov lattice model (Bazhanov, Mangazeev and Sergeev 2007, 2008), and it accords with the exact sine-Gordon results of Lukyanov and Zamolodchikov (1997). Crucially, however, assumption (ii) is justified only self-consistently — the bound $\frac{1}{N}\log|\Delta_{\gamma,n}|<0$ is verified using the thermodynamic-limit results themselves — and the author states explicitly that the derivation rests on conjectures, that no numerical results on the auxiliary zeros exist, and that the finite-$N$ bootstrap is a theorem of existence only.

Problem Statement

The problem is to replace the conjectural thermodynamic limit of the spectral equations of the modular sinh-Gordon model (Sergeev, arXiv:2303.17725) by a rigorous argument, or to refute it within the model's stated domain.

Start from the finite-$N$ spectral equations $\lim_{x\to x_\nu}Q_1(x)/Q_2(x)=\xi$, $\nu=1,\dots,N$, where $x_\nu$ (multiplicatively $u_\nu=e^{2\pi bx_\nu}$) are the zeros of the $q$-difference Wronskian of the holomorphic blocks $\chi_\pm$, and $Q_{1,2}$ depend on $\{x_\nu\}$ through $\chi_\pm$, the $t^2$-bootstrap, and the auxiliary zeros $u_{\nu,m}$, $m\ge 1$, of the zero decomposition of $\chi_\pm$. Establish — by proof, or by a derivation with explicit finite-$N$ error estimates — the $N\to\infty$ ground-state limit in the regime of real $x$, real $\mu$, $0<\theta<\pi/2$, $|t^2|<1$, and external inhomogeneity densities $P_A,P_B$ (positive, normalized, finite second moments, means $\pm\mu$). Concretely:

1. Prove that the empirical distribution of the ground-state Wronskian zeros $\{x_\nu\}$ — the solution branch with all $x_\nu$ real, whose existence at finite $N$ is itself part of what must be established — converges as $N\to\infty$ to an analytic density $P$ on the real axis.
2. Prove the auxiliary-zero convergence: writing $\gamma$ for a generic ground-state zero index, the auxiliary zeros $x_{\gamma,m}$, $m\ge 1$, of the zero decomposition of $\chi_\pm$ satisfy $x_{\gamma,m}\to x_\gamma$ as $N\to\infty$, with the contour-straightening property that their limiting contours can be analytically deformed to the real axis carrying the same density — for which it suffices to establish a priori, independently of the thermodynamic-limit results, the bound $\lim_{N\to\infty}\frac{1}{N}\log|\Delta_{\gamma,n}|<0$ of the source (eq. (8.7)), where $\Delta_{\gamma,n}$, $n\ge 1$, are the coefficients of the auxiliary-zero asymptotics $u'_{\gamma,n}/u_\gamma=1+t^{2n}\Delta_{\gamma,n}+O(t^{2n+2})$ (the companion family of $\chi_+$ satisfies $u_\gamma/u_{\gamma,n}=1+t^{2n}\Delta'_{\gamma,n}+O(t^{2n+2})$, and the source's bound refers to both families); the source verifies this bound only self-consistently, inside the conjectural limit itself.
3. Justify the interchange of the $N\to\infty$ limit with the $t^2$-bootstrap expansion, i.e. the dominance of the leading $t^2$-orders in the passage from the spectral equations to their integral form.

Then conclude, with error control at finite $N$, the ground-state density equation

$$P(x_0+i\eta)+P(x_0-i\eta)=P_A(x_0)+P_B(x_0),\qquad \eta=\frac{b+b^{-1}}{2},$$

with solution $P=K*(P_A+P_B)$, $K(x)=[4\eta\cosh(\pi x/2\eta)]^{-1}$, normalization $\int P\,dx_0=1$ and $\int P\,x_0\,dx_0=0$, and second moment $S_P=\eta^2+(S_A+S_B)/2$.

A valid answer may alternatively refute the conjectures: exhibit, within the stated parameter domain, a regime where the ground-state root distribution fails to converge to an analytic density, the auxiliary-zero convergence fails, or the limit equation above is violated — established by proof, or by computation certified with error bounds uniform in $N$ (a finite-$N$ numerical deviation alone does not refute an asymptotic claim).

The further directions listed there — closed-form finite-$N$ expressions for the bootstrap coefficients beyond $N=1$, systematic numerical evaluation of the auxiliary zeros (which would constitute supporting evidence but is not required here), the continuum theory of low-energy excitations above the ground state, and extension to analytically continued regimes with non-unimodal densities — remain open but lie outside this problem's completion boundary.

References

1. Sergey M. Sergeev, Spectral equations and ground state for the modular sinh-Gordon model, arXiv:2303.17725 (2023), DOI 10.48550/arXiv.2303.17725, https://arxiv.org/abs/2303.17725
2. Sergey Sergeev, Functional Bethe Ansatz for a sinh-Gordon model with real q, Symmetry 16(8), 947 (2024), DOI 10.3390/sym16080947, https://arxiv.org/abs/2404.00503
3. Vladimir V. Bazhanov and Sergey M. Sergeev, A distant descendant of the six-vertex model, arXiv:2310.08427 (2023), DOI 10.48550/arXiv.2310.08427, https://arxiv.org/abs/2310.08427
4. Rinat M. Kashaev and Sergey M. Sergeev, Spectral equations for the modular oscillator, Reviews in Mathematical Physics 30(7), 1840009 (2018), DOI 10.1142/S0129055X18400093, https://arxiv.org/abs/1703.06016
5. Sergei Lukyanov and Alexander Zamolodchikov, Exact expectation values of local fields in the quantum sine-Gordon model, Nuclear Physics B 493(3), 571-587 (1997), DOI 10.1016/S0550-3213(97)00123-5, https://doi.org/10.1016/S0550-3213(97)00123-5
6. Vladimir V. Bazhanov, Vladimir V. Mangazeev and Sergey M. Sergeev, Faddeev-Volkov solution of the Yang-Baxter equation and discrete conformal symmetry, Nuclear Physics B 784(3), 234-258 (2007), DOI 10.1016/j.nuclphysb.2007.05.013, https://doi.org/10.1016/j.nuclphysb.2007.05.013
7. Vladimir V. Bazhanov, Vladimir V. Mangazeev and Sergey M. Sergeev, Exact solution of the Faddeev-Volkov model, Physics Letters A 372(10), 1547-1550 (2008), DOI 10.1016/j.physleta.2007.10.053, https://doi.org/10.1016/j.physleta.2007.10.053
8. L. D. Faddeev, Discrete Heisenberg-Weyl group and modular group, Letters in Mathematical Physics 34(3), 249-254 (1995), DOI 10.1007/BF01872779, https://doi.org/10.1007/BF01872779