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arXiv:2603.00652

Synchronous Tunneling in a Multidimensional Quartic Potential: Competing Instanton Pathways and $D_4$ Symmetry Melting

Pervez Hoodbhoy, M. Haashir Ismail, M. Mufassir

quant-phhep-thmath-phnlin.PSphysics.chem-phSubmitted Feb 28, 2026• Announced Mar 3, 2026• Revised May 11, 2026 (v3)
v1 · Feb 28, 2026v2 · Mar 11, 2026v3 · May 11, 2026v4 · May 16, 2026v5 · Aug 7, 2026
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Abstract

Semi-classical analysis is used to investigate synchronous quantum tunneling in a multidimensional potential energy surface (PES) characterized by four degenerate minima, serving as a foundational model for coupled vibrational modes. The primary challenge in such systems is the non-linear ``locking" of trajectories where degrees of freedom must traverse their respective barriers synchronously. Starting from the Feynman path integral in imaginary time, we analytically identify longitudinal, transverse, and diagonal instanton configurations that mediate competing tunneling pathways between minima. The translational zero mode for each trajectory is treated rigorously by transforming to a comoving rotating frame. By applying the Gelfand-Yaglom method to the functional determinant and utilizing graph theory to sum the multi-flavor dilute instanton gas , we derive coherent Rabi-type oscillations and exact ground-state tunneling splittings. Crucially, we identify a critical coupling regime where the discrete $D_4$ spatial symmetry of the minima undergoes a topological 'melting' transition into a continuous $O(2)$ rotational symmetry. These analytical results, validated against high-precision numerical diagonalization, provide a rigorous benchmark for multidimensional computational techniques, such as Ring Polymer Instanton (RPI) theory, particularly in the strongly coupled regime where standard discrete instanton approximations break down.

Comments
35 pages, 9 figures, This manuscript supersedes and substantially expands upon the earlier, withdrawn preprint arXiv:2504.15607, featuring a fully revised analytic treatment of the functional determinant and multi-flavor dilute gas summation

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Semi-classical analysis is used to investigate synchronous quantum tunneling in a multidimensional potential energy surface (PES) characterized by four degenerate minima, serving as a foundational model for coupled vibrational modes. The primary challenge in such systems is the non-linear ``locking" of trajectories where degrees of freedom must traverse their respective barriers synchronously. Starting from the Feynman path integral in imaginary time, we analytically identify longitudinal, transverse, and diagonal instanton configurations that mediate competing tunneling pathways between minima. The translational zero mode for each trajectory is treated rigorously by transforming to a comoving rotating frame. By applying the Gelfand-Yaglom method to the functional determinant and utilizing graph theory to sum the multi-flavor dilute instanton gas , we derive coherent Rabi-type oscillations and exact ground-state tunneling splittings. Crucially, we identify a critical coupling regime where the discrete D4D_4D4​ spatial symmetry of the minima undergoes a topological 'melting' transition into a continuous O(2)O(2)O(2) rotational symmetry. These analytical results, validated against high-precision numerical diagonalization, provide a rigorous benchmark for multidimensional computational techniques, such as Ring Polymer Instanton (RPI) theory, particularly in the strongly coupled regime where standard discrete instanton approximations break down.
We review the euclidean path-integral formalism in connection with the one-dimensional non-relativistic particle.The configurations which allow construction of a semiclassical approximation classify themselves into either topological (instantons) and non-topological (bounces) solutions.The quantum amplitudes consist of an exponential associated with the classical contribution multiplied by the fluctuation factor which is given by a functional determinant.The eigenfunctions as well as the energy eigenvalues of the quadratic operators at issue can be written in closed form due to the shape-invariance property.Accordingly, we resort to the zeta-function method to compute the functional determinants in a systematic way.The effect of the multi-instantons configurations is also carefully considered.To illustrate the instanton calculus in a relevant model, we go to the double-well potential.The second popular case is the periodic-potential where the initial levels split into bands.The quantum decay rate of the metastable states in a cubic model is evaluated by means of the bounce-like solution.
We consider the quantum tunneling phenomenon in a well-behaved triple-well potential. As required by the semiclassical approximation we take into account the quadratic fluctuations over the instanton which represents as usual the localised finite-action solution of the euclidean equation of motion. The determinants of the quadratic differential operators at issue are evaluated by means of the Gelfang-Yaglom method. In doing so the explicit computation of the conventional ratio of determinants takes as reference the harmonic oscillator whose frequency is the average of the individual frequencies derived from the non-equivalent minima of the potential. Eventually the physical effects of the multi-instanton configurations are included in this approach. As a matter of fact we obtain information about the energies of the ground-state and the two first excited levels of the discrete spectrum at issue.
Starting from trace formulae for the tunnelling splittings (or decay rates) analytically continued in the complex time domain, we obtain explicit semiclassical expansions in terms of complex trajectories that are selected with appropriate complex-time paths. We show how this instanton-like approach, which takes advantage of an incomplete Wick rotation, accurately reproduces tunnelling effects not only in the usual double-well potential but also in situations where a pure Wick rotation is insufficient, for instance dynamical tunnelling or resonant tunnelling. Even though only one-dimensional autonomous Hamiltonian systems are quantitatively studied, we discuss the relevance of our method for multidimensional and/or chaotic tunnelling.
We review the euclidean path-integral formalism in connection with the one-dimensional non-relativistic particle. The configurations which allow to construct a semiclassical approximation classify themselves into either topological (instantons) and non-topological (bounces) solutions. The quantum amplitudes consist on an exponential associated with the classical contribution multiplied by the fluctuation factor which is given by a functional determinant. The eigenfunctions as well as the energy eigenvalues of the quadratic operators at issue can be written in closed form due to the shape-invariance property. Accordingly we resort to the zeta-function method to compute the functional determinants in a systematic way. The effect of the multi-instantons configurations is also carefully considered. To illustrate the instanton calculus in a relevant model we go to the double-well potential. The second popular case is the periodic-potential where the initial levels split into bands. The quantum decay rate of the metastable states in a cubic model is evaluated by means of the bounce-like solution.
Starting from trace formulas for the tunneling splittings (or decay rates) analytically continued in the complex time domain, we obtain explicit semiclassical expansions in terms of complex trajectories that are selected with appropriate complex-time paths. We show how this instantonlike approach, which takes advantage of an incomplete Wick rotation, accurately reproduces tunneling effects not only in the usual double-well potential but also in situations where a pure Wick rotation is insufficient, for instance dynamical tunneling or resonant tunneling. Even though only one-dimensional autonomous Hamiltonian systems are quantitatively studied, we discuss the relevance of our method for multidimensional and/or chaotic tunneling.
We present an alternative way to compute and interpret quantum tunneling in a one-dimensional double-well potential. For large transition time we show that the quantum action functional gives an analytical expression for tunneling amplitudes. This has been confirmed by numerical simulations giving relative errors in the order of ${10}^{\ensuremath{-}5}$. In contrast to the classical potential, the quantum potential has a triple well if the classical wells are deep enough. Its minima are located at the positions of extrema of the ground state wave function. The striking feature is that a single trajectory with a double instanton reproduces the tunneling amplitude. This is in contrast to the standard instanton approach, where infinitely many instantons and anti-instantons have to be taken into account. The quantum action functional is valid in the deep quantum regime in contrast to the semiclassical regime where the standard instanton approach holds. We compare both approaches via numerical simulations. While the standard instanton picture describes only the transition between potential minima of equal depth, the quantum action may give rise to instantons also for asymmetric potential minima. Such a case is illustrated by an example.