--- title: Polyakov's Energy Bifurcation Explained url: https://www.emergentmind.com/topics/polyakov-s-energy-bifurcation type: topic --- # Polyakov's Energy Bifurcation Explained Searching arXiv for the cited works and closely related context on Polyakov’s energy bifurcation. Polyakov’s energy bifurcation denotes, in its most explicit recent usage, the non-perturbative lifting of a degenerate ground-state manifold into distinct low-lying levels once tunneling trajectories connecting symmetry-related minima are included. In the quantum Rabi model this is formulated as the splitting between $E_0$ and $E_1$ induced by instanton-like trajectories in an effective double-well potential, $\Delta E = E_1-E_0 = \hbar \omega_a \exp[-S_{\rm euc}(q_{\rm cl})/\hbar]$ [2605.23413]. In the compactified Polyakov model on $T^2\times\mathbb R$ with background ’t Hooft/GNO flux, an analogous mechanism appears as the splitting of $N$ classically degenerate fractional-flux vacua by monopole instantons that survive the Born–Oppenheimer reduction at arbitrarily small torus area [2112.02483]. A further, more interpretive usage appears in the holonomous-plasma literature, where holonomy causes a discontinuous split between diagonal and off-diagonal $O(g^3)$ plasmon contributions [2002.00968]. These usages are not identical. This suggests a family resemblance rather than a single universally standardized definition. ## 1. Terminological scope The most precise definition currently available is the one given in the 2026 Perspective on macroscopic quantum tunneling: Polyakov’s energy bifurcation is “the lifting of a degenerate ground-state manifold into two distinct levels—the ground state and the first excited state—when tunneling trajectories (instanton-like configurations) that connect two symmetry-related minima are included” [2605.23413]. Neglecting tunneling in a symmetric double-well yields a degenerate vacuum and spontaneous symmetry breaking; including instanton trajectories restores symmetry and splits the degeneracy. In the 2021 study of the Polyakov model in a background ’t Hooft flux, the same structural motif appears in a gauge-theoretic setting. Standard compactification on small $T^2\times\mathbb R$ has a unique perturbative vacuum and flux sectors separated parametrically by a classical gap, so instanton effects do not survive in the Born–Oppenheimer approximation. With a background magnetic GNO flux in the co-weight lattice, however, $N$-degenerate vacua appear at small torus, and $N-1$ types of flux-changing instantons connect them [2112.02483]. The resulting low-energy quantum mechanics is a double-well for $SU(2)$ and a multi-well system for $SU(N)$. In the high-temperature holonomy problem, the phrase is more interpretive. There the relevant split is not a lowest-doublet tunneling problem but a discontinuity in the $O(g^3)$ free-energy contribution: the off-diagonal plasmon sector switches off for any nonzero holonomy, while the diagonal sector varies continuously as holonomy vanishes [2002.00968]. A concise comparison is as follows. | Setting | Object that splits | Mechanism | |---|---|---| | Quantum Rabi model | Ground-state doublet $E_0,E_1$ | Instanton-like tunneling between two minima | | Polyakov model on $T^2\times\mathbb R$ with flux | $N$ fractional-flux vacua | Monopole-instanton tunneling in BO QM | | Holonomous plasma | $O(g^3)$ plasmon free energy by sector | Holonomy-induced loss of off-diagonal static channel | ## 2. Double-well formulation in the quantum Rabi model The quantum Rabi model is defined by $$ H_{\rm QR}=\frac{\hbar \omega_a}{2}\sigma_z+\hbar \omega_c\left(a^\dagger a+\frac12\right)+\hbar g\,\sigma_x(a+a^\dagger), $$ with exact parity symmetry generated by $$ P=\sigma_z(-1)^{a^\dagger a},\qquad [H_{\rm QR},P]=0 $$ [2605.23413]. Introducing $$ \hat x=\sqrt{\frac{\hbar}{2\omega_c}}(a+a^\dagger),\qquad \hat p=-i\sqrt{\frac{\hbar\omega_c}{2}}(a-a^\dagger), $$ and performing the unitary rotation $U_0$, one obtains $$ U_0^\ast H_{\rm QR}U_0= -\frac{\hbar\omega_a}{2}\sigma_x+H_{\rm MS}^{\rm ho}(\omega_c,0)+g\sqrt{2\hbar\omega_c}\,\sigma_z\hat x. $$ Completing the square yields two harmonic wells, $$ H_{\rm MS}^{\rm ho}(\omega_c,0)\pm g\sqrt{2\hbar\omega_c}\,\hat x = \frac12\hat p^2+\frac{\omega_c^2}{2}(\hat x\pm x_0)^2-\hbar g^2/\omega_c, $$ centered at $$ x_0=\frac{g}{\omega_c}\sqrt{\frac{2\hbar}{\omega_c}}. $$ This mapping makes the bifurcation mechanism explicit. The wells have identical curvature $\omega_c^2$, their separation grows linearly with $g$, and the off-diagonal tunneling operator carries a weighted phase factor that becomes highly oscillatory with increasing $g$, suppressing tunneling in the semiclassical limit. In the transformed parity sectors one obtains $$ E_\pm=\frac{\hbar\omega_c}{2}-\frac{\hbar g^2}{\omega_c} \mp \frac{\hbar\omega_a}{2} \exp\!\left[-2g^2/\omega_c^2\,(G(g)+1)\right], $$ with $-1\le G(g)\le 0$ [2605.23413]. Identifying $E_0=E_+$ and $E_1=E_-$, the renormalized energies are $$ E_0^{\rm ren}=\frac{\hbar\omega_c}{2}-\frac{\hbar\omega_a}{2}e^{-S_{\rm euc}/\hbar},\qquad E_1^{\rm ren}=\frac{\hbar\omega_c}{2}+\frac{\hbar\omega_a}{2}e^{-S_{\rm euc}/\hbar}, $$ with $$ S_{\rm euc}(q_{\rm cl})=2\hbar g^2/\omega_c^2\,(G(g)+1). $$ Hence $$ \Delta E=E_1^{\rm ren}-E_0^{\rm ren} =\hbar\omega_a e^{-S_{\rm euc}(q_{\rm cl})/\hbar}. $$ The associated physical interpretation is sharply stated. In the strict limits $g\to\infty$ at fixed $\omega_a,\omega_c$ or $\omega_a\to 0$ at fixed $\omega_c$, tunneling terms vanish, the transformed Hamiltonian converges in the norm-resolvent sense to a free boson Hamiltonian with two degenerate ground states, and a $\mathbb Z_2$ symmetry is spontaneously broken. For finite $g/\omega_a$, instanton-like contributions restore parity and split the doublet. There is no finite critical coupling $g_c$; the phenomenon is a crossover controlled by $g/\omega_c$ and $g/\omega_a$ [2605.23413]. ## 3. Flux-induced bifurcation in the Polyakov model on $T^2\times\mathbb R$ The three-dimensional $SU(N)$ Polyakov model contains an adjoint Higgs field $\phi$ that breaks $SU(N)\to U(1)^{N-1}$, with low-energy dynamics described by dual photons $\sigma\in \mathbb R^{N-1}/(2\pi\Gamma_w)$. Abelian duality reads $$ \star_3\mathbf F=\frac{g^2}{2\pi}\,d\mathbf\sigma, $$ and the leading non-perturbative saddles are monopole-instantons $$ {\cal M}_i(x)=K e^{-S_0}e^{i\mathbf\alpha_i\cdot \mathbf\sigma(x)} \equiv \xi\,e^{i\mathbf\alpha_i\cdot\mathbf\sigma(x)}, \qquad i=1,\dots,N-1, $$ with $S_0=\frac{4\pi v}{g_3^2}f(\lambda)$ and $\xi\sim K e^{-S_0}$ [2112.02483]. Their induced potential gives the non-perturbative dual-photon masses $$ m_k^2=m_\gamma^2\sin^2\!\Big(\frac{\pi k}{2N}\Big), \qquad m_\gamma^2\sim \frac{\xi}{g^2}, $$ which is Polyakov’s mass gap. The compactified theory on $T^2\times\mathbb R$ behaves differently depending on background flux. Without background flux, the perturbative vacuum is unique and higher magnetic-flux sectors are separated by a classical gap $\Delta E\sim (2\pi/g)^2/A_{T^2}$, so monopole effects do not survive in the low-energy quantum mechanics. With a background ’t Hooft/GNO flux in the co-weight lattice, by contrast, the spectrum shifts so that multiple perturbative minima become classically degenerate even at finite area. For $SU(N)$ with background flux $\mu_1$, the theory has $N$ distinct fractional-flux states $\nu_1,\dots,\nu_N$, all classically degenerate, while the $N-1$ simple-root monopoles change the flux by $$ \Delta\Phi=\frac{2\pi}{g}\,\mathbf\alpha_a,\qquad a=1,\dots,N-1, $$ and connect nearest-neighbor vacua [2112.02483]. In the Born–Oppenheimer limit, the low-energy Hilbert space is spanned by these $N$ degenerate states. The effective Hamiltonian is a nearest-neighbor tight-binding problem, $$ H=E_{\rm bg}\,\mathbf 1_N+\widetilde H,\qquad \widetilde H=-\sum_j \xi\Big(|\nu_{j+1}\rangle\langle \nu_j|+{\rm h.c.}\Big). $$ For $SU(2)$ this is exactly a double-well, $$ \widetilde H=-\begin{pmatrix}0&\xi\\ \xi&0\end{pmatrix}, \qquad E_{0,1}=\mp \xi, $$ so the level splitting is $$ \Delta E=E_1-E_0=2\xi=2K e^{-S_0}. $$ For $SU(N)$ one obtains an open-chain tridiagonal Hamiltonian with eigenvalues $$ E_k=-2\xi\cos\!\Big(\frac{\pi(k+1)}{N+1}\Big), \qquad k=0,\dots,N-1, $$ producing a narrow band of width $O(\xi)$ [2112.02483]. This is the paper’s “memory” effect. The background flux creates perturbative degeneracy at arbitrarily small $A_{T^2}$, so QFT monopole instantons survive the quantum-mechanical reduction and split the degenerate multiplet non-perturbatively. The vacuum structures of QFT and QM are therefore continuously connected despite the absence of a mixed anomaly between the 1-form center symmetry and time reversal. ## 4. Holonomy and bifurcation in a holonomous plasma At nonzero temperature, a constant diagonal background $$ \overline A_0=\frac{2\pi T}{g}\,{\rm diag}(q_1,\dots,q_N),\qquad \sum_i q_i=0, $$ generates Polyakov-loop eigenvalues $e^{i\theta_a}$ with $\theta_a=2\pi q_a$. Off-diagonal adjoint gluons experience shifted Matsubara frequencies $$ K_0^{ab}=2\pi T(n+q_{ab}),\qquad q_{ab}=q_a-q_b, $$ whereas diagonal gluons retain $K_0^{aa}=2\pi nT$ [2002.00968]. The one-loop Weiss potential is $$ V_1(\{q_a\})= -\frac{\pi^2T^4}{45}(N^2-1) +\frac{2\pi^2T^4}{3}\sum_{i