Periodicity-driven revision of the phase diagram of the generalized Baxter-Wu model with asymmetric complex couplings

Abstract: The conventional self-dual lines of the generalized Baxter-Wu (GBW) model with asymmetric complex couplings are known to be $\sinh(2K)=\pm \cos(2φ)$, where $K$ and $φ$ are the real and imaginary parts of the coupling. We demonstrate that these lines are incomplete: the periodicity of the partition function, encoded in the cosine factor of the bundled Boltzmann weight, generates additional self-dual lines $\sinh(2K)=\pm \sin(2φ)$. Guided by the complete set of self-dual candidates, we perform Monte Carlo simulations using brute-force reweighting (Metropolis) and the Wang-Landau methods. Simulations indicate that the self-dual lines at the partition-function minima $φ<em>{\mathcal{Z}</em>{\min}}=(2n+1)π/8$ constitute a critical threshold. They are genuine critical boundaries for $$|K| \ge \frac{1}{2}\operatorname{arsinh}(\cos(π/4)) \approx 0.32924$$, while for smaller $|K|$ they are not. At $φ<em>{\mathcal{Z}</em>{\min}}$, the sign problem is most severe and finite-size scaling corrections are largest; the local peak observed below the phase boundary in the temperature scan is thus a finite-size artifact, not a genuine new phase. We further clarify the capability and limitations of the average sign and its derivatives for detecting phase transitions. In particular, the negative peak of the average sign at $φ<em>{\mathcal{Z}</em>{\min}}$ does not correspond to a genuine phase transition. We also evaluate the Wang-Landau method, which, despite formally circumventing the sign problem, still faces the exponential barrier.

• The paper incorporates the partition function’s π/4 periodicity to derive a second family of self-dual lines, then uses Monte Carlo and Wang–Landau simulations to identify genuine phase boundaries.
• The revised phase diagram shows that critical segments satisfy |K| ≥ ½ arsinh(cos(π/4)) ≈ 0.32924 and exhibit four-state Potts critical behavior with ν = 2/3 and η = 1/4.
• The paper finds that average-sign minima can falsely indicate transitions and that Wang–Landau sampling retains an exponential-in-volume cost because sign cancellations amplify statistical errors.

The generalized Baxter–Wu (GBW) model with mutually complex-conjugate couplings $K_{\rm up}=K+i\phi$, $K_{\rm down}=K-i\phi$ has served as a controlled setting for studying both complex-coupled phase transitions and the sign problem. Prior work established self-dual lines $\sinh(2K)=\pm\cos(2\phi)$ and used them to benchmark sign-based phase-transition probes. This paper shows that those lines are incomplete: the $\pi/4$ periodicity of the partition function, inherited from the cosine factor of the bundled Boltzmann weight, generates an additional family of self-dual candidates. Guided by the complete set of candidates, the authors perform unbiased Monte Carlo simulations—brute-force reweighting with Metropolis sampling and a two-dimensional Wang–Landau (WL) algorithm—to determine which candidate segments are genuine phase boundaries, to re-examine sign-based probes against the corrected phase diagram, and to test whether WL circumvents the exponential barrier.

Periodicity and the complete set of self-dual lines

Pairing each spin configuration $\Gamma$ with its $\pi$-rotated partner removes the imaginary part of the Boltzmann weight, leaving a real bundled weight $$W(\tilde\Gamma,T)=2e^{KH_K(\Gamma)}\cos(\phi H_{i\phi}(\Gamma))$$. Because a single spin flip changes $H_{i\phi}$ by multiples of 8, the partition function is periodic in $\phi$ with period $\pi/4$—verified by exact enumeration of a $K_{\rm down}=K-i\phi$0 system. This periodicity directly contradicts the conventional self-dual phase diagram, which has period $K_{\rm down}=K-i\phi$1 in $K_{\rm down}=K-i\phi$2.

Re-examining the duality analysis via the mapping onto a square-lattice two-state Potts model with diagonal bonds, the authors substitute $K_{\rm down}=K-i\phi$3 into the self-duality condition $K_{\rm down}=K-i\phi$4, obtaining the complete relation $K_{\rm down}=K-i\phi$5. This splits into two families:

$K_{\rm down}=K-i\phi$6

SDA recovers the conventional line; SDB is new. Since self-duality only nominates candidates, Monte Carlo verification is required to decide which segments are genuine critical boundaries.

Monte Carlo verification of the revised phase diagram

Using the four-state Potts-type order parameter and the fourth-order Binder ratio $K_{\rm down}=K-i\phi$7, the authors scan temperature paths crossing the candidate lines. Along $K_{\rm down}=K-i\phi$8, which intersects SDA at $K_{\rm down}=K-i\phi$9, clear Binder crossings appear at $\sinh(2K)=\pm\cos(2\phi)$0, and data collapse with $\sinh(2K)=\pm\cos(2\phi)$1 and $\sinh(2K)=\pm\cos(2\phi)$2 confirms the four-state Potts universality class, consistent with earlier transfer-matrix results. Along $\sinh(2K)=\pm\cos(2\phi)$3, crossings occur at SDB ($\sinh(2K)=\pm\cos(2\phi)$4) but not at the SDA intersection ($\sinh(2K)=\pm\cos(2\phi)$5), establishing that SDB contains genuine boundaries while parts of SDA do not. Fixing $\sinh(2K)=\pm\cos(2\phi)$6 and scanning $\sinh(2K)=\pm\cos(2\phi)$7 yields a single transition on the solid segment of the self-dual lines, again collapsing with $\sinh(2K)=\pm\cos(2\phi)$8, $\sinh(2K)=\pm\cos(2\phi)$9.

The decisive result concerns the partition-function minima $\pi/4$0, where the sign problem is most severe. There, the self-dual lines form a critical threshold: segments with $\pi/4$1 are genuine critical boundaries, while segments with smaller $\pi/4$2 are not. At $\pi/4$3, the Binder ratio data at $\pi/4$4 are too noisy for reliable extrapolation, so the crossing location is instead pinned using the scaled quantity $\pi/4$5, whose intersections approach the SDA–SDB intersection point. WL simulations of $\pi/4$6 corroborate this: at $\pi/4$7 the negative peak shifts toward the self-dual line with increasing $\pi/4$8, and no signal appears on the excluded segments.

A spurious signal—a local peak in $\pi/4$9 and $\Gamma$0 growing with system size at $\Gamma$1 along the second temperature path—is identified as a finite-size artifact rather than a new phase. The argument is that near $\Gamma$2 positive and negative signs occur with nearly equal probability, so any apparent order arises from superposition of opposite-sign configurations and cannot represent long-range order from a single configuration. The authors concede that small accessible sizes ($\Gamma$3 at this point due to the sign problem) prevent a rigorous finite-size exclusion; the conclusion remains preliminary but strongly supported by the decay of the average sign toward zero.

Sign-based probes revisited

Because $\Gamma$4 mixes the original and reference partition functions, its behavior as a transition probe is representation-dependent. Extending previous exact-enumeration results ($\Gamma$5) to $\Gamma$6 via WL, the paper reaches two conclusions. First, along the path crossing genuine SDA, the minimum of $\Gamma$7 converges to the true critical point—but this is coincidental with the structure of $\Gamma$8: the maximum of $\Gamma$9 reflects inflection points of both free energies, which occur at distinct temperatures ($\pi$0 for $\pi$1, $\pi$2 for $\pi$3). Second, along the path through $\pi$4, $\pi$5 develops a second minimum at $\pi$6 where no transition exists; it originates from a minimum of $\pi$7 coinciding with a maximum of $\pi$8, a mechanism not covered by the three scenarios catalogued by Ma et al., providing a new concrete failure mode of the average-sign probe.

By contrast, extrema of $\pi$9 coincide with those of $$W(\tilde\Gamma,T)=2e^{KH_K(\Gamma)}\cos(\phi H_{i\phi}(\Gamma))$$0 in both paths, confirming that the derivative of the average sign tracks the transition of whichever partition function varies more rapidly—in this model, the reference system. The practical implication is that sign-based probes cannot stand alone as transition detectors; they function only as cross-checks against direct thermodynamic observables.

Exponential barrier in the Wang–Landau algorithm

Although WL samples the positive-definite density of states $$W(\tilde\Gamma,T)=2e^{KH_K(\Gamma)}\cos(\phi H_{i\phi}(\Gamma))$$1 and thus formally avoids the sign problem, the paper demonstrates that the barrier reappears. The average sign on the critical line decays as $$W(\tilde\Gamma,T)=2e^{KH_K(\Gamma)}\cos(\phi H_{i\phi}(\Gamma))$$2, with $$W(\tilde\Gamma,T)=2e^{KH_K(\Gamma)}\cos(\phi H_{i\phi}(\Gamma))$$3 linear in $$W(\tilde\Gamma,T)=2e^{KH_K(\Gamma)}\cos(\phi H_{i\phi}(\Gamma))$$4. Comparing 100 independent WL runs at $$W(\tilde\Gamma,T)=2e^{KH_K(\Gamma)}\cos(\phi H_{i\phi}(\Gamma))$$5, observables computed for the original system carry substantially larger statistical errors than for the reference system sharing the identical density of states—the difference attributable entirely to cancellations amplifying fluctuations in the ratio $$W(\tilde\Gamma,T)=2e^{KH_K(\Gamma)}\cos(\phi H_{i\phi}(\Gamma))$$6.

The error analysis decomposes the single-measurement error as $$W(\tilde\Gamma,T)=2e^{KH_K(\Gamma)}\cos(\phi H_{i\phi}(\Gamma))$$7, where $$W(\tilde\Gamma,T)=2e^{KH_K(\Gamma)}\cos(\phi H_{i\phi}(\Gamma))$$8 is set by the termination condition. Compensating the sign amplification requires $$W(\tilde\Gamma,T)=2e^{KH_K(\Gamma)}\cos(\phi H_{i\phi}(\Gamma))$$9, and since tightening the termination costs $H_{i\phi}$0, the per-run cost scales as $H_{i\phi}$1—exponential in volume. The exponential barrier is therefore governed by the decay of the average sign, not by the choice between reweighting and density-of-states sampling, consistent with the model dependence observed in LLR applications (polynomial for the $H_{i\phi}$2 model, exponential for the hexagonal Hubbard model). A practical limitation of the WL implementation itself is the rapid growth of the state space: the number of accessible $H_{i\phi}$3 pairs scales as $H_{i\phi}$4 (e.g., 10,718 pairs at $H_{i\phi}$5), versus $H_{i\phi}$6 for the pure Baxter–Wu model, restricting simulations to $H_{i\phi}$7 despite windowed sampling with 75% overlap.

Limitations and open questions

Several caveats qualify the results. Finite-size scaling is limited to $H_{i\phi}$8 by the sign problem, precluding resolution of the logarithmic corrections expected near the four-state Potts fixed point. At $H_{i\phi}$9, accessible sizes drop to $\phi$0, so exclusion of long-range order below the threshold—and definitive convergence of the Binder crossing to the SDA–SDB intersection—remains unestablished. The interpolation formula for the number of accessible $\phi$1 pairs is validated only up to $\phi$2. Finally, whether the periodicity-driven revision generalizes to other complex-coupled models depends on microscopic details and is left open, as is the development of sign-problem-free schemes exploiting the Hermitian transfer matrix and implied PT symmetry of this model.

Conclusion

This work corrects the phase diagram of the GBW model with asymmetric complex couplings by incorporating the $\phi$3 partition-function periodicity into the self-duality analysis, yielding a new family of self-dual lines and identifying, via unbiased simulation, the threshold $\phi$4 separating genuine critical segments from non-critical ones, all in the four-state Potts universality class. It further demonstrates that minima of the average sign can be non-critical artifacts of the cosine factor's periodicity, that the derivative of the average sign probes the reference-system transition, and that the WL algorithm—despite formally bypassing signed sampling—retains the exponential barrier through the interplay of density-of-states precision and sign amplification. The results establish periodicity as a factor that must be accounted for when constructing phase diagrams from duality arguments in complex-coupled systems.