The quantum Baxter–Wu model (BWTF) — spin-1/2 degrees of freedom on the triangular lattice with ferromagnetic three-spin interactions perturbed by a transverse magnetic field — has a phase diagram in which a second-order thermal transition line in the four-state Potts universality class meets a first-order line emanating from the zero-temperature quantum transition at hc≈2.4Jh_c\approx 2.4J, at a tricritical point bracketed only roughly as h∗/J∈(2.25,2.35)h^*/J\in(2.25,2.35), T∗/J∈(0.85,1.15)T^*/J\in(0.85,1.15) (best estimate T∗≈JT^*\approx J). The problem is to determine this tricritical point precisely: its location, its tricritical critical exponents, and its universal critical behavior (the conformal field theory describing it), while disentangling genuine tricritical asymptotics from strong first-order-like finite-size crossover effects. The question was posed explicitly by Capponi, Jahromi, Alet, and Schmidt in 2014; it remains unanswered.

Background

The classical Baxter–Wu model (Baxter and Wu, 1973) places Ising spins σ=±1\sigma=\pm1 on the sites of a triangular lattice and couples the three spins of every elementary triangle through a three-spin interaction −Jσiσjσk-J\sigma_i\sigma_j\sigma_k with J>0J>0. It is one of the few exactly solved interacting lattice models: its critical temperature is known exactly (Tc=2J/ln⁡(1+2)≈2.269JT_c=2J/\ln(1+\sqrt2)\approx 2.269J at zero field, a self-dual point, with kB=1k_B=1 so that temperatures are measured in units of JJ), the ground state is fourfold degenerate because the ordering pattern is fixed only on each of the three sublattices of the triangular lattice, and the thermal transition lies in the universality class of the four-state Potts model but — remarkably — without the logarithmic corrections that afflict the standard four-state Potts model. The quantum extension studied by Capponi, Jahromi, Alet, and Schmidt (2014) adds a transverse magnetic field, H=−J∑⟨ijk⟩σizσjzσkz−h∑iσixH=-J\sum_{\langle ijk\rangle}\sigma_i^z\sigma_j^z\sigma_k^z-h\sum_i\sigma_i^x (the Baxter–Wu model in a transverse field, BWTF), where a single σx\sigma^x flip changes the eigenvalues of the six triangle ('face') operators around a site, so hh tunes quantum fluctuations between the fourfold-degenerate Baxter–Wu order at h=0h=0 and a fully xx-polarized paramagnet at large hh. The model has a second physical motivation: it is isospectral, up to a constant, to the sector with trivial plaquette eigenvalues of the topological color code (a two-dimensional stabilizer quantum-error-correcting code on the dual honeycomb lattice whose plaquette operators commute and can be measured simultaneously) subjected to a uniform magnetic field parallel to the code's qubit axis, so the BWTF phase diagram encodes the robustness of that quantum memory. Using stochastic-series-expansion quantum Monte Carlo (a sampling method that expands the partition function in powers of the Hamiltonian) and perturbative continuous-unitary-transformation series expansions (a controlled high-order perturbative method about the low- and high-field limits), the 2014 work established the (h,T)(h,T) phase diagram: a second-order thermal line in the four-state-Potts universality class (again without logarithmic corrections) persists from h=0h=0 up to h≈2.25Jh\approx 2.25J; the zero-temperature transition at hc≈2.4Jh_c\approx 2.4J is first order — expected, since the T=0T=0 quantum problem maps to the three-dimensional classical Baxter–Wu model, and the three-dimensional classical version was later confirmed numerically to be first order — and this first-order character extends to finite temperature. Where the two boundaries merge, a tricritical point (a multicritical point at which a line of continuous transitions terminates on a line of discontinuous ones) is predicted at approximately (h∗≈2.3J,T∗≈J)(h^*\approx 2.3J, T^*\approx J), but only a window h∗/J∈(2.25,2.35)h^*/J\in(2.25,2.35), T∗/J∈(0.85,1.15)T^*/J\in(0.85,1.15) could be bracketed, because finite lattices in this regime display first-order-like artifacts — negative Binder cumulants (the dimensionless ratio UL=1−⟨m4⟩/(3⟨m2⟩2)U_L=1-\langle m^4\rangle/(3\langle m^2\rangle^2), built from a sublattice magnetization, taking negative values on finite lattices) and double-peaked energy histograms that vanish only in the thermodynamic limit — while the effective correlation-length exponent drifts from ν≈0.6\nu\approx0.6 at h=2.25Jh=2.25J to ν≈0.52\nu\approx0.52 at h=2.35Jh=2.35J. Tricritical points occupy a central place in Potts-model physics since the renormalization-group solution of Nienhuis, Berker, Riedel, and Schick (1979): in two dimensions the tricritical qq-state Potts family terminates at q=4q=4 in a marginal situation whose critical behavior is described by a conformal field theory (the scale-invariant field theory governing two-dimensional critical points; its central charge c=1c=1 fixes the universal finite-size scaling of the free energy), so the identification of any new q=4q=4-related tricriticality is theoretically delicate. The BWTF itself has no known exact solution or self-duality, so no analytic anchor exists for the tricritical location or its exponents; the model's finite-temperature transitions are governed by effective two-dimensional classical fixed points, which is why a conformal-field-theory description is expected to exist at the tricritical point. The classical analogue — the spin-1 (dilute) Baxter–Wu model in a crystal field, whose annealed vacancies generate a closely related second-order-to-first-order changeover — has been studied intensively by Monte Carlo and transfer-matrix methods through 2026 and still exhibits unsettled multicritical behavior with c≈1c\approx1 and systematically drifting scaling dimensions, underscoring the difficulty of the quantum question.

Problem Statement

For the Baxter–Wu model in a transverse magnetic field on the triangular lattice, H=−J∑⟨ijk⟩σizσjzσkz−h∑iσixH=-J\sum_{\langle ijk\rangle}\sigma_i^z\sigma_j^z\sigma_k^z-h\sum_i\sigma_i^x with ferromagnetic J>0J>0, determine the tricritical point at which the second-order thermal transition line (in the four-state-Potts universality class for h≲2.25Jh\lesssim 2.25J) meets the first-order line extending from the zero-temperature quantum transition at hc≈2.4Jh_c\approx 2.4J. The characterization consists of three jointly required deliverables: (i) the location (h∗,T∗)(h^*,T^*) with controlled uncertainties, improving on the currently bracketed window h∗/J∈(2.25,2.35)h^*/J\in(2.25,2.35), T∗/J∈(0.85,1.15)T^*/J\in(0.85,1.15) (best estimate (h∗,T∗)≈(2.3J,J)(h^*,T^*)\approx(2.3J,J)); (ii) the tricritical critical exponents (for the correlation length, order parameter, susceptibility, and specific heat, satisfying the exact two-dimensional scaling relations); and (iii) the universal critical behavior of the point — its universality class, expressed as the conformal field theory (central charge and relevant operator content) that describes it. Any answer must demonstrate that the claimed tricritical behavior is the asymptotic thermodynamic-limit behavior and not a finite-size crossover artifact, given the documented first-order-like signatures (negative Binder cumulants, double-peaked energy histograms, drifting effective exponents) that appear on approaching the tricritical region at accessible lattice sizes.

References

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