Kosterlitz–Thouless critical reduced field of the triangular Ising antiferromagnet in a field: controlled finite-bond-dimension extrapolation for $C_3$-symmetric CTMRG and the asymptotics of the three-state Potts critical line

The triangular-lattice Ising antiferromagnet (TIAF) in a magnetic field is a canonical frustrated classical spin model whose zero-temperature line, in reduced variables $H=h/T$, hosts a field-driven Kosterlitz–Thouless (KT) transition: the constrained model has a critical phase for $0\le H<H_{KT}$ and a $\sqrt{3}\times\sqrt{3}$ ordered phase beyond, the transition occurring where the correlation-decay exponent reaches $\eta=4/9$. Transfer-matrix studies locate $H_{KT}=0.266\pm0.01$, but only after applying finite-size logarithmic and power-law corrections, whereas the $C_3$-symmetric honeycomb CTMRG (corner transfer matrix renormalization group) study of Nyckees, Rufino, Mila and Colbois finds $H_{KT}=0.305\pm0.006$ from finite-bond-dimension scaling, with a drift of the fitted exponent on the bond-dimension window that makes this an upper bound. The open problem is to settle $H_{KT}$ decisively — and with it the predicted square-root approach $K_c^{-1}\propto(H_c-H_{KT})^{1/2}$ of the finite-temperature three-state Potts critical line to the KT point — which requires a controlled finite-$\chi$ corrections-to-scaling framework for CTMRG near KT transitions (analogous to transfer-matrix finite-size corrections) and/or access to temperatures below $K^{-1}=0.25$, where the physical correlation length $\xi\sim e^{2J/T}$ exceeds any representable $\xi_\chi=\chi^{\kappa}$. Auxiliary goals identified by the source work are an optimal (Corboz-type) isometry and a multisite, arbitrary-unit-cell generalization of the honeycomb CTMRG restoring access to the true transfer matrix in symmetry-broken phases.

Background

The triangular-lattice Ising antiferromagnet (TIAF) is defined by the Hamiltonian $\mathcal{H}=J\sum_{\langle i,j\rangle}\sigma_i\sigma_j-h\sum_i\sigma_i$ with Ising spins $\sigma_i=\pm 1$, antiferromagnetic nearest-neighbor couplings $J>0$ on the triangular lattice, and field $h$. Because a triangular plaquette cannot satisfy all three of its antiferromagnetic bonds simultaneously, the zero-field ground state found by Wannier is macroscopically degenerate and critical: spin correlations $G(r)\sim r^{-\eta}$ decay algebraically with exponent $\eta=1/2$, and for any $T>0$ the correlation length is finite but diverges exponentially, $\xi\sim e^{2J/T}$, as $T\to 0$ (Stephenson; the point $h=T=0$ is often called the Villain–Stephenson point).

A positive field favors configurations in which every triangle carries two spins up and one down (the UUD rule). This selects the three-sublattice $\sqrt{3}\times\sqrt{3}$ state of the $m=1/3$ magnetization plateau, stable for $0<h<6J$; in the limit $T\to0$, $h\to6J$ at fixed $(6J-h)/T$ the near-saturation configurations form Baxter's solved hard-hexagon model (a lattice gas of particles with nearest-neighbor exclusion, solved exactly), whose critical activity $\zeta_c=(11+5\sqrt{5})/2$ fixes the low-temperature asymptote of the Potts critical line bounding the plateau. At finite temperature the plateau order melts through a transition in the three-state Potts universality class — the standard class for spontaneous breaking of a threefold ($Z_3$) symmetry, with order-parameter exponent $\beta=1/9$, correlation-length exponent $\nu=5/6$, $\eta=4/15$ and central charge $c=4/5$.

The low-temperature structure is best expressed in reduced variables $H=h/T$ and $K=J/T$. On the line $K^{-1}=0$ (zero temperature at fixed reduced field) the partition function reduces to the constrained model, in which every triangle obeys the UUD rule. This model maps exactly onto a triangular solid-on-solid (SOS) model — a lattice model of a discrete height variable whose rough phase realizes algebraic correlations — and admits a Coulomb-gas renormalization-group (RG) treatment (a field-theoretic description of two-dimensional criticality as a gas of logarithmically interacting charges). The reduced field acts as a threefold-symmetric perturbation of the critical phase; by the Kosterlitz–Thouless RG criterion it is irrelevant while $\eta>4/9$ and becomes relevant at $\eta=4/9$, where a Kosterlitz–Thouless (KT) transition — an infinite-order transition with an essential (exponential) divergence of the correlation length — separates the critical phase ($0\le H<H_{KT}$) from the ordered phase ($H>H_{KT}$). The RG does not predict the location $H_{KT}$.

The location and the associated line shape have a forty-year numerical history. Kinzel and Schick conjectured that the Potts critical line approaches the zero-field critical point with infinite slope; the Coulomb-gas RG of Nienhuis, Hilhorst and Blöte instead predicts a finite slope. Transfer-matrix studies located the KT point at $H_{KT}=0.266\pm0.01$ (Blöte, Nightingale, Wu and Hoogland; Blöte and Nightingale; Qian, Wegewijs and Blöte) — but only after applying both logarithmic and power-law finite-size corrections in the finite-size scaling analysis; Qian et al. moreover gave RG and transfer-matrix evidence that the Potts line terminates on the KT point with a square-root shape, $K_c^{-1}\propto(H_c-H_{KT})^{1/2}$.

On the numerical-method side, the corner transfer matrix renormalization group (CTMRG) — introduced by Nishino and Okunishi by combining Baxter's corner transfer matrix (an exact decomposition of a two-dimensional partition function into four quarter-plane blocks) with White's density-matrix renormalization group (a variational truncation method for low-dimensional quantum lattice models) — contracts infinite two-dimensional tensor networks by iteratively growing corner and edge tensors and truncating the boundary (bond) dimension to $\chi$. Finite $\chi$ induces a finite-entanglement scaling: at a critical point the induced correlation length grows as $\xi_\chi\sim\chi^{\kappa}$ with $\kappa=6/(c(\sqrt{12/c}+1))$, while the order parameter and boundary entanglement entropy scale as $\psi\sim\chi^{-\kappa\eta/2}$ and $S\sim(c\kappa/6)\log\chi$ (Tagliacozzo et al.; Pollmann et al.). These power laws carry corrections in $\chi$ whose form near KT transitions is not known — in contrast both to the well-developed finite-size correction theory of transfer-matrix scaling and to BKT analyses in one-dimensional quantum (matrix-product-state) settings.

Nyckees, Rufino, Mila and Colbois introduced a $C_3$-symmetric CTMRG that contracts honeycomb-lattice tensor networks natively — the natural geometry of the dual tensor network of the TIAF — thereby preserving the threefold lattice symmetry. With it they recovered the Potts line down to $K^{-1}=0.25$ with much smaller error bars than transfer matrices, and at $K^{-1}=0$ located the KT transition via the $\eta=4/9$ criterion within finite-$\chi$ scaling, obtaining $H_{KT}=0.305\pm0.006$. However, near $H\approx 0.29$–$0.30$ the fitted exponent drifts with the bond-dimension window, so the authors regard their estimate as an upper bound; they state explicitly that it is unclear how to perform corrections analogous to the transfer-matrix finite-size logarithmic and power-law corrections within finite-$\chi$ scaling. Relatedly, their measured $\kappa\approx1.23$ at the Villain–Stephenson point disagrees with the theoretical prediction, which they attribute to uncontrolled bond-dimension corrections. Temperatures below $K^{-1}=0.25$ were inaccessible: the physical correlation length $\xi\sim e^{2J/T}$ would require exponentially large $\chi$, since the representable correlation length is bounded by $\xi_\chi=\chi^{\kappa}$, and in the constrained model the CTMRG environment spontaneously breaks translation symmetry (converging modulo three), blocking the usual transfer-matrix diagnostics of correlation length.

Problem Statement

Determine, with controlled and quantified error treatment, the zero-temperature Kosterlitz–Thouless critical reduced field $H_{KT}$ of the triangular Ising antiferromagnet in a field — i.e. of the constrained model on the line $K^{-1}=0$ — together with the asymptotic shape of the finite-temperature three-state Potts critical line where it terminates at this KT point. Concretely, the problem posed by the source work is: (1) establish a controlled finite-bond-dimension extrapolation or corrections-to-scaling framework for CTMRG near KT transitions, analogous to the finite-size logarithmic and power-law corrections of transfer-matrix finite-size scaling, and use it — or an equally controlled alternative method — to decide whether $H_{KT}$ is the transfer-matrix value $0.266\pm0.01$, the finite-$\chi$ CTMRG value $0.305\pm0.006$ (regarded as an upper bound), or some other value, by an independently obtained determination of $H_{KT}$ (not a restatement of either published value) whose own error budget excludes at least one of the two competing values; and (2) determine whether the Potts line approaches the KT point with the predicted square-root form $K_c^{-1}\propto(H_c-H_{KT})^{1/2}$ as $H_c\to H_{KT}^{+}$, down to temperatures below the currently accessible $K^{-1}=0.25$, where the physical correlation length $\xi\sim e^{2J/T}$ exceeds any representable $\xi_\chi=\chi^{\kappa}$. Auxiliary methodological components identified by the source work in direct service of this objective are the formulation of an optimal (Corboz-type) isometry — the choice of truncation map used when enlarging the corner tensors — for the honeycomb CTMRG, and a multisite (arbitrary-unit-cell) generalization that restores access to the true transfer matrix, and hence to correlation lengths, in the symmetry-broken phase; both would reduce the systematic errors entering the determination above, but neither is required: the objective is resolved by (1) and (2) with or without these two auxiliary components.

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