The row transfer matrix of a two-dimensional classical spin model can be written as the exponential of a one-dimensional quantum spin Hamiltonian. The standard construction (the strong-anisotropy or Hamiltonian limit) requires both expansion parameters - txt_x, which multiplies the diagonal in-row part of the transfer matrix, and tyt_y, which multiplies its row-flip part - to be small, and neglects the non-commutativity of the two parts. Zaburannyi (arXiv:1006.1608, 2010) introduced a strong-interaction limit (SIL) that keeps txt_x finite: expanding to linear order in tyt_y and resumming all even nested commutators in txt_x yields, for the 2D Ising model, a transverse-Ising chain with three-spin terms that is still exactly solvable and yields the same critical temperature as the strong-anisotropy limit, and, for the six-vertex model, a spin-1/2 chain whose nearest-neighbour XY exchange is multiplied by an operator-valued four-spin factor. For this multispin chain the paper states that no exact results are available, so the SIL prediction for the six-vertex critical temperature was left undetermined, along with the size of the omitted O(ty2)O(t_y^2) terms and the range of validity of the nested-commutator periodicity assumption underlying the construction. This problem asks for the central of these completions: the phase diagram and quantum critical line of the six-vertex SIL Hamiltonian.

Background

Transfer matrices and the classical-quantum correspondence. For a two-dimensional classical lattice model the partition function can be written Z=Tr TMZ=\mathrm{Tr}\,T^{M}, where TT is the row-to-row transfer matrix acting on the 2N2^{N}-dimensional space of configurations of one row of NN sites; the free energy per site is fixed by the largest eigenvalue of TT. Writing H=−β−1ln⁡TH=-\beta^{-1}\ln T (with β\beta the classical inverse temperature) turns the classical model into a one-dimensional quantum spin chain: the classical free energy per site equals the ground-state energy per site of HH in the limit of infinitely many rows. This is the standard bridge between two-dimensional classical criticality and one-dimensional quantum criticality [Kogut 1979]. The usual construction, the strong-anisotropy or Hamiltonian limit [Suzuki 1971], splits T=etxX^(1+tyY^)etxX^+O(ty2)T=e^{t_x\hat X}(1+t_y\hat Y)e^{t_x\hat X}+O(t_y^{2}) into a diagonal part X^\hat X and an off-diagonal row-flip part Y^\hat Y and takes both txt_x and tyt_y small with fixed ratio, so that T≈e2txX^+tyY^T\approx e^{2t_x\hat X+t_y\hat Y}, with the non-commutativity of X^\hat X and Y^\hat Y neglected at order tx2t_x^{2}. In this limit the 2D Ising model maps onto the transverse-field Ising chain (a chain of spins with nearest-neighbour Ising coupling along zz and a field along xx that flips single spins) and the six-vertex model onto the spin-1/2 XXZ chain (nearest-neighbour exchange sjxsj+1x+sjysj+1ys^{x}_{j}s^{x}_{j+1}+s^{y}_{j}s^{y}_{j+1} plus an anisotropic sjzsj+1zs^{z}_{j}s^{z}_{j+1} coupling), both exactly solvable: the former by the Jordan-Wigner transformation (a mapping of spins to fermions that diagonalizes quadratic Hamiltonians), the latter by Bethe ansatz (an exact construction of interacting one-dimensional many-body wavefunctions as superpositions of plane-wave states).

The six-vertex model itself is a two-dimensional model of arrows on lattice edges obeying the ice rule - at each vertex exactly two arrows point in and two point out, leaving six allowed vertex configurations - with vertex energies e0e_{0}, exe_{x}, eye_{y}, e∞e_{\infty} in the parameterization used by the source paper. It is exactly solved [Lieb 1967; Sutherland 1970]; its classical critical line is e−βex+e−βey=1e^{-\beta e_{x}}+e^{-\beta e_{y}}=1. The strong-anisotropy XXZ correspondence above approximates this line by e−βey=βexe^{-\beta e_{y}}=\beta e_{x} (the isotropy condition of the XXZ chain, Eq. (47) of the source).

The strong-interaction limit. Zaburannyi (2010) observed that demanding tx→0t_x\to0 is unnecessarily restrictive: the exact relation T=etxX^(1+tyY^)etxX^+O(ty2)T=e^{t_x\hat X}(1+t_y\hat Y)e^{t_x\hat X}+O(t_y^{2}) can be evaluated for arbitrary txt_x if the non-commutativity is kept. Writing T=exp⁡(2txX^+ty(Y^+Z^))+O(ty2)T=\exp\bigl(2t_x\hat X+t_y(\hat Y+\hat Z)\bigr)+O(t_y^{2}), the correction operator is a series over even nested commutators, Z^=∑k≥1A2k(2k)!tx2k[X^,Y^]2k\hat Z=\sum_{k\ge1}\frac{A_{2k}}{(2k)!}t_x^{2k}[\hat X,\hat Y]_{2k} with [X^,Y^]k=[X^,[X^,Y^]k−1][\hat X,\hat Y]_{k}=[\hat X,[\hat X,\hat Y]_{k-1}], whose coefficients are generated by A(x)=x/sinh⁡x−1\mathcal{A}(x)=x/\sinh x-1 (a function whose Taylor coefficients are rational multiples of the Bernoulli numbers). The resummation goes through whenever the even nested commutators close into a periodic sandwich form - the ansatz of Eq. (A10) of the source. For the Ising model, where tx=βJx/2t_x=\beta J_{x}/2 and ty=e−βJy/2t_y=e^{-\beta J_{y}/2}, the resulting SIL Hamiltonian (Eq. (24) of the source) is a transverse-Ising chain with a renormalized transverse field and additional three-spin interactions sj−1zsjxsj+1zs^{z}_{j-1}s^{x}_{j}s^{z}_{j+1}; this is still a member of Suzuki's exactly solvable generalized XY chains (a family of chains with one- and multi-spin XY-type interactions that remain exactly solvable by the Jordan-Wigner method), and its critical point coincides with the strong-anisotropy one, which in turn reproduces the exact Onsager critical temperature (from Onsager's exact solution of the 2D Ising model) in the extreme-anisotropic regime. For the six-vertex model, where tx=−βex/2t_x=-\beta e_{x}/2 and ty=e−βeyt_y=e^{-\beta e_{y}}, the SIL Hamiltonian is

Hsil=2∑j=1N[ex sjzsj+1z−e−βeyβ(1+A(βex)2(1−4sj−1zsj+2z))(sjxsj+1x+sjysj+1y)],H_{\mathrm{sil}}=2\sum_{j=1}^{N}\Bigl[e_{x}\,s^{z}_{j}s^{z}_{j+1}-\frac{e^{-\beta e_{y}}}{\beta}\Bigl(1+\frac{\mathcal{A}(\beta e_{x})}{2}\bigl(1-4s^{z}_{j-1}s^{z}_{j+2}\bigr)\Bigr)\bigl(s^{x}_{j}s^{x}_{j+1}+s^{y}_{j}s^{y}_{j+1}\bigr)\Bigr],

with sjαs^{\alpha}_{j} spin-1/2 operators, up to an additive constant. Here the source's printed Eq. (43) omits the bond sum and the overall factor 2 on the exchange term; the form above is the intended reading, fixed by the source's general formula Eq. (14) with the appendix resummation, in which the periodicity parameter of Eq. (A10) is z^=2\hat z=2 and the argument of A\mathcal{A} is 2tx=−βex2t_{x}=-\beta e_{x} (the even function A\mathcal{A} therefore makes A(βex)\mathcal{A}(\beta e_{x}) as printed). The bracket modulates the XY exchange by a four-spin factor: it equals 1 when the flanking spins at sites j−1j-1 and j+2j+2 are parallel and 1+A(βex)1+\mathcal{A}(\beta e_{x}) when they are antiparallel. The chain conserves total SzS^{z} but is interacting: after a Jordan-Wigner transformation the four-spin term becomes a correlated-hopping term in which the amplitude for a fermion to hop between sites jj and j+1j+1 depends on the occupations of sites j−1j-1 and j+2j+2. The free-fermion methods that settled the Ising case therefore do not apply, and no exact solution is known. The paper's conclusions state verbatim that the strong-interaction critical temperature 'requires a study of the quantum spin chain described by Hamiltonian (43) for which no exact results are available'. The source also observes that on the exact critical line tx=ln⁡(1−ty)/2=−ty/2+O(ty2)t_x=\ln(1-t_y)/2=-t_y/2+O(t_y^{2}), so that both expansion parameters are small there, from which it expects the strong-interaction and strong-anisotropy predictions to agree near criticality - an expectation it states but does not verify. The SIL construction could thus be benchmarked on the Ising model but not on the six-vertex model, and the size of the neglected O(ty2)O(t_y^{2}) terms and the range of validity of the commutator-closure ansatz were left unquantified.

Problem Statement

The problem is to complete the strong-interaction-limit (SIL) correspondence between the two paradigmatic 2D classical spin systems and their 1D quantum chains, so that the one quantitative prediction the construction could not deliver - the critical behaviour of the six-vertex SIL chain - is determined and its accuracy assessed. Concretely, with HsilH_{\mathrm{sil}} as defined in the Background, determine the ground-state phase diagram of HsilH_{\mathrm{sil}} in the plane of its dimensionless couplings (equivalently, of the classical parameters βex\beta e_{x} and βey\beta e_{y}): the phase boundaries, the critical line or lines, and the universality classes of the transitions. These three components are all required, and the comparison that follows forms part of the critical-line determination rather than replacing the phase boundaries and universality classes. In particular, settle the SIL prediction for the six-vertex critical temperature, which the source left undetermined because no exact results are available for this chain: establish whether the set of classical parameters at which HsilH_{\mathrm{sil}} is critical coincides with the strong-anisotropy prediction e−βey=βexe^{-\beta e_{y}}=\beta e_{x} - as the Ising SIL chain reproduces the strong-anisotropy result - or deviates from it, and how the line so obtained compares with the exact classical six-vertex critical line e−βex+e−βey=1e^{-\beta e_{x}}+e^{-\beta e_{y}}=1, which serves as the accuracy benchmark.

References

  1. Oles Zaburannyi, 'Strong-interaction approximation for transfer matrix method', arXiv:1006.1608 [cond-mat.stat-mech] (2010), DOI: 10.48550/arXiv.1006.1608, https://arxiv.org/abs/1006.1608
  2. Masuo Suzuki, 'Relationship among Exactly Soluble Models of Critical Phenomena. I', Progress in Theoretical Physics 46(5), 1337-1359 (1971), DOI: 10.1143/PTP.46.1337, https://doi.org/10.1143/PTP.46.1337
  3. Elliott H. Lieb, 'Exact Solution of the Two-Dimensional Slater KDP Model of a Ferroelectric', Physical Review Letters 19(3), 108-110 (1967), DOI: 10.1103/PhysRevLett.19.108, https://doi.org/10.1103/PhysRevLett.19.108
  4. Bill Sutherland, 'Two-Dimensional Hydrogen Bonded Crystals without the Ice Rule', Journal of Mathematical Physics 11(11), 3183-3186 (1970), DOI: 10.1063/1.1665111, https://doi.org/10.1063/1.1665111
  5. John B. Kogut, 'An introduction to lattice gauge theory and spin systems', Reviews of Modern Physics 51(4), 659-713 (1979), DOI: 10.1103/RevModPhys.51.659, https://doi.org/10.1103/RevModPhys.51.659
  6. O. Zaburannyi, 'Discrete lattice surface models and their quantum spin counterparts', Journal of Physical Studies 10(2), 98-107 (2006), DOI: 10.30970/jps.10.98, https://doi.org/10.30970/jps.10.98