Completing the string-order-parameter and dual-Landau characterization of topological quantum phase transitions in the generalized Kitaev brick-wall model

The generalized Kitaev spin-1/2 model on brick-wall lattices maps, via a Jordan-Wigner transformation, to free Majorana fermions in a static $\mathbb{Z}_2$ gauge field, and its zero-temperature quantum phase transitions are of topological type: they are characterized by nonlocal string order parameters that, under a spin duality transformation, become local Landau order parameters of dual transverse-field Ising/XY chains (Feng, Zhang, and Xiang, Phys. Rev. Lett. 98, 087204 (2007)). Three issues were left open and remain open in whole or in part: (1) explicit closed-form string order parameters for the transverse-momentum sectors $q\neq 0,\pi$ of multi-row ladders, where the sector Hamiltonians are coupled and the known constructions fail; (2) the source's tentative remark that the gapless phase of the two-dimensional limit "may have a complicated non-Abelian topological structure", although the source itself notes that transitions from any gapped phase into the gapless phase are already characterized by string order parameters, its phase boundaries being fully determined by the transition lines of the decoupled $q=0$ and $q=\pi$ chains; (3) the conjecture that the correspondence between topological transitions and the condensation of nonlocal operators that are local order parameters in a dual representation holds generically for interacting systems without exact solvability, for which only partial results exist. The problem is to complete this dual-Landau characterization of the model family's phase transitions; the generality of the mechanism beyond exactly solvable models is stated by the source as an open direction for further exploration.

Background

Landau theory describes continuous phase transitions in terms of a local order parameter whose expectation value distinguishes an ordered phase with broken symmetry from a disordered one. A quantum phase transition (QPT), driven by a change of a Hamiltonian parameter at zero temperature, can however occur between two phases that break no symmetry; the canonical modern examples are transitions between topologically ordered phases, which admit no conventional local order parameter. The spin-1/2 Kitaev model on the honeycomb lattice is the cleanest exactly solvable laboratory for this phenomenon: nearest-neighbor spins interact through bond-dependent Ising couplings, $J_1\sigma^x_{j,l}\sigma^x_{j+1,l}$, $J_2\sigma^y_{j-1,l}\sigma^y_{j,l}$, and $J_3\sigma^z_{j,l}\sigma^z_{j,l+1}$, according to the direction of the bond. Feng, Zhang, and Xiang studied this Hamiltonian on the brick-wall lattice, which is the honeycomb lattice redrawn as staggered horizontal rows and which makes a row-by-row Jordan-Wigner transformation natural. The Jordan-Wigner transformation is a nonlocal mapping of spin-1/2 operators to spinless fermions; splitting each complex fermion into two Majorana fermions (hermitian fermion operators with $c=c^\dagger$ and $c^2=1$), they expressed the Hamiltonian exactly as free Majorana fermions hopping in a static $\mathbb{Z}_2$ gauge field: the bond variables $D_{j,l}=i d_{j,l}d_{j,l+1}$ on vertical bonds commute with the Hamiltonian, act as discrete local gauge fields, and label superselection (flux) sectors. Unlike Kitaev's original four-Majorana representation, this construction carries no redundant degrees of freedom.

For a brick-wall ladder of $2M$ rows, Fourier transformation along the rows decomposes the problem into sectors of transverse momentum $q=2\pi m/M$. For $q=0$ (and for $q=\pi$ when $M$ is even) the sector operators $c_{j,q}$ are self-conjugate Majoranas and each sector is equivalent to a two-leg ladder; the excitation gap closes on critical lines such as $J_- = J_3$ with $J_\pm=J_1\pm J_2$, where the second derivative of the ground-state energy density diverges logarithmically. To characterize these transitions, Feng-Zhang-Xiang introduced nonlocal string order parameters (SOPs), expectation values of spin operators multiplied along a semi-infinite string,

$\Delta_x=\lim_{j\to\infty}\Big\langle\prod_{k=1}^{2j}\sigma^x_k\Big\rangle,$

a direct analogue of the string order that detects hidden (dilute antiferromagnetic) order in the Haldane phase of spin-1 chains. Their central result is that under a spin duality transformation, for example $\sigma^x_j=\tau^x_{j-1}\tau^x_j$ and $\sigma^y_j=\prod_{k\ge j}\tau^y_k$, the model maps onto a transverse-field Ising or anisotropic XY chain in dual spin variables $\tau$, and the SOP becomes a local order parameter of the dual model: $\hat\Delta_x=\tau^x_0\tau^x_{2j}$ has a finite long-range limit exactly where the dual Ising model is ordered, and closed-form expressions follow from the known Ising/XY solutions. The topological QPT is thereby the condensation of a nonlocal string operator that is a conventional local order parameter in a dual representation, so that Landau's concepts survive in the dual space. In the two-dimensional limit $M\to\infty$ the critical lines merge into a gapless phase, reproducing Kitaev's phase diagram: gapped Abelian phases, a gapless phase, and, when a weak magnetic field breaks time-reversal symmetry and gaps the gapless phase, a gapped phase supporting non-Abelian anyons (quasiparticles whose exchange operations do not commute), equivalent to Ising-model conformal blocks and characterized by the Chern number (an integer topological invariant of the occupied Majorana band) of magnitude one.

This framework, however, was established only for the momentum sectors $q=0,\pi$, for limiting regimes of the remaining sectors, and within a Hamiltonian that is exactly solvable. The open problem is the completion and generalization of this dual-Landau characterization.

Problem Statement

The problem is to complete the topological order-parameter characterization of the quantum phase transitions of the generalized Kitaev model on brick-wall ladders. Concretely, the source's stated open issue is:

(Generic momentum sectors) Derive explicit, accurate closed-form expressions for the string order parameters of the transverse-momentum sectors $q\neq 0,\pi$ of the $2M$-row brick-wall lattice ($M\ge 3$), where $c_{j,q}$ and $c_{j,-q}$ combine into ordinary (Dirac) fermions and the sector Hamiltonians $H_q$ and $H_{-q}$ are coupled, so that the constructions used for $q=0$ and $q=\pi$ do not apply. Expectation values are taken in the source's zero-flux ground-state sector $D_{j,l}=(-)^l$; the ground state of the $2M$-row lattice is $2^{2M}$-fold degenerate, and the source notes that its conclusions generalize to the other degenerate states by unitary transformations. The source shows that, in the limiting regimes $J_1\gg J_2$ and $J_1\ll J_2$, two string order parameters defined from combinations of $c_{j,q}$ and $c_{j,-q}$ exist, but displays no explicit formulas for them; the sought closed-form expressions must be consistent with this limiting-regime existence assertion. The solution must characterize the condensation of the string order parameters across the gap-closing lines of the multi-row phase diagram, of which there are $M+2$ for even $M$ and $M+1$ for odd $M$ (source, Fig. 5; analytic lines are given in the source only for the decoupled $q=0,\pi$ sectors), consistently with the source's explicit $q=0$ results (condensation at $J_-=\pm J_3$) and $q=\pi$ results (at $J_+=J_3$ and $J_-=0$), where $J_\pm=J_1\pm J_2$.

Beyond this model-family completion, the source closes with the conjecture that the correspondence it demonstrated, namely a topological quantum phase transition described as the condensation of a nonlocal string operator that is a local order parameter in an appropriate dual representation, "might be valid generically" beyond the specific solvable Hamiltonian, and states that "further exploration of this problem is desired". The generality of this mechanism for interacting quantum systems lacking exact solvability is not part of this task. The source's tentative remark that the gapless phase of the two-dimensional limit may have a complicated non-Abelian topological structure likewise concerns a broader open direction and is not part of this task.

References

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