Constructing Landau framework for topological order: quantum chains and ladders

Abstract

We studied quantum phase transitions in the antiferromagnetic dimerized spin-1 2 XY chain and two-leg ladders. From analysis of several spin models we present our main result: the framework to deal with topological orders and hidden symmetries within the Landau paradigm. After mapping of the spin Hamiltonians onto the tight-binding models with Dirac or Majorana fermions and, when necessary, the mean-field approximation, the analysis can be done analytically. By utilizing duality transformations the calculation of nonlocal string order parameters is mapped onto the local order problem in some dual representation and done without further approximations. Calculated phase diagrams, phase boundaries, order parameters and their symmetries for each of the phases provide a comprehensive quantitative Landau description of the quantum critical properties of the models considered. Complementarily, the phases with hidden orders can also be distinguished by the Pontryagin (winding) numbers which we have calculated as well. This unified framework can be straightforwardly applied for various spin chains and ladders, topological insulators and superconductors. Applications to other systems are under way.

String and conventional order parameters in the solvable modulated quantum chain

“…Using judiciously chosen duality transformations we show that a SOP can be identified as a local order parameter of some dual Hamiltonian. Sometimes this can help to easily calculate the SOP in the dual framework, 13,14 , sometimes not. But it is important to understand as a matter of principle.…”

“…In the limit h = h a = 0 the above determinants can be evaluated exactly by the standard technique, 49 reproducing the earlier results for nonvanishing O x/y,e/o obtained from duality mappings. 13 When h = 0 and/or h a = 0 we were unable to derive analytical results for asymptotics of these Toeplitz determinants. Numerical results show that in the presence of fields all SOPs O x/y,e/o die off in the thermodynamic limit N → ∞.…”

“…In the related recent work we have systematically demonstrated how the Landau formalism can be extended to deal with nonconventional quantum orders. 13,14 The key point is to incorporate nonlocal string operators, 15 string correlation functions, and string order parameters (SOPs). The appearance of nonlocal SOP is accompanied by a hidden symmetry breaking.…”

“…Ox,e(m)Ox,e(n)Ox,o(m − 1)Ox,o(n − 1) = In analytical work it is not convenient to deal with O y,e/o as strings of dual spins (B9), (B10). Instead,13 it is easier to apply the duality transformations(3,4) with the interchange x ↔ y. Then the r.h.s.…”

Local and nonlocal order parameters in the Kitaev chain

“…In our recent work we made a strong claim that the Landau formalism, although extended, remains instrumental even for nonconventional quantum orders. 7 The formalism needs to be extended to incorporate nonlocal (string) operators, 8,9 string correlation functions, and string order parameters (SOPs). The appearance of nonlocal SOP is accompanied by a hidden symmetry breaking.…”