Long-range Ising and Kitaev models: phases, correlations and edge modes

Abstract

We analyze the quantum phases, correlation functions and edge modes for a class of spin-1/2 and fermionic models related to the one-dimensional Ising chain in the presence of a transverse field. These models are the Ising chain with anti-ferromagnetic long-range interactions that decay with distancer as a r 1 , as well as a related class of fermionic Hamiltonians that generalize the Kitaev chain, where both the hopping and pairing terms are long-range and their relative strength can be varied. For these models, we provide the phase diagram for all exponents α, based on an analysis of the entanglement entropy, the decay of correlation functions, and the edge modes in the case of open chains. We demonstrate that violations of the area law can occur for  a 1, while connected correlation functions can decay with a hybrid exponential and power-law behavior, with a power that is α-dependent. Interestingly, for the fermionic models we provide an exact analytical derivation for the decay of the correlation functions at every α. Along the critical lines, for all models breaking of conformal symmetry is argued at low enough α. For the fermionic models we show that the edge modes, massless for  a 1, can acquire a mass for a < 1. The mass of these modes can be tuned by varying the relative strength of the kinetic and pairing terms in the Hamiltonian. Interestingly, for the Ising chain a similar edge localization appears for the first and second excited states on the paramagnetic side of the phase diagram, where edge modes are not expected. We argue that, at least for the fermionic chains, these massive states correspond to the appearance of new phases, notably approached via quantum phase transitions without mass gap closure. Finally, we discuss the possibility to detect some of these effects in experiments with cold trapped ions.

Edge insulating topological phases in a two-dimensional superconductor with long-range pairing

“…Independent theoretical works have concluded that LR systems can exhibit many interesting and unusual properties, essentially due to the violation of locality (see e.g. [13][14][15][16][17][18][19][20]). In particular, one-dimensional LR models can host new phases, manifesting striking properties, not present in the SR limit.…”

“…In particular, one-dimensional LR models can host new phases, manifesting striking properties, not present in the SR limit. [15][16][17][21][22][23][24][25][26][27][28][29][30].…”

“…In particular, one-dimensional LR models can host new phases, manifesting striking properties, not present in the SR limit. [15][16][17][21][22][23][24][25][26][27][28][29][30].In spite of these relevant achievements, full comprehension and classification phases emerging from Hamiltonians with LR terms are still not available.In [29] a partial solution to these problems has been discussed, exploiting one-dimensional topological superconductive chains as playgrounds, but inferring nontrivial properties also for LR topological insulators/superconductors in higher dimensions. …”

“…The same feature justifies the evolution of the edge modes described in the following. The two-point correlations are computed from the Bogoliubov transformations diagonalizing the Hamiltonian (1) [17,52]. Both correlations asymptotically display an algebraic decay, also if the mass gap is nonvanishing.…”

“…The stability of the MEMs against a local disorder parallels that of the LR phases discussed above. The possibility of a hybridization mechanism (of the massless edge states at α 1) generating the MEMs, as in one dimension [17,25,57], remains a likely but open issue in the mixed case, while an edge reconstruction mechanism [58] seems ruled out by the change of n passing through α = 1.…”