Asymptotic Correlations in Gapped and Critical Topological Phases of 1D Quantum Systems

Abstract

Topological phases protected by symmetry can occur in gapped and-surprisingly-in critical systems. We consider the class of non-interacting fermions in one dimension with spinless timereversal symmetry. It is known that the phases in this class are classified by a topological invariant ω and a central charge c. Here we investigate the correlations of string operators in order to gain insight into the interplay between topology and criticality. In the gapped phases, these non-local operators are the string order parameters that allow us to extract ω. More remarkable is that the correlation lengths of these operators show universal features, depending only on ω. In the critical phases, the scaling dimensions of these operators serve as an order parameter, encoding both ω and c. More generally, we derive the exact long-distance asymptotics of these correlation functions using the theory of Toeplitz determinants. We include physical discussion in light of the mathematical results. This includes an expansion of the lattice operators in terms of the operator content of the relevant conformal field theory. Moreover, we discuss the spin chains which are dual to these fermionic systems. Contents

Topological quantum phase transitions and criticality in a longer-range Kitaev chain

“…WN always corresponds to the number of localized edge modes of the topological gapped phases. Recently there are some works which show the localized edge modes even at the criticality [36][37][38][39][40][41][42][43][44][45][46][47][48][49]. Hence it is clear and meaningful to find the WN around criticality and physically it is possible to find the corresponding edge modes.…”

“…In general, BC becomes nonanalytic at these points and WN becomes ill-defined. However, there are efforts which show the localized edge modes even at the gapless region [36,38,[41][42][43][44][45][46][47] especially in longer-range models [37,39,40,48,49].…”

Formation probabilities and statistics of observables as defect problems in free fermions and quantum spin chains

“…It is very useful to put the coupling constants as the coefficients of the following holomorphic function f (z) = r t r z r . Then the Hamiltonian can be diagonalized by going to the Fourier space and then Bogoliubov transformation as follows, see for example 68 :…”

Conformal Boundary Conditions of Symmetry-Enriched Quantum Critical Spin Chains

“…This brings up the notion of gapless SPT [4,5] or "symmetry-enriched" quantum critical states [6]. Robust degenerate edge states persist in some quantum critical states [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20], which are secured by the symmetry-flux (disorder) operators in the bulk carrying nontrivial symmetry charges [6]. This signifies a novel bulk-boundary correspondence.…”

“…In this work, we study two families of quantum spin chains, which generalize the 1D Ising and the three-state Potts models [17,18,40], respectively. Each family contains quantum critical states that are described by the same CFT but cannot be smoothly connected.…”