Symmetry protected phases in inhomogeneous spin chains

Abstract

It has been shown recently that inhomogenous spin chains can exhibit exotic phenomena such as the breaking of the area law of the entanglement entropy. An example is given by the rainbow model where the exchange coupling constants decrease from the center of the chain. Here we show that by folding the chain around its center, the long-range entanglement becomes short-range which can lead to topological phases protected by symmetries (SPT). The phases are trivial for bond-centered foldings, and non trivial for site-centered ones. In the latter case, the folded spin 1/2 chain with U (1) symmetry belongs to the Su-Schrieer-Heeger class, while the folded chain with SU (2) symmetry is in the Haldane phase. Finally, we extend these results to higher spin chains where we find a correspondence between the symmetry protection of gapped and gapless phases.

Entanglement as geometry and flow

“…Indeed, this suggestive connection stems from the area law: the entanglement entropy of blocks of lowenergy states of local Hamiltonians is frequently proportional to the measure of the boundary separating the block from its environment [13][14][15][16], with at most logarithmic corrections [17][18][19]. Yet, there are relevant exceptions to the area law, such as the rainbow state [20][21][22][23][24][25][26][27] and the Motzkin state [28][29][30][31][32][33][34]. In some of these cases, as we will show, an area law is indeed fulfilled for a geometry that differs from the geometry defined by the local structure of the Hamiltonian.…”

Exotic correlation spread in free-fermionic states with initial patterns

“…Therefore, the pattern repeats itself after exactly P = 4q sites. Next, we consider the rainbow state [22][23][24][25][26][27][28][29][30], which is formed by a concentric set of bonds and presents maximal entropy between its left and right halves, as shown in Fig. 1(e).…”

“…The rainbow state has received a great deal of attention because it can be built as the ground state (GS) of a deformed local Hamiltonian in the limit in which the inhomogeneity is large. We should stress that the form (11) describes the GS of some spin chains, such as the XX, XXZ, or Ising chains [25,28,31], after a Jordan-Wigner (JW) transformation has been applied, as can be shown making use of the strong disorder renormalization group devised by Dasgupta and Ma [32]. The alternating character of the signs of its bonds can be understood in terms of the nonlocal nature of the JW transformation.…”

“…The behavior is also found to be imprinted in the growth of the EE where we report more than one linear stage of growth, with different slopes, corresponding to the passage of the different types of quasiparticles. We also extend the set of initial states, considering cases with short-range correlations, such as the dimerized state and a few more complex relatives, but also initial states with long-range correlations, such as the rainbow state and its variants [22][23][24][25][26][27][28][29][30][31]. By extending the formalism in the continuum limit, we show that in all the cases the structure of the correlation matrix away from the light cone presents universal signatures: the correlations along the light cone decay as t −1/3 .…”

Link representation of the entanglement entropies for all bipartitions

“…This last conjecture derives from the so-called area law, which states that the entanglement entropy (EE) of a block of a low-lying energy eigenstate of a local Hamiltonian is proportional to the measure to its boundary [11][12][13][14], sometimes presenting logarithmic corrections [15][16][17][18]. Interestingly, there are relevant exceptions to the area law, such as the rainbow state [19][20][21][22][23][24][25][26][27][28]. In this case, despite the locality of the Hamiltonian, the ground state (GS) establishes long-range bonds between opposite sites of a chain, suggesting the possibility that the geometric structure described by the entanglement differs from the one associated to the Hamiltonian.…”