Topological incommensurate magnetization plateaus in quasi-periodic quantum spin chains

Uncovering topologically nontrivial states in nature is an intriguing and important issue in recent years. While most studies are based on the topological band insulators, the topological state in strongly correlated low-dimensional systems has not been extensively explored due to the failure of direct explanation from the topological band insulator theory on such systems and the origin of the topological property is unclear. Here we report the theoretical discovery of strongly correlated topological states in quasi-periodic Heisenberg spin chain systems corresponding to a series of incommensurate magnetization plateaus under the presence of the magnetic field, which are uniquely determined by the quasi-periodic structure of exchange couplings. The topological features of plateau states are demonstrated by the existence of non-trivial spin-flip edge excitations, which can be well characterized by nonzero topological invariants defined in a two-dimensional parameter space. Furthermore, we demonstrate that the topological invariant of the plateau state can be read out from a generalized Streda formula and the spin-flip excitation spectrum exhibits a similar structure of the Hofstadter's butterfly spectrum for the two-dimensional quantum Hall system on a lattice.

S ince the discovery of topological insulators 1-3 nearly ten years ago, topological states have attracted great interests in condensed matter physics both theoretically and experimentally [4][5][6] . A hallmark feature of these exotic phases is the appearance of gapless edge states which is robust against local perturbations as long as the bulk gap is not closed. To characterize these states, global topological invariants rather than local order parameters should be introduced. Although topological states based on band theory have been well understood, till today the goal of searching topological states in strongly correlated systems remains fascinating and challenging [7][8][9][10][11][12][13][14] .
While most of previous studies on topological states focus on either two-dimensional (2D) or three-dimensional materials, recent researches on one-dimensional (1D) periodic and quasi-periodic systems have revealed these systems exhibit non-trivial topological properties 15,16 due to a nontrivial link between these 1D systems and 2D topological insulators [15][16][17][18][19][20][21] . Experimentally, using the propagation of light in photonic waveguides, topologically protected boundary states 16 and phase transition are also observed 18 . The 1D quasi-periodic crystal can be viewed as the simplest realization of a topologically nontrivial insulator. A crucial question is: for more general 1D systems which inevitably suffer from strong quantum fluctuations, can topological states induced by the quasiperiodic modulation survive in the strong correlated regime? If these states exist, they are undoubtedly the strongly correlated topological states being persistently sought by condensed matter physicists. The existence of powerful numerical methods for 1D correlated systems, e.g., the density matrix renormalization group (DMRG) method, permits us to explore novel correlated topological states in a numerically exact way.
In this paper, we investigate the paradigmatic strongly correlated model, i.e., quantum Heisenberg model on a 1D quasi-periodic lattice. We report the findings of a series of non-trivial incommensurate magnetization plateaus as consequence of the existence of large excitation gaps in quasi-periodic quantum spin chains. Quite surprisingly, these incommensurate plateaus will approach to specific non-trivial irrational values which are uniquely determined by the quasi-periodic modulation parameter in thermodynamic limit. The nontrivial topological properties of the incommensurate plateaus are unveiled by using two independent methods, i.e., calculating the edge excitations and topological invariants, both of which are well established and have been widely adopted in the study of topological states 4,5,22 . Under open boundary conditions (OBC), we find that these non-trivial plateaus can host robust edge spin-flip excitations which connect the lower and upper excitation bands. Different plateau states can be well characterized by topologically invariant Chern numbers, which are defined in a 2D parameter space and are related to the height of plateaus via a generalized Streda formula. It is interesting that the spin-flip excitation spectrum of the quasi-periodic Heisenberg model exhibits a butterfly-like structure, which resembles the Hofstadter spectrum of the 2D quantum hall system.

Results
Incommensurate plateaus. We consider a general Heisenberg spin-S chain with quasi-periodic geometry which is described by where we take the quasi-periodic modulation parameter a g (0, 1) as an irrational number. The exchange strength J i is quasi-periodic with modulation strength l and phase factor d. The special case with l 5 0 reduces the Hamiltonian to the homogenous Heisenberg model. In this work, we focus on the anti-ferromagnetic (AFM) couplings, i.e., J . 0 and jlj , 1. For convenience, J 5 1 is taken as the unit of energy. First we study the magnetization process under magnetic field h which couples to the z component of spins 23,24 . The magnetization per spin is defined as m z 5 S z /L with S z~X L i S z i being the z component of the total spin and L denoting the lattice size. In Fig. 1 The above incommensurate magnetization plateaus for finite-size systems, which approximate to the special values 2S 1 (a, 2a, 1 2 2a, 1 2 a), will tend exactly to these values in thermodynamic limit L R '. For brevity, we mark these irrational plateaus from bottom to top as P a ,P 2a ,P 22a ,P 2a . Take a specific plateau P a as an example. As a is irrational, La is not an integer. Denote N l and N u as the nearest lower and upper bound integer with N l , La , N u . In our spin model, magnetization for the system with N l or N u up spins is m l ; 21/2 1 N l /L or m u ; 21/2 1 N u /L, respectively. Obviously, we have m l , P a , m u . For a finite chain with the length L, our DMRG results show that the magnetization plateau is located at either m l or m u for a given magnetic field as illustrated in Fig. 2. When the length L increases, positions of magnetization plateaus exhibit damped oscillations. For different a, e.g., (a) a~ffi , the positions of plateaus will definitely tend to P a 5 21/2 1 a as lim L R ' m l 5 lim L R ' m u 5 P a . In thermodynamic limit, these incommensurate plateaus will eventually evolve into irrational magnetization plateaus P a . Further we define a l : N l L and a u : a. I f we consider these rational a l and a u modulation of spin chains with length L, commensurate plateaus at m l and m u appear 25 . Based on the above discussion, we can conclude that magnetization plateaus for the quasi-periodic spin chain are totally determined by the irrational modulation parameter a.    Fig. 3(a). Further, these in-gap modes under OBC are edge modes considering the spin-flip distributions are mainly localized at two ends of the chain as illustrated in Fig. 3(b) and Fig. 3(c). Our numerical results show that once the edge modes touch the bulk band, the distributions will change side. With no gap closing on path, the cyclical change of d leads to windings of edge modes around excitation gaps 27 . The winding numbers for these four non-trivial plateau states are 1, 2, 22, and 21, respectively.
Quasi-periodic spin-1 chains. We have demonstrated the nontrivial edge excitations of incommensurate plateau states for quasiperiodic spin-1 2 chains. In this part, we extend the study to the quasiperiodic spin-1 Heisenberg model. As has been noticed by Haldane 28,29 based on the low-energy effective field theory, halfodd-integer and integer spin chain systems exhibit quite different behaviors. The well known Haldane's conjecture for homogeneous Heisenberg AFM model states that the low energy excitation is gapless for half-odd-integer spins while gapped for integer spins. Here m DMRG represents magnetization plateau calculated by the DMRG method. The black guidelines denote the corresponding irrational values of magnetization plateaus in thermodynamic limit.  from bottom to top. The existence of the middle zero-plateau is a reminiscence of Haldane gap 28 . Other nontrivial incommensurate plateaus will approach to 21 1 (a; 2a; 2 2 2a; 2 2 a) in thermodynamical limit. Denote these plateaus as P a , P 2a , P 22a , P 2a . In Fig. 4 Topological invariants. According to the bulk-edge correspondence for topological states, the existence of non-trivial edge states is generally attributed to the non-trivial topology of bulk states 27 . Such a correspondence holds true even for topologically nontrivial interacting systems [30][31][32][33] . As both spin-1 2 and spin-1 quasi-periodic chains display nontrivial edge excitations, we can summarize that, for a general spin-S chain with quasi-periodic modulation parameter a, the nontrivial magnetization plateaus in thermodynamic limit should appear at the following specific values: 6(S 2 a, S 2 2a, …, S 2 na…) as long as S 2 na . 0. The differences for various non-trivial plateau states are the number of edge states and the winding pattern with respect to the phase factor d. A natural question is how to define topological invariants to characterize different plateau states. As a totally determines the position of plateaus, the adiabatical evolution of d produces a family of systems with quite similar magnetization curves, we can define the topological invariants for the plateau states associated with excitation gaps in a 2D manifold spanned by (h, d) 26,34,35 where h is the twist angle introduced by applying the twist boundary condition to the many body wave function y. For an arbitrary site j, the twist boundary condition is y(j 1 L, d) 5 e ih y(j, d), which has been widely used in spin systems 36,37 . The Chern number is defined as the integral of Berry curvature 34  and spin-1 systems in Fig. 5(a) and Fig. 5(b), respectively. It is amazing that the spin-flip excitation spectrum of the quasiperiodic Heisenberg model exhibits the similar structure of the Hofstadter's butterfly spectrum 39  Haldane gap in the middle of the excitation spectrum for the spin-1 system, and finally six large bands are formed with the evolution of a.
Another notable feature is the existence of in-gap states once La is not an integer under PBC. Excitation spectrum of quasi-periodic systems with irrational a always has in-gap states, which are the origin of slightly change of plateaus in the magnetization curves. While no in-gap states exist for the periodic commensurate system with aL 5 integer, the existence of in-gap states for the incommensurate chain under PBC is due to the mismatch of exchange coupling between the first and N-th site. To see clearly the distribution of the in-gap state, we display real space profiles rS z of plateau states at

Discussion
By investigating the paradigmatic Heisenberg model with quasi-periodic geometry, we find a series of incommensurate magnetization plateaus. Under OBC, these non-trivial plateaus can host continuous edge states which connect the lower and upper excitation bands. The Chern numbers defined in a 2D parameter space to characterize different plateau states describe the winding patterns of edge modes. The topological properties of the magnetization plateaus are coded in a generalized Streda formula and the butterfly-like excitation spectrum. Our work unifies the quantum hall conductivity plateaus and quantized plateau states for quasi-periodic spin models in the scheme of strongly correlated topological insulators.
Our conclusion can be directly extended to the general XXZ spin models, with the spin exchange term S x i S x iz1 zS y i S y iz1 zDS z i S z iz1 . For AFM couplings, we find that the anisotropic exchange interactions do not destroy but stabilize these incommensurate plateaus as the plateau width has a positive correlation with D in the whole regime of D $ 0. The sweeping of the anisotropy parameter D produces a series of Hamiltonians which exhibit non-trivial a-dependent magnetization plateaus. Particularly, for the quasi-periodic spin-1 2 chains, the XX model with D 5 0 is exactly solvable and the non-trivial topological properties can be understood based on single-particle band theory of free fermions via a Jordan-Wigner transformation, where the latter can be exactly mapped to the famous 2D Hofstadter problem 39 . The extension to the anisotropic high-spin system is straightforward and the general rules we summarized remain valid.

Methods
The magnetization curves are determined by using the DMRG method which is the most powerful numerical tool for studying 1D strongly correlated systems. For the 6(S 2 a) +1 6(S 2 a) +1 6(S 2 a*) 61 6(S 2 2a) +2 spin-1 6(S 2 a) +1 6(S 2 a) +1 6(S 2 a*) 61 6(S 2 2a) +2 6(S 2 3a) +3 6(S 2 2a)  considered systems in the paper, the total S z is a good quantum number. Ground state energies in different subspace are compared to determine the magnetization under the specific magnetic field h. Our DMRG simulations are rather reliable. The error truncation of the reduced density matrix is up to 10 28 to 10 212 . We utilize four to fifteen sweeps to reach the convergence of the eighth digit for ground state energy per site. We have checked the accuracy of the DMRG algorithm by comparing the results from the exact diagonalization method on systems with lengths up to L 5 24. The calculation of Chern numbers is settled in a 2D parameter space (h, d). The Chern number is well defined for the ground state which is protected by a finite gap under PBC. Numerically, the continuous 2D space are divided into a discrete manifold 40 . For (h, d) g [0, 2p] 3 [0, 2p], we have analyzed different partitions: 5 3 5, 10 3 10, and 20 3 20 of the manifold and found the Chern numbers stay unchanged as long as we are considering the plateau states.