{'fraction_non_alphanumeric': 0.0386962962962963, 'fraction_numerical': 0.013807407407407407, 'mean_word_length': 4.435104669887279, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 2, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Edge states and topological phases in one-dimensional optical superlattices

We show that one-dimensional quasi-periodic optical lattice systems can exhibit edge states and topological phases which are generally believed to appear in two-dimensional systems. When the Fermi energy lies in gaps, the Fermi system on the optical superlattice is a topological insulator characterized by a nonzero topological invariant. The topological nature can be revealed by observing the density profile of a trapped fermion system, which displays plateaus with their positions uniquely determined by the ration of wavelengths of the bichromatic optical lattice. The butterfly-like spectrum of the superlattice system can be also determined from the finite-temperature density profiles of the trapped fermion system. This finding opens an alternative avenue to study the topological phases and Hofstadter-like spectrum in one-dimensional optical lattices.

We show that one-dimensional quasi-periodic optical lattice systems can exhibit edge states and topological phases which are generally believed to appear in two-dimensional systems. When the Fermi energy lies in gaps, the Fermi system on the optical superlattice is a topological insulator characterized by a nonzero topological invariant. The topological nature can be revealed by observing the density profile of a trapped fermion system, which displays plateaus with their positions uniquely determined by the ration of wavelengths of the bichromatic optical lattice. The butterfly-like spectrum of the superlattice system can be also determined from the finite-temperature density profiles of the trapped fermion system. This finding opens an alternative avenue to study the topological phases and Hofstadter-like spectrum in one-dimensional optical lattices. Introduction.-In recent years ultracold atomic systems have proven to be powerful quantum simulators for investigating various interesting physical problems including many-body physics [1] and topological insulators [2]. In comparison with traditional condensed matter systems, cold-atom systems provide more control in constructing specific optical lattice Hamiltonians by allowing both tunable hopping and trapping potential to be adjusted as needed, thus optical lattices populated with cold atoms offer a very promising alternative avenue to build topological insulating states [2]. Several schemes exploring topological insulators with or without Landau levels in optical lattices have been proposed [3][4][5][6]. So far all these schemes focus on two-dimensional (2D) systems, as onedimensional (1D) systems without additional symmetries are generally thought as lack of topological nontrivial phases [7].
In this work, we study properties of trapped fermions on 1D quasi-periodic optical lattices and show that these systems display non-trivial topological properties, which share the same physical origins of topological phases of 2D quantum Hall effects on periodic lattices [8]. The quasi-periodic optical lattices can be generated by superimposing two 1D optical lattices with commensurate or incommensurate wavelengths [9][10][11], which has led to experimental observation of Anderson localization of a noninteracting BEC of 39 K atoms in 1D incommensurate optical lattices [10]. Motivated by the experimental progress, one-dimensional optical superlattices have been theoretically studied [12][13][14][15][16]. For a quasi-periodic superlattice, the single-particle spectrum is organized in bands. When fermions is loaded in the superlattice, the system shall form insulators if the chemical potential (Fermi energy) lies in the gaps. For the open boundary system, localized edge states are found to appear in the gap regimes. The appearance of edge states is a signature indicating that the bulk states are topological insulators characterized by nonzero Chern number. We show that the topological invariant Chern number can be detected from the density distribution of a trapped fermion system, which displays plateaus in the local average density profiles with positions of plateaus uniquely determined by the ration of wavelengths of the bichromatic optical lattice. Through the analysis of universal scaling behaviors in quantum critical regimes of conductor-toinsulator transitions, we further display that both the positions and widths of plateaus can be read out from the finite-temperature density distributions, which provides us an alternative way to study topological phases and Hofstadter-like spectrum by using 1D optical superlattices. We notice that localized boundary states in 1D incommensurate lattices have been experimentally observed very recently by using photonic quasi-crystals [17].
Quasi-periodic lattices.-We consider a 1D polarized Fermi gas loaded in a bichromatic optical lattice [10,11], which is described by where L is the number of the lattice sites,ĉ † i (ĉ i ) is the creation (annihilation) operator of the fermion, and n i =ĉ † iĉ i . The hopping amplitude t is set to be the unit of the energy (t = 1), and V is the strength of commensurate (incommensurate) potential with α being a rational (irrational) number and δ an arbitrary phase whose effect shall be illustrated lately. Suppose that the n-th eigen-state of a single particle in the 1D lattice is given by |ψ n = i u i,n c † i |0 , the eigenvalue equation H|ψ n = E n |ψ n leads to the following Harper equation: where u i,n is the amplitude of the particle wave function at i-th site with V i the on-site diagonal potential and E n the n-th single particle eigenenergy. The ground state wave function of the N spinless free fermionic system can be written as The solution to Eq. (3) is closely related to the structure of the potential V i . For the incommensurate case V i = V cos(2παi) with irrational α, Eq.(3) is the wellknown Aubry-André model [18], which has been showed that when V < 2 all the single particle states are extended and when V > 2 all the single particle states are the localized states. However, for a periodic V i , all the single particle states are extended band states according to Bloch's theorem. Next we shall consider the commensurate potential V i with a rational α given by α = p/q with p and q being integers which are prime to each other. Since the potential V i is periodic with a period q, the wave functions take the Bloch form, which fulfills u i+q = e ikq u i , for the lattice under the periodic boundary condition. Taking Since φ j+q = φ j , the problem of solving the Harper equation Eq.(4) reduces to solving the eigenvalue equation, Solving the eigenvalue problem, we get q eigenvalues: E α (k) with α = 1, · · · , q. Consequently, the energy spectrum consists of q bands. As an example, the energy spectrum for a commensurate lattice with α = 1/3 and L = 99 under periodic boundary condition is given in Fig.1a. For the lattice with open boundary condition, the momentum k is no longer a good quantum number. There appear edge states in gap regimes as indicated by symbols of "star" located at gap regimes in Fig.1b. As shown in Fig.1c, these edge states are localized in the left and right boundaries in contrast to extended states marked by symbols of "+" in band regimes. As the phase δ varies from 0 to 2π, the spectrum for a given α = p/q changes periodically. The position of the edge states in the gaps also varies continuously with the change of δ. In Fig.2 we show the spectrum of the quasi-periodic systems with α = 1/3 and 1/4 versus δ under the open boundary condition. The shade regimes correspond to the band regimes and the lines between bands are the spectra of edge states.
Topological invariant.-Appearance of edge states is generally attributed to the nontrivial topological properties of bulk systems [2,19]. Next we explore the topological properties of states under the periodic boundary condition. The topological property of the system can be understood in terms of Berry phase in k space, which is defined as γ = A k dk, where A k is the Berry connection A k = i φ(k)|∂ k |φ(k) and φ(k) the occupied Bloch state [20]. Adiabatically varying the phase δ from 0 to 2π, we get a manifold of Hamiltonian H(δ) in the space of parameter δ. Similarly, we can define the Berry connection A δ = i φ(k, δ)|∂ δ |φ(k, δ) . For eigenstates φ(k, δ) of H(δ), we may use the Chern number to characterize their topological properties. The Chern number is a topological invariant which can be calculated via To calculate it, we need work on a discretized Brillouin zone. Here we follow the method in Ref. [21] to directly perform the lattice computation of the Chern number. For the system with α = 1/3, we find that the Chern number for fermions in the lowest filled band (with 1/3 filling) is 1, while the Chern number for states with the second band fully filled (with 2/3 filling) is −1.
To understand the topological origin of fermions in the 1D quasi-periodic lattices, we explore connection of the present model to the well-known Hofstadter problem [8,22], which describes electrons hopping on a 2D square lattice in a perpendicular magnetic field, with the Hamiltonian given by jĉ i e i2πφij , where the summation is over nearest-neighbor sites and the magnetic flux through each plaquette given by φ = plaquette φ ij . Taking the Landau gauge, the eigenvalue problem is described by the Harper equation [8]: where t x (t y ) is the hopping amplitude along the x (y) direction. If we make substitutions of t → t x , V → −2t y , α → φ and δ → −k y , the current 1D problem can be mapped to the lattice version of the 2D integer Hall effect problem. For the latter case, it was shown that when the chemical potential lies in gap regimes, the Hall conductance of the system is quantized [23] and the corresponding Hofstadter insulating phase is a topological insulator characterized by nonzero Chern number. Keeping this connection in mind, it is not strange to see that the energy spectrum of the 1D quasi-periodic system versus different α has similar structure as the spectrum of the 2D Hofstadter butterfly. In Fig.3a and b, we show the spectrum for the system with phase δ = 0 and δ = π/4, respectively. The basic structure is quite similar for different phases except for some minor differences, for example, the spectrum for α = π/4 is continuous around E = 0 for δ = 0, but separated for δ = π/4 (see also Fig2.b). The familiar Hofstadter spectrum is actually a summation over all the 1D spectra with different phases δ from 0 to 2π. Despite the existence of the mapping between the 1D quasi-periodic system and the 2D Hofstadter system, we note that the edge modes in the 1D system are localized and would not be used for dissipationless transport as in high-dimensional systems.
Experimental detection.-In realistic ultracold atom experiments, we need consider the effect of an external confining potential, i.e., V i in Eq.(1) is given by where V H is the strength of the additional harmonic trap with i 0 being the position of trap center. The density distribution for the trapped system can be calculated via n i = Ψ G F |n i |Ψ G F with Ψ G F the ground state wavefuction. In Fig.4, we have shown the local average density distribution of fermions trapped in both commensurate and incommensurate optical lattices with a harmonic trap. Here, in order to reduce the oscillations in density profiles induced by the modulation of potentials, we have defined the local average densitȳ n i = M j=−M n i+j /(2M + 1), where 2M + 1 is the length to count the local average density with M ≪ L [24]. After counting on the locations of plateaus, we find that heights of plateaus are completely decided by α with valuesn(α) = α, 1 − α, 2α, 1 − 2α, 4α, 1 − 4α, ..., if the values are in the range of (0, 1). The plateaus withn i = α, 1 − α are the widest ones corresponding to the largest gap regimes in the butterfly spectrum, whilen i = 2α, 1 − 2α correspond to the smaller gap regimes. The width of plateau is associated with the size of gap.
One can also understand the appearance of plateaus under the local-density approximation (LDA), in which a local chemical potential is defined as where V H (i) is the harmonic trap potential and µ is determined by i n(µ(i), T ) = N . The LDA is applicable provided that the number of fermions is large, and the variation of the trap potential is slow. Within the LDA, the local chemical potential µ(i) decreases parabolically away from the center of the trap. When the local chemical potential lies in one of the gaps, there appear plateaus in the density profile. The discernible number of plateaus is related to the size of the energy gaps. For instance, in Fig. 4, the plateau with n = 1 corresponds to the band insulator with completely filled band, which is topologically trivial with zero Chern number. For α = 1/3 the chemical potential passes through two gap regions which produces two plateaus with n = 1/3 and n = 2/3, whereas for α = 1/5 the chemical potential passes through four gap regions which gives four plateaus with n = 1/5, 2/5, 3/5, 4/5, respectively. The Chern number can be obtained from the density by using the Streda formula [3,25], which is valid when the chemical potential lies in a gap. From the Streda formula C = ∂n(α) ∂α , we can get C = 1, −1 forn i = α, 1 − α and C = 2, −2 forn i = 2α, 1 − 2α.
In principle, all gaps in the spectrum, including the nontrivial smaller gaps in the butterfly, can be determined via observing the corresponding plateaus. Widths of plateaus are proportional to sizes of corresponding energy gaps. Experimentally it becomes increasingly Here we take M = q for the rational case of (a) and M = 20 for the irrational case of (b). . harder to observe these smaller gaps, as the corresponding plateaus are very narrow which needs more precisely experimental measurement of the density distribution. For the realistic detection, we need consider the effect of temperature. The finite-temperature density distribution can be calculated via n i (T ) = 1 Z Ns n=1 e −En/kB T Ψ n F |ĉ † iĉ i |Ψ n F , where N s = L!/(L − N )!N ! is the number of states, E n is the energy of eigenstate Ψ n F , and Z = Ns n=1 e −En/kB T is the canonical par-tition function [26]. In Fig.5a, we display the local average density profiles at different temperatures. It is shown that plateaus already become invisible for T > 0.3t. Despite the fact that the obvious plateaus are smeared out by temperature fluctuations, we demonstrate that the position and the width of zero-temperature plateaus can be uniquely mapped out from finite-temperature density distributions of the trapped optical lattice system, which fulfil some universal scaling relations [27] near the zerotemperature phase transition point µ = µ c : where n = n(T, µ) represents the density distribution for fermions with temperature T and chemical potential µ, n r (µ, T ) is the regular part of the density, Ω(x) is a universal function describing the singular part of density near criticality, d = 1 the dimensionality of the system, ν the correlation length exponent, and z the dynamical exponent. It was shown that z = 2 and ν = 1/2 for the metal-insulator transition of 1D free fermions [27]. From Fig.5b, we see that the different curves of [n(µ, T ) − 2/3]/T 3/2 intersect at two points. From intersecting points, we can determine µ c1 and µ c2 , and thus the corresponding gap in the spectrum can be inferred from µ c2 − µ c1 . In terms of the scaled chemical potential µ = (µ − µ c )/T , curves for different temperatures collapse into a single one as shown in the inset of Fig.5b. In order to connect to the real experiment in cold atoms, we refer to the parameters in Ref. [28] in which 40 K atoms were used in an optical lattice with spacing a = 413 nm, the deep of the potential V 0 = 5E R , and recoil energy E R = 45.98 kHz, thus the hopping magnitude t = 0.066E R = 3.035 kHz. Correspondingly, the temperature T in Fig. 5 in unit t/k B , where k B is the Boltzmann constant, is of the order of 10 nK, e.g., T = 0.1 corresponds to 2.3084 nK.
Summary.-In summary, we explored edge states and topological nature of Fermi systems confined in 1D optical superlattices. Our study reveals that the topological invariant can be detected from plateaus of density profiles of the trapped lattice systems. Our results also clarify the connection of 1D superlattice systems to 2D Hofstadter systems and will be useful for studying topological phases and observing the Hofstadter spectrum by using 1D optical lattices.
SC would thank Prof. S. Q. Shen for helpful discussions on edge states of the dimerized chain which stimulates our interest in the 1D topological insulator. While we are preparing the present paper, we become aware of the experiment for the observation of edge states in 1D quasi-crystals [17]. This work has been supported by NSF