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Topological phase diagram for the Kitaev chain with long-range pairing (8). The wavy lines at the border of certain phases indicate that they extend endlessly. Fractional topological numbers highlight the appearance of an unconventional topological phase with massive non-local Dirac edge states. The topological characterisation of the crossover phase is discussed in the main text and the Appendix.
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We discover novel topological effects in the one-dimensional Kitaev chain
modified by long-range Hamiltonian deformations in the hopping and pairing
terms. This class of models display symmetry-protected topological order
measured by the Berry phase of the ground state and the winding number of the
Hamiltonians. For exponentially-decaying hopping a...
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... always present [see Fig. 2(a)]. The function f α (k) is not divergent and we can compute the winding number ω of Eq. (4) and the Berry phase Φ B of Eq. (5) obtaining Φ B = πω = π. The lower band eigen- vector u − k , thus, shows a U (1) phase discontinuity at k = 0. The corresponding topological phase is depicted in blue in the phase diagram of Fig. ...
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... for the new unconventional topological phase is ω = +1/2 if µ < 1. The semi-integer character of ω is associated to the integrable divergence at k = 0, which modifies the continuous mapping S 1 −→ S 1 . Notwithstanding, in this region there is still a jump of one unit between the two topologically different phases, ∆ω = ω top − ω trivial = 1 (see Fig. 3). Moreover, the topological indicators take on the same value within the whole phase until the bulk gap closes at µ = 1, giving rise to a topological phase transition, and the new massive topological edge states disappear. Therefore, we can still establish a bulk-edge ...
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... Fig. 3, we present a complete phase diagram sum- marising the different topological phases of the model as a function of µ and α. 5. Outlook and Conclusions. We have found that finite- range and long-range extensions of the one-dimensional Kitaev chain can be used as a resource for enhanc- ing existing topological properties and for unveiling ...
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... Fig. A3(a) we perform a finite-size scaling for the masses of the MZMs for the Majorana sector. Within the topological sector µ ∈ (−1, 1), the edge mass gap clearly goes to zero with L as we expected. In Fig. A3(b) we per- form the same finite-size scaling analysis for the massive Dirac sector. In this case, there are edge states for µ < 1. As we ...
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... Fig. A3(a) we perform a finite-size scaling for the masses of the MZMs for the Majorana sector. Within the topological sector µ ∈ (−1, 1), the edge mass gap clearly goes to zero with L as we expected. In Fig. A3(b) we per- form the same finite-size scaling analysis for the massive Dirac sector. In this case, there are edge states for µ < 1. As we can see, the masses of the edge states depend on both µ and α, and go to a finite value even in the thermo- dynamic limit. This proves that the topological nature of the non-local massive Dirac fermions ...
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... the other hand, Fig. A3(c) and Fig. A3(d) show the finite-size scaling for the edge mass gap within the crossover sector. Although there are edge states all over µ < 1, they can be either massive or massless depending on the chemical potential µ. If −1 < µ < 1 the edge states are massless as shown in Fig. A3(c), whereas for µ < −1 the edge states become massive ...
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... the other hand, Fig. A3(c) and Fig. A3(d) show the finite-size scaling for the edge mass gap within the crossover sector. Although there are edge states all over µ < 1, they can be either massive or massless depending on the chemical potential µ. If −1 < µ < 1 the edge states are massless as shown in Fig. A3(c), whereas for µ < −1 the edge states become massive as shown in ...
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... the other hand, Fig. A3(c) and Fig. A3(d) show the finite-size scaling for the edge mass gap within the crossover sector. Although there are edge states all over µ < 1, they can be either massive or massless depending on the chemical potential µ. If −1 < µ < 1 the edge states are massless as shown in Fig. A3(c), whereas for µ < −1 the edge states become massive as shown in Fig. A3(d). Hence, this sector displays a mixed character between a Majorana and a massive Dirac ...
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... and Fig. A3(d) show the finite-size scaling for the edge mass gap within the crossover sector. Although there are edge states all over µ < 1, they can be either massive or massless depending on the chemical potential µ. If −1 < µ < 1 the edge states are massless as shown in Fig. A3(c), whereas for µ < −1 the edge states become massive as shown in Fig. A3(d). Hence, this sector displays a mixed character between a Majorana and a massive Dirac ...
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... |ψ(0)| 2 for the lowest-energy single-particle eigenstate at one of the edges. Fig. A4 shows |ψ(0)| 2 for different µ as a function of α. The results can be summarised as follows: i/ for µ > 1 (purple line) there are no edge states regardless of α. ii/ If −1 < µ < 1 (blue line) there is always a finite edge-state density. In addition, from Fig. A3 we obtain that if α > 1 the edge states are massless, whereas if α < 1 they are massive. iii/ If µ < −1 (green line) there are edge states if α < 3 2 . Then, from Fig. A3 we conclude that they are always massive in this case. Actually, we can even monitor how one of the Dirac bulk states gets transmuted into a non-local massive Dirac ...
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... as follows: i/ for µ > 1 (purple line) there are no edge states regardless of α. ii/ If −1 < µ < 1 (blue line) there is always a finite edge-state density. In addition, from Fig. A3 we obtain that if α > 1 the edge states are massless, whereas if α < 1 they are massive. iii/ If µ < −1 (green line) there are edge states if α < 3 2 . Then, from Fig. A3 we conclude that they are always massive in this case. Actually, we can even monitor how one of the Dirac bulk states gets transmuted into a non-local massive Dirac edge mode by lowering ...
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... results are in complete agreement with Fig.3 from the main text and the finite size scaling analysis in Fig. A3. ...
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... results are in complete agreement with Fig.3 from the main text and the finite size scaling analysis in Fig. ...
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... analytical solution might be in- volved, this process can be easily programmed in a com- puter as a set of linear equations. In Fig. A5, we compute the finite-size scaling of ∆ M within the crossover sector for −1 < µ < 1 where we expect MZMs. Up to our nu- merical precision, ∆ M goes to zero in perfect accordance with the phase diagram of Fig. 3 in the main text and the finite-size scaling for the energy of the edge modes in Fig. ...
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... a com- puter as a set of linear equations. In Fig. A5, we compute the finite-size scaling of ∆ M within the crossover sector for −1 < µ < 1 where we expect MZMs. Up to our nu- merical precision, ∆ M goes to zero in perfect accordance with the phase diagram of Fig. 3 in the main text and the finite-size scaling for the energy of the edge modes in Fig. ...
Citations
... On the other hand, it is the fermionized version of the well-known one-dimensional transverse-field Ising model [45], which is one of the simplest solvable models that exhibit quantum criticality and phase transition with spontaneous symmetry breaking [46]. Several studies have been conducted with a focus on long-range Kitaev chains, in which the superconducting pairing term decays with distance as a power law [47][48][49][50][51][52][53]. ...
We investigated the topological pumping charge of a tetramerized Kitaev chain with spatially modulated chemical potential, which hosts nodal loops in parameter space and violates particle number conservation. In the simplest case, with alternatively assigned hopping and pairing terms, we show that the model can be mapped into the Rice–Mele model by a partial particle-hole transformation and subsequently supports topological charge pumping as a demonstration of the Chern number for the ground state. Beyond this special case, analytic analysis shows that the nodal loops are conic curves. Numerical simulation of a finite-size chain indicates that the pumping charge is zero for a quasiadiabatic loop within the nodal loop and is ±1 for a quasiadiabatic passage enclosing the nodal loop. Our findings unveil the topology of Kitaev chains in parameter space.
... From an experimental point of view, it is inevitable to consider the influence of LR interactions when implementing nontrivial topological phases in state-of-the-art quantum simulators. To date, various studies [63][64][65] have demonstrated qualitative changes of physical properties under LR interactions in gapped topological phases, including the emergence of new massive edge modes [16,17,66] and novel bulk-boundary correspondence [67,68]. However, it is not immediately clear < l a t e x i t s h a 1 _ b a s e 6 4 = " W L u l S n 8 O k g 6 u p P I O 6 4 v R s w b d g M Q = " > A A A C R X i c b V B L S w M x E M 7 W V 6 2 v q k c v w S I I 6 r J b a 9 V b 0 Y t H B W u F 7 l q y a d q G Z h 8 k s 2 J d 9 s 9 5 8 e 7 N f + D F g y J e N X 2 A 2 j q Q 8 M 3 3 z U w m n x c J r s C y n o 3 M 1 P T M 7 F x 2 P r e w u L S 8 k l 9 d u 1 J h L C m r 0 l C E 8 t o j i g k e s C p w E O w 6 k o z 4 n m A 1 r 3 v a 1 2 u 3 T C o e B p f Q i 5 j r k 3 b A W 5 w S 0 F Q j 7 z j A 7 m A ...
Topology in condensed matter physics is typically associated with a bulk energy gap. However, recent research has shifted focus to topological phases without a bulk energy gap, exhibiting nontrivial gapless topological behaviors. In this letter, we explore a cluster Ising chain with long-range antiferromagnetic interactions that decay as a power law with the distance. Using complementary numerical and analytical techniques, we demonstrate that long-range interactions can unambiguously induce an algebraic topological phase and a topological Gaussian universality, both of which exhibit nontrivial gapless topological behaviors. Our study not only provides a platform to investigate the fundamental physics of quantum many-body systems but also offers a novel route toward searching for gapless topological phases in realistic quantum simulators.
... Recently, there has been a surge of interest in the LR version of an integrable topological fermionic chain, featuring a LR superconducting pairing or hopping term [72-81], which exhibits exotic properties such as massive edge modes [12][13][14][15], anomalous behavior of correlation functions [13], and novel bulk-boundary correspondence [82]. Unlike the SR case, the LR fermionic and spin chains are two independent models that cannot be mapped to each other through the Jordan-Wigner transformation. ...
... Therefore, it is fundamentally important to study LR interacting models both theoretically and experimentally. However, over the past few decades, most research efforts have been focused on the stability of edge modes in gapped topological phases with LR interactions [12][13][14][15]83]. For gapless topological phases, such as topologically nontrivial quantum critical points (QCPs), it remains unclear how the critical edge modes remain stable against LR interactions. ...
... In recent years, there has been an investigation into the overall picture of novel phenomena in classical and quantum systems with long-range (LR) interactions [5,6]. Increasing the range of interactions ∼ 1/ (or equivalently reducing the power exponent ) can fundamentally alter the "basic laws" of statistical and condensed matter physics, including the failure of quantum-classical correspondence [7][8][9], breakdown of the Mermin-Wagner theorem [10,11], emergence of massive edge modes [12][13][14][15], and new LR universality classes [16][17][18][19][20][21][22][23], among others. Moreover, the experimental realization of quantum systems with LR interactions, such as cold atomic gases in cavities [24][25][26], trapped ions [27,28], as well as programmable Rydberg quantum simulators [29][30][31][32], has spurred significant motivation to explore the novel properties of such systems. ...
The long-range interaction can fundamentally alter properties in gapped topological phases such as emergent massive edge modes. However, recent research has shifted attention to topological nontrivial critical points or phases, and it is natural to explore how long-range interaction influences them. In this work, we investigate the topological behavior and phase transition of extended Kitaev chains with long-range interactions, which can be derived from the critical Ising model via the Jordan-Wigner transformation in the short-range limit. Specifically, we analytically find the critical edge modes at the critical point remain stable against long-range interaction. More importantly, we observe these critical edge modes remain massless even when long-range interactions become substantially strong. As a byproduct, we numerically find that the critical behavior of the long-range model belongs to the free Majorana fermion universality class, which is entirely different from the long-range universality class in usual long-range spin models. Our work could shed new light on the interplay between long-range interactions (frustrated) and the gapless topological phases of matter.
... In the context of criticality, the study of topological state of matter becomes important as it is the platform for the emergence of exotic particles, unlike fermions and bosons. There are a number of examples which signals the emergence of Majorana zero modes 8 , massive edge modes 9 , and chiral edge modes in the topological systems 10 . Under this scenario, the area has become interesting both from experimental and theoretical perspective. ...
Extended-range models are the interesting systems, which has been widely used to understand the non-local properties of the fermions at quantum scale. We aim to study the interplay between criticality and extended range couplings under various symmetry constraints. Here, we consider a two orbital Bernevig–Hughes–Zhang model in one dimension with longer (finite neighbor) and long-range (infinite neighbor) couplings. We study the behavior of model using scaling laws and universality class for models with Hermitian, parity-time (PT\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr{P}\mathscr{T}$$\end{document}) symmetric and broken time-reversal symmetries. We observe the interesting results on multi-criticalities, where the universality class of critical exponent is different than the normal criticalities. Also, the results can be generalized by considering the interplay between criticalities and different symmetry classes of Hamiltonian. Also, with the introduction of extended-range of coupling, there occurs different criticalities, and we provide the analogy to characterize their universality classes. We also show the violation of Lorentz invariance at multi-criticalities and evaluation of short-range limit in long-range models as the highlights of this work.
... It is the fermionized version of the well-known one-dimensional transverse-field Ising model [11], which is one of the simplest solvable models when the translational symmetry is imposed that exhibits quantum criticality and phase transition with spontaneous symmetry breaking [12,13]. In addition, several studies have been conducted with a focus on long-range Kitaev chains [14][15][16][17][18]. Although system we concerned in this work contains only the nearest neighbor hopping and pairing terms, this method be applied to the system involving long-range terms under certain conditions. ...
We propose an extended Bogoliubov transformation in real space for spinless fermions, based on which a class of Kitaev chains of length $2N$ with zero chemical potential can be mapped to two independent Kitaev chains of length $N$. It provides an alternative way to investigate a complicated system from the result of relatively simple systems. We demonstrate the implications of this decomposition by a Su-Schrieffer-Heeger (SSH) Kitaev model, which supports rich quantum phases. The features of the system, including the groundstate topology and nonequilibrium dynamics, can be revealed directly from that of sub-Kitaev chains. Based on this connection, two types of Bardeen-Cooper-Schrieffer (BCS)-pair order parameters are introduced to characterize the phase diagram, showing the ingredient of two different BCS pairing modes. Analytical analysis and numerical simulations show that the real-space decomposition for the ground state still holds true approximately in presence of finite chemical potential in the gapful regions.
... As discussed in the literature, there is an ambiguity in the definition of the winding number when β ≤ α and β < 1. It has been claimed that the winding number takes half-integer values, which is accompanied by the appearance of so-called massive Dirac edge states, which have a non-vanishing energy for any length of the Kitaev chain [15,17,18,29,30]. It has been even argued that in this case the ten-fold way classification of topological insulators and superconductors is not applicable, and that the bulk-boundary correspondence is weakened [28]. ...
... In Ref. [29], the boundaries of the topological phases for the following model with short-range pairing and long-range hopping with exponential falloff with distance have been discussed: ...
Majorana edge states in Kitaev chains possessing an effective time reversal symmetry with one fermionic site per unit cell are studied. For a semi-infinite chain the equations for the wave functions of Majorana zero modes can be reduced to a single Wiener-Hopf equation, which has an exact analytical solution. We use this solution to determine the asymptotic behaviors of the Majorana wave functions at large distances from the edge of the chain for several infinite-range models described in the literature with focus on a model with slow power-law falloff of pairing and hopping amplitudes. For these models we also determine the asymptotic behavior of the energy of the fermionic state composed of two Majorana modes in the limit of long (finite) Kitaev chains.
... In recent years, an extended version, namely the long-range Kitaev (LRK) chain, has become significant from both theoretical and experimental points of view. Long-range chain, which allows pairing and hopping terms to couple electrons from non-adjacent sites with strength decaying algebraically with the distance between the sites, shows stability against external perturbations [16][17][18][19][20][21]. Moreover, the long-range interactions induce novel correlations and entanglement behavior [22], delocalization of Majorana states in 1D LRK chain [23]. ...
... Generally speaking, this long-range paring and hopping affect MBSs and the gap-closing phenomenon near the topological phase transition (TPT) point. Zero energy MBSs of the standard (short-range) Kitaev chain transformed into massive Dirac fermions when the long-range interaction is switched on as this causes the wavefunctions of the two boundary Majorana modes to hybridize by overlapping [17,23,[31][32][33]. Moreover, the long-range interaction affects the spatial extent of the eigenfunctions of the LRK chain, and also modifies the density of states. ...
... There are some interesting studies of the Fano factor related to the electrical current, which is the ratio of shot noise and the electrical current, as a function of voltage bias in [33], which emphasizes the contrasting behavior of the same for LRK and SRK chains. We find that the study of electrical, thermal, and thermoelectrical currents as a function of different types of biases (voltage/temperature) is also interesting for LRK chain as these observables are greatly influenced by the delocalized states, the modified density of states, the finite mass of subgap edge states, and the absence of gap closure at TPT which are categorically different from the characteristics of SRK chain [17]. Hence the current characteristic at TPT point can be a good indicator of formation of massive Dirac edge mode [17]. ...
We study electrical, thermal and thermoelectric transport in a hybrid device consisting of a long-range Kitaev chain coupled to two metallic leads at two ends. Electrical and thermal currents are calculated in this device under both voltage and thermal bias conditions. We find that the transport characteristics of the long-range Kitaev chain are distinguishably different from its short-range counterpart, which is well known for hosting zero energy Majorana edge modes under some specific range of values of the model parameters. The emergence of massive Dirac fermions, the absence of gap closing at the topological phase transition point and some special features of the energy spectrum which are unique to the long-range Kitaev chain, significantly alter electrical/thermal current vs voltage/temperature bias characteristics compared with that of the short-range Kitaev chain. These novel transport characteristics of the long-range Kitaev model can be helpful to understand nontrivial topological phases of the long-range Kitaev chain.
... originating in long-range systems can induce unconventional phases in ground states and quantum dynamics. The consequences of tailoring long-range couplings have attracted a variety of research interests, including quantum Hall effect in topological Haldane model [1][2][3], localized edge modes in Kitaev's chain [4][5][6][7], and roton-like dispersion in metamaterials [8,9]. The long-range couplings naturally arise in diverse physical platforms [10], such as trapped ions, Rydberg atoms, and neutral atoms in cavities. ...
Recent years have witnessed a surge of research on the non-Hermitian skin effect (NHSE) in one-dimensional lattices with finite-range couplings. In this work, we show that the long-range couplings that decay as 1/lα at distance l can fundamentally modify the behavior of NHSE and the scaling of quantum entanglement in the presence of nonreciprocity. At α=0, the nonlocality of couplings gives rise to the scale-free skin modes, whose localization length is proportional to the system size. Increasing the exponent α drives a complex-to-real spectral transition and a crossover from a scalefree to constant localization length. Furthermore, the scaling of nonequilibrium steady-state entanglement entropy exhibits a subextensive law due to the nonlocality and the complex spectrum, in contrast to an area law arising from NHSE. Our results provide a theoretical understanding on the interplay between long-range couplings and non-Hermiticity.
... Some properties have been discussed in comparison with short-range systems. Notable examples are the existence (or absence) of an area law of entanglement [10][11][12][13], the algebraic decay of two-point correlators out of criticality [14][15][16], the spreading of correlations [17], the existence of Majorana modes [18] and topological properties [19]. ...
... (21) The susceptibility can be computed analytically from Eq. (19) for a translation invariant model [42] (See. App. ...
... From Eq. (19) we have ...
We present an exact analytical solution for quantum strong long-range models in the canonical ensemble by extending the classical solution proposed in Campa et al., J. Phys. A 36, 6897 (2003). Specifically, we utilize the equivalence between generalized Dicke models and interacting quantum models as a generalization of the Hubbard-Stratonovich transformation. To demonstrate our method, we apply it to the Ising chain in transverse field and discuss its potential application to other models, such as the Fermi-Hubbard model, combined short and long-range models and models with antiferromagnetic interactions. Our findings indicate that the critical behaviour of a model is independent of the range of interactions, within the strong long-range regime, and the dimensionality of the model. Moreover, we show that the order parameter expression is equivalent to that provided by mean-field theory, thus confirming the exactness of the latter. Finally, we examine the algebraic decay of correlations and characterize its dependence on the range of interactions in the full phase diagram.
... The breaking of symmetry may violate the above property 44 . With the infinite neighbor coupling, one can obtain long-range models which are interesting from the perspective of massive edge modes and breaking of Lorentz invariance [44][45][46][47][48][49][50] . The long-range models have been experimentally realized in trapped ions [51][52][53][54] , atom coupled to multi-mode cavities 55 , magnetic impurities 56,57 and simulated circuits 58 . ...
... They have a advantage of suppressing the finite sized effect over to short-range models 59 . The massive edge modes are found to be an effective qubits in topological computations 48 . On the other hand, non-Hermitian system exhibit sensitivity towards the boundary conditions, and there exists non-Hermitian skin effect which creates an extra localization of eigen states in the open boundary condition. ...
... If the encircling is only around one of them yields fractional and neither of them yields W = 0 respectively. (48) χ re x (k) = −χ im y (k)and χ re y (k) = χ im x (k) or www.nature.com/scientificreports/ Generalized Brillouin zone and topological invariant. ...
Geometric phase is an important tool to define the topology of the Hermitian and non-Hermitian systems. Besides, the range of coupling plays an important role in realizing higher topological indices and transition among them. With a motivation to understand the geometric phases for mixed states, we discuss finite temperature analysis of Hermitian and non-Hermitian topological models with extended range of couplings. To understand the geometric phases for the mixed states, we use Uhlmann phase and discuss the merit-limitation with respect extended range couplings. We extend the finite temperature analysis to non-Hermitian models and define topological invariant for different ranges of coupling. We include the non-Hermitian skin effect, and provide the derivation of topological invariant in the generalized Brillouin zone and their mixed state behavior also. We also adopt mixed geometric phases through interferometric approach, and discuss the geometric phases of extended-range (Hermitian and non-Hermitian) models at finite temperature.
