Dual-mapping and quantum criticality in off-diagonal Aubry-André models
DDual-mapping and quantum criticality in off-diagonal Aubry-Andr´e models
Tong Liu ∗ and Xu Xia † School of Science, Nanjing University of Posts and Telecommunications, Nanjing 210003, China Chern Institute of Mathematics and LPMC, Nankai University, Tianjin 300071, China (Dated: January 12, 2021)We study a class of off-diagonal quasiperiodic hopping models described by one-dimensional Su-Schrieffer-Heeger chain with quasiperiodic modulations. We unveil a general dual-mapping relationin parameter space of the dimerization strength λ and the quasiperiodic modulation strength V ,regardless of the specific details of the quasiperiodic modulation. Moreover, we demonstrated semi-analytically and numerically that under the specific quasiperiodic modulation, quantum criticalitycan emerge and persist in a wide parameter space. These unusual properties provides a distinctiveparadigm compared with the diagonal quasiperiodic systems. PACS numbers: 71.23.An, 71.23.Ft, 05.70.Jk
I. INTRODUCTION
Since the publication of Anderson’s seminal paper ,the metal-insulator transition has been studied in a widerange of quantum disordered systems. According to thescaling theory , there is no metal-insulator transition inone-dimensional (1D) systems with random on-site po-tentials. However, the eigenstate near the band centerin the off-diagonal random hopping disorder exhibitsthe delocalization. This model has a chiral symmetrywhich is not present in the standard Anderson case withdiagonal/on-site disorder. At the band center E = 0,the conductance exhibits strong fluctuations and displaysan algebraically decaying mean value, and the density ofstate has a logarithmically divergent scaling behavior,which is different from the standard localized state .On the other hand, 1D quasiperiodic models whichcan host localized, extended or critical eigenstates alsoattracts much interest in view of its rich physics. TheAubry-Andr´e (AA) model is an important paradigm of1D quasiperiodic systems. It can be derived from the re-duction of a two-dimensional quantum Hall system in themagnetic field. Due to recent advances in experimentaltechniques, the AA model has been realized in ultracoldatoms and photonic crystals . The phase diagramof the AA model has been well understood with extensiveresearches , and many different variations of the AAmodel were studied .Beyond the diagonal AA model, the off-diagonal AAmodel may also exhibit an abundant physical phenom-ena. The commensurate off-diagonal AA model canhost the zero-energy topological edge modes , whereasthe incommensurate cases can bring the localization-delocalization phase transition. Inspired by the the off-diagonal random hopping model, we wondered that doesit exist the novel physical phenomena in the off-diagonalquasiperiodic hopping model.In this paper, we explore a class of off-diagonalquasiperiodic hopping models described by one-dimensional Su-Schrieffer-Heeger chain with quasiperi-odic modulation, we unveil a general dual-mapping relation in the parameter space. Moreover we discoverthere exists a wide critical region under the specificquasiperiodic modulation. Critical states, also known asmultifractal states, i.e., the non-ergodic extended statesdo exist at the phase transition point of 3D Andersonmodel and 1D AA model. The emergence of the conceptof many-body localization (MBL) inspires enormousinterest of searching the non-ergodic phase. However,due to the complexity of many-body systems it isworthwhile to explore and demonstrate existence of thenon-ergodic phase of the single-particle system in a wideparameter space (not only a phase transition point).References have numerically demonstrate suchphase exists in a finite range of the disorder strengthon the random regular graph (a treelike graph withoutboundary). Our work provides another paradigm whichpossesses this distinct phase in a wide parameter space. II. MODEL AND DUAL-MAPPING
We consider a class of off-diagonal quasiperiodic hop-ping models, more intuitively, it can be viewed as a Su-Schrieffer-Heeger (SSH) chain with quasiperiodic modu-lation. The Hamiltonian of the model is expressed as Ea n = ( t + V m ) b n − + ( t + V m ) b n ,Eb n = ( t + V m ) a n + ( t + V m ) a n +1 , (1)In a disorder-free lattice with intra- and inter-hoppingamplitudes, the SSH lattice exhibits two topologicallydistinct phases, characterized by a distinct winding num-ber Q = 0 for t > t and Q = 1 for t < t . Here weconsider a SSH chain in the topological nontrivial phase t = 1 − λ and t = 1 + λ , where λ is the dimerizationstrength. a n ( b n ) are the wave function on sublattice A(or B) of the n -th unit cell, and there exists an extraquasiperiodic modulation V n = V cos(2 παn + θ ). A typ-ical choice of the parameters is α = ( √ − /
2. Forconvenience, t = 1 is set as the energy unit.It should be noted that, the quasiperiodic modula-tion in Eq. (1) can take various sequences. Here, we a r X i v : . [ c ond - m a t . d i s - nn ] J a n FIG. 1: (Color online) (a) Schematic of a SSH chain com-prising N unit cells with the intra-hopping amplitude t , theinter-hopping amplitude t and the quasiperiodic perturba-tion {{ V n − , V n , V n , V n } (model I). (b) Schematic of a SSHchain comprising N unit cells with the intra-hopping ampli-tude t , the inter-hopping amplitude t and the quasiperi-odic perturbation { V n − , V n − , V n − , V n } (model II). (c)MIPR of model I in the ( λ, V ) parameter space. Two red linesrepresent two extended-nonextended phase transition lines V = 0 . − λ ) (0 ≤ λ ≤
1) and V = 0 . λ − /λ ) (1 ≤ λ ).(d) MIPR of model II in the ( λ, V ) parameter space. Twored lines represent two extended-nonextended phase transi-tion lines V = 1 − λ (0 ≤ λ ≤
1) and V = λ − ≤ λ ). Thetotal number of unit cells is set to be L = 1000 choose two common sequences, { V m , V m , V m , V m } = { V n − , V n , V n , V n } (model I) illustrated in Fig. 1(a)and { V m , V m , V m , V m } = { V n − , V n − , V n − , V n } (model II) illustrated in Fig. 1(b). The Hermitian na-ture of two systems still remain.In order to investigate the localization nature of twomodels, we numerically solving Eq. (1), and obtain twocomponents a n and b n of the wave functions. The inverseparticipation ratio (IPR) is usually used to study thelocalization-delocalization transition . For any givennormalized wave function, the corresponding IPR is de-fined as IPR n = (cid:80) Lj =1 (cid:16) | a n,j | + | b n,j | (cid:17) , which mea-sures the inverse of the number of sites being occupiedby particles. It is well known that the IPR of an extendedstate scales like L − which approaches zero in the ther-modynamic limit. However, for a localized state, sinceonly finite number of sites are occupied, the IPR is fi-nite even in the thermodynamic limit. The mean of IPRover all the 2 L eigenstates is dubbed the MIPR which isexpressed as MIPR = L (cid:80) Ln =1 IPR n . In Fig. 1 (c) and (d), we plot MIPR of model I andmodel II as a function of λ and V , respectively. Thedark blue regions represents the low amplitudes of MIPR,correspond to the extended phase, other colour regionscorrespond to nonextended phase, and the extended-nonextended phase transition lines (red lines) can be de-termined numerically. Interestingly, the phase transitionlines in regions (0 ≤ λ ≤
1) and (1 ≤ λ ) seems to be a FIG. 2: (Color online) (a) MIPR as a function of the in-verse Fibonacci index 1 /m for the various ( λ, V ) parametersin model I. (b) MIPR as a function of the inverse Fibonacciindex 1 /m for the various ( λ, V ) parameters in model II. connection, i.e., V = 0 . − λ ) ↔ V = 0 . λ − /λ ) formodel I, and V = 1 − λ ↔ V = λ − H (1 , λ, V, θ ) can be expressed explicitly Ea n = (1 + λ + V m ) b n − + (1 − λ + V m ) b n ,Eb n = (1 − λ + V m ) a n + (1 + λ + V m ) a n +1 , (2)We exchange the positions of 1 and λ , ˆ H (1 , λ, V, θ ) be-come ˆ H ( λ, , V, θ ) Ea n = (1 + λ + V m ) b n − + ( − λ + V m ) b n ,Eb n = ( − λ + V m ) a n + (1 + λ + V m ) a n +1 , (3)continue to transform Eq. (3), we can obtain Ea n = (1 + λ + V m ) b n − + [1 − λ + V m (+ π )]( − b n ) ,Eb n = [1 − λ + V m (+ π )]( − a n ) + (1 + λ + V m ) a n +1 , (4)Thus, ˆ H ( λ, , V, θ ) is equivalent to ˆ H (1 , λ, V, θ + π ∗ κ ),where κ represents the parity of lattice dependence. Notethe phase factor θ in V n = V cos(2 παn + θ ) doesn’t affectspectrum properties of the system, so we can ignore π ∗ κ ,and conclude ˆ H ( λ, , V, θ ) and ˆ H (1 , λ, V, θ ) should havethe same spectrum properties, i.e, localization properties.Consequently, Hamiltonian before exchange ˆ H (1 , λ, V )has the same localization properties with Hamiltonianafter exchange ˆ H ( λ, , V ), i.e. λ ˆ H (1 , λ , Vλ ). Thus, thepoint ( λ, V ) in the parameter space dually maps to thepoint ( λ , Vλ ). The above steps can be summarized asfollowsˆ H (1 , λ , Vλ , θ ) ≡ ˆ H (1 , λ, V, θ + π ∗ κ ) ↔ ˆ H (1 , λ, V, θ ) . (5)To support Eq. (5), we implement numerical calcu-lation of MIPR. The size of the system L is chosen asthe m th Fibonacci number F m . The advantage of thisarrangement is that the golden ratio can be approxi-mately replaced by the ratio of two successive Fibonaccinumbers, i.e., α = ( √ − / m →∞ F m − /F m .In Fig. 2 (a) we plot the trend of MIPR for dualpoints of model I as the system size increases. For { ( λ, V ) = (0 . , . , (1000 , } in extended phase, { ( λ, V ) = (0 . , . , (2 , } and { ( λ, V ) = (0 . , , (2 , } in localized phase, MIPR of dual points conform to eachother. However, for { ( λ, V ) = (0 . , , (5 , } in local-ized phase, MIPR of dual points don’t match up.Here, we emphasize that the phase factor θ only doesnot affect the spectrum properties of the system, thisdoes not mean that other physical quantities remain un-change. Regard to absolutely continuous spectrum (ex-tended phase), the MIPR of ˆ H (1 , λ, V, θ ) should tend tobe consistent with the MIPR of ˆ H (1 , λ , Vλ , θ ), due to bothMIPR’s being zero in the thermodynamic limit. How-ever, Regard to point spectrum (localized phase), theMIPR of ˆ H (1 , λ, V, θ ) should not necessarily tend to beconsistent with the MIPR of ˆ H (1 , λ , Vλ , θ ), due to bothMIPR’s just being nonzero in the thermodynamic limit,actually it just guarantees both MIPR’s stay in the sameorder of magnitude.In Fig. 2 (b), for { ( λ, V ) = (0 . , . , (1000 , } of model II in extended phase, the trend of MIPR alsoconverges to zero. However, more interestingly, for thedual points in non-extended phase, the trend of MIPRin model II is not linearly extrapolated to a finite valuelike the localized phase in model I, distinctly, it obeythe power-law curve a ∗ ( m ) b . In the thermodynamiclimit m → ∞ , the trend of MIPR approach zero, butthe convergence rate is slower than that of the extendedphase, this indicates that these dual points may belongto the critical phase.From the point of view of dual-mapping, we can ex-plicitly explain the relation of two red lines in Fig. 1 (c)and (d). The region V ≤ . − λ ) is dually map-ping to the region V ≤ . λ − /λ ) for model I due to( λ, V ) ↔ ( λ , Vλ ). In the same way, the region V ≤ − λ is dually mapping to the region V ≤ λ − III. QUANTUM CRITICALITY
By comparing the value of MIPR of various V’s formodel I, it is very clearly that the self-mapping regionof the model I is localized. However, The log trend ofMIPR indicates the self-mapping region of the model II(
V > − λ and V > λ −
1) is neither extended norlocalized, but critical. However, it is difficult to study thelocalization nature of this model quantitatively. Here, wesemi-analytically determines the emergence of quantumcriticality in the self-mapping region of the model II.In the first step, we prove that V = 1 is a quantum critical point when λ = 0. Thus, Eq. (1) of model II canbe expressed as Eψ n = (1 + V n − ) ψ n − + (1 + V n ) ψ n +1 , (6)where V n = V cos(2 παn + θ ) denotes the off-diagonalquasiperiodic modulation.We can study the localization properties of Eq. (6)through the Lyapunov exponent (LE) . To compute theLE γ ( E ), we use Avila’s global theory of quasiperiodicoperators. Writing T n ( θ ) the total transfer matrix of themodel (6), the LE can be computed as γ (cid:15) ( E ) = lim n →∞ πn (cid:90) π ln (cid:107) T n ( θ + i(cid:15) ) (cid:107) dθ, where (cid:107) M (cid:107) represents the norm of the matrix M . Thecomplexification of the phase ( θ → θ + i(cid:15) ) is crucial here,since our computation relies on Avila’s global theory .First note that the transfer matrix can be written as T ( θ ) = (cid:18) E V n − V n − V n (cid:19) = 11 + V cos(2 παn + θ ) B ( θ ) ,B ( θ ) = (cid:18) E − − V cos[2 πα ( n −
1) + θ ]1 + V cos(2 παn + θ ) 0 (cid:19) , then LE can be expressed as γ (cid:15) ( E ) = lim n →∞ n (cid:90) ln (cid:107) B n ( θ + i(cid:15) ) (cid:107) dθ + (cid:90) ln |
11 + V cos(2 παn + θ + i(cid:15) ) | dθ = (cid:26) γ (cid:15) ( E ) − ln √ − V if V < γ (cid:15) ( E ) − ln | V | − π(cid:15) if V ≥ . (7)where γ (cid:15) ( E ) = lim n →∞ n (cid:82) ln (cid:107) B n ( θ + i(cid:15) ) (cid:107) dθ . Let usthen complexify the phase, and let (cid:15) goes to infinity, di-rect computation of B ( θ + i(cid:15) ) yields B ( θ + i(cid:15) ) = e π(cid:15) e − i π ( θ − α ) (cid:18) − V e iπα
V e − iπα (cid:19) + o (1) . Thus we have γ (cid:15) ( E ) = 2 π(cid:15) + log | V | + o (1) . By the ex-tended A. Avila’s global theory to jacobi matrix , γ (cid:15) ( E ) = 2 π(cid:15) + log | V | . Consequently, we obtain γ (cid:15) ( E ) = (cid:26) log | V √ − V | + 2 π(cid:15) if V < if V ≥ . (8)Thus, if the energy E lies in the spectrum, we have γ ( E ) = ln | V √ − V | or 0. If V <
1, we haveln | V √ − V | <
0, in this place Avila dubbed it as sub-crtical, and he has prove there is only absolutely con-tinuous spectrum . If V ≥ V cos(2 παn + θ ) alwayshas singularity, by the paper , we know there is Thereis no absolutely continuous spectrum. At the same time,through the paper , we can know that there is no pointspectrum. Therefore in this case, we only singular con-tinuous spectrum. Consequently, we analytically demon-strate the emergence of quantum criticality in the self-mapping region ( λ = 0 and V ≥ λ = 1 is the degen-eracy point of energy level when V = 0. Thus, Eq. (1) ofthe Hamiltonian reduces to the SSH model Ea n = (1 + λ ) b n − + (1 − λ ) b n ,Eb n = (1 − λ ) a n + (1 + λ ) a n +1 , (9)When λ = 1, Eq. (9) become Ea n = 2 b n − ,Eb n = 2 a n +1 , (10)Thus, we obtain a degenerate energy spectrum E = 2.Here, λ = 1 is not conventional quantum critical point,this is no topological phase transition. However, the de-generacy of energy levels are commonly accompanied bythe emergence of quantum criticality.Reference studies the phase diagram of the trans-verse field Ising model under both the transverse mag-netic field and the finite temperature and experimentallyconfirms the quantum critical behavior. In that model,quantum fluctuations controlled by the nonthermal pa-rameter g lead to a phase transition at a critical value g c , the so called quantum critical point, already at zerotemperature. And there exists a thermal phase transitionat a critical temperature T c . The interplay of quantumfluctuations g c and thermal fluctuations T c opens up aprogressively broader, V -shaped quantum critical regionextending much above the zero temperature. Analogi-cally, under the critical dimerization strength λ c = 1 andthe critical quasiperiodic modulation strength V c = 1 inour model, a similar shaped quantum critical region canemerge, thus we explain the emergence of quantum crit-icality of the model II qualitatively.To characterize the wave functions and validate ourconjecture of critical states, we use the multifractal the-ory, which has been widely applied to standard quasiperi-odic models, such as the Aubry-Andr´e model . Foreach wave function ψ jn = ( a jn , b jn ) T , a scaling exponent β jn can be extracted from the n th on-site probability P jn = | ψ jn | ∼ (1 /F m ) β jn . According to the multifractaltheorem, when the wave functions are extended, the max-imum of P jn scales as max( P jn ) ∼ (1 /F m ) , i.e., β jmin =min( β jn ) = 1. On the other hand, when the wave func-tions are localized, P jn peaks at very few sites and nearlyzero at the other sites, yielding max( P jn ) ∼ (1 /F m ) and β jmin = min( β jn ) = 0. As for the critical wave func-tions, the corresponding β jmin is located within the inter-val (0 , β jmin forthe critical Aubry-Andr´e model fluctuates sharply withsystem size L . In order to reduce finite-size effects, wethus use the average of β jmin , i.e. β min = L (cid:80) Lj =1 β jmin over wave functions of the system so as to distinguish thecritical state in practical numerical calculations. FIG. 3: (a) β min as a function of the inverse Fibonacci index1 /m for the various ( λ, V ) parameters in model II. The brownmarkers correspond to the critical phase, and the black mark-ers correspond to the extended phase. (b) the representativewave function in ( λ, V ) = (0 . ,
2) is extended. (c) the repre-sentative wave function in ( λ, V ) = (0 . ,
1) is critical. (d) therepresentative wave function in ( λ, V ) = (0 . ,
5) is critical.The system size is set to be L = F m = 4181. Figure 3(a) plots β min as a function of the inverse Fi-bonacci index 1 /m . It clearly shows that β min is between0 and 1 in the large L limit for the self-mapping region { ( λ, V ) = (0 . , , (2 , , (0 . , , (2 , } , hence suggest-ing that wave functions in the self-mapping region arecritical. Conversely, for { ( λ, V ) = (0 . , . , (2 , . } , β min asymptotically tends to 1 in the thermodynamiclimit, indicating that the corresponding wave functionsare extended. Figure 3 [panels (b), (c), and (d)] plotsthe representative wave functions of the extended andself-mapping region, respectively. An inspection of thefigure clearly indicates that the wave function is extendedfor { ( λ, V ) = (0 . , . } [panels (b)]. In contrast, thewave functions of { ( λ, V ) = (0 . , , (0 . , } in the self-mapping region [panels (c) and (d)] are neither localizednor extended over the whole space. Instead, they dis-play clear self-similarities, which is the characteristic ofcritical states. IV. CONCLUSION
In summary, We study two off-diagonal quasiperiodicmodels and unveil a general dual-mapping relation in pa-rameter space of the dimerization strength λ and thequasiperiodic modulation strength V . Moreover, wedemonstrated semi-analytically and numerically that aspecific off-diagonal quasiperiodic model can host a broadquantum critical region. This is different from the cus-tomary diagonal quasiperiodic models, in which quan-tum criticality can only emerge at the quantum phasetransition point. Our discovery will enrich the abundantlocalization phenomena in the quasiperiodic systems. Acknowledgments
Tong Liu thank Pei Wang for fruitful discussions. TongLiu acknowledges Natural Science Foundation of Jiangsu Province (Grant No. BK20200737) and NUPTSF (GrantNo. NY220090 and No. NY220208). X. Xia is supportedby Nankai Zhide Foundation. ∗ [email protected] † P. W. Anderson, Phys. Rev. , 1492 (1958). E. Abrahams, P. W. Anderson, D. C. Licciardello, and T.V. Ramakrishnan, Phys. Rev. Lett. , 673 (1979). L. Fleishman and D. C. Licciardello, J. Phys. C , L125(1977). M. Inui, S. A. Trugman, and E. Abrahams, Phys. Rev. B , 3190 (1994). T. A. L. Ziman, Phys. Rev. B , 7066 (1982). M. Kohmoto and D. Tobe, Phys. Rev. B , 134204 (2008). C. Gramsch and M. Rigol, Phys. Rev. A , 053615 (2012). T. Liu, H. Guo, Y. Pu, and S. Longhi, Phys. Rev. B ,024205 (2020). T. Liu and H. Guo, Phys. Rev. B , 104201 (2018). S. Aubry and G. Andr´e, Ann. Israel Phys. Soc. (133), 18(1980). J. Billy, V. Josse, Z. Zuo, A. Bernard, B. Hambrecht, P.Lugan, D. Cl´ement, L. Sanchez-Palencia, P. Bouyer, andA. Aspect, Nature , 891 (2008). G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M.Zaccanti, G. Modugno, M. Modugno, and M. Inguscio, Na-ture , 895 (2008). Y. Lahini, R. Pugatch, F. Pozzi, M. Sorel, R. Morandotti,N. Davidson, and Y. Silberberg, Phys. Rev. Lett. ,013901 (2009). Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, and O. Zil-berberg, Phys. Rev. Lett. , 106402 (2012). H. Yin, J. Hu, A.-C. Ji, G. Juzeliunas, X.-J. Liu, and Q.Sun, Phys. Rev. Lett. , 113601 (2020). Y. Wang, L. Zhang, S. Niu, D. Yu, and X.-J. Liu, Phys.Rev. Lett. , 073204 (2020). Y. Wang, X. Xia, L. Zhang, H. Yao, S. Chen, J. You, Q.Zhou, and X.-J. Liu, Phys. Rev. Lett. , 196604 (2020). T. Liu, G. Xianlong, S. Chen, and H. Guo, Phys. Lett. A , 3683 (2017). T. Liu, H. Yan, and H. Guo, Phys. Rev. B , 174207(2017). S. Ganeshan, J. H. Pixley, and S. Das Sarma, Phys. Rev.Lett. , 146601 (2015). J. Biddle and S. Das Sarma, Phys. Rev. Lett. , 070601 (2010). J. Wang, X.-J. Liu, G. Xianlong, and H. Hu, Phys. Rev. B , 104504 (2016). S. Ganeshan, K. Sun, and S. Das Sarma, Phys. Rev. Lett. , 180403 (2013). F. Liu, S. Ghosh, and Y. D. Chong, Phys. Rev. B ,014108 (2015). J. C. C. Cestari, A. Foerster, and M. A. Gusm˜ao, Phys.Rev. B , 205441 (2016). A. De Luca, B. L. Altshuler, V. E. Kravtsov, and A.Scardicchio, Phys. Rev. Lett. , 046806 (2014). B. L. Altshuler, E. Cuevas, L. B. Ioffe, and V. E. Kravtsov,Phys. Rev. Lett. , 156601 (2016). D. J. Thouless, Phys. Rep. , 93 (1974). M. Kohmoto, Phys. Rev. Lett , 1198 (1983). A. Avila, J. You , Q. Zhou, Sharp phase transitions for thealmost Mathieu operator, Duke. Math. J. , 166 (2017). A. Avila, Global theory of one-frequency Schr¨odinger op-erators, Acta. Math. , 215, (2015). S. Jitomirskaya, C. A. Marx, Erratum to: Analytic quasi-perodic cocycles with singularities and the Lyapunov Ex-ponent of Extended Harper’s Model, Commun. Math.Phys, 2013, 317: 269-271. S. Jitomirskaya, I. Krasovsky, Critical almost Mathieu op-erator: hidden singularity, gap continuity, and the Haus-dorff dimension of the spectrum. arXiv:1909.04429, 2019. A. Avila, Almost reducibility and absolute continuity,arXiv preprint arXiv:1006.0704, 2010. A. Avila, Global theory of one-frequency Schr¨odinger op-erators, Acta Mathematica, 2015, 215(1): 1-54. R. Han, Absence of point spectrum for the self-dual ex-tended Harper’s model, International Mathematics Re-search Notices, 2018, 2018(9): 2801-2809. A. W. Kinross, M. Fu, T.J. Munsie, H. A. Dabkowska,G.M. Luke, S. Sachdev, and T. Imai, Phys. Rev. X ,031008 (2014). M. Kohmoto and D. Tobe, Localization problem in aquasiperiodic system with spin-orbit interaction, Phys.Rev. B77