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交错跃迁Hofstadter梯子的量子流相

刘彪 周晓凡 陈刚 贾锁堂

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交错跃迁Hofstadter梯子的量子流相

刘彪, 周晓凡, 陈刚, 贾锁堂

Current phases in Hofstadter ladder with staggered hopping

Liu Biao, Zhou Xiao-Fan, Chen Gang, Jia Suo-Tang
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  • 为玻色Hofstadter梯子模型引入交错跃迁, 来扩展模型支持的量子流相. 基于精确对角化和密度矩阵重整化群计算发现, 无相互作用时, 系统中包含横流相、涡旋相和纵流相; 横流相来自均匀跃迁时Hofstadter梯子模型的Meissner相, 纵流相是交错跃迁时才可见的流相. 强相互作用极限下系统的超流区也包含横流相、纵流相和涡旋相, 但存在更多的相变级数; 超流区的横流相、纵流相之间存在相变但Mott区的不存在, 把Mott区的“横、纵流相”称为Mott-均匀相, 在Mott区只存在均匀相和涡旋相. 跃迁的交错会压缩涡旋相存在的区域, 使Mott区最终只剩下均匀相; 跃迁的交错不仅能驱动Mott-超流相变, 还使磁通的改变也能够驱动系统的Mott-超流相变. 对这一系统的研究丰富了磁通系统中的量子流相, 同时为研究拓扑流特性提供了模型支持.
    Hofstadter ladder describes a Boson ladder under a uniform magnetic field and supports nontrivial energy band and fractional quantum Hall states. Staggered hopping is illuminated from the SSH model and proved to have non-trivial effects on current phases. We introduce staggered hopping on Hofstadter ladder to study the novel current phases. Exact diagonalization (ED) and density matrix renormalization group (DMRG) methods have been employed to study the current phases of the ladder in noninteraction and strong interaction (hard core boson) cases. By observing energy singularities and the new flux patterns when increasing the staggered hopping strength, we extend Meissner and vortex phase to horizontal current phase, vertical current phase and vortex phase. The horizontal current phase has stronger chiral currents in horizontal direction, which is the long direction of the ladder. The vertical current phase has stronger chiral currents in vertical direction. The above two phases do not break translational invariance while the vortex phase does. The current patterns of horizontal current phase are proved to be continuously deformed form the Meissner phase, and the vortex phase has similar signatures. The vertical current phase is only visible when the hopping is staggered. These phases generally exist in noninteraction regimes and interacting superfluid regimes. We have defined new quantities (i.e. current inhomogeneity and nearest overlap) to characterize different quantum phases. In noninteraction case, the horizontal current phase go through the vortex phase to enter the vertical current phase by second order phase transitions, but in strong interaction case such a change can be directly made in a first order phase transition. The direct transition is made in higher fillings with almost identical flux. Surprisingly, the three phases turn into only two phases in Mott regimes, and the phase transition between the horizontal current phase and the vertical current phase has disappeared. We call the new phase as Mott-homogenous phase. The staggered hopping has exotic effects in strong interaction case. For n = 0.25 filling, the staggered hopping shrinks the region of vortex phases and produces Mott-SF transition. When the staggered hopping is weak, the system achieves Mott-SF transition just by varying the flux. This research can enrich current phases in lattice systems and illuminate further studies on chiral currents.
      通信作者: 周晓凡, xiaofanlaohu@163.com ; 陈刚, chengang971@163.com
    • 基金项目: 国家级-国家重点研发计划(2017YFA0304203)
      Corresponding author: Zhou Xiao-Fan, xiaofanlaohu@163.com ; Chen Gang, chengang971@163.com
    [1]

    Thouless D J, Kohmoto M, Nightingale M P, Dennijs M 1982 Phys. Rev. Lett. 49 405Google Scholar

    [2]

    Kane C L, Mele E J 2005 Phys. Rev. Lett. 95 146802Google Scholar

    [3]

    Hasan M Z, Kane C L 2010 Rev. Mod. Phys. 82 3045Google Scholar

    [4]

    Qi X L, Zhang S C 2011 Rev. Mod. Phys. 83 1057Google Scholar

    [5]

    Hügel D, Paredes B 2014 Phys. Rev. A 89 023619Google Scholar

    [6]

    Zupancic P, Preiss P M, Ma R C, Lukin A, Tai M E, Rispoli M, Islam R, Greiner M 2016 Opt. Express 24 13881Google Scholar

    [7]

    Tai M E, Lukin A, Rispoli M, Schittko R, Menke T, Borgnia D, Preiss P M, Grusdt F, Kaufman A M, Greiner M 2017 Nature 546 519Google Scholar

    [8]

    Atala M, Aidelsburger M, Lohse M, Barreiro J T, Paredes B, Bloch I 2014 Nat. Phys. 10 588Google Scholar

    [9]

    Aidelsburger M, Atala M, Lohse M, Barreiro J T, Paredes B, Bloch I 2013 Phys. Rev. Lett. 111 185301Google Scholar

    [10]

    Miyake H, Siviloglou G A, Kennedy C J, Burton W C, Ketterle W 2013 Phys. Rev. Lett. 111 185302Google Scholar

    [11]

    Aidelsburger M, Lohse M, Schweizer C, Atala M, Barreiro J T, Nascimbène S, Cooper N R, Bloch I, Goldman N 2015 Nat. Phys. 11 162Google Scholar

    [12]

    Jiménez-García K, LeBlanc L J, Williams R A, Beeler M C, Perry A R, Spielman I B 2012 Phys. Rev. Lett. 108 225303Google Scholar

    [13]

    Struck J, Ölschläger C, Weinberg M, Hauke P, Simonet J, Eckardt A, Lewenstein M, Sengstock K, Windpassinger P 2012 Phys. Rev. Lett. 108 225304Google Scholar

    [14]

    Celi A, Massignan P, Ruseckas J, Goldman N, Spielman I B, Juzeliūnas G, Lewenstein M 2014 Phys. Rev. Lett. 112 043001Google Scholar

    [15]

    Mancini M, Pagano G, Cappellini G, Livi L, Rider M, Catani J, Sias C, Zoller P, Inguscio M, Dalmonte M, Fallani L 2015 Science 349 1510Google Scholar

    [16]

    Cooper N R, Rey A M 2015 Phys. Rev. A 92 021401Google Scholar

    [17]

    Pagano G, Mancini M, Cappellini G, Lombardi P, Schäfer F, Hu H, Liu X J, Catani J, Sias C, Inguscio M, Fallani L 2014 Nat. Phys. 10 198Google Scholar

    [18]

    Stuhl B K, Lu H I, Aycock L M, Genkina D, Spielman I B 2015 Science 349 1514Google Scholar

    [19]

    Aidelsburger M 2015 Artificial Gauge Fields with Ultracold Atoms in Optical Lattices (Switzerland: Springer) pp27–44

    [20]

    Azbel M Y 1964 Sov. Phys. JETP 19 634

    [21]

    Harper P G 1955 Proc. Phys. Soc. London, Sect. A 68 874Google Scholar

    [22]

    Hofstadter D R 1976 Phys. Rev. B 14 2239

    [23]

    Giamarchi T 2004 Quantum Physics in One Dimension (Clarendon: Oxford) p29

    [24]

    Dolfi M, Bauer B, Keller S, Kosenkov A, Ewart T, Kantian A, Giamarchi T, Troyer M 2014 Comput. Phys. Commun. 185 3430Google Scholar

    [25]

    White S R 1992 Phys. Rev. Lett. 69 2863Google Scholar

    [26]

    Schollwöck U 2011 Ann. Phys. 326 96Google Scholar

    [27]

    Tsui D C, Stormer H L, Gossard A C 1982 Phys. Rev. Lett. 48 1559Google Scholar

    [28]

    Palmer R N, Jaksch D 2006 Phys. Rev. Lett. 96 180407Google Scholar

    [29]

    Hafezi M, Sørensen A S, Demler E, Lukin M D 2007 Phys. Rev. A 76 023613Google Scholar

    [30]

    Cooper N R, Dalibard J 2013 Phys. Rev. Lett. 110 185301Google Scholar

    [31]

    Strinati M C, Cornfeld E, Rossini D, Barbarino S, Dalmonte M, Fazio R, Sela E, Mazza L 2017 Phys. Rev. X 7 021033

    [32]

    Cornfeld E, Sela E 2015 Phys. Rev. B 92 115446Google Scholar

    [33]

    Petrescu A, Piraud M, Roux G, McCulloch I P, Le Hur K 2017 Phys. Rev. B 96 014524Google Scholar

    [34]

    Petrescu A, Le Hur K 2013 Phys. Rev. Lett. 111 150601Google Scholar

    [35]

    Petrescu A, Le Hur K 2015 Phys. Rev. B 91 054520Google Scholar

    [36]

    Piraud M, Heidrich-Meisner F, McCulloch I P, Greschner S, Vekua T, Schollwöck U 2015 Phys. Rev. B 91 140406Google Scholar

    [37]

    Geim A K, Dubonos S V, Lok J G S, Henini M, Maan J C 1998 Nature 396 144Google Scholar

    [38]

    Bardeen J 1955 Phys. Rev. 97 1724Google Scholar

    [39]

    Abrikosov A A 1957 Sov. Phys. JETP 511 74

    [40]

    Su W P, Schrieffer J R, Heeger A J 1979 Phys. Rev. Lett. 42 1698Google Scholar

    [41]

    Su W P, Schrieffer J R, Heeger A J 1983 Phys. Rev. B 28 1138

    [42]

    Zhang J M, Dong R X 2010 Eur. J. Phys. 31 591Google Scholar

    [43]

    Calabrese P, Cardy J 2009 . Phys. A 42 504005Google Scholar

    [44]

    Holzhey C, Larsen F, Wilczek F 1994 Nucl. Phys. B 424 443Google Scholar

    [45]

    Vidal G, Latorre J I, Rico E, Kitaev A 2003 Phys. Rev. Lett. 90 227902Google Scholar

    [46]

    Roux G, Capponi S, Lecheminant P, Azaria P 2009 Eur. Phys. J. B 68 293Google Scholar

    [47]

    Kühner T D, White S R, Monien H 2000 Phys. Rev. B 61 12474Google Scholar

    [48]

    Asbóth J K, Oroszlány L, Pályi A 2016 Lect. Notes Phys. 919 87

    [49]

    Sheikhan A, Brennecke F, Kollath C 2016 Phys. Rev. B 94 061603

    [50]

    Feng Y L, Zhang K, Fan J T, Mei F, Chen G, Jia S T 2018 Sci. China: Phys. Mech. 61 123011

  • 图 1  哈密顿量中跃迁项示意图

    Fig. 1.  Sketch of the hopping term of the Hamiltonian.

    图 2  (a) 用来识别相变的基态能导数的突变, 数据取自${t_1}/K = 1$, ${t_2}/K = 0.3$; (b) 无相互作用相图; (c1)−(c3) 区分三种相的流图案. 图中箭头表示流, 点表示密度, 箭头的方向和粗细代表流的方向和强度, 点的大小表示密度的大小, 箭头的粗细和点的大小都已经除以当组最大的流或密度归一. 数据取自${t_1}/K = 1$, ${t_2}/K = 0.3$, $\phi /{\text{π}} = $0.1, 0.6, 0.9, 精确对角化

    Fig. 2.  (a) Singularities in derivative of ground state energy indicate phase transitions. Data is from ${t_1}/K = 1$, ${t_2}/K = 0.3$; (b) phase diagram for non-interacting case; (c1)−(c3) the current pattern used to distinguish the 3 phases. Direction and thickness of an arrow indicate the direction and strength of the current on the plotted bond. The strengths are normalized by the strongest local current. The sizes of the points indicate the density strengths and are normalized by the largest density. The current patterns are from ED calculations for $\phi /{\text{π}} = 0.1, \;0.6, \;0.9$ when ${t_1}/K = 1$, ${t_2}/K = 0.3$.

    图 3  (a) 无相互作用时扫描$\phi $计算出的最近邻内积, 数据取自${t_1}/K = 1, \;{t_2}/K = 0.3$; (b) 保持${t_1}/K = 1$, 逐渐改变${t_2}/K$时, 横流相和纵流相的态和${t_1}/K = {t_2}/K = 1$时态的内积; (c) 一般的, 系统在$\phi = {\text{π}}$处整个流图案的流会逆转方向, 交错跃迁会让流换向以一阶相变的形式进行, 数据取自${t_1}/K = 1$

    Fig. 3.  (a) Nearest overlaps from ED calculations. Data is from ${t_1}/K = 1, {t_2}/K = 0.3$; (b) when ${t_1}/K = 1$, as ${t_2}/K$ decays, the overlap with ${t_1}/K = {t_2}/K = 1$ state decays from 1 smoothly; (c) generally all local currents in the system reverse sign in $\phi = {\text{π}}$. The staggered hopping makes the process the first order phase transition. Data is from ${t_1}/K = 1$.

    图 4  (a) ${t_1}/K = 1, \;{t_2}/K = 0.3$强相互作用极限(HCB)相图, $n = N/(2 L)$为填充; (b) 该相图中三个相对应的纠缠熵分布与拟合结果, $\phi /{\text{π}} = 0.3, 0.64, 0.8$分别取自横流相、涡旋相和纵流相, 散点是有限尺寸系统中计算的纠缠熵(S), 实线和中心荷(c)是将散点用公式(7)拟合的结果. 交错跃迁导致了纠缠熵空间分布的起伏, 数据取自周期边界, $L = 64$, $N = 12$; (c) 横流相-纵流相相变时横向手性流和纵向手性流的突变, 突变后二者强弱交换; 数据取自$L = 64, N = 25$, 在手性流的计算中为了减小开边界带来的边界效应, 只截取了中间$L/2$部分; (d) 横流相-涡旋相-纵流相相变; 扫描$\phi $时, 最近邻内积进入涡旋相时会突变, 并且数值上类似于无相互作用时一样振荡; 数据取自$N = 4$

    Fig. 4.  (a) Phase diagram for HCB when ${t_1}/K = 1$, ${t_2}/K = 0.3$, $n = N/(2 L)$; (b) VN entropy and the corresponding fitted central charge for three phases in the phase diagram above. $\phi /{\text{π}} = 0.3, 0.64, 0.8$ are from horizontal current, vortex and vertical current phase respectively. The points are VN entropy data in PBC finite system. Solid lines and central charge are fitted from formula(7). The staggered hopping has made the VN entropy also staggered. Data is from $L = 64$, $N = 12$; (c) horizontal current phase to vertical current phase transition for$L = 64, N = 25$. The average horizontal current and average vertical current will swap their strong and weak relations in the transition point. In order to reduce the boundary effect in average currents, we used the$L/2$part in the middle of the ladder; (d) horizontal current to vortex and vortex to vertical current phase transition. The nearest overlap shows that in horizontal and vertical current phases the ground state changes smoothly, and numerically oscillates in the vortex phase. Data is from$N = 4$.

    图 5  (a) $n = 0.25$相图, ${t_1}/K = 1$, 虚线圈出的是vortex相; (b) 相图中均匀相与涡旋相是用量化的流的周期2不均匀性区分的, 数据取自${t_1}/K = 1$, ${t_2}/K = 0.9$; (c) ${t_1}/K = 1$, ${t_2}/K = 0.3$$n = 0.25$ Mott线, 系统基态平缓地从横流相变成纵流相, 这个过程中流图案的周期一直为2

    Fig. 5.  (a) Phase diagram for $n = 0.25$ filling when${t_1}/K = 1$; (b) the homogenous phase and vortex phase are distinguished from the inhomogeneity of densities and currents. Data is from ${t_1}/K = 1$, ${t_2}/K = 0.9$; (c) the $n = 0.25$ Mott line for ${t_1}/K = 1$, ${t_2}/K = 0.3$. The average vertical current surpasses the average horizontal current smoothly, during the process the current patterns have perfect periodicity of 2 and the density is homogenous.

    表 1  在有代表性的区域取点, 精确计算中心荷(c)来验证各个流相的存在, 系统取周期边界来减小纠缠熵振荡. 表格列出了散点在相图中的位置,对这些点的电荷能隙和流相性质的判断,以及对应的中心荷判据,L = 64—100, t1/K = 1

    Table 1.  PBC central charges (c) are accurately checked in several points in typical regions. The table has listed the points' position in the phase diagram, the judged charge gap and current phase pro-perties, and the corresponding central charge evidence, L = 64–100, t1/K = 1.

    $\left( {{t_2}/K, \;\phi /{\text{π}}} \right)$Region in phase diagramc
    $(0.9000, ~0.1000)$$\varDelta = 0$, not vortex$1.00$
    $(0.9000,~ 0.8000)$$\varDelta = 0$, vortex $2.00$
    $(0.8500,~0.5000)$$\varDelta > 0$, not vortex$0.01$
    $(0.8500,~0.6125)$$\varDelta > 0$, vortex $0.98$
    $(0.8000,~ 0.9750)$$\varDelta = 0$, not vortex$1.02$
    下载: 导出CSV
  • [1]

    Thouless D J, Kohmoto M, Nightingale M P, Dennijs M 1982 Phys. Rev. Lett. 49 405Google Scholar

    [2]

    Kane C L, Mele E J 2005 Phys. Rev. Lett. 95 146802Google Scholar

    [3]

    Hasan M Z, Kane C L 2010 Rev. Mod. Phys. 82 3045Google Scholar

    [4]

    Qi X L, Zhang S C 2011 Rev. Mod. Phys. 83 1057Google Scholar

    [5]

    Hügel D, Paredes B 2014 Phys. Rev. A 89 023619Google Scholar

    [6]

    Zupancic P, Preiss P M, Ma R C, Lukin A, Tai M E, Rispoli M, Islam R, Greiner M 2016 Opt. Express 24 13881Google Scholar

    [7]

    Tai M E, Lukin A, Rispoli M, Schittko R, Menke T, Borgnia D, Preiss P M, Grusdt F, Kaufman A M, Greiner M 2017 Nature 546 519Google Scholar

    [8]

    Atala M, Aidelsburger M, Lohse M, Barreiro J T, Paredes B, Bloch I 2014 Nat. Phys. 10 588Google Scholar

    [9]

    Aidelsburger M, Atala M, Lohse M, Barreiro J T, Paredes B, Bloch I 2013 Phys. Rev. Lett. 111 185301Google Scholar

    [10]

    Miyake H, Siviloglou G A, Kennedy C J, Burton W C, Ketterle W 2013 Phys. Rev. Lett. 111 185302Google Scholar

    [11]

    Aidelsburger M, Lohse M, Schweizer C, Atala M, Barreiro J T, Nascimbène S, Cooper N R, Bloch I, Goldman N 2015 Nat. Phys. 11 162Google Scholar

    [12]

    Jiménez-García K, LeBlanc L J, Williams R A, Beeler M C, Perry A R, Spielman I B 2012 Phys. Rev. Lett. 108 225303Google Scholar

    [13]

    Struck J, Ölschläger C, Weinberg M, Hauke P, Simonet J, Eckardt A, Lewenstein M, Sengstock K, Windpassinger P 2012 Phys. Rev. Lett. 108 225304Google Scholar

    [14]

    Celi A, Massignan P, Ruseckas J, Goldman N, Spielman I B, Juzeliūnas G, Lewenstein M 2014 Phys. Rev. Lett. 112 043001Google Scholar

    [15]

    Mancini M, Pagano G, Cappellini G, Livi L, Rider M, Catani J, Sias C, Zoller P, Inguscio M, Dalmonte M, Fallani L 2015 Science 349 1510Google Scholar

    [16]

    Cooper N R, Rey A M 2015 Phys. Rev. A 92 021401Google Scholar

    [17]

    Pagano G, Mancini M, Cappellini G, Lombardi P, Schäfer F, Hu H, Liu X J, Catani J, Sias C, Inguscio M, Fallani L 2014 Nat. Phys. 10 198Google Scholar

    [18]

    Stuhl B K, Lu H I, Aycock L M, Genkina D, Spielman I B 2015 Science 349 1514Google Scholar

    [19]

    Aidelsburger M 2015 Artificial Gauge Fields with Ultracold Atoms in Optical Lattices (Switzerland: Springer) pp27–44

    [20]

    Azbel M Y 1964 Sov. Phys. JETP 19 634

    [21]

    Harper P G 1955 Proc. Phys. Soc. London, Sect. A 68 874Google Scholar

    [22]

    Hofstadter D R 1976 Phys. Rev. B 14 2239

    [23]

    Giamarchi T 2004 Quantum Physics in One Dimension (Clarendon: Oxford) p29

    [24]

    Dolfi M, Bauer B, Keller S, Kosenkov A, Ewart T, Kantian A, Giamarchi T, Troyer M 2014 Comput. Phys. Commun. 185 3430Google Scholar

    [25]

    White S R 1992 Phys. Rev. Lett. 69 2863Google Scholar

    [26]

    Schollwöck U 2011 Ann. Phys. 326 96Google Scholar

    [27]

    Tsui D C, Stormer H L, Gossard A C 1982 Phys. Rev. Lett. 48 1559Google Scholar

    [28]

    Palmer R N, Jaksch D 2006 Phys. Rev. Lett. 96 180407Google Scholar

    [29]

    Hafezi M, Sørensen A S, Demler E, Lukin M D 2007 Phys. Rev. A 76 023613Google Scholar

    [30]

    Cooper N R, Dalibard J 2013 Phys. Rev. Lett. 110 185301Google Scholar

    [31]

    Strinati M C, Cornfeld E, Rossini D, Barbarino S, Dalmonte M, Fazio R, Sela E, Mazza L 2017 Phys. Rev. X 7 021033

    [32]

    Cornfeld E, Sela E 2015 Phys. Rev. B 92 115446Google Scholar

    [33]

    Petrescu A, Piraud M, Roux G, McCulloch I P, Le Hur K 2017 Phys. Rev. B 96 014524Google Scholar

    [34]

    Petrescu A, Le Hur K 2013 Phys. Rev. Lett. 111 150601Google Scholar

    [35]

    Petrescu A, Le Hur K 2015 Phys. Rev. B 91 054520Google Scholar

    [36]

    Piraud M, Heidrich-Meisner F, McCulloch I P, Greschner S, Vekua T, Schollwöck U 2015 Phys. Rev. B 91 140406Google Scholar

    [37]

    Geim A K, Dubonos S V, Lok J G S, Henini M, Maan J C 1998 Nature 396 144Google Scholar

    [38]

    Bardeen J 1955 Phys. Rev. 97 1724Google Scholar

    [39]

    Abrikosov A A 1957 Sov. Phys. JETP 511 74

    [40]

    Su W P, Schrieffer J R, Heeger A J 1979 Phys. Rev. Lett. 42 1698Google Scholar

    [41]

    Su W P, Schrieffer J R, Heeger A J 1983 Phys. Rev. B 28 1138

    [42]

    Zhang J M, Dong R X 2010 Eur. J. Phys. 31 591Google Scholar

    [43]

    Calabrese P, Cardy J 2009 . Phys. A 42 504005Google Scholar

    [44]

    Holzhey C, Larsen F, Wilczek F 1994 Nucl. Phys. B 424 443Google Scholar

    [45]

    Vidal G, Latorre J I, Rico E, Kitaev A 2003 Phys. Rev. Lett. 90 227902Google Scholar

    [46]

    Roux G, Capponi S, Lecheminant P, Azaria P 2009 Eur. Phys. J. B 68 293Google Scholar

    [47]

    Kühner T D, White S R, Monien H 2000 Phys. Rev. B 61 12474Google Scholar

    [48]

    Asbóth J K, Oroszlány L, Pályi A 2016 Lect. Notes Phys. 919 87

    [49]

    Sheikhan A, Brennecke F, Kollath C 2016 Phys. Rev. B 94 061603

    [50]

    Feng Y L, Zhang K, Fan J T, Mei F, Chen G, Jia S T 2018 Sci. China: Phys. Mech. 61 123011

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出版历程
  • 收稿日期:  2019-12-25
  • 修回日期:  2020-02-10
  • 刊出日期:  2020-04-20

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