Dynamical quantum phase transitions on cross-stitch flat band networks
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] M a r Dynamical quantum phase transitions on cross-stitch flat band networks
Tong Liu and Hao Guo ∗ Department of Physics, Southeast University, Nanjing 211189, China (Dated: April 2, 2019)We study the quench dynamics on cross-stitch flat band networks by a sudden change of the inter-cell hoppingstrength J . For quench processes with J changing as J = → J ,
0, we give the analytical expression to theLoschmidt echo which possesses a series of zero points at critical times t ∗ , indicating where the dynamicalquantum phase transitions occur. We further study the converse quench process with J , → J =
0, and finda non-trivial example that the pre-quench quantum state is not an eigenstate of the post-quench Hamiltonian,whereas the Loschmidt echo L ( t ) ≡ PACS numbers: 03.65.Ge, 03.65.Vf, 03.75.Kk, 05.70.Ln, 71.10.Fd
I. INTRODUCTION
Phase transition, the transformation from one (equilibrium)physical state to another, is a central research topic in con-densed matter physics. The dynamical quantum phase transi-tion (DQPT) , a generalization of this fundamental conceptto the nonequilibrium quantum evolution, has been studied in-tensively in recent years . It has been confirmed that DQPTsare directly connected to the underlying equilibrium phasetransitions of the systems in broken-symmetry phases .For noninteracting topological systems , it has been ver-ified on general grounds that two topologically di ff erent equi-librium ground states necessarily impose the existence ofDQPTs. Inspired by the underlying phase transitions in theseequilibrium systems, DQPTs have also been connected to theinhomogeneous systems, including the Anderson model andthe incommensurate Aubry-Andr´e model . Recently, DQPTshave been experimentally observed in two types of quantumsimulating platforms, the trapped-ion system in which the dy-namics of transverse-field Ising models is synthesized, andthe ultracold-atom system in which the dynamical topologicalquantum phase transitions are observed .Generically, it is argued that the occurrence of DQPTs re-quires that the quench process is ramped through a quantumcritical point. For quenches not belonging to these classes,the so called “accidental” DQPTs can still occur, requiring afine-tuning of the Hamiltonian. In this paper we propose aquench scheme on flat band networks in which DQPTs canoccur without ramping through a quantum critical point.Flat band networks are translationally invariant tight-binding lattices which support at least one dispersionless bandin the energy spectrum. This system has usually been consid-ered as an ideal playground to explore the strong correlationphenomena due to the complete quenching of the kineticenergy. For example, a nearly flat band with non-trivial topo-logical properties was proposed to simulate fractional Cherninsulators . For single-particle systems, Ref. argued thatthere exist three criteria to determine the topological proper-ties of flat bands in two-dimensional lattices, exactly flat band,non-zero Chern number, and local hopping. The authors havedemonstrated that only two criteria can be simultaneously sat- isfied. The fact that all three criteria can not be satisfied simul-taneously indicates that the topology of the strictly flat bandof real materials (short-ranged hopping) is trivial. Thus, thetheory of DQPTs in topological band systems can not be ap-plied to flat band systems. However, inspired by the theory ofDQPTs in inhomogeneous systems , we notice a remark-able feature of the flat band, the so-called compact localizedstates (CLSs) , which are strictly localized eigenstates inreal space. The CLSs can be considered as a Wannier functionof which the amplitude is finite only in very limited regions,and vanishes identically outside. Contrast to the Andersonlocalization in which the exponentially localized states are in-duced by disorders, the CLSs typically occur in perfectly pe-riodic systems, originated from the destructive interference bythe hopping processes of specific lattices.Among a wide variety of flat band networks, the classifica-tion of flat bands is useful for choosing the appropriate modelto realize DQPTs. A first attempt to classify flat bands by theproperties of CLSs was discussed in Ref. . The authors clas-sify the CLSs by the number U of unit cells occupied by aCLS. And a very recent work developed another classifica-tion scheme of flat band systems from the perspective of theBloch wave function’s singularity. For the U = , indicat-ing that a single CLS is disentangled from the rest of unit cells,such as the cross-stitch network. However, for generic U > , and do notform a complete set of bases spanning the whole network intwo dimension, such as the Lieb lattice , where compact lo-calized lines must be added to form a complete set. The or-thogonality and completeness of CLSs are crucial for provingthe existence of zeros of Loschmidt echo in our quench pro-tocol, which is the main motivation for choosing cross-stitchnetworks.The rest of the paper is organized as follows. In Sec. II, weintroduce the model and study the quench dynamics with theinter-cell hopping strength J changing as J = → J , J changing as J , → J = FIG. 1. (Color online) (a) The cross-stitch geometry with the inter-cell hopping strength J , V inthe unit cell. The red and blue filled circles denote an antisymmetri-cal CLS | f m i = ( − , T δ n , m / √ A , B ). (b) Theisolated two-site geometry with the inter-cell hopping strength J = V . The red filled circles denote ansymmetrical CLS | φ − i = (1 , T δ n , m / √ FIG. 2. (Color online) (a) Single-particle dispersion with the corre-sponding eigenstates | f m i and | k i on the cross-stitch lattice ( J =
1) asa function of quasimomentum k . (b) Single-particle dispersion withthe corresponding eigenstates | φ + i and | φ − i on the isolated two-sitelattice ( J =
0) as a function of the quasimomentum k . Here we applythe periodic boundary condition and choose V =
4. The total numberof unit cells is set to be L =
0. We find that there exists a non-trivial example that the pre-quench quantum state is not an eigenstate of the post-quenchHamiltonian, whereas the value of Loschmidt echo is always 1as time varies. We conclude and discuss possible experimentalrealizations in Sec. IV.
II. MODEL AND DQPTS
As defined in the seminal paper , a key quantity within thetheory of DQPTs is the Loschmidt amplitude G ( t ) = h Ψ i | Ψ i ( t ) i = h Ψ i | e − i ˆ H f t | Ψ i i , (1) where | Ψ i i denotes the pre-quench quantum state and ˆ H f thepost-quench Hamiltonian. The Loschmidt echo L ( t ) is de-fined as the squared modulus of the Loschmidt amplitude L ( t ) = |G ( t ) | . Analogous to the equilibrium phase transitiontheory, the Loschmidt amplitude can be viewed as a boundarypartition function along the complex temperature. And theinitial state | Ψ i i plays the role of a boundary condition in timeinstead of space. Thus, a dynamical free energy density canbe defined as f ( t ) = − lim L →∞ L ln L ( t ) , (2)where L is the overall degrees of freedom of the system. Sim-ilar to the emergence of Fisher zeros in the equilibrium phasetransition, DQPTs can occur at some critical times t ∗ , where L ( t ) vanishes, and the corresponding dynamical free energy f ( t ) exhibits divergent behavior in the thermodynamic limit.To illustrate our quantum quench protocol, cross-stitch net-works consisted of two interconnected chains are plotted inFig. 1(a), the unit cell of which is given by two lattice sites( A , B ), and the wave function at the n -th unit cell is denotedby Ψ n . The stationary Schr¨odinger equation ˆ H Ψ n = E Ψ n isexpressed as ˆ ǫ n Ψ n − ˆ V Ψ n − ˆ T ( Ψ n − + Ψ n + ) = E Ψ n , (3)with ˆ ǫ n = ǫ an ǫ bn ! , ˆ V = VV ! , ˆ T = J JJ J ! . (4)where J is the inter-cell hopping strength and V is the intra-cell hopping strength. In the absence of the potential, ǫ an = ǫ bn =
0, there is exactly one flat band E FB = V , associatedwith an antisymmetrical CLS | f m i = ( − , T δ n , m / √
2, andone dispersive band E ( k ) = − J cos( k ) − V , associated witha Bloch wave function | k i = e ikn u k ( n ) with u k ( n ) being theperiodic envelope function, as shown in Fig. 2(a). Here wechoose J = V = J = H ( k ) = ˆ V , independent of the quasimomen-tum k . As shown in Fig. 2(b) ( J = V = E ± = ± V with the corresponding eigenstates | φ ± i = ( ∓ , T δ n , m / √ / symmetricalCLSs respectively.We first consider the quench process with J changing as J = → J ,
0, a single particle is initially prepared in theground state | Ψ g i of ˆ H ( J = | Ψ g i = | φ − i = (1 , T δ n , m / √ m -th unit cell. Performing a sudden quench,the Loschmidt amplitude can be written as G ( t ) = h Ψ g | e − i ˆ H ( J , t | Ψ g i = X m ′ e − iVt |h f m ′ | Ψ g i| + X k e i (4 J cos( k ) + V ) t |h k | Ψ g i| . (5)Due to the fact that the CLS | f m ′ = m i is antisymmetrical while | Ψ g i is symmetrical, we have the relation h f m ′ = m | Ψ g i = | f m ′ , m i , recalling the existence of a complete or-thogonal set of the CLSs on the cross-stitch lattice, then wehave | f m ′ , m i are orthogonal to | Ψ g i , i.e., h f m ′ , m | Ψ g i =
0. Sothe first term on the right-hand-side of Eq.(5) vanishes, i.e., P m ′ e − iVt |h f m ′ | Ψ g i| = u k ( n ) varies with the di ff erentquasimomentum k . So, only |h k | Ψ g i| with minimum k can beapproximated as L , while others can not. However, accordingto the fact |h k | Ψ g i| = P k |h k | Ψ g i| L = L , we approximately obtain G ( t ) ≈ X k e i (4 J cos( k ) + V ) t |h k | Ψ g i| = L X k e i (4 J cos( k ) + V ) t . (6)In the large L limit, since the quasimomentum k continuouslydistributes within (0 , π ), we can replace the summation byintegration G ( t ) = π Z π e i (4 J cos( k ) + V ) t dk = e iVt J (4 Jt ) , (7)where J (4 Jt ) is the zero-order Bessel function. It is knownthat J ( x ) has a series of zeros x i with i = , , , · · · , whichindicates that the Loschmit echo becomes zero at times t ∗ i = x i J . (8)To strengthen the validity of our analytical results, we nu-merically study the Loschmidt echo and the dynamical freeenergy. The initial state is set to be the ground state ofˆ H ( J = J is switched on at t =
0. Accord-ing to the theory of DQPTs, the occurrence of a series of zerosin the Loschmidt echo can be recognized as the signatures ofDQPTs, and we focus our attention on it first. Without loss ofgenerality, we choose a symmetrical CLS | φ − i and calculate L ( t ) with di ff erent J ’s. As shown in Figs. 3(a) and (b), theLoschmidt echo does become zero at some critical times ofwhich the values agree very well with the analytic predictionEq.(8). This demonstrates that Eq.(6) is a good approxima-tion.To show the zeros of L ( t ) more reliably, we also calculatethe dynamical free energy f ( t ), which diverges at the dynami-cal critical time. As shown in Figs. 3(c) and (d), the numericaland analytical results are in good agreement with each other, f ( t ) does exhibit obvious peaks at t = t ∗ i . We also implementcalculations for various J ’s and obtain similar results as ex-pected.Here we emphasize that (anti)symmetric properties of theCLSs and the existence of a complete orthogonal set of theCLSs are both crucial to ensure P m ′ e − iVt |h f m ′ | Ψ g i| =
0, nei-ther one can be absent. In addition, the approximate substi-tution |h k | Ψ g i| → |h k | Ψ g i| requires a simple structure of thereal space wave function | k i . On the cross-stitch lattice, theamplitudes of | k i on two sites ( A , B ) are equal, so this substi-tution is a good approximation. However, in other U = , the amplitudesof | k i on three sites of the unit cell are di ff erent from eachother, this approximation is no longer valid, and DQPTs donot occur as expected. FIG. 3. (Color online) The panels (a) and (b) plot the Loschmidt echo L ( t ) for di ff erent quench parameters J . At a critical time t ∗ i = x i J , L ( t ) (red hollow circle) reaches the zero point, which agrees with thebehaviors of the analytic result | e iVt J (4 Jt ) | (blue solid line). Thepanels (c) and (d) plot the dynamical free energy f ( t ) for di ff erentquench parameters J . At a critical time t ∗ i = x i J , f ( t ) (red double dashline) exhibits a sharp peak, which also agrees with the behaviors ofthe analytic result − ln | e iVt J (4 Jt ) | (blue solid line). Here we applythe periodic boundary condition and choose V =
4. The total numberof unit cells is set to be L = FIG. 4. (Color online) The Loschmidt echo L ( t ) in the quench pro-cess from ˆ H ( J =
1) to ˆ H ( J = V = L = III. A NON-TRIVIAL EXAMPLE OF L ( t ) ≡ In this section we study the converse quench process with J , → J =
0. No DQPTs are found, however we find aninteresting feature in this process. In general the Loschmidtecho L ( t ) gradually decreases to zero as the time is longenough, except in the special case that the pre-quench quan-tum state is an eigenstate of the post-quench Hamiltonian. Itcan be easily proved that if ˆ H f | Ψ i i = E i | Ψ i i , we must have G ( t ) = h Ψ i | e − i ˆ H f t | Ψ i i = P n e − iE n t |h Ψ n | Ψ i i| = e − iE i t . Thus,the Loschmidt echo L ( t ) ≡
1. Now, here comes a naturalquestion whether it can be deduced from L ( t ) ≡ | k i = e ikn u k ( n ). We emphasizethat the structure of | k i plays a crucial role in the time evolu-tion, which will be demonstrated later in details. Under theFourier transformation, the real space Hamiltonian ˆ H ( J , H k = X k ~ C † k H k ~ C k , (9)where the “spinor” ~ C k = [ c A , k , c B , k ] T represents the two sitesin the unit cell and H k = " − J cos( k ) − V − J cos( k ) − V − J cos( k ) − J cos( k ) . (10)By diagonalizing the Hamiltonian H k , we get one flatband with the corresponding eigenstate | ψ FB i = [ − , T and one dispersive band with the corresponding eigenstate | ψ k i = [1 , T . This means the amplitudes of the wavefunction | ψ k i on two sites ( A , B ) in the momentum spaceare equal. By applying the inverse Fourier transforma-tion of | ψ k i , the real space wave function | k i has the form[ · · · , u n − , u n − , u n , u n , , u n + , u n + , · · · ] T , ≤ n ≤ L .Now performing a sudden quench, the Loschmidt ampli-tude can be written as G ( t ) = h k | e − i ˆ H ( J = t | k i = X k e − iVt |h φ + | k i| + X k e iVt |h φ − | k i| . (11)The overlap between an antisymmetrical CLS | φ + i = ( − , T δ n , m / √ | k i must vanish, i.e., |h φ + | k i| = | φ − i = (1 , T δ n , m / √ | k i sums to 1, i.e., P k |h φ − | k i| =
1. Finallywe obtain G ( t ) = e iVt . (12)Thus, our example shows that the pre-quench quantum stateis not an eigenstate of the post-quench Hamiltonian, whereas the Loschmidt echo L ( t ) ≡
1. We also numerically verify ouranalytic prediction in Fig. 4, where the numerical results agreewell with the analytic results.
IV. CONCLUSIONS
In summary, we have studied the quench dynamics oncross-stitch flat band networks by preparing the initial stateas an eigenstate of the initial Hamiltonian H ( J i ) and then per-forming a sudden quench to the final Hamiltonian H ( J f ). Forthe quench process changing as J = → J ,
0, we cal-culate the Loschmidt echo both analytically and numerically.We find there exist a series of zero points at critical times t ∗ ,at which the DQPTs occur. We further study the conversequench process with J , → J =
0, and find that Loschmidtecho L ( t ) ≡ , which hosts a variety of novelphenomena when interactions are introduced, has been real-ized as the prototypical model for exploring flat band in theultracold-atom system since it is relatively simple to transferatoms into the flat band. The cross-stitch lattice has not beenrealized experimentally yet, which is partly due to the di ffi -culty in transferring atoms into the flat band (the upper energyband). In our quantum quench protocol, the initial quantumstate is prepared as the eigenstate of the lower energy band,which is more accessible in the ultracold-atom experiment.The quench operations can be realized by drastically increas-ing or decreasing the spacing of unit cells, which leads either J = J ,
0. By using time- and momentum-resolvedfull state tomography methods, the dynamical evolution of thewave function in optical lattices can be monitored, hence theobservation of DQPTs on the cross-stitch lattice can be real-ized experimentally.
ACKNOWLEDGMENTS
This work was supported by the National Natural ScienceFoundation of China (Grant No. 11674051), the FundamentalResearch Funds for the Central Universities, and PostgraduateResearch & Practice Innovation Program of Jiangsu Province(Grant No. KYCX18 0057). ∗ [email protected] M. Heyl, Rep. Prog. Phys. , 054001 (2018). M. Heyl, Phys. Rev. Lett. , 205701 (2014). M. Heyl, Phys. Rev. Lett. , 140602 (2015). F. Andraschko and J. Sirker, Phys. Rev. B , 125120 (2014). E. Canovi, P. Werner, and M. Eckstein, Phys. Rev. Lett. ,265702 (2014). C. Karrasch and D. Schuricht, Phys. Rev. B 87, 195104 (2013). M. Marcuzzi, E. Levi, S. Diehl, J. P. Garrahan, and I. Lesanovsky,Phys. Rev. Lett. , 210401 (2014). J. M. Hickey, S. Genway, and J. P. Garrahan, Phys. Rev. B ,054301 (2014). J. Lang, B. Frank, and J. C. Halimeh, Phys. Rev. Lett. , 130603(2018). M. E. Fisher, in Boulder Lectures in TheoreticalPhysics(University of Colorado, Boulder, 1965) Vol 7. M. Heyl, A. Polkovnikov, and S. Kehrein, Phys. Rev. Lett. ,135704 (2013). K. Brandner, V. F. Maisi, J. P. Pekola, J. P. Garrahan, and C. Flindt,Phys. Rev. Lett. , 180601 (2017). U. Marzolino and T. Prosen, Phys. Rev. B , 104402 (2017). S. Vajna and B. D´ora, Phys. Rev. B , 161105 (2014). S. Vajna and B. D´ora, Phys. Rev. B , 155127 (2015). J. C. Budich and M. Heyl, Phys. Rev. B , 085416 (2016). Z. Huang and A. V. Balatsky, Phys. Rev. Lett. , 086802 (2016). C. Yang, L. Li, and S. Chen, Phys. Rev. B , 060304(R) (2018). M. Schmitt and S. Kehrein, Phys. Rev. B , 075114 (2015). N. Sedlmayr, P. Jaeger, M. Maiti, and J. Sirker, Phys. Rev. B ,064304 (2018). H. Yin, S. Chen, X. Gao, and P. Wang, Phys. Rev. A , 033624(2018). C. Yang, Y. Wang, P. Wang, X. Gao, and S. Chen, Phys. Rev. B , 184201 (2017). P. Jurcevic, H. Shen, P. Hauke, C. Maier, T. Brydges, C. Hempel,B. P. Lanyon, M. Heyl, R. Blatt, and C. F. Roos, Phys. Rev. Lett. , 080501 (2017). N. Fl¨aschner, D. Vogel, M. Tarnowski, B. S. Rem, D.-S.L¨uhmann, M. Heyl, J. C. Budich, L. Mathey, K. Sengstock, andC. Weitenberg, Nat. Phys. , 265 (2018). S. Flach, D. Leykam, J. D. Bodyfelt, P. Matthies, and A. S. Desy-atnikov, Europhys. Lett. , 30001 (2014). W. Maimaiti, A. Andreanov, H. C. Park, O. Gendelman, and S.Flach, Phys. Rev. B , 115135 (2017). A. Ramachandran, A. Andreanov, and S. Flach, Phys. Rev. B ,161104(R) (2017). J. D. Bodyfelt, D. Leykam, C. Danieli, X. Yu, and S. Flach, Phys. Rev. Lett. , 236403 (2014). R. Khomeriki and S. Flach, Phys. Rev. Lett. , 245301 (2016). C. Gneiting, Z. Li, and F. Nori, Phys. Rev. B , 134203 (2018). J. Vidal, R. Mosseri, and B. Doucot, Phys. Rev. Lett. , 5888(1998). J. Vidal, B. Doucot, R. Mosseri, and P. Butaud, Phys. Rev. Lett. , 3906 (2000). Z. Liu, E. J. Bergholtz, H. Fan, and A. M. L¨auchli, Phys. Rev.Lett. , 186805 (2012). L. Chen, T. Mazaheri, A. Seidel, and X. Tang, J. Phys. A: Math.Theor. , 152001 (2014). C. Danieli, J. D. Bodyfelt, and S. Flach, Phys. Rev. B , 235134(2015). A. R. Kolovsky, A. Ramachandran, and S. Flach, Phys. Rev. B ,045120 (2018). C. Danieli, A. Maluckov, and S. Flach, J. Low Temp. Phys. ,678 (2018). M. R¨ontgen, C. V. Morfonios, and P. Schmelcher, Phys. Rev. B , 035161 (2018). N. Perchikov and O. V. Gendelman, Phys. Rev. E , 052208(2017). M. Johansson, U. Naether, and R. A. Vicencio, Phys. Rev. E ,032912 (2015). B. Real and R. A. Vicencio, Phys. Rev. A , 053845 (2018). J.-W. Rhim and B.-J. Yang, Phys. Rev. B , 045107 (2019). S. D. Huber and E. Altman, Phys. Rev. B , 184502 (2010). E. H. Lieb, Phys. Rev. Lett. , 1201 (1989). H. Ozawa, S. Taie, T. Ichinose, and Y. Takahashi, Phys. Rev. Lett. , 175301 (2017). A. Julku, S. Peotta, T. I. Vanhala, D. H. Kim, and P. T¨orm¨a, Phys.Rev. Lett.117