Dynamical Phase Transitions in the Two-Dimensional Transverse-Field Ising Model
We investigate two separate notions of dynamical phase transitions in the two-dimensional nearest-neighbor transverse-field Ising model on a square lattice using matrix product states and a new hybrid time-evolving block decimation algorithm. Starting in an ordered ground state, quenches below the dynamical critical point give rise to a ferromagnetic long-time steady state with the Loschmidt return rate exhibiting anomalous cusps even when the order parameter never crosses zero. Quenching above the dynamical critical point leads to a paramagnetic long-time steady state with the return rate exhibiting a regular cusp for every zero crossing of the order parameter. Additionally, our simulations indicate that quenching slightly above the dynamical critical point leads to a coexistence region where both anomalous and regular cusps appear in the return rate. Quenches from the disordered phase further confirm our main conclusions. Our work supports the recent finding that anomalous cusps arise only when local spin excitations are the energetically dominant quasiparticles. Our results are accessible in modern Rydberg experiments.
Criticality is deeply dependent on dimensionality, which comprises one of the three integral constituents of an equilibrium universality class alongside range of interactions and kind of symmetries (equivalently, number of components of the order parameter) Cardy (1996); Sachdev (2001); Ma (1985). For example, in the one-dimensional (1D) nearest-neighbor Ising model, it was first shown in 1924 by Ernst Ising that there is no thermal phase transition – i.e., magnetic order can only exist at zero temperature, and any infinitesimal temperature would destroy it Ising (1925). Even though Ising incorrectly surmised from this result that his eponymous model would have no thermal phase transition in any dimension, Onsager’s exact solution for the two-dimensional (2D) nearest-neighbor Ising model in 1944 Onsager (1944) established the existence of a thermal phase transition at a critical temperature of , with the spin-coupling constant. Both the 1D and 2D nearest-neighbor Ising models have the same symmetry and range of interactions, but the difference in spatial dimension leads to fundamentally different physics. In 1958, Landau and Lifshitz proved that long-range order is impossible in 1D systems with short-range interactions Landau and Lifshitz (2013), thus generalizing Ising’s original result. Subsequently in 1969, Thouless and Dyson Thouless (1969); Dyson (1969) showed that in 1D Ising chains with ferromagnetic power-law interaction profiles , with inter-spin distance and , long-range order can persist at finite temperature if and only if , while for there is no finite-temperature phase transition.
In recent years, the field of dynamical phase transitions (DPT) in quantum many-body physics has witnessed a surge of activity, not least because of significant advancements in ultracold-atom Levin et al. (2012); Yukalov (2011); Bloch et al. (2008); Greiner et al. (2002) and ion-trap Porras and Cirac (2004); Kim et al. (2009); Jurcevic et al. (2014) experiments that made possible achieving evolution times long enough to adequately investigate dynamical criticality in such models. Given a Hamiltonian with an experimentally accessible control parameter, the most common setup has involved preparing the system in its equilibrium thermal state under some initial Hamiltonian , and then abruptly switching the value of . The consequent dynamics due to this quantum quench can then host critical phenomena dependent on both and . One notion of dynamical criticality resembles the Landau paradigm of phase transitions in equilibrium, where nonanalytic or scaling behavior is sought in the dynamics of the order parameter or two-point correlation and response functions Moeckel and Kehrein (2008, 2010); Sciolla and Biroli (2010, 2011); Gambassi and Calabrese (2011); Sciolla and Biroli (2013); Maraga et al. (2015); Chandran et al. (2013); Smacchia et al. (2015); Mori et al. (2018); Zhang et al. (2017); Chiocchetta et al. (2015); Marcuzzi et al. (2016); Chiocchetta et al. (2017); Nicklas et al. (2015); Halimeh et al. (2017); Halimeh and Zauner-Stauber (2017). For example, if the system is prepared in an ordered initial state with a finite order parameter, then a dynamical critical point is defined as the value of at which the long-time steady state goes from being ordered to being disordered with a vanishing order parameter. We refer to this type of DPT as DPT-I, and it has been investigated in the transverse-field Ising chain with power-law interaction profiles Halimeh and Zauner-Stauber (2017); Žunkovič et al. (2018), and the fully connected () transverse-field Ising model at zero Sciolla and Biroli (2011); Smacchia et al. (2015); Homrighausen et al. (2017) and finite Lang et al. (2018a, b) temperature.
The second notion of dynamical criticality, DPT-II, rests on an intuitive analogy. Restricting our discussion to zero temperature for simplicity, we quench the ground state of with and construe the overlap as a dynamical analog of the equilibrium thermal partition function, where now the inverse temperature is complexified time . Consequently, the return rate
with the system size, is now a dynamical analog of the thermal free energy. Just as nonanalyticities in the latter denote the existence of a thermal phase transition at a critical temperature, nonanalyticities in the return rate indicate dynamical quantum phase transitions at critical evolution times Heyl et al. (2013); Heyl (2014, 2015). In the last five years significant research effort in DPT-II has culminated in various theoretical studies Heyl et al. (2013); Heyl (2014, 2015); Andraschko and Sirker (2014); Vajna and Dóra (2014); Budich and Heyl (2016); Bhattacharya et al. (2017); Heyl and Budich (2017) and experimental realizations Jurcevic et al. (2017); Fläschner et al. (2018). In the seminal work of Ref. Heyl et al., 2013, two dynamical phases were discovered in the nearest-neighbor transverse-field Ising chain (TFIC), which can be exactly solved by a Jordan-Wigner transformation; see Supplemental Material (SM) SM1 . The first, which we refer to as the trivial dynamical phase, is for quenches within the same equilibrium phase where no cusps appear in the return rate. This coincides with the order parameter going asymptotically to zero without crossing it Calabrese et al. (2011, 2012). The second is the regular phase, which occurs for quenches across the critical point and where cusps appear at equally spaced critical times, with each cusp corresponding to a zero crossing of the order parameter. DPT-II was also investigated in higher dimensions such as in the integrable two-dimensional Kitaev honeycomb Schmitt and Kehrein (2015) and three-dimensional Weidinger et al. (2017) models. However, the original picture of two dynamical phases – one where the return rate is smooth and a second where it is nonanalytic – persisted. The two-dimensional nearest-neighbor transverse-field Ising model (TFIM) was also considered in exact diagonalization (ED) Heyl et al. (2018); De Nicola et al. (2018) and using a stochastic nonequilibrium approach De Nicola et al. (2018), albeit for a few sites ( and sites, respectively), which rendered a valid characterization of critical behavior, present inherently in the thermodynamic limit (TL), impractical.
Recently, it was shown that in 1D transverse-field Ising models with certain interaction profiles beyond nearest-neighbor range Halimeh and Zauner-Stauber (2017); Zauner-Stauber and Halimeh (2017); Homrighausen et al. (2017); Lang et al. (2018a, b); Halimeh et al. (2018), a third anomalous phase can occur for certain quenches ending below the dynamical critical point, in which a new kind of cusps appear in the return rate that are not related to any zero crossings of the order parameter. These anomalous cusps occur when local excitations in the quench Hamiltonian are energetically favorable to two-domain-wall excitations, whereas when the opposite is true, the return rate is smooth Halimeh et al. (2018). Unlike TFIC, in TFIM local spin excitations are energetically favorable to two-domain-wall states because of higher connectivity due to increased dimensionality even when the interactions are still nearest-neighbor. In particular, domain walls in 2D are always energetically unbounded because they scale as the square root of system size. This in principle suggests that the type of DPT-II criticality in TFIM should significantly differ from that of TFIC.
In this work, we show that TFIM hosts dynamical criticality that is fundamentally different from that of TFIC, and further validate the conclusions in Ref. Halimeh et al., 2018 of a quasiparticle origin of the anomalous phase. To the best of our knowledge, our work comprises the first numerically exact study of dynamical phase transitions in nonintegrable higher-dimensional quantum many-body systems in TL.
Model.–The Hamiltonian of TFIM is
where and are lattice vectors, indicates nearest-neighbor interactions where each bond is counted only once, and are the Pauli matrices on site . We build in the framework of the infinite density matrix renormalization group method (iDMRG) McCulloch (2008); Ian an effective square lattice in TL on a cylindrical geometry of infinite length and a six-site circumference along which periodic boundary conditions are enforced. Let us first consider ferromagnetic interactions () – as we will see later, this leads to no loss of generality. In the full TL, the square-lattice TFIM has an equilibrium quantum critical point du Croo de Jongh and van Leeuwen (1998); Blöte and Deng (2002) and a critical temperature Onsager (1944). However, since in our cylinder geometry TL is achieved only along the axial direction, finite-size fluctuations due to the six-site circumference lead to a smaller equilibrium quantum critical point . The quantum equilibrium phase diagram of our model is shown in Fig. 1, where the ground-state calculation is based on iDMRG.
Results and discussion.–We now present our matrix product state (MPS) results for the time evolution of the Loschmidt return rate (1) and the longitudinal and transverse magnetizations
respectively, upon quenching the fully ordered () and fully disordered () ground states of TFIM. Our time-evolution results are computed with the hybrid time-evolving block decimation (HTEBD) algorithm, the implementation of which can be found in the Matrix Product Toolkit mpt . Details on this novel approach are provided in SM SM2 , and the full description of its implementation and benchmarking results will appear in Ref. Has, . Our results reach overall convergence at maximum bond dimension and a time step .
Let us first consider as initial state the fully -up-polarized ground state () of TFIM. We proceed to quench this state with , and then calculate the corresponding Loschmidt return rate and order parameter . The behavior of the latter critically depends on the value of to which we quench. Indeed, we find that for quenches below a dynamical critical point the order parameter neither crosses nor decays to zero, but rather seems to go asymptotically to a finite value as indicated in the evolution-time intervals that we can access numerically; see Fig. 2. This behavior is reminiscent of the 1D transverse-field Ising model with power-law ferromagnetic interactions Halimeh and Zauner-Stauber (2017); Zauner-Stauber and Halimeh (2017); Žunkovič et al. (2018); Homrighausen et al. (2017); Lang et al. (2018a, b). For sufficiently long-range interactions (), the latter is expected to go into a ferromagnetic steady state in the long-time limit for small quenches due to the model hosting a finite-temperature phase transition. Even when it has no finite-temperature phase transition (), due to domain-wall confinement Liu et al. (2018) this system can even settle into a long-lived prethermal state Halimeh et al. (2017), which is absent only in the integrable case of nearest-neighbor interactions Calabrese et al. (2011, 2012) where two-domain-wall states are always energetically favorable to local excitations in the ordered phase Liu et al. (2018); Halimeh et al. (2018). Therefore, just like long-range interactions in 1D quantum Ising models give rise to fundamentally different DPT-I criticality, dimensionality in the case of TFIM leads to a ferromagnetic steady state for small quenches that does not exist in the case of TFIC. Indeed, in the latter one can still define a DPT-I by the manner in which the order parameter decays to zero. Specifically, quenches in TFIC within the ordered phase lead to the order parameter decaying asymptotically to zero without ever crossing it Calabrese et al. (2011, 2012). On the other hand, if the quench is across the equilibrium critical point, the order parameter will cross zero and form an envelope that asymptotically vanishes at long times Calabrese et al. (2012). However, in both cases the long-time steady state is paramagnetic. In contrast, DPT-I in TFIM separates a ferromagnetic steady state for small quenches from a paramagnetic one for large quenches.
In all cases, in Fig. 2 exhibits cusps, and, in fact, resembles the return rates due to small quenches in the 1D power-law and exponential-decay interaction models for sufficiently long-range interactions Halimeh and Zauner-Stauber (2017); Halimeh et al. (2018). Indeed, we find that the first cycle of the return rate is smooth without any nonanalyticities. This behavior persists even for quenches right below where the order parameter barely scrapes zero but does not cross it; see bottom panel of Fig. 2. This behavior is fundamentally different from that in TFIC for quenches within the ordered phase where the return rate is analytic SM1 . This showcases the crucial effect of dimensionality on DPT-II criticality as well.
We now consider quenches to shown in Fig. 3. Here the order parameter makes zero crossings and its envelope seems to asymptotically go to zero in the long-time limit. The larger is, the larger the oscillation frequency of the order parameter. At large (bottom two panels of Fig. 3), the return rate exhibits a cusp in each cycle such that the periodicity of the cusps is double that of the order-parameter zero crossings. At the smallest value of (top panel of Fig. 3) where the order parameter still exhibits zero crossings, the cusps appear to be anomalous rather than regular. This may be because of one of two reasons. The first is that DPT-I and DPT-II may simply not share a common dynamical critical point. The second reason is that there is possibly a coexistence region of both anomalous and regular cusps similar to the case of sufficiently long-range interactions in the 1D case, when the dynamical critical point separating a ferromagnetic steady state from a paramagnetic one is smaller than the crossover value of the transverse field below which local spin excitations are energetically favorable to two-domain-wall states Halimeh et al. (2018). Indeed, in TFIM as domain walls are energetically unbounded in 2D, and thus local spin flips will always be the energetically dominant quasiparticles in the ordered phase. The quench where is exactly when the coexistence region forms for in the dynamical phase diagram of Ref. Halimeh et al., 2018. For this to be rigorously confirmed though, we must access in the interval longer evolution times than our code is currently able to achieve in order to discern anomalous from regular cusps. However, lending support to the existence of a coexistance region in the interval is the result in Fig. 3 for , where cusps appear at earlier times. There we see the return rate hosting what resembles both regular and anomalous cusps. The first cycle shows a cusp, as is the case in the regular phase, but at the same time the cusps are not evenly spaced in time, which is one of the characteristics of the anomalous phase. This return rate is in great qualitative agreement with those of Ref. Halimeh et al., 2018 for quenches from to .
The overall picture drawn from the results of Figs. 2 and 3 strongly suggests, therefore, that the anomalous (regular) phase coincides with a ferromagnetic (paramagnetic) long-time steady state. This is again in remarkable agreement with the cases of the fully connected and 1D long-range quantum Ising models, where in the former that overlap was numerically shown to be full Homrighausen et al. (2017); Lang et al. (2018a) while in the latter this could not be completely ascertained within the precision of determining the dynamical critical point for either DPT Halimeh and Zauner-Stauber (2017), though seems very likely.
We now consider quenches starting in the fully disordered () ground state of TFIM, and quench to various values of . The dynamical critical point in this case is . We see two main cases displayed in Fig. 4. For quenches within the disordered phase, the return rate shows no cusps, while for quenches to the ordered phase, the return rate displays cusps that are not evenly spaced in time. This is similar to the case of TFIC, except in the latter the cusps always appear at evenly spaced times that are multiples of an analytically determined critical time; cf. SM SM1 . This is qualitatively identical to what is observed for the same quench in the 1D quantum Ising model with exponentially decaying interactions when the spectrum of the quench Hamiltonian has local spin excitations energetically favorable to two-domain-wall states, in which a coexistence region emerges in Halimeh et al. (2018).
Finally, we note that the above quenches for the antiferromagnetic case () yield the same behavior qualitatively and quantitatively. This is obvious from the bipartite lattice in a nearest-neighbor model, where ferromagnetic-antiferromagnetic symmetry is up to a spin flip on every second lattice site. This is also the case of TFIC, where the sign of is inconsequential to the emergent dynamics; see SM SM1 for analytical proof.
Summary.–We have presented matrix product state results for two notions of dynamical phase transitions in the two-dimensional transverse-field Ising model with nearest-neighbor interactions. When the initial state is ordered, cusps always appear in the return rate regardless of quench distance. Large quenches lead to regular cusps with a periodicity double that of the zero crossings of the order parameter. For small quenches, anomalous cusps appear that do not show periodicity. In a small interval above the dynamical critical point separating a ferromagnetic steady state from a paramagnetic one, our results indicate the formation of a coexistence region in which both anomalous and regular cusps appear in the return rate. This supports results found in Ref. Halimeh et al., 2018 for the 1D long-range case, where a crossover value of the transverse field – below which local spin excitations dominate – is greater than the DPT-I critical point, as is the case in TFIM. Moreover, our simulations show that the anomalous phase overlaps with a ferromagnetic steady state, while the coexistence region and regular phase coincide with a paramagnetic steady state. Quenches from the fully disordered state show no cusps within the disordered phase. When the quench ends in the ordered phase, the return rate shows both regular and anomalous cusps, i.e., the return rate displays the coexistence region, which is found for the same quenches in Ref. Halimeh et al., 2018 for quenches from the fully disordered state to values of the transverse-field strength below the crossover point.
Our results confirm the quasiparticle origin of anomalous cusps Halimeh et al. (2018), are experimentally accessible in modern Rydberg experiments Zeiher et al. (2016); Gross and Bloch (2017), and usher in the possibility of discerning the long-time steady state properties of a system from the short-time behavior of the return rate.
Acknowledgements.–J.C.H. acknowledges stimulating discussions with Bernhard Frank, Christian Gross, Markus Heyl, Johannes Lang, David J. Luitz, and Daniele Trapin. I.P.M. acknowledges support from the ARC Future Fellowships scheme, FT140100625.
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— Supplemental Material —
Dynamical Phase Transitions in the Two-Dimensional Transverse-Field Ising Model Tomohiro Hashizume, Ian P. McCulloch, and Jad C. Halimeh
I 1D nearest-neighbor transverse-field Ising chain
The 1D nearest-neighbor transverse-field Ising chain (TFIC) is described by the Hamiltonian
We employ the Jordan-Wigner transformation
where are fermionic annihilation and creation operators, respectively, obeying the canonical anticommutation relations and . This renders (S1) in the form
Inserting the Fourier transformation , with the number of sites, into (S5), the Hamiltonian in momentum space takes the form
The Bogoliubov transformation
diagonalizes (S6) leading to the dispersion relation
Let us now prepare our system in the ground state of :
is the ground state of , and and are the eigenstates of , with the superscript ‘i’ referring to the initial Hamiltonian () and ‘f’ to the quench Hamiltonian (). We can now express the Loschmidt amplitude as
This then leads to the Loschmidt return rate
It is therefore clear from (S12) that nonanalyticities can only occur at critical momenta
where , i.e., when there is equal probability of occupying both levels in the momentum sector . These nonanalyticities occur at well-specified (periodic) critical times
if and only if and are on different sides of the equilibrium critical point , otherwise , and therefore , are not well-defined.
Already from (S14) we see a fundamental difference from the case of the two-dimensional transverse-field Ising model (TFIM) discussed in the main text. Whereas here cusps can only occur when crossing a dynamical critical point, which for TFIC coincides with its equilibrium critical point, in the 2D case the cusps occur at any as long as , the equilibrium critical point of TFIM. The anomalous cusps present for quenches within the ordered phase and below the dynamical critical point in TFIM are due to an underlying quasiparticle spectrum crossover where at small values of the transverse-field strength spin-flip excitations are energetically favorable to two-domain-wall states, as discussed in the main text. This crossover is absent in TFIC, in which two-domain-wall states are always energetically dominant. Moreover, (S14) indicates a clear periodicity in the return rate after quenches in TFIC, and even though this is also the case for TFIM for quenches deep in the regular phase, in the anomalous phase and coexistence region of the return rate we see cusps that are not evenly spaced in time (see main text). Nevertheless, there is one feature that both models share in that it does not matter whether the interactions are ferromagnetic or antiferromagnetic, the dynamics will be equivalent so long as the interactions are nearest-neighbor. In fact, plugging (S13) into (S8), it is clear that the sign of has no effect on the value of , which means that the critical times (S14) are the same for .
Ii Hybrid time-evolving block decimation algorithm
In this section, we introduce the hybrid time-evolving block decimation (HTEBD) algorithm that is used for the time evolution of the states. HTEBD performs a global time evolution through the Suzuki-Trotter expansion Trotter (1959); Suzuki (1976) and a local time evolution with a method of choice. Here we choose the Krylov subspace expansion method Noack and Manmana (2005); Hochbruck and Lubich (1997); García-Ripoll (2006) for the local time evolution.
In the ordinary time-evolving block decimation algorithm Orús and Vidal (2008), a wave function in the thermodynamic limit is described with one pair of and matrices (Vidal’s notation in Ref. Vidal, 2004). Therefore, with this method, only Hamiltonians with nearest-neighbor interactions can be evolved. Although swap gates can be used to force sites to be nearest-neighbor, this is a cumbersome approach that does not readily extend to three- or more-site interactions or exponetially decaying long-range interactions. Here we extend this method so that we can time-evolve Hamiltonians with long-range interactions. By introducing a unit cell of sites, with pairs of and (), we can study systems with dimension greater than one.
To evolve a state, we first construct two Hamiltonians and such that describes interactions confined within sites to on the unit cell, and describes all interactions between site on one unit cell to site of the next unit cell with a constraint , where is the original Hamiltonian with long-range interactions such that interaction ranges going beyond are truncated. By making use of the second-order Suzuki-Trotter formula, we can decompose the infinitesimal time-evolution operator , with , into a product of local operators that act independently on all of the parts of the infinitely long chain:
The local time evolution operators and can then be calculated by one’s choice of MPS algorithm. For the calculations that are done in this paper, the Krylov subspace expansion algorithm Noack and Manmana (2005); Hochbruck and Lubich (1997); García-Ripoll (2006) is used. Due to the leading error of order from the second-order Suzuki-Trotter expansion, only three Krylov vectors are calculated for each of the local time-evolution operator. This is because with three Krylov vectors the leading error is of the order of . The unit cell in the HTEBD algorithm can be quite large with no loss of efficiency, which allows for the simulation of long-range interacting models such as the Ising model with power-law decaying interactions Has (a), and the method applies naturally in finite and infinite settings, as well as infinte boundary conditions Phien et al. (2012).
For our numerical simulations, we find that all results converge at maximum bond dimension and time-step . In Fig. S1, we show the converged return rate for a quench on the fully -polarized state with .
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