Quasiparticle Origin of Dynamical Quantum Phase Transitions
Considering nonintegrable quantum Ising chains with exponentially decaying interactions, we present matrix product state results that establish a connection between low-energy quasiparticle excitations and the kind of nonanalyticities in the Loschmidt return rate. When local spin excitations are energetically favorable to two-domain-wall excitations in the quench Hamiltonian, anomalous cusps appear in the return rate regardless of initial state, except in the nearest-neighbor limit. This illustrates that models in the same equilibrium unversality class can still exhibit fundamentally distinct out-of-equilibrium criticality. Our results are accessible to current ultracold-atom and ion-trap experiments.
It is no overstatement that critical phenomena are among the most intriguing and actively investigated subjects in physics, and have been extensively studied theoretically and experimentally for decades. Theoretical understanding has been established by the renormalization-group method, which relates criticality to scale invariance, universality, and a characteristic set of critical exponents Sachdev (2001); Ma (1985); Cardy (1996). A natural question, motivated by substantial technological advancement on the experimental side, concerns extending the frontier of criticality to include its out-of-equilibrium properties.
Various concepts of dynamical criticality have been studied in classical Hohenberg and Halperin (1977) and quantum Täuber (2014) out-of-equilibrium physics. In recent years, dynamical quantum phase transitions (DQPT) Heyl et al. (2013); foo () have come under considerable theoretical Heyl (2014); Andraschko and Sirker (2014); Vajna and Dóra (2014); Heyl (2015); Vajna and Dóra (2015); Budich and Heyl (2016); Heyl (2018) and experimental Fläschner et al. (2018); Jurcevic et al. (2017) invetigation. DQPT is based on nonanalyticities in the Loschmidt return rate, a dynamical analog of the equilibrium free energy, where quenches below or above a dynamical critical point lead to different phases characterized by the absence, presence, and kind of nonanalyticities in the return rate.
Though first characterized in free fermionic two-band models Heyl et al. (2013), the study of DQPT has been extended to systems with long-range interactions such as transverse-field Ising chains (TFIC) with power-law interactions , with interspin distance and Halimeh and Zauner-Stauber (2017); Zauner-Stauber and Halimeh (2017); Žunkovič et al. (2018), and in its mean-field limit () Homrighausen et al. (2017); Lang et al. (2018a, b). These studies went beyond the standard DQPT picture Heyl et al. (2013) in which only two kinds of dynamical phases exist: (i) the trivial phase (TDP) for quenches before the dynamical critical point where no nonanalyticities (cusps) appear in the return rate; and (ii) the regular phase (RDP) where quenching across the dynamical critical point leads to temporally equidistant cusps. Indeed, Ref. Halimeh and Zauner-Stauber, 2017 showed that starting in an ordered state in TFIC with sufficiently long-range power-law interactions, the trivial phase is replaced by the so-called anomalous dynamical phase (ADP) for , where, even though one still quenches below the dynamical critical point, cusps arise in the return rate albeit only after its first minimum. Interestingly, anomalous cusps do not correspond to zero crossings of the order parameter, whereas regular cusps do when the initial state is ordered Heyl (2014); Halimeh and Zauner-Stauber (2017); Homrighausen et al. (2017). It was also shown that anomalous cusps belong to a different group of Fisher zeros relative to their regular counterparts Halimeh and Zauner-Stauber (2017); Zauner-Stauber and Halimeh (2017). ADP was then later found to exist in the integrable limit of full connectedness of TFIC at zero Homrighausen et al. (2017) and finite Lang et al. (2018a, b) temperature. Additionally, ADP seems to coincide with a long-time ordered steady state Halimeh and Zauner-Stauber (2017); Zauner-Stauber and Halimeh (2017); Homrighausen et al. (2017); Lang et al. (2018a, b). These studies have explored a rich phenomenology of ADP, but leave open whether the origin of ADP lies in the presence of sufficiently long-range interactions, the existence of a finite-temperature phase transition, or yet another physical mechanism.
In a seemingly unrelated direction, many efforts have been devoted towards understanding quench dynamics in terms of the ballistic propagation of quasiparticle excitations Calabrese and Cardy (2006, 2007). These efforts also considered the case of long-range interactions, where the nonlocal nature of the latter can lead to divergences in the quasiparticle group velocity and, consequently, the super-ballistic propagation of information through the system Hauke and Tagliacozzo (2013); Maghrebi et al. (2016); Vanderstraeten et al. (2018). In one dimension, long-range interactions can have even more drastic effects on the quasiparticle spectrum: whereas a domain-wall excitation is often the quasiparticle with the lowest energy, long-range interactions across the domain wall lead to a significant increase in its energy and a local excitation can become energetically favorable. In a recent study Vanderstraeten et al. (2018) of the power-law interacting TFIC, it was shown that this scenario leads to a crossover from the ‘local’ regime, where (topologically nontrivial) domain walls are the low-energy quasiparticles, to the ‘long-range’ regime where (topologically trivial) local excitations abound at low energies. This scenario of domain-wall confinement was recently exploited for observing confined dynamics in long-range interacting spin systems Liu et al. (2018). Interestingly, the crossover region in the power-law interacting TFIC was situated around , around the value below which ADP arises.
In this study, we bring these different directions together for exploring the physical origin of ADP. We provide analytic and numerical evidence that truly long-range interactions are, in fact, not a necessary condition, therefore ruling out a finite-temperature equilibrium phase transition as the origin of ADP. Instead, we present evidence suggesting that the actual origin of ADP is, indeed, the existence of an underlying quasiparticle spectrum crossover.
Model.– In previous works, TFIC with power-law decaying interactions has been studied, and the existence of ADP has been established Halimeh and Zauner-Stauber (2017); Zauner-Stauber and Halimeh (2017). In this paper, we study the case of TFIC with exponentially decaying interactions, given by the Hamiltonian
where are the Pauli matrices on site , is the spin-coupling constant, is the transverse-field strength, and . The model exhibits a quantum phase transition from a symmetry-broken ground state (small ) to a polarized state (large ), where the critical field shifts as decreases (see Fig. 1). The physics of the exponentially decaying interactions can be adiabatically connected to the nearest-neighbor case (), and we expect that the phase transition is in the same universality class. Moreover, this excludes a finite-temperature phase transition, as is confirmed by a simple Landau-Lifshitz argument Landau and Lifshitz (2013); Thouless (1969); SM1 ().
In the short-range model (large ), the symmetry-broken phase hosts domain-wall excitations that interpolate between the two ferromagnetic ground states. Upon decreasing , the domain walls become more massive (because of the interactions between different ground-state configurations across the domain walls) and a local excitation goes down in energy. In the limit , these two different types of quasiparticles can be understood as, on the one hand, a bare domain wall and, on the other, a single spin flip on one of the two ground states. In this limit, their energies can be derived as
respectively. A crossover between these two excitations at occurs at . In the presence of a magnetic field (), the quasiparticles will become dressed by quantum fluctuations and these energies will start to shift, crossing at . We have used a variational method Vanderstraeten et al. (2018) to compute the excitation gaps in the two sectors in the quantum regime that confirm this picture, see Fig. 2: In the ‘local’ regime the lowest-lying excitation in the topologically trivial sector is a two-domain-wall scattering state, whereas in the ‘long-range’ regime there is a stable local excitation that is below the two-domain-wall continuum. We define as the value of the transverse-field strength separating the two regions, cf. Fig. 1.
Results and discussion.–We now present our numerical results, where we simulate the quench dynamics using uniform matrix product states (MPS) and the time-dependent variational principle Haegeman et al. (2011, 2016). The Loschmidt return rate is defined as
with the ground state of , and can be computed in MPS from the return-rate branches, which are the (negative of the) logarithms of the eigenvalues of the mixed MPS transfer matrix Zauner-Stauber and Halimeh (2017). Nonanalyticities in the return rate emerge when the two lowest-lying branches intersect, thereby making the detection of cusps straightforward. Simple observables such as
are readily evaluated using MPS. Here, is the Landau order parameter. Post-quench dynamics of the order parameter have been studied in the nearest-neighbor TFIC Calabrese et al. (2011, 2012), the XXZ chain Barmettler et al. (2009); Heyl (2014), and the Bose-Hubbard model Altman and Auerbach (2002). For quenches from an ordered initial state, the order parameter makes zero crossings (changes sign) only for quenches across a dynamical critical point, while asymptotically going to zero in all cases in the absence of a finite-temperature phase transition. In our model, which is nonintegrable and in the short-range universality class SM1 (), is therefore expected to go to zero asymptotically for all quenches, and to make zero crossings only for quenches above the dynamical critical point . Moreover, these zero crossings have been shown to correspond to regular cusps in the return rate for Ising-like models Heyl (2014); Halimeh and Zauner-Stauber (2017).
Let us prepare our system in the fully -polarized state, a ground state of , and quench with . First we look at the case of shown in Fig. 3. For quenches below the dynamical critical point , the return rate displays anomalous cusps that do not correspond to any zero crossings in the order parameter. For quenches to we observe (regular) cusps that correspond to zero crossings in in addition to (anomalous) cusps that do not, signifying a sort of coexistence of both ADP and RDP. Our results indicate that this ‘coexistence region’ of ADP and RDP happens only for where local-spin excitations are not only energetically favorable to two-domain-wall, but also single-domain-wall excitations at small . On the other hand, when , only regular cusps exist, and for large quenches they are always evenly spaced in time, consistent with RDP. The picture qualitatively changes for shown in Fig. 4. Here , which leads to anomalous cusps for , regular cusps for , and, interestingly, a smooth return rate for . Indeed, the return rate shows a cusp in its eighth peak at , which then smoothens out at , contrary to the property of cusp proliferation with increasing when ADP borders RDP Halimeh and Zauner-Stauber (2017); Homrighausen et al. (2017); Lang et al. (2018a).
These results suggest that the occurence of cusps in the return rate is connected to the stability of local excitations in the system. Indeed, in the local regime where unbound domain-wall excitations dominate, we observe the dynamical properties of the nearest-neighbor case, but in the long-range regime the domain walls cannot propagate as they are confined into a stable local excitation, and only then ADP emerges. This connection implies that ADP always exists in our model for finite positive , albeit it shrinks ( gets smaller) with increasing and completely disappears for .
In order to further confirm this picture, we repeat the above quench procedure but starting in the fully -polarized state, the ground state of . For the case of in Fig. 5, the return rate shows cusps for quenches only across the equilibrium critical point , in agreement with the nearest-neighbor limit. However, even though only regular cusps appear for that are evenly spaced in time, for larger quenches we observe both anomalous cusps, which are unevenly spaced in time, and regular cusps. The results for in Fig. 6 and larger values SM2 () are qualitatively the same. It is worth noting that even though the first cusps may appear regular for quenches to , the higher branch-cut segments are qualitatively different from the case of . In Figs. 5 and 6 we also show . We see roughly a common periodicity between the inflection points of this observable and the regular cusps, although not much can be deduced from this, because is not the order parameter. The latter is here always zero because both and possess symmetry.
Finally, we note that for our numerical simulations we have used a maximum bond dimension and a time-step , at which convergence is achieved for all our results. Since we work in the thermodynamic limit directly, no finite-size errors are present.
Summary.– We have provided numerical evidence linking the existence of a quasiparticle spectrum crossover between local and two-domain-wall excitations in the topologically trivial quasiparticle sector to anomalous criticality in the return rate, which for quenches within the ordered phase does not correspond to any zero crossings in the order parameter. This is demonstrated in the transverse-field Ising chain with exponentially decaying interactions, where anomalous criticality arises regardless of the initial state, only disappearing in the integrable nearest-neighbor limit. As a consequence, our results show that models in the same equilibrium universality class can host drastically different out-of-equilibrium properties: for any finite positive , the dynamical phase diagram is qualitatively different from that of the nearest-neighbor quantum Ising chain. Moreover, our study resolves the outstanding question as to whether anomalous cusps are associated with truly long-range interactions or a finite-temperature phase transition, as here we observe anomalous cusps in a short-range model that has neither. Our results should be experimentally accessible in modern ultracold-atom Fläschner et al. (2018) and ion-trap Jurcevic et al. (2017) setups, which have already detected regular cusps.
Acknowledgements.– The authors gratefully acknowledge stimulating discussions with Bernhard Frank, Markus Heyl, Johannes Lang, Achilleas Lazarides, Francesco Piazza, and Matthias Punk. This work is supported by the Research Foundation Flanders, ERC grants QUTE (647905) and ERQUAF (715861).
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— Supplemental Material —
Quasiparticle Origin of Dynamical Quantum Phase Transitions
Jad C. Halimeh, Maarten Van Damme, Valentin Zauner-Stauber, and Laurens Vanderstraeten
Appendix A Landau-Lifshitz argument
We consider the fully -polarized ground state of the Ising chain described by the Hamiltonian
where we take the thermodynamic limit and . We now add a “droplet” Landau and Lifshitz (2013); Thouless (1969) of opposite spin polarization along neighboring sites. The energy cost of this droplet with respect to the ground state is
Since there are sites on which this droplet can be positioned, the entropy change is . Thus, the change in the Gibbs free energy upon adding this droplet is
In the case of nearest-neighbor interactions, , and the change in free energy becomes . It is therefore clear that for large , for , which means that the ordered state cannot be the equilibrium state unless for the nearest-neighbor Ising chain.
Let us now consider exponentially decaying interactions , which means that
which shows that in the limit , will be negative at irrespective of the value of . Hence, an ordered state cannot be the equilibrium state except at for any . Therefore, in the case of exponentially decaying interactions there is no finite-temperature phase transition in the Ising chain.
Appendix B Results for larger
In this section we provide additional results supporting the conclusions in the main text. Our initial state is the fully -polarized state, the ground state of , which we quench with of (1) with . Just as in the main results of Figs. 5 and 6, in the case of and in Fig. S1 we also see three distinct dynamical phases in the return rate (4). For quenches above the equilibrium critical point , i.e., within the same paramagnetic phase, the return rate is smooth. On the other hand, when is across but still above , the return rate shows regular cusps that are evenly spaced in time. Finally, when , the return rate fundamentally changes with respect to the nearest-neighbor case, where regular cusps appear alongside anomalous cusps that are not evenly spaced in time.