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 domain walls in the spectrum of the quench Hamiltonian are energetically favored to be bound rather than freely propagating, anomalous cusps appear in the return rate regardless of the initial state. In the nearest-neighbor limit, domain walls are always freely propagating, and anomalous cusps never appear. As a consequence, our work illustrates that models in the same equilibrium universality 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 renormalizationgroup method, which relates criticality to scale invariance, universality, and a characteristic set of critical exponents [1][2][3].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 [4] and quantum [5] out-of-equilibrium physics.In recent years, dynamical quantum phase transitions (DQPT) [6,7] have come under considerable theoretical [8][9][10][11][12][13][14] and experimental [15,16] 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 [6], the study of DQPT has been extended to systems with long-range interactions such as transverse-field Ising chains (TFIC) with power-law interactions ∝ 1/r α , with r interspin distance and α > 0 [17][18][19], and in its mean-field limit (α = 0) [20][21][22].These studies went beyond the standard DQPT picture [6] 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. 17 showed that starting in an ordered state in TFIC with sufficiently longrange power-law interactions, the trivial phase is replaced by the so-called anomalous dynamical phase (ADP) for α 2.3 − 2.4, where, even though one still quenches be-low 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 [8,17,20].It was also shown that anomalous cusps belong to a different group of Fisher zeros relative to their regular counterparts [17,18].ADP was then later found to exist in the integrable limit of full connectedness of TFIC at zero [20] and finite [21,22] temperature.Additionally, ADP seems to coincide with a long-time ordered steady state [17,18,[20][21][22].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 [23,24].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 [25][26][27].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 [27] 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 [28].Interestingly, the crossover region In this study, we bring these different directions together for exploring the physical origin of ADP.We provide analytic and numerical evidence that truly longrange 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 [17,18].In this paper, we study the case of TFIC with exponentially decaying interactions, given by the Hamiltonian Ĥ where σ{x,y,z} j are the Pauli matrices on site j, J > 0 is the spin-coupling constant, h is the transverse-field strength, and λ > 0. The model exhibits a quantum phase transition from a symmetry-broken ground state (small h) to a polarized state (large h), where the critical field h e c shifts as λ decreases (see Fig. 1).The physics of the exponentially decaying interactions can be adiabatically connected to the nearest-neighbor case (λ → ∞), 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 [1,2,31].
In the short-range model (large λ), the symmetrybroken phase hosts domain-wall excitations that interpolate between the two ferromagnetic ground states.Upon decreasing λ, 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 h = 0, 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 h = 0 occurs at λ c = log 2 ≈ 0.69.In the presence of a magnetic field (h > 0), the quasiparticles will become dressed by quantum fluctuations and these energies will start to shift, crossing at λ < λ c .We have used a variational method [27] 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 lowestlying excitation in the topologically trivial sector is a twodomain-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 h cross (λ) as the value of the transverse-field strength separating the two regions, cf.Fig. 1. Results and discussion.-Wenow present our numerical results, where we simulate the quench dynamics using uniform matrix product states (MPS) and the timedependent variational principle [32,33].The Loschmidt return rate is defined as with |ψ i the ground state of Ĥ(h i ), 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 [18].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, M z (t) is the Landau order parameter.Post-quench dynamics of the order parameter have been studied in the nearest-neighbor TFIC [34,35], the XXZ chain [8,36], and the Bose-Hubbard model [37].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 [31], M z (t) is therefore expected to go to zero asymptotically for all quenches, and to make zero crossings only for quenches above the dynamical critical point h d c .Moreover, these zero crossings have been shown to correspond to regular cusps in the return rate for Ising-like models [8,17].
Let us prepare our system in the fully z-polarized state, a ground state of Ĥ(h i = 0), and quench with Ĥ(h f ).First we look at the case of λ = 0.4 shown in Fig. 3.For quenches below the dynamical critical point h d c , the return rate displays anomalous cusps that do not correspond to any zero crossings in the order parameter.For quenches to h f ∈ (h d c , h cross ) we observe (regular) cusps that correspond to zero crossings in M z (t) 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 λ λ c where local-spin excitations are not only energetically favorable to two-domain-wall, but also single-domain-wall excitations at small h.On the other hand, when h f > h cross , only regular cusps exist, and for large quenches they are always evenly spaced in time, consistent with RDP.The picture qualitatively changes for λ = 0.8 shown in Fig. 4.Here h cross < h d c , which leads to anomalous cusps for h f < h cross , regular cusps for h f > h cross , and, interestingly, a smooth return rate for h cross < h f < h d c .Indeed, the return rate shows a cusp in its eighth peak at h f = 1.35J, which then smoothens out at h f = 1.65J, contrary to the property of cusp proliferation with increasing h f when ADP borders RDP [17,20,21].
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 λ, albeit it shrinks (h cross gets smaller) with increasing λ and completely disappears for λ → ∞.
In order to further confirm this picture, we repeat the above quench procedure but starting in the fully xpolarized state, the ground state of Ĥ(h i → ∞).For the case of λ = 0.4 in Fig. 5, the return rate shows cusps for quenches only across the equilibrium critical point h e c , in agreement with the nearest-neighbor limit.However, even though only regular cusps appear for h cross < h f < h e c that are evenly spaced in time, for larger quenches h f < h cross we observe both anomalous cusps, which are unevenly spaced in time, and regular cusps.The results for λ = 0.8 in Fig. 6 and larger λ values [38] are qualitatively the same.It is worth noting that even though the first cusps may appear regular for quenches to h f < h cross , the higher branch-cut segments are qualitatively different from the case of h cross < h f < h d c .In Figs. 5 and 6 we also show M x (t).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 M x (t) is not the order parameter.The latter is here always zero because both |ψ i and Ĥ(h f ) possess Z 2 symmetry.
Finally, we note that for our numerical simulations we have used a maximum bond dimension D max = 350 and a time-step dt = 10 −3 /J, 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 λ, the dynamical phase diagram is qualitatively different from that of the nearestneighbor 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 [15] and ion-trap [16] setups, which have already detected regular cusps.

RESULTS FOR LARGER λ
In this section we provide additional results supporting the conclusions in the main text.Our initial state is the fully x-polarized state, the ground state of Ĥ(h i → ∞), which we quench with Ĥ(h f ) of (1) with J > 0. Just as in the main results of Figs. 5 and 6, in the case of λ = 2 and λ = 3 in Fig. S1 we also see three distinct dynamical phases in the return rate (4).For quenches above the equilibrium critical point h e c , i.e., within the same paramagnetic phase, the return rate is smooth.On the other hand, when h f is across h e c but still above h cross , the return rate shows regular cusps that are evenly spaced in time.Finally, when h f < h cross , 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.[S1] L.

Figure 1 .
Figure 1.(Color online).Dynamical phase diagrams of (1) in the (λ, h f ) plane with hi = 0 (top panel) and hi → ∞ (bottom panel).The different regions and transitions are explained in the text.

Figure 2 .
Figure 2. (Color online).The energy gap in the topologically nontrivial (dashed lines) and topologically trivial (solid lines) sectors for (1) λ = 0.4 (blue) and λ = 0.8 (red) as function of transverse-field strength h.The dotted line indicates the edge of the two-domain-wall continuum:if we find an excitation below this continuum, this corresponds to a stable local excitation.Both gaps approach zero at the equilibrium critical point.There is no topological sector in the paramagnetic phase, so we cannot define a crossover there.Note how there is a crossover between local-spin and single-domain-wall excitations only when λ < λc (see text).

Figure 3 .
Figure 3. (Color online).Loschmidt return rates after quenching a fully z-polarized state with Ĥ(h f ) for λ = 0.4.Dotted lines indicate sections of the branch cut above the return rate, and cusps form when they intersect.Here two phases appear separately as ADP for small quenches (top panel) and RDP for large quenches (bottom panel), but the return rate also exhibits both cusps for intermediate quenches (middle panel).

Figure 4 .
Figure 4. (Color online).Same as Fig. 3 but for λ = 0.8.The return rate shows one of three distinct phases: ADP for small quenches (top panel), TDP for intermediate quenches (middle panel), and RDP for large quenches (bottom panel).

Figure 5 .
Figure 5. (Color online).Same as Fig.3but starting from a fully x-polarized state (hi → ∞), and while showing Mx(t) rather than the order parameter, which is always zero here due to both the initial state and quenching Hamiltonian being Z2-symmetric.The return rate shows TDP (top panel), RDP (middle panel), and both regular and anomalous cusps (bottom panel).

Figure 6 .
Figure 6.(Color online).Same as Fig. 5 but for λ = 0.8.The dynamical behavior is qualitatively the same as at other values λ for this quench.

Figure
Figure S1.(Color online).Loschmidt return rates after quenching a fully x-polarized state with Ĥ(h f ) for λ = 2.0 (left panels) and λ = 0.3 (right panels).Dotted lines indicate sections of the branch cut above the return rate, and cusps form when they intersect.Just as in the main text, we see three distinct dynamical phases: TDP, RDP, and a coexistence region of ADP and RDP.