Introduction.—Phase transitions and critical phenomena, along with the resulting features of universality and scaling, are well-understood concepts in equilibrium. In a far-from-equilibrium setting, a unified framework of these notions is still missing. The pursuit of an overarching theory of far-from-equilibrium quantum many-body criticality has recently led to different concepts of nonequilibrium phase transitions [1], [2]. The first one is related to the dynamics of the equilibrium Landau order parameter, which is connected to the spontaneous breaking of a global symmetry in the ground state [3]. Upon quenching a given symmetry-broken initial state, if the long-time steady state exhibits a nonzero (zero) order parameter, then the system is in a symmetry-broken (symmetry-preserved) dynamical phase [4], [5]. The value of the quench parameter separating these two phases is the dynamical quantum critical point. This type of dynamical phase transition has been dubbed DPT-I, and has been studied in various systems, including mean-field models [6]–[14], the Hubbard model [15]–[17], \(O(N)\) model [18]–[22], long-range quantum spin chains [23], [24], among others [25]–[27].

Another approach to dynamical criticality encompasses the construction of a dynamical analog of the thermal free energy. This becomes straightforward when recognizing the overlap of the time-evolved wave function with the initial state as a boundary partition function where evolution time stands for a complex inverse temperature [28]–[30]. By taking the negative of the logarithm of this overlap in the thermodynamic limit, one obtains the return rate, which is the sought-after dynamical analog of the thermal free energy. Nonanalyticities in the return rate are thus dynamical quantum phase transitions (DQPTs) at critical evolution times. DQPTs are also referred to as DPT-II, and have been extensively studied in nonintegrable short-range quantum spin systems [31]–[33], long-range quantum many-body models [8], [9], [12], [13], [24], [34]–[40], topological systems [41]–[49], higher-dimensional models [42], [50]–[56], systems initialized in thermal ensembles [10], [11], [57]–[60], high-energy models known as lattice gauge theories [61]–[70], non-Hermitian systems [71]–[76], short-range interacting systems with broken time-translation symmetry [77], [78], and disordered models [79], [80]. Furthermore, they have been the subject of several successful experiments [81]–[83].

A different source of criticality beyond the ground state is given by excited-state quantum phase transitions (ESQPTs) [84], [85]. They consist in a generalization of quantum phase transitions to excited states, typically manifested as a singularity of the density of states and the level flow. Notwithstanding, the main consequences are dynamical, like huge decoherence [86], [87], singularities in quench dynamics [88]–[92], feedback control in dissipative systems [93], quantum work statistics [94], symmetry-breaking equilibrium states [95], [96], dynamical instabilities [97], irreversibility without energy dissipation [98], and reversible quantum information spreading [99]. It has been recently shown that they may give rise to a phase diagram composed by different dynamical phases, each one characterized by a set of (generally noncommuting) constants of motion [100]–[102].

A pertinent question is how these different concepts of nonequilibrium quantum phase transitions are related to one another. When it comes to DPT-I and DPT-II, a connection has been established between the phase of the former and the type of the latter [9]: Starting in a symmetry-broken initial state, if the long-time steady state breaks (preserves) the global symmetry of the quench Hamiltonian, then the DPT-II will be of the anomalous (regular) type. This connection also persists at finite temperature [10], [11]. This potentially allows drawing connections between far-from-equilibrium critical exponents arising in both these DPTs [103], [104]. Nevertheless, the connection between ESQPTs and DPTs, in particular at finite temperature, is still ambiguous. Given the potential of understanding DPT criticality from that of ESQPTs, it is therefore important to investigate if a direct connection exists. This is the purpose of this Letter.

Model.— Although our arguments are general, we chose a collective model, allowing us to reach large system sizes, as an illustration. It is the transverse-field Ising model with infinite-range interactions, which coincides with a version of the Lipkin-Meshkov-Glick (LMG) Hamiltonian [105]–[107], \[\label{eq:ham} \hat{H}=-\frac{\lambda}{N}\hat{J}_{x}^{2}+h\hat{J}_{z}\tag{1}\] The total collective spin operator commutes with the Hamiltonian, \([\hat{H},\mathbf{J}^{2}]=0\), allowing us to separate spin sectors labeled by the eigenvalues of \(\mathbf{J}^{2}\), \(j(j+1)\), the dimension of each being \(D(j)=2j+1\). Hamiltonian 1 also has a discrete \(\mathbb{Z}_2\) symmetry generated by a \(\pi\)-rotation around the \(z\)-axis. In each \(j\)-sector it is represented by a parity operator, \(\hat{\Pi}=\textrm{e}^{i \pi (\hat{J}_z + j)}\), which allows to classify the Hamiltonian eigenstates according to \(\hat{\Pi}\ket{E_{n,\pm}}=\pm\ket{E_{n,\pm}}\), \(n=0,1,\ldots\)

The full model displays two critical phenomena. At \(\lambda_c=\lambda\) there is a quantum phase transition (QPT): for \(\lambda>\lambda_c\) the ground state is ferromagnetic and the symmetry generated by \(\hat{\Pi}\) is broken; for \(\lambda<\lambda_c\) the ground state is symmetric. In the first case, there exists also a thermal phase transition with a critical inverse temperature given by \(\beta_c = 2h^{-1} \textrm{arctanh} (h/\lambda)\); at lower temperatures, the system is ferromagnetic, and the \(\mathbb{Z}_2\) symmetry is broken.

To make a connection between these facts and ESQPTs and DPTs, we work with all \(j\)-sectors, \(j=0,1,...,N/2\), each with a degeneracy factor of \(g(N,j)=\frac{1+2j}{1+j+N/2}\binom{N}{N/2-j}\), so \(\sum_{j=0}^{N/2}g(N,j)D(j)=2^{N}\). As noted in [108] for a similar fully connected model, each \(j\)-sector is completely independent of the others, and therefore it can be described by Hamiltonian 1 with an effective coupling constant given by \(\lambda_{\textrm{eff}}=2 j \lambda /N\). This means that each \(j\)-sector has its own critical points. For the QPT, it is \(\lambda_c (j)=hN/(2j)\). Thus, above the critical point for the global QPT, \(\lambda_c=h\), some of the \(j\)-sectors are in the ferromagnetic ground-state phase, and some others are in the paramagnetic ground-state phase; they are separated a critical value \(j_c(\lambda)=Nh/(2\lambda)\). If \(j>j_c\), the corresponding sector is in the ferromagnetic phase, and the opposite occurs if \(j<j_c\). This argument is important to understand the behavior of ESQPTs:

(i) If \(\lambda<\lambda_c\), all the \(j\)-sectors are in the paramagnetic phase. Therefore, there are no critical energies and their ground-state energies are \(\varepsilon_{\textrm{GS}} (j) = - 2 h j/N\), where \(\varepsilon=2 E /N\).

(ii) If \(\lambda>\lambda_c\), the behavior is more involved:

(a) If \(j>j_c(\lambda)\), the \(j\)-sector is in the ferromagnetic ground-state phase. Therefore, it has a critical energy below which all its energy levels are pairwise degenerate in the infinite-size limit. This is the ESQPT energy: \[\label{eq:critj} \varepsilon_c (j) = - \frac{2h j}{N}.\tag{2}\] The corresponding ground-state energy is \[\label{eq:gsj} \varepsilon_{\textrm{GS}} (j) = -\left[ 2 \lambda \left( \frac{j}{N} \right)^2 + \frac{h^2}{2 \lambda} \right].\tag{3}\] Note that Eqs. 2 and 3 coincide if \(j=j_c\), \(\varepsilon_c(j_c)=-h^2/\lambda\).

(b) If \(j<j_c\), the sector is in the paramagnetic phase. Therefore, none of its eigenlevels are degenerate, with a ground-state energy of \(\varepsilon_{\textrm{GS}} (j) = - 2 h j/N\).

The main consequence of these facts is that we can define two critical energies for the full Hamiltonian, \(\varepsilon_{c1}=-h\) and \(\varepsilon_{c2}=-h^2/\lambda>\varepsilon_{c1}\). If \(\varepsilon < \varepsilon_{c1}\), all the energy levels are degenerate in pairs; and if \(\varepsilon_{c1} \leq \varepsilon < \varepsilon_{c2}\), pairwise degenerate and non-degenerate energy occur simultaneously. Above \(\varepsilon_{c2}\), there are no degeneracies.

Equivalence of ESQPTs and DPTs.—All these features refer to static or equilibrium properties of the LMG Hamiltonian. However, DPTs are nonequilibrium phenomena. To establish a link between them, we focus on a dynamical property of a class of ESQPTs. In Refs. [100]–[102] it is shown that for a wide class of models to which the LMG belongs, there are two additional constants of motion below the critical energy of the ESQPT related to the order parameter of the QPT and the operator generating the \(\mathbb{Z}_2\) symmetry: \(\hat{\mathcal{C}}=\textrm{sign}(\hat{J}_x)\) and \(\hat{\mathcal{K}}=(i/2) [\hat{\mathcal{C}},\hat{\Pi}]\). Thus, if a nonequilibrium protocol leads the system into an energy region below the ESQPT, then the dynamics is restricted by the conservation of \(\hat{\mathcal{C}}\) and \(\hat{\mathcal{K}}\). As a consequence, quenching an initial symmetry-breaking state polarized along the ferromagnetic axis cannot lead the order parameter \(\langle \hat{J}_x \rangle\) to change sign in its dynamics. On the contrary, there are no restrictions if the energy is above the ESQPT. These dynamical features are also expected for noncollective models with an equilibrium symmetry-breaking phase [102], [109].

From these facts, we propose the main conclusion of this Letter: there are only two possible dynamical phases (DPs) starting from an equilibrium symmetry-breaking initial state:

DPa. A constant value for \(\hat{\mathcal{C}}\) and \(\hat{\mathcal{K}}\), together with the order parameter \(\langle \hat{J}_x \rangle\) oscillating around a nonzero value, without changing sign.

DPb. Both \(\hat{\mathcal{C}}\), \(\hat{\mathcal{K}}\), and \(\langle \hat{J}_x \rangle\) oscillating around zero. These two dynamical phases are separated by the critical energy of the ESQPT.

To illustrate this picture, we show in Fig. 1 the density of states for \(\lambda=0.5\), \(h=0.1\), and different values of \(j\). All of them verify \(j>j_c(\lambda)\), and therefore exhibit ESQPTs. It is clearly seen that the corresponding critical energy, identified by the logarithmic divergence in the density of states, shifts to lower energies as \(j\) is increased. This means that the populated sectors of \(j\) play a fundamental role in the dynamics. Let us suppose that we prepare a state with \(\epsilon=-0.07\). This value is above the critical energy of panels (a)-(c), and below the one on panel (d). If the expectation value of \(\mathbf{J}^2\) in our state, which is a conserved quantity, is narrowly picked around \(j/N>0.35\), then neither \(\hat{\mathcal{C}}\) nor \(\hat{\mathcal{K}}\) are constants, and therefore the order parameter \(\langle J_x \rangle\) must oscillate around zero. On the contrary, from Eqs. 2 and 3 , we can conclude that if \(\sqrt{0.06}<j/N<0.35\), then both \(\hat{\mathcal{C}}\) and \(\hat{\mathcal{K}}\) are constant, and therefore \(\langle \hat{J}_x \rangle\) cannot cross \(\langle \hat{J}_x \rangle=0\).

In Fig. 1 (e) we represent the consequences of the previous facts for the full Hamiltonian. As discussed in [100], the constancy of \(\hat{\mathcal{C}}\) requires that \(\bra{\varepsilon_{n,-}} \hat{\mathcal{C}} \ket{\varepsilon_{n,+}}= \pm 1\) for degenerate energy levels \(\varepsilon_{n,-}=\varepsilon_{n,+}\). The results indicate that this is globally fulfilled if \(\varepsilon < \varepsilon_{c2}\) in the thermodynamic limit. The reason is that, the lower the value of \(j\), the larger the degeneracy factor, \(g(N,j)\); therefore, only the lowest possible value of \(j\), which gives rise to the highest critical energy \(\varepsilon_c(j)\), contributes to the dynamics of \(\hat{\mathcal{C}}\) in the thermodynamic limit. Therefore, we can expect symmetry-broken thermal states if \(\varepsilon<\varepsilon_{c2}\), whose associated temperature is below the critical temperature of the phase transition, \(T<T_{c}\). However, this is not enough to determine the dynamics of a thermal state subjected to a nonequilibrium process. As \(\hat{\mathbf{J}}^2\) is conserved, the population of each \(j\)-sector must be taken into account to determine whether the final state is above or below the critical energies of the corresponding ESQPTs. It is worth to note that the same qualitative result shown in Fig. 1 (e) has been observed in the transverse-field Ising model with long-range interactions [102], in which \(\hat{\mathbf{J}}^2\) is not conserved. Hence, the same classification in two dynamical phases is expected for noncollective models.

To test our hypothesis, we have performed a set of numerical experiments on the LMG model. In all of them, we prepare an initial state in the ferromagnetic phase, with \(\varepsilon<\varepsilon_{c1}\). As \(\hat{\mathbf{J}}^2\), \(\hat{\Pi}\), \(\hat{\mathcal{C}}\), \(\hat{\mathcal{K}}\) are conserved under these circumstances, the most general equilibrium state is \[\label{eq:initialstate} \hat{\rho} = \frac{1}{Z} \textrm{e}^{-\beta \hat{H} - \mu_c \hat{\mathcal{C}}- \mu_k \hat{\mathcal{K}}- \mu_{\pi} \hat{\Pi} -\mu_j \hat{\mathbf{J}}^2},\tag{4}\] where \(Z\) is the partition function ensuring that \(\textrm{Tr}[\hat{\rho}]=1\), and \(\mu_c\), \(\mu_k\), \(\mu_{\pi}\), \(\mu_{j}\in\mathbb{R}\) are free parameters linked to the initial values of \(\langle \hat{\mathcal{C}} \rangle\), \(\langle \hat{\mathcal{K}} \rangle\), \(\langle \hat{\Pi} \rangle\), and \(\langle \hat{\mathbf{J}}^2 \rangle\). To study the dynamics, we start from an initial state \(\hat{\rho}_{i}\) of the form 4 with \(\mu_k = \mu_{\pi} = \mu_j=0\), \(\mu_c=100\) and \(\beta=5\), though our conclusions also hold for other values (see Supplemental Material [110]). The initial Hamiltonian, \(\hat{H}_{i}\), has parameters \(\lambda=0.5\) and \(h_{i}=0\). This choice gives rise to a polarized thermal state, with \(\langle \hat{J}_x \rangle < 0\). We then quench the initial state with a final Hamiltonian, \(\hat{H}_{f}\), with different \(h_{f}=0.1,0.15,0.2,0.3\) and \(\lambda=0.5\). The time-evolved density operator at time \(t\) is \(\hat{\rho}_{f}(t)=e^{-i\hat{H}_{f}t}\hat{\rho}_{i}e^{i\hat{H}_{f}t}\). Since \(\hat{\mathbf{J}}^{2}\) is conserved by Eq. 1 , the distribution \(P(j)\) of populated \(j\)-sectors remains unchanged in the wake of the quench. The dynamics will be dominated by \(j\)-sectors with large \(P(j)\).

Figure 2 illustrates the dynamical effects of these quenches. We focus first on the largest system size, \(N=1600\). For \(h_f=0.1\) and \(h_f=0.15\), the average quench energy is below the critical energy of the most-populated \(j\)-sector. We can see that \(\langle \hat{J}_x \rangle\) oscillates around a nonzero value (note that the larger the system, the longer the oscillating behavior remains), and \(\langle \hat{\mathcal{C}} \rangle\) is perfectly constant; therefore, the system is in the dynamical phase DPa. On the contrary, for \(h_f=0.2\) and \(h_f=0.3\) the average quench energy is above the critical one, and the dynamics is consistent with the dynamical phase DPb: both \(\langle \hat{J}_x \rangle\) and \(\langle \hat{\mathcal{C}} \rangle\) oscillate around zero. It is worth to remark that the critical quench separating regular and anomalous DPTs-II is given by \(h_f^c \approx 0.1776\) [10]. Therefore, our numerical results show that DPa leads to anomalous DPTs-II, and DPb to regular ones.

Notwithstanding, the picture is not so clear for smaller system sizes. For \(h_f=0.15\) and \(N=100\), \(200\), \(400\), and \(800\), \(\langle \hat{J}_x \rangle\) oscillates around a non-zero value, but \(\langle \hat{\mathcal{C}} \rangle\) is clearly not constant. To explain this behavior and to understand what is expected to occur in the thermodynamic limit (TL), we perform a finite-size scaling. Results are given in Table ¿tbl:table2?. We focus there on two quantities: the energy width, \(\sigma_{\varepsilon} = \sqrt{\langle \hat{H}^2 \rangle - \langle \hat{H} \rangle^2}\), and a range of critical energies obtained from the population of the different \(j\)-sectors (see caption for details). The key point is the overlap between these two intervals. For a non-empty overlap, we expect a mixture of DPa and DPb. For those sectors in which \(\langle \varepsilon \rangle_j < \varepsilon_c (j)\), \(\langle \hat{\mathcal{C}} \rangle_j\) is constant, and \(\langle \hat{J}_x \rangle_j\) oscillates around a nonzero value (\(\langle \bullet \rangle_j\) stands for the expectation value of an observable in the projection of the state onto the eigenspace in which \(\mathbf{\hat{J}}^2\) is equal to \(j(j+1)\)). And for those sectors in which \(\langle \varepsilon \rangle_j > \varepsilon_c (j)\), both \(\langle \hat{\mathcal{C}}\rangle_j\) and \(\langle \hat{J}_x \rangle_j\) oscillate around zero. Therefore, when putting together all the \(j\)-sectors, we obtain the intermediate picture observed for \(h_f=0.15\) and \(N=100\), \(200\), \(400\), and \(800\). On the contrary, if the intervals do not overlap, the dynamical phases are either DPa or DPb.

To extrapolate these result to the TL, we study how the energy width and the range of critical energies change with system size. A least-squares fit of the data shown in Table ¿tbl:table2? provides \(\sigma_{\varepsilon} \propto N^{-0.516(5)}\) and \(\left| \varepsilon_{c,\textrm{max}} - \varepsilon_{c, \textrm{min}} \right| \propto N^{-0.54(2)}\). This means that the only possible nonequilibrium dynamics in the TL is either DPa or DPb, and that an initial state gives rise to either one or the other depending on whether its average quench energy is below or above the critical energy of the ESQPT.

Discussion and outlook.—Through analytic arguments and numerical simulations, we have shown that ESQPTs and two major types of DPTs have a direct connection to each other in the LMG model. When the quench energy is below (above) the ESQPT critical points, the long-time steady-state falls in the ferromagnetic (paramagnetic) phase of DPT-I. Given that DPT-I and DPT-II have been shown to be directly connected to each other in the LMG model [9], [10], this means that ESQPTs are also directly connected to DPT-II.

Demonstrating such a direct connection between ESQPTs and DPTs provides evidence that varying concepts of criticality beyond that of the ground state are may be intimately related. This is promising in the pursuit of an overarching framework for far-from-equilibrium quantum many-body universality.

Our conclusions should be valid in other mean-field models where large enough system sizes are accessible in order to faithfully probe criticality. An interesting question is whether our findings also hold for nonintegrable models where access to the full spectrum is only possible for small system sizes that cannot reasonably discern criticality. This makes it hard to adequately study ESQPTs in such systems, although a direct connection between DPT-I and DPT-II is well-established in them [111].

Another interesting venue for future work entails connecting the critical exponents extracted from ESQPTs and DPTs. For example, it is known that DPT-I and DPT-II have seemingly disparate critical exponents, but since both DPTs have been shown to coincide [10], [111], it is likely that their critical exponents have a direct relation.

In this Supplemental Material we show the quench dynamics generated by initial states with inverse temperatures \(\beta\) and \(h_{f}\) different from those in the main text. The qualitative picture for \(\beta=10,7,4.5\) in Figs. 3, 4, and 5 is essentially the same as in Fig. 2 of the main text. As an exception to this behavior, in Fig. 6 we focus on \(\beta=3.5<\beta_{c}\), for which the time evolution of the relevant observables always oscillates around zero irrespective of \(h_{f}\). This is because for \(\beta<\beta_{c}\), the most populated \(j\)-sector is always below the critical \(j_{c}\), and therefore there is no ESQPT critical energy. As such, this does not contradict our conclusions, but rather reinforces them, because also for \(\beta<\beta_c\) there is no DPT for quenches in \(h\). Tables ¿tbl:tablebeta10?, ¿tbl:tablebeta7?, and ¿tbl:tablebeta4955? compare the average energy of the quenched state with the range of ESQPT critical energies associated to the 95% most populated \(j\)-sectors in the final Hamiltonian. Note that for the quench where \(\beta<\beta_c\), Fig. 6 has no corresponding table as there is no ESQPT critical energy.

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