Quantum Phase Transitions

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The lecture will give an introduction to the field from the perspective of condensed-matter physics. It will cover concrete model systems for quantum phase transitions, universal aspects of classical and quantum critical phenomena, theoretical methods such as quantum field theories and the renormalization group, as well as advanced topics including related aspects in high-energy physics. Relevant experimental observations will be discussed as well. The lecture is suitable for Master students and PhD students, as well as for Bachelor students who are familiar with second quantization.

Basic knowledge in many-particle theory Green's functions, mean-field theory, diagrammatics is helpful. Skip to main navigation. Skip to secondary navigation. Info This document is not available in the language you requested. It is therefore shown in German. Quantum phase transitions of bosons and fermions Description Quantum Phase Transitions have become an important research area of modern condensed matter physics.

Quantum phase transitions in highly crystalline two-dimensional superconductors.

If, however, the perturbation breaks the symmetry of the system, stability is not guaranteed, as has also been seen numerically for particular cases [ 41 , 81 ]. In this sense, the robustness of DQPTs seems to follow similar principles to conventional phase transitions for the examples in the literature. It is the goal of the following discussion to give a physical interpretation of DQPTs in terms of a dynamical analog to conventional quantum phase transitions QPT [ 55 ], see figure 3 for an illustration.

This interpretation aims to provide a general argument of how DQPTs can control the dynamics of other observables. Since the Loschmidt amplitude is a projection of the full time-evolved many-body wave function onto one single state in Hilbert space the initial state , and therefore only retrieves partial information from , one might wonder to what extent the overlap will be important for understanding the dynamics of the full wave function.

Overall, this amounts to the question of whether this single overlap represents a singular point or whether also other states in Hilbert space show similar properties such that in this sense the properties of the Loschmidt amplitude can spread out to larger portions of Hilbert space. In the following, we argue that an analogous picture can also emerge for DQPTs, and be made concrete and even measurable, see section 4. A schematic illustration of the analogy between dynamical quantum phase transitions and conventional quantum phase transitions.

The Loschmidt amplitude probes the ground-state manifold of the initial Hamiltonian energy density. While the nonanalytic behavior can disappear for excited energy densities , where local observables acquire their dominant contribution, there can still be an extended region white space controlled by the underlying dynamical critical point. As already anticipated before, the Loschmidt amplitude is a projection of the time-evolved state back onto the initial state , which is always chosen as the ground state of the initial Hamiltonian.

From this perspective, Loschmidt amplitudes probe the asymptotic low-energy properties of when measuring energies with the initial and not the final Hamiltonian [ 55 ]. This interpretation naturally leads to the general picture in figure 3. In this picture, the DQPT occurs along the line of vanishing energy density upon choosing the zero of energy accordingly at a critical time t c. Due to the quantum quench, however, energy is pumped into the system and the dominant contribution to local observables or correlation functions originates from a narrow shell in the vicinity of the mean energy density [ 55 ] beyond the ground-state manifold.

The central question, in the end, becomes whether there exists a dynamical analog to a quantum critical region controlled by the DQPT and whether crosses that region or not. For certain examples, this ascribed analogy to QPT can be made concrete [ 55 ] and even measured experimentally [ 38 ], as will be discussed in more detail in section 4. Whether, however, such a dynamical analog of a quantum critical region exists for any DQPT is not known and has to be checked on a case-by-case basis.

In the equilibrium case, QPTs cannot be observed directly in experiments because of the third law of thermodynamics. DQPTs also exhibit an analog to the third law in the sense that it is not possible to experimentally observe them without further theoretical input, as it was used in the recent trapped ion experiment [ 38 ]. This is because of the exponential suppression of the Loschmidt echo , for example, with the number of degrees of freedom N:.

Importantly, is independent of N in the thermodynamic limit, implying that it becomes exponentially challenging and therefore asymptotically impossible to measure experimentally. Since DQPTs only occur for , observing them in an experiment becomes exponentially hard. It has been observed in many cases that DQPTs share a close connection to the underlying equilibrium QPTs of the respective studied model. It therefore appears as a central question to ask how these two phase transition phenomena are related to each other.

For topological systems of noninteracting fermions the connection is by now particularly clear for two-band models [ 61 , 82 ], as will also be discussed in more detail in section 3. Whenever a topological phase transition is crossed by a quantum quench in 1D, a DQPT necessarily has to emerge.

In 2D, the situation is a bit more subtle and requires the absolute value of the Chern number of the underlying equilibrium ground states to change. A similar phenomenology has been observed for the XY chain in a transverse field [ 62 ], which is also mappable to a system of noninteracting fermions in 1D.

In addition, however, it was found that for this model that it is also possible for no DQPT to arise, even though the quench has crossed an underlying equilibrium QPT. This particular property can be traced back to a kinetic constraint, as also observed, for example, for the ferromagnetic XXZ chain [ 64 ]. This kinetic constraint is a symmetry due to particle or magnetization conservation, which does not allow the particle number or magnetization sectors to be dynamically adopted.

Without a coupling to a grand-canonical bath with particle-number exchange, the system is trapped therefore in a fixed sector which lifts, in general, the connection between the dynamical and equilibrium phase transitions. All these examples are related to systems which do not exhibit nonzero-temperature phase transitions, such that order only exists in the ground states of the respective models.

How order in excited states, which is relevant for the so-called excited state phase transitions [ 83 — 89 ], affects DQPTs is a much more intricate question, and the situation is much less clear at this point, mainly because there exist only a few studied models in the literature with such properties [ 36 , 56 , 58 — 60 , 63 ]. For quenches in the 2D transverse-field Ising model it has been found that DQPTs emerge with the same nonanalytic structures as the free energy at the equilibrium nonzero-temperature critical point of the classical 2D Ising model [ 36 ].

This suggests again a close connection of DQPTs to the underlying equilibrium phase transitions. In long-range interacting Ising models, on the other hand, it has been found that the various observed DQPTs are not related to the underlying equilibrium phase transition [ 56 , 58 — 60 ], i. While a connection between DQPTs and another nonequilibrium phase transition in the long-time steady state after the quench has been observed [ 56 , 58 ], evidence for DQPTs in a weak quench regime has also been found [ 58 — 60 ], and termed 'anomalous' accordingly [ 58 ].

These anomalous DQPTs also connect to an observation in other models, where it has been shown that DQPTs can even occur without crossing an underlying equilibrium transition [ 62 , 64 , 65 ]. Summarizing, in many cases a strong connection appears between DQPTs and the underlying equilibrium phase diagram especially in low dimensions. However, the results in the literature also show that a substantial number of counterexamples exist, which suggest that DQPTs constitute a genuine nonequilibrium phenomenon.

The theory of DQPTs in noninteracting topological systems, also termed dynamical topological quantum phase transitions, has reached a rather extensive understanding in recent years [ 30 , 32 — 35 , 61 , 65 , 82 , 90 ]. Interestingly, DQPTs in these models are strongly connected to the underlying equilibrium topological properties: Moreover, DQPTs in such topological systems can be characterized by dynamical order parameters [ 30 , 32 — 34 ], which are capable of distinguishing the two 'dynamical phases' separated by a DQPT.

As will be discussed in section 4. Overall, the quantum quenches in these topological systems provide an instructive example, offering both intuitive explanations and a straightforward mathematical understanding of the nature and occurrences of DQPTs. Many of the discussed formal properties—as long as they are not of topological origin—also directly extend to other fermionic systems or spin models that are mappable to fermionic ones [ 39 , 92 — 94 ], which are alternatively summarized in the anticipated recent review [ 39 ].

For the sake of simplicity we study DQPTs for two-band topological systems. For extensions to multiple bands we refer to [ 82 ]. Consider noninteracting fermions exhibiting translational invariance and particle—hole symmetry. Such systems exhibit a compact representation of the Hamiltonian.

Here, denotes a spinor which has different representations depending on the microscopic details of the studied model system. This spinor can be of the form for insulators with A and B referring, for example, to two sublattices, or the spinor can acquire the form for superconductors.

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The properties of the particular model are fully specified by the Hermitian matrices h k , which can be represented in terms of Pauli matrices , due to the particle—hole symmetry in the following form:. The Hamiltonian can be diagonalized for each momentum sector k separately yielding the two Bloch states and with energies and , respectively, with. Because the different momentum sectors are decoupled, ground states as well as other eigenstates exhibit a factorization property:.

Moreover, any nonequilibrium protocol which induces a time-dependence in without coupling of the momentum sectors preserves this property, yielding. Accordingly, let us denote the corresponding Bloch states by and the energies via. Based on the considerations of the previous section, it is straightforward to study DQPTs in the Loschmidt amplitude. Due to the factorizing property of the quantum many-body state in equation 24 , also factorizes.

Expanding in the Bloch states of the final Hamiltonian one obtains. As discussed in section 2. Because of equation 25 , a zero in is equivalent to finding at least one critical momentum and one critical time t c where. According to equation 27 , such a zero can occur whenever there is a mode with [ 29 ].

Using this insight, the question remains under which conditions is possible. Formally, it means that the two-level system at is maximally mixed, i. In general, the occurrence of a critical momentum depends on the details of the studied problem. However, in many systems the existence of a is ensured whenever the system is quenched across an underlying equilibrium quantum critical point, as discussed in the following. Why the crossing of an underlying equilibrium quantum phase transition can lead to the appearance of DQPTs can be seen most directly by invoking general Landau—Zener arguments [ 95 ].

For this purpose, let us first consider the more general scenario of a parameter ramp for our general Hamiltonian , which will be used afterwards to argue about the quantum quench case. Let us focus on the situation where the ramp crosses an underlying equilibrium quantum critical point with a gap closing. Since all momenta k are decoupled from each other, we can study the parameter ramp problem for each k separately. For each k , we are dealing with a two-level system such that we can define a momentum-dependent gap. Starting with a slow ramp, the threshold for the breakdown of adiabaticity, and therefore the excitement of the upper Bloch band, is [ 96 ].

For the momentum k 0 exhibiting the gap closing, this threshold is crossed with certainty and the excitation probability in the higher of the two levels approaches unity, i. On the other hand, in most cases there are modes k that are only weakly excited with. By using continuity, there then has to be at least one momentum for which accordingly, which is the required condition for the presence of DQPTs. It appears that making the ramp faster and therefore exciting the system more strongly does not typically change the final occupation of the k 0 mode, which has already reached its maximum value.

What can happen is a shift of the critical mode which, however, only modifies the time scales of the appearance of the DQPTs—see equation 29 —but not the principle occurrence. Consequently, a gap closing, i. This argument might not be applicable to cases where all k modes are strongly excited with , which occurs, for example, when inverting the complete band structure. In most generic situations, however, this scenario does not occur and occupation inversion happens only for a subset of modes.

Generally, any state living in a single momentum sector k can be expanded in the eigenbasis of a set of Bloch states according to:. Consequently, admits a representation as a point on the Bloch sphere with denoting the polar and denoting the azimuthal angle. For an illustration, see figure 4. The relative Bloch sphere representation of a single-momentum state fully specified by the azimuthal and polar angle.

The north south pole corresponds to the lower upper band of the initial Hamiltonian. Real-time evolution describes a trajectory indicated by the orange line with time-dependent angles. Such a representation is not only valid in equilibrium but also within dynamical processes. When fixing some particular basis set we can write:.

This particular representation of the state is referred to as the 'relative Bloch sphere' [ 30 ]. This means that is located at the south pole. We have seen that for DQPTs to occur it is sufficient for at least one mode to reach the south pole on the relative Bloch sphere. The considerations from section 3. It is the goal of the following to summarize the rigorous results on the occurrence of DQPTs in topological systems. On general grounds it has been shown in [ 61 ] under which conditions the equilibrium ground state topology necessarily imposes the existence of DQPTs.

In 1D the situation is particularly clear. For any quantum quench between two topologically different Hamiltonians, as characterized by their winding number, there exists at least one critical momentum. In 2D there is a richer phenomenology. There, it can be shown that it is not sufficient to change the ground state topology, as measured by the Chern number, in order to be guaranteed to get DQPTs. It is rather relevant whether the absolute value of the Chern number differs for the two Hamiltonians appearing in the quench; only in these cases do DQPTs necessarily appear.

Thus, for quenches in 1D between Hamiltonians with different equilibrium topological properties, or in 2D with a different absolute value of the Chern number, DQPTs are robust and are therefore termed 'topological' or 'symmetry-protected' [ 82 ]. For quenches not falling into these classes, DQPTs can still occur [ 30 , 61 , 82 , 90 ]. In these cases they, however, require a fine-tuning of the Hamiltonian. These DQPTs have acquired the notion of 'accidental'.

DQPTs in topological systems come with interesting structures in the dynamics of certain phase profiles. These include the azimuthal angle of the relative Bloch sphere, see equation 32 , and the so-called Pancharatnam geometric phase [ 30 ]. The phase profile of the azimuthal angle for a 2D system has been measured experimentally, as summarized in more detail in section 4.

It is straightforward to see on general grounds why DQPTs have a strong impact on the azimuthal angle. Due to the unitarity of time evolution, the dynamics for each momentum on the Bloch sphere describes a smooth trajectory. In figure 5 b this phase profile is shown for a quantum quench in the Haldane model [ 97 ], which in the context of equation 22 exhibits the following representation:.

Illustration of the vortex dynamics for a quantum quench in the Haldane model [ 97 ]. One of the pairs we track by enclosing it in orange circles. Here, m denotes an energy offset between two A and B sublattices on the considered honeycomb lattice, see figure 5 a for the respective real-space structure. V is the nearest-neighbor hopping amplitude and is the complex hopping amplitude within the same sublattice.

For the definition of the lattice vectors a j and b j , connecting the nearest and next-to-nearest neighbor lattice sites we refer to [ 97 ]. As one can see, there appears a critical time where suddenly pairs of vortices are created which start to move through the Brillouin zone.

An alternative phase profile to characterize DQPTs in topological systems makes use of the concept of the Pancharatnam geometric phase [ 98 , 99 ], which extends the notion of Berry's phase [ , ] to general unitary evolution with nonorthogonal initial and final states. Importantly, this phase is naturally contained in the Loschmidt amplitude.

Let us introduce a polar decomposition of at a given momentum k:. The phase contains a geometric part —the Pancharatnam geometric phase—by subtracting a dynamical contribution. Interestingly, from the dynamics of this winding number it is, in principle, possible to distinguish accidental from symmetry-protected DQPTs [ 30 ]. The definition of these dynamically created or annihilated vortices for the Pancharatnam geometric phase can also be generalized to the case of mixed states [ 33 , 35 ], see also section 6. Recently, DQPTs have been observed in two experiments performed on quantum simulators [ 32 , 38 ], which are summarized in the following.

We do not attempt to discuss the experimental details, for which we refer to the respective articles, but rather focus on the main findings and implications. While these two experiments have observed DQPTs with tailored methods, on a general level, a protocol has recently been introduced which allows Loschmidt amplitudes to be accessed in systems of ultracold atoms [ , ]. Moreover, in systems where the complete quantum state can be reconstructed with full state tomography, such as in trapped ions or superconducting qubits, the Loschmidt amplitudes are accessible directly.

Systems of trapped ions can synthesize the dynamics of transverse-field Ising models of the form [ 9 , 38 , — ]. Here, with denote the Pauli operators on the lattice site , where N is the total number of spins. The coupling J lm is approximately of long-ranged form [ ]. In the trapped ion experiment on DQPTs [ 38 ] a quantum quench across the ferromagnetic to paramagnetic equilibrium phase transition has been realized—a situation where generically DQPTs are expected.

Initially, the system is prepared in the fully polarized state. After this stage of preparation, the subsequent dynamics of the system is driven by H in a transverse field h sufficiently large such that in equilibrium the system would reside in a paramagnetic phase. This property has important implications for the rate function of the full. In particular, for is always dominated by one of the two contributions or such that [ 38 , 55 , 56 ]. This is the central tool for predicting DQPTs from the experiment where the , , can be measured individually.

Let us define a function , which for a finite system is different from but converges to the same in the thermodynamic limit. And let us suppose that one can reach system sizes where the individual can be considered as converging with negligible finite-size corrections. Then, we obtain that and the finite-size data can already be used to predict the behavior in the thermodynamic limit. Of course, this procedure implies an additional theoretical input to the experiment. The measured data for the individual rate functions is shown in figures 6 a and b.

Quantum Phase Transitions

As one can see, the have already almost converged, at least for times within the first half of the shown data. Consequently, one can use the to construct , which due to almost converging with N is equivalent to , so that in the following we do not have to distinguish from.

Since there appear points in time where the two intersect, develops a kink in the thermodynamic limit. With this theoretical input, the experimental data implies nonanalytic real-time dynamics in particular. Without this theoretical input of the minimum construction, one could have studied instead. In contrast to , is a smooth function for a finite system, which, however, becomes sharper at a critical time for increasing the system size, eventually leading to the nonanalytic behavior of in the thermodynamic limit, as also discussed in [ 38 ].

Dynamical quantum phase transitions in the trapped ion experiment [ 38 ]. The colored data points show , obtained by taking its dominant contribution , whereas the grey data points refer to the respective subleading ones. This sharp feature fades out to , eventually crossing the mean energy density , included as the red line, where local observables attain their dominant contribution.

Reprinted figure with permission from [ 38 ], Copyright by the American Physical Society. In the experiment, not only have DQPTs been observed but the relation to other observables has also been systematically studied. This includes, in particular, a quantitative approach to the analogy between DQPTs and conventional equilibrium QPTs, discussed on general grounds in section 2. Due to the measurement capabilities in trapped ions it is also possible to experimentally access this quantity, which is shown in figure 6 c.

As one can clearly see, there appears to be a temporal analog of the quantum critical region in the energy- density—time plane, which is controlled by the DQPT occurring at upon choosing the zero of energy accordingly at a critical time t c. Moreover, the experiment has studied entanglement production and observed a close connection, see also section 6. The ultra-cold atom experiment on DQPTs [ 32 ] has observed dynamical topological quantum phase transitions.

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While the general theory of such DQPTs has already been discussed in section 3 , it is the goal of the present section to outline and discuss the experimental aspects. This experiment synthesized a quantum quench in a system of noninteracting fermionic degrees of freedom on a hexagonal lattice, see figure 7 for an illustration. For the initial preparation, a large energetic offset between two sublattices A and B was imposed, such that the system realizes a simple insulating ground state of a two-band system at half filling to a very good approximation with the particles localized on the lattice sites of the B sublattice.

An illustration of the setup used in the observation of dynamical quantum phase transitions in ultra-cold atom experiments [ 32 ]. Reprinted figure with permission from [ 32 ], Copyright by the American Physical Society. As discussed already in section 3 , the dynamics in such a topological system can be decomposed into contributions from all the crystal momenta k of the Brillouin zone. Moreover, for each momentum k the wave function for such a two-band model is reduced to an effective two-level system admitting a representation on the Bloch sphere with two associated angles: Using full-state tomography techniques for two-band noninteracting fermionic systems [ , ], the experiment obtained access to both of these angles.

Of particular interest in the context of DQPTs is the azimuthal , whose dynamics has already been anticipated in section 3. Monitoring the dynamics of in the Brillouin zone one can observe that there appear points in time where pairs of vortices are created or annihilated. Importantly, such a sudden creation or annihilation of vortex-antivortex pairs is in one-to-one correspondence with an underlying DQPT independent of the model details [ 32 ], meaning that the associated number of dynamical vortices can change if and only if the system experiences a DQPT—as long as it can be considered as noninteracting.

Consequently, tracking this vortex number over time is equivalent to tracking DQPTs. This is of particular importance since the experiment only provides access to a discrete set of points in time, as it is realized as a Floquet system and is therefore only monitored stroboscopically. While from the full-state tomography, in principle, the Loschmidt echo rate function can also be reconstructed [ 32 ], it is not possible to uniquely identify nonanalytic real-time structures from a fixed time grid.

In this context the dynamical vortex number is appealing due to its quantized nature, in that a change in this number can only happen nonanalytically. Even further, the vortex number is not only to be viewed as a way of detecting DQPTs but can also be interpreted as a dynamical order parameter for the DQPTs in this model [ 32 ].

The nonanalyticities at the DQPTs and the formal similarity of Loschmidt amplitudes to equilibrium partition functions suggest a close connection between DQPTs and conventional phase transitions. Equilibrium transitions, however, entail many further key properties beyond the mere nonanalytic character of thermodynamic potentials.

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It is one of the major challenges in the theory of DQPTs to identify the proper extensions of such characteristic principles to the far-from-equilibrium regime. It is the main purpose of this section to summarize and discuss results on the theory of DQPTs that address such fundamental questions. It is important to emphasize, however, that the current understanding rather represents a first step towards a comprehensive theory for nonequilibrium phase transitions. The summarized results are supposed to be seen as a starting point for further investigations towards this major goal.

Let us start by discussing to what extent the concepts of scaling and universality, which in equilibrium are caused by a divergent correlation length, can be applied to DQPTs. While a general understanding has not yet been reached, for the Ising model these concepts can be extended to the dynamical regime [ 36 ].

It is the goal of the following section to summarize the main idea and to discuss the implications. Here, denotes a summation over the nearest-neighboring lattice sites. At the moment, let us not be restricted to a particular dimension or graph. While this quench is very specific, it represents a fruitful starting point for approaching the problem on a general level. Within this nonequilibrium scenario, the system is initialized in the fully polarized state along transverse-field direction:.

It is the central observation that in this case the Loschmidt amplitude can be mapped onto a conventional partition function. The only difference with the equilibrium case is that the coupling appearing in is complex. The key property of the particular considered quantum quench is that the boundary conditions can be fully absorbed into the bulk.

The equivalence between and can be seen straightforwardly when recognizing that the initial state is an equally weighted superposition of all spin configurations. Because the Hamiltonian H governing the time evolution is diagonal in the spin basis, i. The major advantage of this construction is that results and strategies known for the equilibrium case can now be applied to Loschmidt amplitudes and thus DQPTs. This is particularly interesting in the 1D and 2D cases as will be discussed now.

Let us start with the 1D Ising chain, where it is possible to construct an exact renormalization group RG transformation allowing for the identification of the exact RG fixed points. Specifically, it is possible to apply conventional decimation RG procedures [ ] to the complex partition function of interest here [ 36 ]. Eliminating every second lattice site, one obtains the following exact recursion relation for the change of the couplings within one RG step:.

This leads to the immediate question of whether the extension of the coupling K into the complex plane can lead to new fixed points. Which fixed points are the DQPTs associated with? Taking the critical coupling , where the system exhibits a DQPT, and applying the RG recursion relation in equation 49 one obtains that maps into the unstable zero-temperature fixed point of the equilibrium Ising model.

This directly implies that these DQPTs obey scaling and universality. As a consequence, one can immediately obtain the universal scaling form of the singular contribution g s t to the dynamical free energy density as: The temporal kink in obtained by the scaling analysis matches precisely the result from the full exact solution, see figure 8.

DQPTs in the Loschmidt echo rate function for quantum quenches in the transverse-field Ising chain for varying final fields h , starting from initial fully polarized states in the transverse field direction [ 36 ]. At the critical times, exhibits a kink as predicted by scaling theory, see equation Reprinted figure with permission from [ 36 ], Copyright by the American Physical Society.

What one can gain from the relation between the DQPT and an unstable fixed point, for example, is that it is now straightforward to systematically study the influence of perturbations on the model. In particular, perturbations that are irrelevant in the RG sense leave the fixed point unchanged, which also implies a certain robustness of DQPTs.

For example, adding a next-to-nearest neighbor spin coupling is always irrelevant under the decimation RG, which holds independently of the associated coupling strength [ ].

Moreover, one can also start studying the influence of a transverse field in the final Hamiltonian. Within a perturbative treatment this leads to an effective classical description with an effective Ising model including weak irrelevant next-to-nearest neighbor interactions [ 36 ]. Interestingly, there is the possibility that the relevance of a perturbation, although appearing potentially with a weak coupling in the Hamiltonian, might depend on time, because the effective couplings appearing in the decimation RG implicitly exhibit a time-dependence.

Let us continue by studying the 2D Ising model on a square lattice, where the identification of the Loschmidt amplitude with a classical partition function is again possible. Although in this case no exact RG transformation can be formulated, the dynamics for the Loschmidt amplitude can still be accessed, extending the solution for the partition function of the 2D Ising model [ — ] to complex couplings [ 36 ].

One finds that this system also exhibits a DQPT. The singular contribution g s t to the Loschmidt amplitude rate function displays nonanalytic behavior according to:. Remarkably, this matches precisely the critical behavior of the free energy density at the thermal critical point of the 2D Ising model, suggesting that scaling and universality also hold for that case. Currently, no other examples are known for which scaling and universality at DQPTs have been established, except the discussed models.

Overall, the DQPTs discussed in this section appear to exhibit scaling which is associated not with the underlying quantum equilibrium phase transition, as also observed for a 1D quantum Potts chain [ 37 ], but rather to the classical one. Investigating to what extent universality and scaling generalize to the other DQPTs, and whether genuine nonequilibrium fixed points can also appear, which are not related to the equilibrium criticality, is a pertinent task for future work on the theory of DQPTs. The robustness of DQPTs has been studied for many models [ 36 , 41 , 42 , 79 , 80 ].

For DQPTs exhibiting scaling and universality as discussed in the previous section, robustness against a large class of perturbations is guaranteed. Whenever the perturbation is weak in the sense of the constructed RG, the structure of the nonanalytic behavior is unchanged, while only nonuniversal aspects such as the critical time of the DQPT might be shifted. An example of such an irrelevant RG perturbation has been provided in [ 36 ]. Upon adding a transverse field to the Ising Hamiltonian discussed in the previous section, the Loschmidt amplitude becomes equivalent to a classical Ising model, including next-to-nearest neighbor interactions.

These longer-ranged couplings are irrelevant and vanish under RG transformation such that the fixed point Hamiltonian is described again by an Ising model, without a transverse field at renormalized couplings, however. Consequently, the DQPTs are robust in the sense that the nonanalytic structure does not change and the sole influence of the transverse field is to shift the critical time. As a further supporting argument for the robustness of DQPTs under symmetry-preserving perturbations, one can resort to the formal similarity of the Loschmidt amplitudes to complex partition functions, which has already been discussed in section 2.

As for equilibrium transitions, DQPTs are linked in a one-to-one correspondence with complex partition function zeros. For equilibrium partition functions it is well known that the structures formed by these zeros in the complex plane are typically robust under weak symmetry-preserving perturbations, which is a different way of seeing the stability of phase transitions without resorting to an RG analysis. These structures might deform but do not immediately melt under the addition of a weak perturbation, which is equivalent to changing the critical value of the control parameter but retaining the structure of the transition.

Assuming that complex partition function zeros also show the same properties in the whole complex plane, DQPTs are then also expected to be robust. This, however, should not been seen as a proof, but rather as a general physical argument. For concrete models, the stability of the DQPTs has been studied both using numerical and analytical approaches [ 36 , 41 , 42 , 79 , 80 ].

Specifically, in these works variants of the transverse-field Ising chain in 1D have been investigated under the inclusion of different perturbations. The robustness under adding a symmetry-preserving next-to-nearest neighbor interaction to the model has been shown on the basis of both numerical simulations using the time-dependent density-matrix renormalization group approach [ 80 ], as well an analytical approach using the flow equation method [ 79 ].

The influence of a magnetic field in the ordering direction of the Ising chain, which constitutes a symmetry-breaking perturbation, can lead to a smearing of the DQPT for a parameter sweep [ 41 ].

Adding such a perturbation to a quantum quench, however, it may happen that the DQPTs still exist [ 42 , 80 ]. Understanding the difference between slow and fast perturbations in this context remains an open question [ 80 ]. All the summarized examples study the robustness of DQPTs on short to intermediate time scales.

A different question is how the DQPTs are influenced in the long-time limit, where it is known from the context of quantum thermalization [ 96 ] that an already vanishingly weak perturbation can have a strong impact on the dynamics. Specifically, an integrable model can be turned into an ergodic one, which implies a drastic change in the asymptotic long-time steady state from a nonthermal to a thermal one. In this light, the robustness of DQPTs might depend on time. This question, however, has not yet been studied, although it might provide an interesting connection with the field of quantum thermalization.

Let us, however, emphasize that DQPTs do not rely on integrability. For example, DQPTs have also been found for genuinely interacting models, such as the Hubbard model, in high dimensions [ 63 ], which is known to exhibit quantum thermalization [ 15 ]. Order parameters are central to the characterization of phase transitions in equilibrium. Therefore, it is important to ask whether this concept can be extended to the considered dynamical regime.

Beyond providing a further element putting DQPTs on comparable footing with equilibrium transitions, dynamical order parameters might help in the understanding of the respective DQPT by, for example, identifying the nature of the two 'dynamical phases' separated by the DQPT. Dynamical order parameters have been formulated and identified for DQPTs in 1D and 2D topological systems [ 30 , 32 — 35 ]. Most notably, the 2D case has also been measured experimentally recently [ 32 ].

All the proposed dynamical topological order parameters share the same property in that they assign quantum numbers to the phase profiles discussed in section 3. Importantly, these quantized integers necessarily jump at DQPTs. For 1D systems [ 30 , 31 ], or along closed 1D paths in the 2D Brillouin zone [ 34 ], these topological order parameters are winding numbers for the Pancharatnam geometric phase as discussed already in section 3.

This winding number is capable of providing insights into the underlying ground state topology of the respective quantum many-body system, although during the nonequilibrium process the system is by no means close to its ground state. Specifically, these winding numbers can, in principle, distinguish topologically protected DQPTs from accidental ones and can thus be used to detect whether a topological quantum phase transition has been crossed with a change in the underlying equilibrium topology [ 30 ].

In 2D, another dynamical order parameter can be constructed by measuring the number of vortices created in phase profiles across the whole Brillouin zone [ 32 , 35 ]. Here, it is possible to choose either the Pancharatnam geometric phase [ 35 ] or the azimuthal phase [ 32 ] of the associated relative Bloch sphere, see section 3.

Notice that the dynamical topological order parameter using the Pancharatnam geometric phase can also be generalized for mixed initial states [ 35 ], as will be discussed in more detail in section 6. In what sense DQPTs can also be characterized by local dynamical order parameters is not yet known. Certainly, it cannot be associated with long-ranged correlations in the conventional sense, because causality in terms of Lieb—Robinson bounds [ — ] prevents the buildup of such long-ranged correlations within a finite time, which is where DQPTs occur.

This, however, does not rule out alternative notions of a divergent correlation length, which one can imagine within the interpretation of DQPTs as the dynamical analogs of equilibrium quantum phase transitions, see section 2. Equilibrium phase transitions are not only manifested in a nonanalytic free energy, but also in other observables such as the order parameter, susceptibilities, or entanglement to name just a few. In particular, it is possible to directly infer measurable quantities such as the specific heat from the free energy by taking derivatives.

At this point a difference between DQPTs and conventional transitions becomes apparent. It is not possible to obtain other measurable quantities from Loschmidt amplitudes in a similar way. How can DQPTs then be related to other quantities? Within this argument, the DQPT is interpreted as a dynamical counterpart to a quantum phase transition in equilibrium. A DQPT can then control the dynamics of observables whenever there exists a dynamical complement to the quantum critical region, which for specific systems has been established both theoretically [ 55 ] as well as experimentally [ 38 ].

This perspective of DQPTs has been successful in explaining the observed signatures in certain observables of systems with symmetry-broken phases [ 29 , 38 , 55 — 57 ]. Since we are dealing with situations where ground states are not unique, it is important to specify the generalization of the Loschmidt amplitude, which is not unique either, see the discussion in section 2.

In the subsequent discussion, we follow the works in [ 29 , 38 , 55 — 57 ] and define the generalization as in equation If alternatively, one would still choose the expression according to equation 6 , DQPTs can also occur [ 26 , 58 — 60 ]. For this choice the connection to other observables, however, is not known, which represents an interesting aspect to study in the future.

Let us now consider a system which is prepared initially in a symmetry-broken ground state with a nonzero value of the order parameter. Monitoring the dynamics after a quantum quench in such systems, it has been found on a rather general level that a DQPT is typically accompanied by a zero of the order parameter and therefore a periodic sequence of DQPTs with an oscillatory decay of the order parameter [ 29 , 36 , 56 , 57 ].

In a quench in the transverse-field Ising chain, for example, the frequency of the oscillatory decay in the longitudinal magnetization matches exactly the periodicity of the DQPTs in the model independent of the details of the quench, see figure 9 for the data shown in [ 29 ]. Importantly, the associated time scale is an emergent nonequilibrium time scale without an equilibrium counterpart [ 29 , ], which within present knowledge only appears in the DQPTs and the anticipated order parameter dynamics, providing strong evidence for a connection between the two quantities.

The underlying mechanism for this connection follows a reasoning already discussed in section 4. For the Ising model, the full return probability to the ground state manifold as defined in equation 11 is given by. As for the Loschmidt echo, the individual probabilities also exhibit a large-deviation scaling, where the rate functions are intensive, independent of the number of degrees of freedom N in the thermodynamic limit.

Consequently, when the two rate functions cross a DQPT occurs in in the form of a kink, as discussed in section 4. Let us emphasize that this kink is not the result of an artificial construction, but rather carries a physical meaning which allows the DQPT to be connected to other observables. At the crossing point of the DQPT, the symmetry in the ground state probability , which was initially broken explicitly by the initial condition, is restored. It is generically found that this symmetry restoration is not just restricted to the ground state manifold, but rather extends to nonzero energy densities, see for example figure 6.

As a consequence, the symmetry is also restored for observables, which implies a vanishing value for the order parameter. While in this discussion we have restricted ourselves to a discrete broken symmetry, the generalization to continuous symmetries is straightforward, yielding a similar connection between the order parameter dynamics and DQPTs [ 57 ]. The decay of the longitudinal magnetization in the transverse-field Ising chain for various initial g i and final g f fields [ 29 ]. When rescaling the time axis by , which is the time of the first DQPT in the dynamics for a given parameter set, at a constant offset , one can identify the periodicity of the oscillatory decay with the periodicity at which DQPTs appear.

Reprinted figure with permission from [ 29 ], Copyright by the American Physical Society. An analogous relation between the order parameter dynamics and DQPTs has been found for certain 1D topological systems with phases characterized by string order parameters [ 30 ]. For some models these observations can be traced back to the previously discussed case of symmetry-broken phases. The Kitaev chain for particular parameter sets, for example, is equivalent to a transverse-field Ising model and the string order parameter maps onto the respective conventional order parameter correlations.

Importantly, however, the ground states of the Kitaev chain and transverse-field Ising model are different [ ], implying that the full ground-state return probability reduces to a conventional Loschmidt amplitude [ 29 ].

Phys. Rev. B 97, () - Quantum phase transition with dissipative frustration

Despite these subtleties, the main phenomenology has been found to be unchanged [ 30 ]. When preparing the system initially in a topological phase characterized by a nonzero string order parameter, the dynamics of the string order parameter after a quantum quench is linked to the DQPTs in the model. Whenever the system experiences DQPTs, which due to the integrability of the model appear periodically see equation Again, the time scale associated with these oscillations is an emergent nonequilibrium scale without an equilibrium counterpart and coincides with the periodicity of DQPTs.

The discussed periodic appearance of DQPTs strongly relies on the integrable nature of the considered models. It is therefore natural to ask how integrability-breaking perturbations might influence both DQPTs and order parameter dynamics. For strong integrability-breaking perturbations it has been found that the connection between DQPTs and order parameter dynamics can become more subtle [ 80 ], since additional DQPTs can appear at longer times which do not become manifest in the order parameter.

A further aspect in the context of the influence of weak perturbations potentially inducing quantum thermalization concerns their behavior over long time scales, as discussed already in section 5. This is not known yet, but represents an interesting and important aspect which would be worthwhile studying in the future. The previous examples summarize the relation between DQPTs and other observables for quantum quenches out of a symmetry-broken or topological phase.

For quenches in the other direction, meaning from a symmetric phase to a parameter set, where the Hamiltonian exhibits symmetry breaking in the ground state, other observables have been found which connect to the DQPTs observed in these models. Since the initial state is symmetric and the Hamiltonian by definition conserves the symmetry, the order parameter has to vanish throughout the dynamics. However, it has been observed that the respective order parameter correlations can exhibit the signatures of the underlying DQPTs for transverse-field Ising models [ 36 , ].

For DQPTs in the transverse-field Ising model for such quenches, it has been found that those in the 1D 2D system are in the same universality class as the critical points of the classical 1D 2D Ising chain. Let us point out that, in general, identifying observables whose dynamics is sensitive to an underlying DQPT can be difficult, because promising candidates might not show the apparent signatures.

One example in this direction has been provided in a system of interacting bosons exhibiting a superfluid to Mott insulator transition in the ground state. For a quantum quench from an initial superfluid state to large interactions, it has been found that DQPTs cannot be identified in the dynamics of the momentum distribution [ ], whose zero momentum peak can be taken as an order parameter for the equilibrium superfluid phase.

A further class of observables that appear to connect to DQPTs are entanglement quantifiers, such as the entanglement entropy [ 38 , ] or spin squeezing parameters [ 38 ], which are also discussed in more detail in section 6. Since the Loschmidt amplitude is the Fourier transform of the energy distribution function [ 49 ], the signatures of the DQPTs have also been identified in the energy and work statistics [ 29 , 51 , , ].

In equilibrium, phase transitions are grouped mainly into two different categories: Continuous phase transitions are further distinguished in terms of universality classes, entailing all those critical points sharing the same set of critical exponents. In view of the results summarized in section 5.

While there exist DQPTs which can be classified in the conventional sense because of their connections to equilibrium criticality [ 36 ], a general classification scheme for them is not known.

In particular, it is likely that there can be new nonequilibrium fixed points for DQPTs that are not accessible within equilibrium dynamics. In the infinite-range transverse-field Ising model, for example, it has been found that the DQPT is related neither to the underlying equilibrium quantum nor the thermal phase transitions [ 56 , 58 , 60 ]. In this sense DQPTs represent a critical phenomenon that is distinct, in general, from equilibrium phase transitions.

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