sign in sign up
<<
Home , Quantum phase transition, Wehrl entropy

Wehrl production rate across a dynamical quantum transition

B. O. Goes,1 G. T. Landi,1 E. Solano,2, 3, 4, 5 M. Sanz,2 and L. C. Céleri6, 2 1Instituto de Física, Universidade de São Paulo, CEP 05314-970, São Paulo, São Paulo, Brazil 2Department of Physical Chemistry, University of the Basque Country UPV/EHU, Apartado 644, 48080 Bilbao, Spain 3IKERBASQUE, Basque Foundation for Science, Maria Diaz de Haro 3, 48013 Bilbao, Spain 4International Center of Quantum Artificial Intelligence for Science and Technology (QuArtist) and Department of , Shanghai University, 200444 Shanghai, China 5IQM, Munich, Germany 6Institute of Physics, Federal University of Goiás, POBOX 131, 74001-970, Goiânia, Brazil The quench dynamics of many-body quantum systems may exhibit non-analyticities in the Loschmidt echo, a phenomenon known as dynamical (DPT). Despite considerable research into the underlying mechanisms behind this phenomenon, several open questions still remain. Motivated by this, we put forth a detailed study of DPTs from the perspective of quantum and entropy production, a key concept in . We focus on the Lipkin-Meshkov-Glick model and use coherent states to construct the corresponding Husimi-Q quasi-probability distribution. The entropy of the Q-function, known as Wehrl entropy, provides a measure of the coarse-grained dynamics of the system and, therefore, evolves non-trivially even for closed systems. We show that critical quenches lead to a quasi-monotonic growth of the Wehrl entropy in time, combined with small oscillations. The former reflects the information scrambling characteristic of these transitions and serves as a measure of entropy production. On the other hand, the small oscillations imply negative entropy production rates and, therefore, signal the recurrences of the Loschmidt echo. Finally, we also study a Gaussification of the model based on a modified Holstein-Primakoff approximation. This allows us to identify the relative contribution of the low sector to the emergence of DPTs. The results presented in this article are relevant not only from the dynamical quantum phase transition perspective, but also for the field of quantum thermodynamics, since they point out that the Wehrl entropy can be used as a viable measure of entropy production.

I. INTRODUCTION Hamiltonian H. The Loschmidt echo is defined as

2 −iHt 2 L(t) = |hψ0|ψti| = |hψ0|e |ψ0i| , (1) The dynamics of closed quantum many-body systems has and therefore, quantifies the overlap between the initial state been the subject of considerable interest in the last decade. and the evolved state at any given time. In other words, it mea- After a sudden quench, for instance, the support of a local op- sures how the support of the wavefunction spreads through the erator will in general spread through all of Hilbert space. The many-body Hilbert space. precise way through which this takes place reveals important In quantum critical systems, the Loschmidt echo in Eq. (1) information about the basic mechanisms underlying the dy- is characterized by sharp recurrences (see Fig.1(a) for an ex- namics. A particularly interesting example is the so-called ample). The nature of these recurrences is more clearly ex- dynamical phase transition (DPT), first discovered in Ref. [1], pressed in terms of the rate function, and subsequently explored in distinct situations. In Ref. [2] it was shown that DPTs have no connection with standard equi- 1 r(t) = − log L(t), (2) librium phase transitions. The case of long-range interacting N system was considered in Ref. [3], while Ref. [4] treated the case of nonintegrable systems. An attempt to classify dynam- where N is the system size. In the thermodynamic limit ical phase transitions (DPTs), leading to the definition of first- (N → ∞), the rate r(t) presents non-analyticities (kinks) at order DPT, was introduced in Ref. [5]. Finally, we can men- certain instants of time (c.f. Fig.1(b)), which are the hall- arXiv:2004.01126v1 [quant-ph] 2 Apr 2020 tion a very interesting connection between these DPTs and mark of DPTs. The Loschmidt echo in Eq. (1) shows a formal quantum speed limits, developed in Ref. [6]. On the experi- relation with a thermal partition function at imaginary time, mental side, we can mention the observation of DPTs in the which allows one to link these non-analyticities to the Lee- Ising model in an optical lattice [7], in the topological phase Yang/Fisher [13, 14] zeros of L(t) (see Ref. [10] for more de- in the Haldane model using an ion trap platform [8] and in tails). a simulation of the Ising model in a quantum computer [9]. Despite considerable progress in our understanding of We refer the reader to the recent reviews [10–12] for further DPTs, several open questions still remain, for instance, re- details about this field. garding their universality and if they can be captured from macroscopic properties [10–12]. In this article we focus on The central quantity in the theory of DPTs, which usually two deeply related questions. The first one concerns the tran- occur in a quenched quantum system, is the Loschmidt echo. sition from quantum to classical, i.e. which aspects of the The basic scenario consists of initially preparing a system in problem are genuinely coherent and which could be under- the ground-state |ψ0i of some Hamiltonian H0. At t = 0 the stood from purely classical equations of motion. Defining this system is then quenched to evolve according to a different transition is not trivial for the great majority of models, such 2 as spin chains. The second issue concerns which sectors of thermodynamic limit. This allows us to address which parts the Hamiltonian contribute to the transition. Indeed, quan- of the spectrum are essential for the description of DPTs. As tum phase transitions depend only on the low energy sector we show, this procedure yields accurate predictions for small (ground-state plus the first few excited states). DPTs, on the quenches, but fails for the strong quenches required to observe other hand, should in principle depend on the entire spectrum. DPTs. The interpretation of L(t) as a measure of how the support of the system spreads in time brings a clear thermodynamic flavor to DPTs. In the language of classical thermodynam- II. DYNAMICAL PHASE TRANSITION IN THE ics, an expanding gas fills all of available space, causing the LIPKIN-MESHKOV-GLICK MODEL entropy to grow monotonically in time. In closed quantum systems, however, the is a constant of Let us consider the Lipkin-Meshkov-Glick (LMG) motion, despite possible information scrambling. This idea model [17–19], described by the Hamiltonian has been explored with certain detail in the context of fluctua- 1 tion theorems and non-equilibrium lag [15]. H = −hJ − γ J2, (3) z 2 j x x One way to reconcile these two views is to move to quan- tum phase space. In Ref. [16], the authors studied the dynam- where j is the total angular momentum, h ≥ 0 is the magnetic ics of the Dicke model in terms of the Husimi-Q quasiprob- field and γx > 0 (critical field hc = γx). This model can be ability function. They showed that the closed (unitary) evo- viewed as the fully connected version of a system of N = 2 j lution yields, notwithstanding, a diffusive-type Fokker-Planck spin 1/2 particles (it therefore presents mean-field exponents). equation in phase-space. It is, of course, a special type of Since it is analytically tractable, it has been the subject of sev- diffusion in order to comply with the fact that the system is eral studies over the last decades [20–32]. closed, and thus energy must be conserved. The Husimi func- tion can be viewed as a convolution of the system’s state with a heterodyne measurement. Consequently, it provides a coarse- A. Brief review of the quantum phase transition grained description of the dynamics, and thus, naturally ac- counts for the scrambling of information. Before describing the dynamical phase transition, let us Motivated by this, we put forth in this article a detailed firstly discuss the regular quantum phase transition for this study of DPTs from the optics of quantum phase space. We model, which occurs in the thermodynamic limit, j → ∞. In focus on the Lipkin-Meshkov-Glick model, describing the dy- order to do this, it is convenient to define the so-called spin namics of a single macrospin [17–32] (Sec.II). This model al- coherent state lows for a well-defined classical limit, which takes place when −iφJz −iθJy the spin j → ∞. In this case, the model reduces to the classi- |Ωi = e e | ji , (4) cal dynamics of a spinning top [26]. In addition, it allows for where | ji is the eigenstate of Jz with eigenvalue j and θ ∈ a neat construction in terms of quantum phase space by us- [0, π] and φ ∈ [0, 2π] are polar coordinates. These states rep- ing the idea of spin coherent states: the corresponding Husimi resent the closest quantum analog of a classical angular mo- function describes a quasiprobability distribution in the unit mentum vector of fixed length j, in the sense that the expecta- sphere. tion values of the spin operators in the state of Eq. (4) take the The entropy associated with the Q-function is known as form Wehrl’s entropy [33, 34] and it can be attributed an opera-   tional meaning in terms of sampling through heterodyne mea- hJxi, hJyi, hJzi = j sin θ cos φ, sin θ sin φ, cos θ . surements [35]. For this reason, it upper bounds the von Neu- h 2i mann entropy, since it encompasses also the extra Heisenberg Moreover, expectation values of higher powers, such as Jx , h i2 uncertainty related to spin coherent states. Moreover, for the differ from Jx only by terms which become negligible in same reason, the Wehrl entropy also evolves non-trivially even the limit of large j. As a consequence, it can be shown that in closed system, unlike the von Neumann entropy, which is the leading order of the ground-state of the LMG model in the constant. It consequently captures the scrambling of informa- thermodynamic limit is a spin coherent state for certain values tion, very much like the Loschmidt echo in Eq. (1), but from of θ and φ [26]. The energy in this limit can be computed as h | | i the phase-space perspective. E = Ω H Ω , resulting in [36] As we show in Sec. III, the dynamics of the Wehrl en- E γ = −h cos θ − x sin2 θ cos2 φ. (5) tropy offers valuable insight about the nature of DPTs. Af- j 2 ter a critical quench, it grows quasi-monotonically, combined with small oscillations. The growth reflects the information The corrections to this behavior will be computed explicitly scrambling characteristic of DPTs. The oscillations, on the in Sec.IV. other hand, mimic the non-analytic behavior of the Loschmidt The ground-state energy is then found by minimizing echo and reflect information backflow to the initial Hilbert Eq. (5) over θ and φ, leading to the set of equations space sector. Finally, in Sec.IV, we also carry out an analysis 2 sin θ(h − γx cos θ cos φ) = 0, using a generalized Holstein-Primakoff transformation that is 2 known to faithfully capture the entire low-energy sector in the γx sin θ cos φ sin φ = 0. 3

For h > γx the only solution is θ = 0, in which case φ is arbitrary. For h < γx, however, two new solutions appear, corresponding to h cos θ = , (6) γx and φ = 0 or φ = π. The magnetization m = sin θ cos φ therefore serves as the order parameter of the model. This p 2 2 is identically zero for h > γx and m = hc − h /γx otherwise. The emergence of these new solutions identifies the critical field hc = γx.

B. Dynamical phase transition

Let us now consider the introduction of quenches in the field h. The system is prepared in the |ψ0i of H0 = H(h0), and at t = 0, it evolved under the final Hamiltonian H = H(h). To quantify the DPT, we use the Loschmidt echo defined in Eq. (1) and the corresponding rate in Eq. (2), with N = 2 j. In this model, a subtlety arises be- cause the ground-state is two-fold degenerate. As discussed in AppendixB, however, this introduces e ffects which become negligible in the thermodynamic limit. For this reason, we henceforth focus only on the analysis starting from one of the ground-states. For the sake of concreteness, we focus on quenches from h0 = 0 to h < γx. Results for the Loschmidt echo and the rate function are shown in Fig.1 for several values of j. The echo (top panel in Fig.1) vanishes for certain periods of time, but presents sharp periodic revivals at certain instants. This is convoluted with a damping, causing the time decay of the magnitude of L(t). The presence of a DPT becomes visible Figure 1. Dynamical phase transition. The top panel shows the in the rate function (bottom panel in Fig.1), which presents Loschmidt echo (1) while the bottom one displays the dynamical be- kinks at certain instants of time, called critical times tc. haviour of the rate function (2) for a quench in the LMG model, from h0 = 0 to h = 0.8. The curves correspond to different values of j. The plateaus that occur in r(t) at short times is a numerical artifact. They happen because L(t) becomes exponentially small in these re- gions (even though this could be fixed with increasing the precision, III. ENTROPIC DYNAMICS IN QUANTUM PHASE SPACE this is not viable in practice because of the exponential dependence).

Let us now address the core part of our article, in which we put forth an analysis of the DPT in Fig.1 from the perspective An operational interpretation of this quantity in terms of of quantum phase space. The spin Husimi-Q function associ- sampling through heterodyne measurements was given in ated with an arbitrary ρ is defined as Ref. [35]. It has also been applied in different contexts, like entanglement theory [38], uncertainty relations [39], and Q(Ω) = hΩ| ρ |Ωi , (7) quantum phase transitions [40], just to name a few. In the con- where |Ωi are the spin coherent states given in Eq. (4). This text of thermodynamics, our interest here, a theory of entropy production for spin systems was put forth in Ref. [41] and the quantity is always non-negativeR and normalized to unity ac- cording to (2 j + 1)/4π dΩQ(Ω) = 1, where dΩ = sin θdθdφ. corresponding bosonic analog in Ref. [42]. It therefore represents a quasi-probability distribution in the In classical thermodynamics, the entropy of a closed system unit sphere, offering the perfect platform to understand the should be monotonically increasing with time. The quantity quantum to classical transition [37]. In our case, ρ = |ψtihψt| |h | i|2 dS Q so the Husimi function simplifies to Q = Ω ψt . := ΠQ (9) The entropy associated with Q(Ω) is known as Wehrl’s en- dt tropy [33, 34], defined as is interpreted as the entropy production rate, since it reflects 2 j + 1 Z the entropy that is irreversibly produced in the system. The S Q = − dΩ Q(Ω) ln Q(Ω). (8) 4π second law then states that ΠQ ≥ 0. In open systems, on the 4 other hand, one has instead

dS Q = Π − Φ , (10) dt Q Q where ΦQ is the entropy flux rate from the system to the en- vironment. In open systems, dS Q/dt does not have a well- defined sign, since the flux ΦQ can be arbitrary. However, one still has ΠQ ≥ 0. In the present case there is no associated flux (ΦQ = 0), since the dynamics of the system is closed. Moreover, it is not guaranteed that dS Q/dt ≥ 0 for all times, since our system is not in the thermodynamic limit. Notwith- standing, one still expects that strong information scrambling, as happens in critical systems, should lead to a ΠQ which is most of the time non-negative. Numerical results for the Wehrl entropy in Eq. (8) and the entropy production rate in Eq. (9) are shown in Fig.2 for the same quench protocol used in Fig.1, which are remark- able. As j increases, the Wehrl entropy presents small oscil- lations enveloping a monotonically increasing behavior. This increase clearly reflects the information scrambling charac- teristic of the DPT. It shows that, as time passes, the coarse- grained nature of Q(Ω) causes the available information about the system’s state to be degraded as a function of time. For long times and j sufficiently large, S Q has a tendency to satu- rate at a constant value. As anticipated, ΠQ oscillates in time, being predominantly positive, but also becoming negative at certain times. These negativities represent the backflow of in- formation, which is characteristic of the recurrences in L(t) (see Fig.1).

Figure 2. Entropy dynamics. The top panel shows the dynamics of the Wehrl entropy (8) while the entropy production rate, (9), is displayed in the bottom panel, for the same quench protocol used in Figure3 shows the entropy production rate ΠQ along with Fig.1. the rate function r(t) for j = 300. As we can see, their be- haviours are clearly linked. In order to make such relation clearer, we present in Fig.4 the relation between the critical times tc where the dynamical quantum phase transitions occur, i.e. the non-analytical points of r(t), and the times tm at which ΠQ present local maxima. From this result it becomes clear that, for large j, one approaches tc ≈ tm, showing how the maxima of ΠQ perfectly correlate with the critical times. This corroborates the idea that the oscillations in ΠQ indeed reflect the DPT. A more detailed analysis is presented in Appendix B.

Finding a closed-form expression for the entropy produc- tion rate (9) is in general not possible. Notwithstanding, we were able to identify which parts of the dynamics, in terms of Figure 3. Rate function r(t), along with the entropy production rate the associated Fokker-Planck equation, contribute to ΠQ. This ΠQ, for j = 300. Other parameters are the same as in Fig.1. analysis, however, is cumbersome and it is thus postponed to AppendixA. 5

large j and keeping only terms which are at most quadratic in a and a†, one finds p 2 j H = E − sin θ(γ cos θ − h)(a + a†) (13) R 2 x γ +a†a(γ sin2 θ + h cos θ) − x cos2 θ(a + a†)2, x 4 where E is the classical energy given in Eq. (5) with φ = 0. The role of φ is trivial, it is not necessary to rotate around the z axis as well. We can now choose θ to eliminate the linear term propor- tional to a + a†. This leads to the same choice that minimized the classical energy in Eq. (6), i.e.  arccos(h/γx), if h < hc = γx θ =  (14) Figure 4. Critical times tc, associated with the non-analytical be- h  0 otherwise. haviour of r(t) and the times tm corresponding to local maxima of ΠQ. Other parameters are the same as in Fig.1. With this choice, Eq. (13) reduces to

2 IV. CONTRIBUTION FROM THE LOW ENERGY SECTOR † h † 2 HR(h) = E + γxa a − (a + a ) , h < hc, (15) 4γx A. Holstein-Primakoff approximation for the LMG model and

† γx † Equilibrium quantum phase transitions are almost entirely HR(h) = E + ha a − (a + a) , h > hc. (16) described by the low energy sector, i.e. ground state plus the 4 first few excited states. This is not true for DPTs which, in Therorefor, the HP method leds to the same classical energy principle, depend on the entire spectrum. In this section we landscape as using spin coherent states [Eq. (5)]. However, ask which aspects of DPTs can nonetheless be captured by it is important to remark that it yields an operator-based rep- the low energy sector. We do this by introducing a Gaussi- resentation of the fluctuations around the ground-state. Note fication procedure based on a generalized Holstein-Primakoff how E ∼ j is extensive, whereas the fluctuations in Eqs. (15) (HP) representation [43]. Finally, we then compare the pre- and (16) are independent of j. Notwithstanding, as we will dictions of this HP model with the full numerics studied in the find below, this does not mean the fluctuations are negligible previous section. when they become significantly close to criticality. The HP method represents spin operators in terms of a sin- We now use the above results to determine the ground-state gle bosonic mode, by means of the non-linear transforma- and the energy gap between the ground-state and the first ex- tion [43] cited state. To do this, we introduce the squeeze operator ξ (a2−(a†)2) † S = e 2 , where for our purposes it suffices to take J = j − a a, (11a) ξ z ξ > 0. This allows us to write Eqs. (15) and (16), up to con- stants, as q † J+ = 2 j − a a a, (11b) † † HR(h) = E + ωhS ha aS h, (17) † where a is a bosonic operator satisfying [a, a ] = 1. Equa- where tion (11a) shows that the excitations of a†a are mapped onto  p 2 2 excitations of Jz, starting from the state | ji, downwards. This  γ − h , if h < hc,  x is reasonable when h > hc. But when h < hc, the ground-state ωh = (18)  p will not be close to | ji at all. To take this into account, we first  h(h − γx), otherwise, rotate the Hamiltonian (3) by an angle θ around the y-axis, be- fore applying the HP transformation. That is, we first consider and S h = S (ξh) with the rotated Hamiltonian  1  2 2 − 4 ln 1 − h /γx , if h < hc, H = eiθS y He−iθS y (12)  R ξh =  (19)  1 γ − ln(1 − γx/h), otherwise. = −h(J cos θ − J sin θ) − x (J sin θ + J cos θ)2, 4 z x 2 j z x The Hamiltonian (17) is now diagonal, so that ωh describes where the value of θ will be fixed below. We now introduce precisely the energy level spacing of the first few excited lev- the HP transformation on HR instead of H. Expanding for els. Whence, we conclude that, at low , the excitations 6

† † −ξh are equally spaced with energy gap ωh. As a feature of quan- D(δα)S = S D(δα˜), where δα˜ = δαe 0 . Finally, we com- h0 h0 tum phase transitions, the gap closes at h = h . Moreover, † c bine the two squeezing operations as S hS = S (δξ), where it does so from both directions and in a manner which is not h0 δξ = ξh − ξh . This leads to symmetric in h. 0 † Finally, to compute the ground-state, we must first go L(t) = |h0|D†(δα˜)S †(δξ)e−iωhta aS (δξ)D(δα˜)|0i|2. (25) back to the original Hamiltonian by undoing the rotation in Eq. (12). In the language of the HP transformation (11b), a ro- This expression can now be evaluated by noting that it repre- tation around y becomes a displacement of the bosonic mode sents the vacuum expectation of a thermal squeezed displaced a Gaussian state at imaginary temperature β = −iωh. The result is therefore iθJ α a†−α∗a e y = D(α ) = e h h , (20) h −2ξ − h0  2 j(θh θh0 )e p exp − δξ 2 where α = − 2 j θ /2 and θ is given in Eq. (14). By com- 1+4e cot (ωht/2) h h h L(t) = q , (26) bining Eqs. (12) and (17), then it yields the original Hamilto- cos2(ω t) + cosh2(2δξ) sin2(ω t) nian in the form h h

† † †  an expression which depends only on the HP parameters θh, H = D S ωha a S hDh, (21) h h ξh and ωh, given by Eqs. (14), (18) and (19), respectively. Note also how the exponent in (26) is extensive in j. As a where Dh = D(αh). The ground-state is now readily found to be consequence, the rate function (2) with N = 2 j becomes, in the thermodynamic limit, † † |ψgs(h)i = DhS h|0i, (22) −2ξ (θ − θ )e h0 r(t) = h h0 . (27) δξ 2 where |0i is the vacuum defined by a. The ground-state is 1 + 4e cot (ωht/2) consequently a displaced squeezed state. The amount of dis- p We can now finally address the main question posed in the placement, α = − 2 jθ /2 is zero when h > h and non- h h c √ beginning of this section. Namely, what is the contribution of zero otherwise. Moreover, the displacement scales as j and the low energy sector to DPTs. To do this, we simply com- thus become extremely large in the thermodynamic limit. This pare Eq. (27) with the full numerics. The results are presented displacement is precisely the prediction using spin coherent in Fig.5. They clearly that DPTs cannot be explained mak- states, but written in a bosonic language. In addition, the state ing use only of the low energy sector. In Fig.5(a), for in- is also squeezed by the amount ξ given in Eq. (19). This h stance, we compare quenches from h = 0 to h/γ = 0.1. therefore represents a correction on top of the spin coherent 0 x In this case, Eq. (27) (in red) faithfully captures the physics state predictions. The squeezing is j independent, but diverges of the full numerical solution (black points). However, for at criticality. Thus, very close to the critical point, the state these small quenches, the non-analyticities of r(t) are not yet should differ significantly from a spin coherent state. present. Conversely, as the value of h increases, the signature kinks of the DPTs start to appear, whereas Eq. (27) remains B. Loschmidt echo perfectly analytical. In fact, Eq. (27) can only become non- analytic when h = γx = 1, in which case ωh → 0 and the cotangent diverges. At this point, the high energy sector be- Armed with the aforementined results, we can now readily comes so important that, albeit non-analytic, Eq. (27) cannot compute both the Loschmidt echo and the rate function within capture at all the physics of the problem. the HP approximation. The system is initially prepared in the ground-state |ψgs(h0)i corresponding to a field h0. After the quench, the field changes to h. Since S h and Dh are unitary, V. CONCLUSION the time-evolution operator can be written as

† † † In summary, we studied the dynamical behaviour of the en- e−iHt = D S e−iωhta aS D . (23) h h h h tropy production in a closed system that undergoes a dynam- The evolved state at time t will then be ical quantum phase transition. Specifically, we considered sudden quenches in the Lipkin-Meshkov-Glick model under † −iHt † † −iωhta a † † the perspective of the Husimi-Q quasi-probability distribu- |ψti = e |ψ0i = DhS he S hDhDh S h |0i. (24) 0 0 tion. Such approach allowed us to define the entropy asso- 2 The Loschmidt echo L(t) = |hψ0|ψti| becomes ciated with the Q-function, known as Wehrl entropy, which is a measure of the coarse-grained dynamics of the system. † † † −iωhta a † † 2 L(t) = |h0|S h Dh D S e S hDhD S |0i| . From such phase space approach, we were able to show 0 0 h h h0 h0 that critical quenches lead to a quasi-monotonic growth of the This can be simplified by exploiting the algebra of displace- Wehrl entropy in time, thus demonstrating the scrambling of ment and squeeze operators, which in this case is facilitated , a characteristic feature of these transi- by the fact that the arguments αh and ξh are all real. First, one tions. Moreover, the entropy rate also presents small oscil- † p has DhD = D(δα), where δα = − 2 j(θh − θh )/2. Second, lations, implying negative entropy production rates at certain h0 0 7

production rate, a key concept in thermodynamics. Acknowledgments — ES, MS and LCC acknowledge sup- port from Spanish MCIU/AEI/FEDER (PGC2018-095113- B-I00), Basque Government IT986-16, the projects QMiCS (820505) and OpenSuperQ (820363) of the EU Flagship on Quantum Technologies and the EU FET Open Grant Quro- morphic and the U.S. Department of Energy, Office of Sci- ence, Office of Advanced Scientific Computing Research (ASCR) quantum algorithm teams program, under field work proposal number ERKJ333. LCC also thanks the Brazil- ian Agencies CNPq, FAPEG. LCC and GTL acknowledge the Brazilian National Institute of Science and Technology of Quantum Information (INCT/IQ) for the financial support. This study was financed in part by the Coordenação de Aper- feiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001. GTL acknowledges the support from the São Paulo Research Foundation FAPESP (grants 2017/07973- 5, 2017/50304-7 and 2018/12813-0).

Appendix A: Entropy production for LMG model Figure 5. Comparison of the rate function r(t) between the full numerics (black points) and the Holstein-Primakoff approximation [Eq. (27)] for j = 200. Each curve describes a quench from h0 = 0 In this appendix we analytically compute the entropy pro- to (a) h/γx = 0.1, (b) h/γx = 0.3, (c) h/γx = 0.4 and (d) h/γx = 0.8 duction rate for the LMG model Eq. (3). To accomplish this [c.f. Fig.1(b)]. task, we use the Schwinger map to transform spin operators into bosonic operators [41, 44]. This map employs two sets of bosonic operators, a and b, such that instants, which signals the recurrences of the Loschmidt echo. 1 † † Jz = (a a − b b) These results are relevant not only from the dynamical quan- 2 (A1) tum phase transition perspective, but also for the field of quan- † † J+ = J− = a b. tum thermodynamics, since they point out that the Wehrl en- tropy can be used as a viable measure of entropy production. To simplify notation we will employ the bosonic coherent Finally, based on a modified Holstein-Primakoff approxi- state representation for the Husimi function. In this way, we mation, a type of Gaussification method, we addressed the define α and β as the complex amplitudes associated with op- contribution of the low energy sector to the dynamical quan- erators a and b, respectively. The correspondence table be- tum phase transition. This procedure, which is known to faith- tween bosonic operator acting on a state ρ and a differential fully capture the entire low energy sector in the thermody- operator acting on the Husimi Q-function Q(α, β) is namic limit, fails to accurately predict the behaviour of the o ρ → (µ + ∂ )Q(µ , µ¯ ) system under dynamical quantum phase transitions. i i µ¯i i i † o ρ → µ¯ iQ(µi, µ¯i) Despite considerable research into the underlying mecha- i (A2) nisms behind this phenomenon, several open questions still ρoi → µiQ(µi, µ¯i) remain and we believe that the phase-space perspective that † → ρoi (¯µi + ∂µi )Q(µi, µ¯i), we put forth here may contribute to deepen our understanding, specially regarding the thermodynamics of such phase transi- where oi stands for a or b and µi represents either α or β,µ ¯i tions, as indicated by the strong connection between critical denotes complex conjugation. Using Eqs. (A1) and (A2) one times, characterizing the critical transition, and the entropy finds,

1 [J , ρ] → J = [(α∂ Q − α∂¯ Q) − (β∂ Q − β∂¯ ¯ Q)], (A3) z z 2 α α¯ β β 8 and, since 2Jx = J+ + J−, it follows that 2 2 ¯ 2 4[Jx , ρ] → 4Jx2 = (2|α| + 1)(β∂β¯ Q − β∂βQ) + (2|β| + 1)(α∂ ¯ α¯ Q − α∂αQ) 2 2 2 2 + 2[(α ¯ β∂β¯ Q − α β∂¯ βQ) + (αβ¯ ∂α¯ Q − αβ¯ ∂αQ)] ¯ + 2(α ¯β∂α¯ ∂β¯ Q − αβ∂α∂βQ) (A4) 2 2 − 2 2 ¯2 2 − 2 2 + [(α ¯ ∂β¯ Q α ∂βQ) + (β ∂α¯ Q β ∂αQ)] J(1) J(2) J(3) J(4) = 4( x2 + x2 + x2 + x2 ).

R J(i) 4 2 2 4 U 2 where we introduced the notation x2 representing the i-th where d µ = d αd β and Λa = d µ a ln Q with a = z, x . line of Eq (A4). The dynamics of our system is governed by From Eqs. (A3) and (A4) one can compute the terms of the von Neumann equation, Eq. (A5). We note that the Schwinger transformation Eq. (A1) maps a bounded set of spin operators into, what is in princi- iγx 2 ple, an unbounded set of bosonic operators. However, the map ∂tρ = −i[Hx2 + Hz, ρ] = ih[Jz, ρ] + [J , ρ], 2 j x introduces a restriction on the set of bosonic operators, which makes the Husimi Q-function and its derivatives vanish for which is mapped into a quantum Fokker-Planck equation for α, β → ±∞ once they are restricted, i.e. non-zero only to a the Husimi Q-function of the bosonic operators introduced by specific region of phase space. Using this fact along with in- the Schwinger map (A1), tegration by parts, we find that Λz = 0 and the only terms that contribute to the entropy rate are those related to diffusion, i.e. ∂tQ(µ) = Ux2 + Uz, the terms with second derivatives in Jx2 . Λx2 can be indepen- dently computed for each term appearing in Eq. (A4). For the where Uz = ihJz and Ux2 = (iγx/8 j)Jx2 . We are interested in the entropy rate, which in the first contribution we have, Schwinger representation is given by Z dS Q 4 = − d µ U(Q) ln Q = −(Λ 2 + Λ ) (A5) dt x z

Z (1) ∝ 4 | |2 ¯ − Λx2 d µ (2 α + 1) ln Q(β∂β¯ Q β∂βQ) Z ! h i∞ ∂β¯ Q ∂βQ = (2|α|2 + 1)β¯Q ln Q − c.c. − d4µ (2|α|2 + 1) ln Q + β¯ − ln Q − β Q −∞ Q Q Z (A6) 4 2 = − d µ (2|α| + 1)(β∂¯ β¯ Q − β∂βQ) Z h i∞ = − (2|α|2 + 1)β¯Q − c.c. + d4µ (2|α|2 + 1)(Q − Q) = 0, −∞ the last term in the first line of Eq. (A4) is structurally the same as the one above and hence it vanishes. The second contribution is given by, Z Z (2) 4 2 2 h 2 i∞ 4 2 2 Λ ∝ d µ (α ¯ β∂¯ Q − α β∂¯ Q) ln Q = Q ln Qα¯ β − c.c. − d µ α¯ β∂¯ Q − α β∂¯ Q x2 β β −∞ β β Z (A7) h 2 i∞ 4 2 2 = − α¯ βQ − c.c. + d µ (∂¯[α ¯ β] − ∂ [α β])Q = 0, −∞ β β and the same result holds for the last contribution of the second line of Eq. (A4). The first non-vanishing contribution comes from the third line, Z Z ! (3) 4 4 ∂α¯ Q ∂β¯ Q Λ ∝ d µ ln Q(α ¯β∂¯ ∂¯ Q − αβ∂ ∂ Q) = − d µ α¯ β¯ − c.c. Q, (A8) x2 α¯ β α β Q Q and finally, Z Z 2 (4) 4 2 2 2 2 h i∞ 4 2 (∂β¯ Q) Λ ∝ d µ ln Q(α ¯ ∂ Q − α ∂ Q) = α¯ ln Q∂¯ Q − c.c. − d µ α¯ − c.c. x2 β¯ β β −∞ Q Z !2 (A9) ∂β¯ Q = − d4µ α¯ Q − c.c. Q 9

J(4) → the second term of x2 is structurally the same as the first, it suffices to substitute α β in Eq. (A9). Hence, we obtain that the non-vanishing contribution to Λx2 is,     Z ∂ Q!2 ∂ Q !2 Z ∂ Q !2 iγx 4  β¯ ∂α¯ Q β¯ ∂α¯ Q  iγx 4  β¯ ∂α¯ Q  Λ 2 = d µ  α¯ + 2α ¯β¯ + β¯ − c.c. Q = d µ  α¯ + β¯ − c.c. Q. (A10) x 8 j  Q Q Q Q  8 j  Q Q 

(1) The contribution due to Λz vanishes because it has exactly the same structure as Λx in Eq. (A6). Now, Eq. (A10) can be written as

* !2+ γx  ∂β¯ Q ∂α¯ Q  Λx2 = Im α¯ + β¯ 4 j  Q Q  (A11) * !2+ γx  ∂βQ ∂αQ  = − Im α + β , 4 j  Q Q  thus leading us to the following expression for the entropy rate

* !2+ dS Q γx  ∂βQ ∂αQ  = −Λx2 = Im α + β . (A12) dt 4 j  Q Q 

In this equation, h...i means an average over the phase-space. For a fixed the value of j, we can go back to the polar representation (θ,φ) using the following relations [44],

−iφ 2 θ h θ ∂φ i α∂β = e cos 2 2 j tan 2 + ∂θ − i sin θ (A13)

iφ 2 θ h θ ∂φ i β∂α = e sin 2 2 j cot 2 − ∂θ + i sin θ (A14)

R 4 R 4 since S Q(µ) = − d µ ln Q(µ)Q(µ) → S Q(Ω) = −(2 j + 1)/4π d Ω ln Q(Ω)Q(Ω) which gives us the following expression for the entropy production in polar coordinates,   Z " # " # !2 dS Ω 2 j + 1  1 −iφ 2 θ θ ∂φ 2 θ θ ∂φ  = γx Im dΩ e cos 2 j tan + ∂θ − i Q(Ω) + Q(Ω) sin 2 j cot − ∂θ + i Q(Ω) dt 16π j  Q(Ω) 2 2 sin θ 2 2 sin θ  (A15)

Appendix B: Numerical studies on the LMG model 1. The dynamical quantum phase transition

In studying the DPT of the LMG model, care must be taken with the fact that the ground-state is two-fold degen- erate. We therefore define an echo for each ground-state, α α 2 Lα = hψ0 |ψt i and consider only the net rate  g  1 X  r (t) = − log  L  , (B1) s N  α α=1 which therefore captures the total return probability. However, as shown [10], in the thermodynamic limit this converges to the minimum among all the contributions 1   rm ≡ lim r(t) = − log min Lα . (B2) N→∞ N α It is for this reason that in the main text it sufficed to consider only the rate starting from a single ground-state. We start by showing the Loschmidt echo for the two ground In this Appendix we describe several numerical studies states of the system in Figs.6 and7. As we can see from these of the LMG model. Specifically, we consider four distinct figures, the behaviour of L regarding the dynamical quantum quench processes, from h0 = 0 to h = 0.1, 0.8, 1.0 and 1.6. phase transition is independent of the considered initial state. Note that the first two quenches occur before the quantum crit- However, it does depend on the quench amplitude and show ical point while the last one occurs after. no sensible difference at the h = 1. 10

Figure 6. Loschmidt echo for the first ground state. From top to Figure 8. Rate function for the total return probability. From top to bottom and from left to right we consider quenches from h0 = 0 to bottom and from left to right we consider quenches from h0 = 0 to h = 0.1, 0.8, 1.0 and 1.6. h = 0.1, 0.8, 1.0 and 1.6.

Figure 7. Loschmidt echo for the second ground state. From top to bottom and from left to right we consider quenches from h0 = 0 to h = 0.1, 0.8, 1.0 and 1.6.

Figure 9. Rate function for the minimum return probability. From top to bottom and from left to right we consider quenches from h0 = 0 to To make this point more clear, in Figs.8 and9 we show h = 0.1, 0.8, 1.0 and 1.6. the rate functions r and rm, respectively. It is clear that the dynamical quantum phase transition depends on the quench amplitude and on the size of the angular momentum. In order to see that rs converges to rm when we approach the thermo- dynamic limit, in Fig. 10 we show both quantities for distinct values of the angular momentum. 11

Figure 10. Comparison between r (solid lines) and rm (dashed lines). Figure 12. Entropy for the second ground state. From top to bottom From top to bottom and from left to right we consider quenches from and from left to right we consider quenches from h0 = 0 to h = 0.1, h0 = 0 to h = 0.1, 0.8, 1.0 and 1.6. 0.8, 1.0 and 1.6.

2. Entropy production

Figures 11, 12 and 13 show the dynamical behaviour of the entropy for three distinct states, the two ground states and the equal superposition of these states. We can observe the same qualitative behaviour for all the considered quenches, except for the smaller one. The entropy approaches a maximum as we approach the thermodynamic limit. The oscillation pattern presented in these plots signal the dynamical quantum phase transitions.

Figure 13. Entropy for the superposition of both ground states. From top to bottom and from left to right we consider quenches from h0 = 0 to h = 0.1, 0.8, 1.0 and 1.6.

In order to show this fact, we consider the entropy produc- tion rate, given by the time derivative of S Q. This is shown in Figs. 14, 15 and 16 for the same three states addressed in the case of the entropy. Again, except for the small quench, the qualitative behaviour of this quantity presents several maxi- mums and we show in the main text and in the next subsec- tion that such maximums are related to the dynamical quan- Figure 11. Entropy for the first ground state. From top to bottom and tum phase transitions. from left to right we consider quenches from h0 = 0 to h = 0.1, 0.8, 1.0 and 1.6. 12

Figure 14. Entropy production rate for the first ground state. From Figure 16. Entropy production rate for the superposition of both top to bottom and from left to right we consider quenches from h0 = ground states. From top to bottom and from left to right we consider 0 to h = 0.1, 0.8, 1.0 and 1.6. quenches from h0 = 0 to h = 0.1, 0.8, 1.0 and 1.6.

3. Entropy production rate and the rate function

Finally, we address here the main message of the present article. Figure 17 shows the comparison between the entropy production rate and the rate function rm for a fixed value of the angular momentum ( j = 500). Except for the case of small quench, where we actually do not have a quantum phase transition since rm is analytical in time, the other considered quenches clearly shows that the entropy production rate is able to sign the dynamical quantum phase transition, as discussed in the main text.

Figure 15. Entropy production rate for the second ground state. From top to bottom and from left to right we consider quenches from h0 = 0 to h = 0.1, 0.8, 1.0 and 1.6.

Figure 17. Comparison between r and rm. From top to bottom and from left to right we consider quenches from h0 = 0 to h = 0.1, 0.8, 1.0 and 1.6. 13

This relation can be deeply stated if we consider the times at which entropy production rate shows a maximum and the crit- ical times, where we have a dynamical quantum phase transi- tion. This is done in Fig. 18, where a clear linear behaviour emerges, thus supporting our claims in the main text. The points highlighted in this figure are the ones shown in Fig.4 of the main text.

Figure 18. Comparison between the maximums of the entropy pro- duction rate and the non-analytical points of the rate function for the quench h0 = 0 to h = 0.8.

[1] M. Heyl, A. Polkovnikov and S. Kehrein. Dynamical quantum Temp. Phys. 42, 971 (2016). phase transitions in the transverse-field Ising model. Phys. Rev. [13] C. Yang and T. Lee. Statistical theory of equations of state and Lett. 110, 135704 (2013). phase transitions. I. Theory of condensation. Phys. Rev. 87, 404 [2] S. Vajna and B. Dóra. Disentangling dynamical phase tran- (1952). sitions from equilibrium phase transitions. Phys. Rev. B 89, [14] M. E. Fisher, in Boulder Lectures in Theoretical Physics (Uni- 161105(R) (2014). versity of Colorado, Boulder, 1965), Vol. 7. [3] J. C. Halimeh and V. Zauner-Stauber. Dynamical phase diagram [15] M. Campisi and J. Goold. Thermodynamics of quantum infor- of quantum spin chains with long-range interactions. Phys. Rev. mation scrambling. Phys. Rev. E 95, 062127 (2017). B 96, 134427 (2017). [16] A. Altland and F. Haake. Quantum chaos and effective thermal- [4] C. Karrasch and D. Schuricht. Dynamical phase transitions after ization. Phys. Rev. Lett. 108, 073601 (2012). quenches in nonintegrable models. Phys. Rev. B 87, 195104 [17] H. Lipkin, N. Meshkov and A. Glick. Nucl. Phys. Validity of (2013). many-body approximation methods for a solvable model: (I). [5] E. Canovi, P. Werner and M. Eckstein. First-order dynamical Exact solutions and perturbation theory. 62, 188 (1965). phase transitions. Phys. Rev. Lett. 113, 265702 (2014). [18] N. Meshkov, A. Glick and H. Lipkin. Validity of many-body ap- [6] M. Heyl. Quenching a quantum critical state by the order pa- proximation methods for a solvable model: (II). Linearization rameter: Dynamical quantum phase transitions and quantum procedures. Nucl. Phys. 62, 199 (1965). speed limits. Phys. Rev. B 95, 060504(R) (2017). [19] A. Glick, H. Lipkin and N. Meshkov. Validity of many-body [7] P. Jurcevic, H. Shen, P. Hauke, C. Maier, T. Brydges, C. approximation methods for a solvable model: (III). Diagram Hempel, B. P. Lanyon, M. Heyl, R. Blatt and C. F. Roos. Di- summations. Nucl. Phys. 62, 211 (1965). rect observation of dynamical quantum phase transitionsin an [20] J. I. Cirac, M. Lewenstein, K. Mølmer and P. Zoller. Quantum interacting many-body system. Phys. Rev. Lett. 119, 080501 superposition states of Bose-Einstein condensates. Phys. Rev. (2017). A 57, 1208 (1998). [8] N. Fläschner, D. Vogel, M. Tarnowski, B. S. Rem, D.-S. Lüh- [21] D. A. Garanin, X. Martínez Hidalgo and E. M. Chudnovsky. mann, M. Heyl, J. C. Budich, L. Mathey, K. Sengstock and C. Quantum-classical transition of the escape rate of a uniaxial Weitenberg. Observation of dynamical vortices after quenches spin system in an arbitrarily directed field. Phys. Rev. B 57, in a system with topology. Nat. Phys. 14, 265 (2018). 13639 (1998). [9] J. Zhang, G. Pegano, P. W. Hess, A. Kyprianidis, P. Becker, H. [22] J. I. Latorre, R. Orús, E. Rico and J. Vidal. Entanglement en- Kaplan, A. V. Gorshkov, Z.-X. Gong and C. Monroe. Observa- tropy in the Lipkin-Meshkov-Glick model. Phys. Rev. A 71, tion of a many-body dynamical phase transition with a 53-qubit 064101 (2005). quantum simulator. Nature 551, 601 (2017). [23] J. Vidal, G. Palacios and R. Mosseri. Entanglement in a second- [10] M. Heyl. Dynamical quantum phase transitions: a review. Rep. order quantum phase transition. Phys. Rev. A 69, 022107 Prog. Phys. 81, 054001 (2018). (2004). [11] M. Heyl. Dynamical quantum phase transition: A brief survey. [24] J. Vidal, R. Mosseri and J. Dukelsky. Entanglement in a first- EPL 125, 26001 (2019). order quantum phase transition. Phys. Rev. A 69, 054101 [12] A. A. Zvyagina. Dynamical quantum phase transitions. Low (2004). 14

[25] P. Ribeiro, J. Vidal and R. Mosseri. Thermodynamical limit of and operational phase-space measurement. I. General formal- the Lipkin-Meshkov-Glick model. Phys. Rev. Lett. 99, 050402 ism. Phys. Rev. A 51, 2575 (1995) (2007). [36] O. Castaños, R. López-Peña, J. G. Hirsch and E. López- [26] P. Ribeiro, J. Vidal and R. Mosseri. Exact spectrum of the Moreno. Classical and quantum phase transitions in the Lipkin- Lipkin-Meshkov-Glick model in the thermodynamic limit and Meshkov-Glick model. Phys. Rev. B 74, 104118 (2006). finite-size corrections. Phys. Rev. E 78, 021106 (2008). [37] K. Takahashi and N. Saitôn. Chaos and Husimi distribution [27] P. Ribeiro and T. Paul. Semiclassical analysis of spin systems function in . Phys. Rev. Lett. textbf55, 645 near critical energies. Phys. Rev. A 79, 032107 (2009). (1985). [28] S. Dusuel and J. Vidal. Finite-size Scaling exponents of the [38] F. Mintert and K. Z. yczkowski. Wehrl entropy, Lieb conjecture, Lipkin-Meshkov-Glick model. Phys. Rev. Lett. 93, 237204 and entanglement monotones. Phys. Rev. A 69, 022317 (2004). (2004). [39] G. Palma. Uncertainty relations with quantum memory for the [29] S. Dusuel and J. Vidal. Continuous unitary transformations Wehrl entropy. Lett. Math. Phys. 108, 2139 (2018). and finite-size scaling exponents in the Lipkin-Meshkov-Glick [40] O. Castaños, M. Calixto, F. Pérez-Bernal and E. Romera. Iden- model. Phys. Rev. B 71, 224420 (2005). tifying the order of a quantum phase transition by means of [30] A. Das, K. Sengupta, D. Sen and B. K. Chakrabarti. Infinite- Wehrl entropy in phase space. Phys. Rev. E 92, 052106 (2015). range Ising ferromagnet in a time-dependent transverse mag- [41] J. P. Santos, L. C. Céleri, F. Brito, G. T. Landi and M. Pa- netic field: Quench and ac dynamics near the quantum critical ternostro. Spin-phase-space-entropy production. Phys. Rev. 97, point. Phys. Rev. B 74, 144423 (2006). 052123 (2018). [31] Y. Hamdouni and F. Petruccione. Time evolution and decoher- [42] B. O. Goes, C. E. Fiore and G. T. Landi. Quantum features of ence of a spin-1/2 particle coupled to a spin bath in thermal entropy production in driven-dissipative transitions. Phys. Rev. equilibrium. Phys. Rev. B 76, 174306 (2007). Res. 2, 013136 (2020). [32] J. Bao, B. Guo, H.-G. Cheng, M. Zhou, Jin Fu, Y.-C. Deng [43] T. Holstein and H. Primakoff. Field dependence of the intrinsic and Z.-Y. Sun. Multipartite nonlocality in the Lipkin-Meshkov- domain magnetization of a ferromagnet. Phys. Rev. 58, 1098 Glick model. Phys. Rev. A 101, 012110 (2020). (1940). [33] A. Wehrl. General properties of entropy. Rev. Mod. Phys. 50, [44] Y. Takahashi and F. Shibata. Spin coherent state representation 221 (1978). in non-equilibrium . J. Phys. Soc. Jpn. 38, [34] A. Wehrl. On the relation between classical and quantum- 656 (1975). mechanical entropy. Rep. Math. Phys. 16, 353 (1979). [35] V. Buzek, C. H. Keitel and P. L. Knight. Sampling