Dynamical critical scaling of long-range interacting quantum magnets
Slow variations (quenches) of the magnetic field across the paramagnetic-ferromagnetic phase transition of spin systems produce heat. In short-ranged systems the heat exhibits a universal power-law scaling as a function of the quench rate, known as Kibble-Zurek (KZ) scaling. Attempts to extend this hypothesis to long-range interacting systems have lead to seemingly contradicting results. In this work we analyse slow quenches of the magnetic field in the Lipkin-Meshkov-Glick model, which describes fully-connected quantum spins. We determine the quantum contribution to the residual heat as a function of the quench rate by means of a Bogoliubov expansion about the mean-field value and calculate the exact solution. For a quench which ends at the quantum critical point we identify two regimes: the adiabatic limit for finite-size chains, where the scaling is dominated by the Landau-Zener tunneling, and the KZ scaling. For a quench symmetric about the critical point, instead, there is no KZ scaling. Here we identify three regimes depending on the velocity of the ramp and on the size of the system: (i) the adiabatic limit for finite-size chains; (ii) the opposite regime, namely, the thermodynamic limit, where the residual heat is independent of the quench rate; and finally (iii) the intermediate regime, which is a crossover between the two solutions. We argue that this behaviour is a property of all-connected spin systems. Our findings agree with previous studies and identify the respective limits in which they hold.
The development of a comprehensive statistical mechanics description of out-of-equilibrium systems is a quest of relevance across disciplines, including biology, physics, computer science and financial markets chou2011 (). A specific, yet relevant question regards the connection between dynamical and equilibrium properties of quantum critical systems polkovnikov2011 (). This would contribute to a systematic understanding of the subtle interplay between time evolution, interactions, quantum, and thermal fluctuations. Moreover, it is important for the development of quantum devices based on quantum annealing, where one aims at preparing many-body quantum states with adiabatic transformations gardas2018 (). Theoretical and experimental studies of many-body critical dynamics after sudden variations of control fields have identified features which are reminiscent of the behavior of thermodynamic functions at transition points heyl2017 (); zhang2017 (). Yet, the relation between dynamical scaling and equilibrium critical phenomena is elusive and often only conjectured.
In this framework, it is believed that the thermodynamics of slow quenches across quantum critical points could be cast in terms of the so-called Kibble-Zurek (KZ) scaling kibble1976 (); zurek1985 (); zurek1996 (); zurek2005 (). The KZ scaling predicts that the heat produced by slow variations (quenches) of control fields across critical points scales with the quench rate through a power law determined by the equilibrium critical exponents dziarmaga2010 (); chandran2012 (); del_campo2014 (). This theory has a strong predictive power and has been experimentally verified for a large variety of physical systems schneider2012 (); ulm2013 (); pyka2013 (); mielenz2013 (); corman2014 (); braun2015 (); navon2015 (); bauerle1996 (); ruutu1996 (); carmi1999 (); monaco2009 (); weller2008 (); yukalov2015 (); chuang1991 (); ducci1999 (). Its validity, though, seems to be limited to systems where the coherence length diverges with a power law at the critical point but is well defined in the critical region. This hypothesis can be explained in a nutshell as follows. Assume a system of interacting spins in presence of a magnetic field , as illustrated in Fig. 1(a). Let the magnitude of the magnetic field be slowly varied from the paramagnetic to the ferromagnetic phase across the critical point . The transformation is adiabatic when the rate of change is larger than the energy gap , while in the other regime non-adiabatic effects are expected. Figure 1(b) displays the energy gap as a function of : The gap vanishes as at the critical point, with and the equilibrium critical exponents. At the times the equality holds. The KZ theory assumes that in the time window the dynamics are frozen and estimates the heat produced by the quench by the scaling , where is the average size of the domains formed at the time in the adiabatic approximation. This yields the scaling . For a quench where the magnetic field varies with time as (), then zurek2005 (); polkovnikov2005 (); dziarmaga2010 (); chandran2012 ()
Although this separation between adiabatic and “impulse regime” may seem oversimplified, it describes the behaviour found in microcanonical systems, where the relaxation time is determined by the instantaneous gap between the ground and the first excited state silvi2016 (). The validity of the KZ scaling (1) has been extensively verified in integrable fermionic systems dziarmaga2005 (); damski2005 (); dutta2017 (); de_grandi2010 (); polkovnikov2005 (). Even at finite temperatures, where one has relevant corrections due to scattering of defects, the KZ scaling is a good working hypothesis biroli2010 (); tomka2018 (). A conceptual problem arises, instead, when one applies the KZ scaling to critical systems where the coherence length is ill-defined campa2014 (); gardas2018 (). This is the case of systems with strong-long-range interactions, where the two-body interaction potential decays as a power law of the distance between its microscopic components such as , with and the spatial dimension. In this case the energy is non-additive campa2014 (); kastner2010 (); kastner2011 () and the scaling becomes meaningless.
In this work we consider a linear quench of the magnetic field across the paramagnetic–to–ferromagnetic transition in the Lipkin-Meshkov-Glick (LMG) model lipkin1965 (). The LMG describes a one dimensional chain of spins and constant all-to-all ferromagnetic interactions in a transverse magnetic field . The system is illustrated in Fig. 1(a) and it can be simulated by chains of trapped ions Richerme2014 (); Jurcevic2014 (). Its Hamiltonian reads
where are the Pauli matrices of spin and the pre-factor in front of the interaction term warrants that the energy is extensive campa2014 (). The parameter scales the energy in units of the interaction strength. From now on energy and time are in units of and , respectively.
When the magnetic field is constant in time, in the thermodynamic limit the LMG model displays a quantum phase transition (QPT) between a symmetric state fully polarized along and a symmetry–broken phase with two degenerate ground states of opposite macroscopic polarization along the direction. The quantum critical point (QCP) is at , the universal behavior is the same as the Dicke model dicke1954 (); Hepp1973 () and is given by a mean-field theory with critical exponents and botet1982 (); botet1983 (); das2006 ().
The dynamical protocol we consider is a continuous ramp of the control field , where is the quench rate and , such that , namely, the quench starts deep in the paramagnetic phase, crosses the QCP at and finally ends far into the symmetry broken phase. Using the critical exponents and in Eq. (1) one would expect the scaling after the quench. This scaling was found using an heuristic application of adiabatic perturbation theory to the Rabi model for a quench to the critical point (), after showing that the Rabi model can be mapped to the Dicke model hwang2015 (). Nevertheless it does seem not agree with the numerical studies reported in Ref. caneva2008 (); acevedo2014 () for symmetric quenches (). Moreover, it is inconsistent with the calculation performed in Ref. bachmann2017 () for a system which is equivalent to the quantum dynamics of the LMG in the strict thermodynamic limit . The works we have just mentioned seem to find mutually contradicting results.
We now provide the solution of the Schrödinger equation governed by Hamiltonian (2) for , which is valid for slow quenches and allows us to determine the spins’ wave function during and after the quench. Our model allows us to show that the predictions of Refs. caneva2008 (); acevedo2014 (); hwang2015 (); bachmann2017 () are consistent with our solution in different regimes. These regimes are identified by analysing the scaling properties at the critical point and are due to the long-range nature of the interactions.
We then perform a expansion around the ground-state of the mean-field model Holstein1940 (); Auerbach1994 (), which is assumed to adiabatically follow the quench. The expansion is obtained by rotating the spin operators to align them with the semiclassical magnetization and then by applying an Holstein-Primakoff transformation up to order 1 Holstein1940 (): , , where the operators and satisfy the bosonic commutation relation . The resulting Hamiltonian is quadratic and in diagonal form reads
where is the thermodynamic mean field energy density, is a constant mean field shift, while the quantum fluctuations are described by the quadratic harmonic oscillator term whose frequency is the gap dusuel2005 (). We focus on the evolution of the quadratic term and observe that the quantum part of Hamiltonian (4) is obtained at leading order in expansion and is thus strictly valid in the thermodynamic limit. Nevertheless, by means of the continuous unitary transformation approach Wegner2000 () the LMG Hamiltonian can be recast into the form (4) even for finite dusuel2005 (); dusuel2004 (). Then, the gap reads dusuel2005 (); dusuel2004 ()
where for large the function for while at the critical point the gap scales as dusuel2005 (); dusuel2004 (). The dynamics governed by Hamiltonian (4) corresponds now to the one of a single harmonic oscillator with the time-dependent frequency . It is exactly solved in terms of the dynamical basis
Here, is the Hermite polynomial of degree , is a phase factor, is the effective frequency and is a time dependent scale factor which obeys the Ermakov-Milne equation leach2008 (); milne1930 (); pinney1950 ()
By integrating Eq. (7) we exactly solve the quantum dynamics.
We consider the time interval with . The time evolution of the initial state , which coincides with the ground state of the instantaneous Hamiltonian at , is to good approximation the wave function , Eq. (6). The evolution is adiabatic when coincides with the instantaneous ground state of Hamiltonian . This corresponds to the fidelity , where and is the overlap integral between and the eigenfunctions of the adiabatic basis of the oscillator with frequency . Specifically, are the solutions of Eq. (6) after setting in Eq. (7) and thus in Eq. (6). The explicit expression for the overlap integral is derived in the Supplementary Material (S.M.) and Ref. Dabrowski2016 () 111See the supplementary materials. The residual energy (heat) at time is proportional to the number of the oscillator’s excitations , namely, , with
In order to verify that our model delivers a reliable description of the LMG model, we compare the predictions of Eqs. (6)-(7) with the results obtained in Ref. acevedo2014 () by numerically integrating the dynamics of spins with Hamiltonian (2). Figure 2(a) displays the time evolution of the fidelity and of the heat obtained from Eqs. (6)-(7) and for the parameters of Ref. acevedo2014 (). The curves in Fig. 2(a) reproduce the ones numerically found in Ref. acevedo2014 () confirming that our approach, based on integrating the Schrödinger equation of a single time-dependent harmonic oscillator, reliably describes the behaviour of slow quenches in the LMG model. As in that work, we observe the collapse of the fidelity at and after the critical point. For the choice of the parameters the dynamics are close to adiabatic with fidelity . In Ref. acevedo2014 (), though, no numerical evidence of KZ scaling was found. It was there conjectured that this may be due to the finite-size universal functions at the critical point. In particular, they introduced the parameter
which was identified on the basis of scaling properties of transition amplitudes in adiabatic perturbation theory. The parameter plays indeed an important role in the dynamics of our model and has a specific physical meaning, as we argue in the following using scaling arguments. For this purpose, we approximate the oscillator frequency with for and for , where the expression is reported apart from scalars which are intensive and not universal. We numerically verified that the remaining terms are irrelevant, since they become sub-leading in the critical stage of the dynamics. The overall effect of finite-size fluctuations is now summarized in an effective finite-size scaling exponent , such that . The full numerical solution of Eq. (7) indicates that finite-size corrections only become important at , therefore in this discussion we assume . We identify the scaling relations by performing the transformation and (note that is the same rescaled time variable as in Fig. 2(c) of Ref. acevedo2014 () apart from the factor ). This transformation leads to the Schrödinger equation governed by the Hamiltonian of a quantum harmonic oscillator with effective frequency , such that
Remarkably, is now the sole physical parameter that depends on the quench rate and is the sole scale which determines the dynamical behaviour at the critical point. We can now identify three regimes: (i) the limit , where the quench rate is much smaller than the gap and thus the dynamics are expected to be adiabatic except for small corrections. This regime is expected to provide the Landau-Zener scaling, where and the corrections to adiabaticity scale with damski2005 (); dziarmaga2010 (); polkovnikov2005 (); de_grandi2010 (). (ii) In the limit , on the other hand, the system approaches the thermodynamic limit where the dynamics are independent of to leading order in an expansion in . In this limit, thus, the dynamics, and in particular the excitations and the fidelity, are expected to be independent of . This result is consistent with the prediction of Ref. bachmann2017 (), which considered a slow quench of the frequency of a single harmonic oscillator, albeit with a different power law in time. The non-analytic regime is expected for intermediate values of .
Figure 2(b) displays the time evolution of and by integrating Eqs. (6)-(7) for values of in the three different regimes. The value of that we extract from these calculations is reported in Fig. 2(c) as a function of for constant. Here we observe the Landau-Zener scaling for , in agreement with our conjecture based on scaling arguments. For the excitation number tends to the constant value predicted by the thermodynamic limit, for the linear quench in the LMG model. Even though in the thermodynamic limit there is no power law scaling, the final number of defects still depends on the scaling of the gap at . It is therefore a universal value, and hints towards a close relation between the out-of-equilibrium dynamics and the equilibrium universal properties. Since the slope of the curve as a function of varies continuously, it contains also an interval of values with scaling . This scaling, which would agree with the KZ prediction, is clearly in a crossover regime and is not found in the thermodynamic limit. We find it instead for a semi-ramp which ends (starts) at the QCP. As we show in the S.M. 222See the supplementary materials, away from the adiabatic regime our model delivers the KZ scaling in the thermodynamic limit. This is in agreement with the predictions of Refs. polkovnikov2005 (); hwang2015 ().
In conclusion, we have shown that the slow quench dynamics presents different qualitative behaviours depending on whether the protocol ends at the critical point (half ramp) or deep in the other phase (full ramp). While the half ramp exhibits KZ scaling in the thermodynamic limit, for the full ramp the residual heat is constant and independent of . These predictions apply also to the Dicke model, which belongs to the same universality class as the LMG model and whose finite-size corrections to the gap have the same scaling properties with vidal2006 (). We remark that our predictions concern the quantum contribution to the heat when the mean-field spin follows adiabatically the magnetic field and thus hold when the non-adiabatic corrections of the mean-field energy are smaller than the quantum heat. Assuming that the semiclassical evolution is analytical in and no work is done on the system in a cyclic process in the adiabatic limit , the semiclassical contribution to the specific heat would scale as zwerger2008 (). The KZ scaling could then already be observed for , while the Landau-Zener scaling just depends on model-specific, non-universal parameters. These observations also suggest that the scaling found in Ref. caneva2008 () for the slower quenches is dominated by the mean-field dynamics, where the quantum contribution to the heat is not yet visible.
Our predictions could be experimentally verified in chains of hundred ions with tailored all-to-all interactions zhang2017 (). The regime corresponding to , where the quantum residual energy tends to a constant, is going to be a very small correction to the mean-field scaling, yet it could be observed in simple systems such as the Rabi model hwang2015 (). Our study, moreover, provides insight into the behavior observed for quenches in systems of ultracold atoms in cavity quantum electrodynamics klinder2015 (); hruby2018 (), where similar results as the ones shown in Fig. 2(a) were reported for the order parameter of the Dicke phase transition. A systematic comparison with these works requires the development of a model including noise and dissipation laguna1997 (); de_chiara2010 (). These are essential features of cavity quantum electrodynamics setups keller2018 () and will be the subject of future work.
Acknowledgements.N.D. acknowledges fruitful discussions with G. Gori, G. M. Graaf, S. Ruffo and A. Trombettoni. G.M. is grateful to L. Hruby, S. Jäger, H. Ritsch, and V. Torggler for stimulating discussions. Financial support by the DFG Collaborative Research Centre “SFB 1225 (ISOQUANT)”, by the DFG DACH project “Quantum crystals of matter and light”, and by the German Ministry of Education and Research (BMBF) via the Quantera project “NAQUAS” is acknowledged.
Appendix A Exact solution and Defect density
Before outlining the solution strategy for our problem a comment is in order here. The dynamics of each oscillator can be solved exactly Lewis1967 (); Lewis1968 (); Lewis1969 () and any dynamical state in the representation of the coordinate can be expressed as
where are time independent constants and the dynamical eigenstates are given by
The effective frequency can be expressed in terms of the effective width as
while is the total phase accumulated at time and reads
The exact time evolution of each harmonic oscillator can be then fully described by a single complex parameter, i.e. the effective width , which satisfies the Ermakov–Milne equation:
If the initial state is a pure state of the basis (A), say, the ground state, then all the coefficients of Eq. (11) vanish except for the coefficient . This holds also at all later times, and thus in the exact dynamical basis (A) no excited states will be generated. However at each time the dynamical pure state will, in general, be different from the instantaneous equilibrium ground state since the effective width will not coincide with the equilibrium basis , whose wave functions read
Therefore, if we decompose any pure state of the dynamical basis using the instantaneous equilibrium basis, it will generally contain a number of equilibrium excitations. Then, assuming to start the evolution in the equilibrium ground state at , the number of excitations in the instantaneous equilibrium basis at time is given by Dabrowski2016 ()
where the coefficients are the transition amplitudes between the dynamical state and the instantaneous equilibrium basis
The definition (17) can be calculated by choosing different basis sets for the evaluation of transition amplitudes rather than the eigenstates given in (16) Dabrowski2016 (). However the basis of the eigenstates in (16) is the most natural choice in the context of the Kibble-Zurek mechanism.
We perform a change of variable and cast the integral as follows:
We then employ the generating function for Hermite polynomials in the integral:
Thus the probability of having excitations in the evolved state at the time is given by
We insert this expression in Eq. (17) and obtain the number of excitations at time :
From this expression we can also determine the ground state fidelity , which reads:
Appendix B Slow quench dynamics starting at the critical point
We here consider the semi ramp case, when the magnetic field is continuously quenched from the critical point far into the symmetry broken phase. The dynamical protocol is still a linear ramp of the magnetic field and the evolution begins at criticality () with the system lying in its instantaneous ground state. For the harmonic oscillator frequency varies as
where, once again, we discarded time dependent finite size corrections and sub-leading terms which do not modify the universal behavior as well as unimportant numerical factors.
It is convenient to employ the rescaling
already introduced in the main text. The Ermakov-Milne equation now reads
with the rescaled frequency given by
where . From now one we will discard the s̃uperscript over rescaled quantities, since they are the only ones appearing in the following calculations. The solution of latter equation can be constructed from the motion of the associated classical Harmonic oscillator
This equation admits the two independent solutions
where we omitted the dependence on . The functions and are Airy functions. The two functions and have the constant and finite Wronskian
It is convenient to rewrite the solutions of equation (26) as a pair of complex conjugate solutions and with
where and are constants. Since Eq. (28) is homogeneous one can rescale the two solution by a constant factor, then, without loss of generality, one can impose . The function
is solution of the Ermakov-Milne equation (15) if
which univocally fixes the imaginary part of . In order to completely define the solution solution one should find the appropriate value of which satisfy the boundary condition
By inverting this expression one readily obtains
When the thermodynamic limit is taken first then the right-hand side (r.h.s.) of this expression diverges. We then consider the limit when can be neglected in the argument of the Airy functions but the r.h.s. of Eq. (36) remains finite. We obtain the expression
Therefore the quantity diverges in the thermodynamic limit for finite values of the ramp velocity . Employing the asymptotic expression for the Airy functions and neglecting oscillatory terms as it was done in polkovnikov2008 (), the asymptotic time limit of the scale parameter is
which, once inserted into Eq. (22), leads to the relation:
which scales as . This expression is equivalent to the one obtained in polkovnikov2008 () for a gapless system and suggests a KZ scaling for large sizes.
Appendix C Slow quench dynamics ending at the critical point
Let us now consider the case of a slow quench starting in the paramagnetic phase and ending at the critical point. The magnetic field is given by the expression , the dynamics starts at in the paramagnetic phase. The magnetic field slowly approaches the critical point at . The solution to Eq. (26) is still given by Eqs. (32) and (33) but with the boundary conditions
These conditions are consistent with the system being in the adiabatic ground state at large . In the limit diverges and one must use the asymptotic expansion for the Airy functions
In order to fulfill conditions (40), the oscillatory terms in the expression for must cancel for large , leading to
Moreover one has to impose the condition
which fully determines the coefficients in Eq. (32)
The resulting expression for the scale factor is
the number of defects is given by the formula
which is identical to Eq. (22), since this quantity is invariant under the rescaling in Eq. (25). The number of defects at the final point of the ramp (which is the critical point) is obtained by evaluating Eq. (49) at . At this instant the rescaled frequency is given by its finite size correction , while the scale factor reads,
Let us consider the thermodynamic limit first . In this case the argument of the Airy functions goes to zero and the terms in the square brackets of Eq. (49) read
which inserted into the defect density in Eq. (49) lead to the result
where we restricted to the leading term in the limit. Therefore the result for the number of excitations diverges in the thermodynamic limit with a power . However the residual heat is finite since it is obtained by multiplying the divergent defect density by the vanishing oscillator frequency leading to
which agrees with the KZ scaling of Ref. hwang2015 (). For a finite size system the slow ramp limit coincides with the limit of Eq. (49) evaluated at . The leading term in this case is generated by the velocity correction to the effective frequency
which substituted into Eq. (49) evaluated at gives
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