6detailed calculation-paraan.pmd Average Work Done in a Ground State Quench 54 SCIENCE DILIMAN (JULY-DECEMBER 2016) 28:2, 54-64 Detailed Calculation of the Average Work Done in a Ground State Quench of the Quantum Ising Model Salvador T. Laurente Jr. University of the Philippines Diliman Francis N.C. Paraan* University of the Philippines Diliman ABSTRACT A detailed derivation of the exact expression for the average work done in a transverse f ield quench of the quantum Ising model ground state is presented. In the thermodynamic limit, it is proved that the average work done generally has inflection points at the critical f ield for the pre-quench quantum phase transition. The f irst divergent f ield derivative i s c a l c u l a t e d a n d i s s h o w n t o d i v e r g e l o g a r i t h m i c a l l y. We a l s o demonstrate that the average work done is equal to the product of the transverse magnetization of the pre-quench ground state and the change in the magnetic f ield. K e y w o r d s : Q u a n t u m q u e n c h , q u a n t u m I s i n g m o d e l , q u a n t u m p h a s e transition _______________ *Corresponding Author ISSN 0115-7809 Print / ISSN 2012-0818 Online INTRODUCTION An initially prepared quantum state evolves unitarily after a rapid change in the Hamiltonian in a quantum quench. The system is isolated from heat and particle baths, and its response time is much longer than the time scale of the Hamiltonian change, so that no adiabatic approximation can be made. Quenches like these generally lead to excited (non-stationary) states, and the resulting non-equilibrium dynamics has been the subject of several investigations on quantum many-body physics in the absence of decohering environmental effects (Polkovnikov et al. 2011; Eisert et al. 2015). Preparing suff iciently isolated states under controllable S.T. Laurente Jr. and F.N.C. Paraan 55 conditions is a technical challenge but experimental progress in the last two decades has allowed quench dynamics to be observed in, for example, ultracold gases in optical lattices (Greiner et al. 2002; Cheneau et al. 2012). Because these isolated quantum systems evolve in time without exchanging matter and heat with a reservoir, conventional statistical mechanical notions, such as temperature, entropy, and equilibriation, are not directly applicable to them. Thus, new theoretical frameworks have been developed to investigate quantum quenches and the novel phenomena that are associated with them. Some recent new discoveries include the emergence of persistent fluctuations despite the absence of energy and matter exchanges with a bath (Rossini et al. 2009; Häppölä et al. 2012; Cosme and Fialko 2014), universal scaling behavior in the vicinity of quantum critical points (Silva 2008; Jacobson et al. 2011; Gambassi and Silva 2012), and the generation of quantum entanglement entropy (Fagotti and Calabrese 2008; Cardy 2011; Daley et al. 2012; Schachenmayer et al. 2013; Alba and Heidrich-Meisner 2014; Alkurtass et al. 2014; Torlai et al. 2014). In this manuscript we consider the case of a zero temperature instantaneous quench of the transverse magnetic f ield h in a quantum Ising model. The Hamiltonian is (1), where is the a-Pauli operator on site j. There are N spins in the chain and periodic boundary conditions are imposed. The system is thermally isolated and prepared in the ground state of the pre-quench Hamiltonian H 0 = H(h 0 ) with f ield h 0 . At time t = 0 + the f ield is suddenly changed to h 1 , and the system subsequently evolves according to the post-quench Hamiltonian H 1 = H(h 1 ). Throughout this work, the subscript index will refer to pre-quench (post-quench) quantities. The quantity of interest in this study is the average work done (or quantum work) in these quenches under the formalism developed by Talkner et al. (2007, 2016). This quantum work W is def ined as the difference between a projective measurement of the system energy at some time T after the quench and the initial ground state energy at time t < 0 . That is, if is the state vector of the system at time T, the work done is (2). ) 1 2 σ σ 1 σ 1 | 0)〉 0 1) | )〉 )| 1 | ) 0)| 0| 0) Average Work Done in a Ground State Quench 56 ) 1 2 0) 1 0 0) (5). For a given measurement of quantum work in this quench protocol, one gets the result W m = E 1 (m) – E 0 , where E 0 is the pre-quench ground state energy and E 1 (m) is the m th energy eigenvalue of the post-quench Hamiltonian. This result occurs with total probability so that the quantum work distribution p(W) is equal to (3). Taking the Fourier transform of this distribution gives the characteristic function (4), whose logarithm in G(u) is the generating function for the cumulants of p(W). From the known properties of this cumulant generating function, the average work done is 〈W〉 = _ilim u→0∂ ln G(u)/∂u and the variance in the work done is ∑ 2 =_lim u→0 ∂2 ln G(u)/∂u2. These statistical measures of the work done in a quenched quantum Ising model have been investigated at zero initial temperature (Silva 2008), non- zero temperatures (Dorner et al. 2012), and the more general anisotropic XY model (Bayocboc Jr. and Paraan 2015). These studies have shown that signatures of the quantum phase transition of the model, such as the vanishing excitation gap, can be revealed through an analysis of the work statistics. In this paper, we focus on providing a detailed derivation of the exact solution for the average work done along the quantum Ising line to complement the previous work of Silva (2008) and Bayocboc Jr. and Paraan (2015). In particular, we will prove that the average work done is not an analytic function of the pre-quench f ield h 0 when the excitation gap closes at the quantum critical point. The characteristic function G(u) is given as an expectation value (4) with respect to the pre-quench ground state . It is convenient to calculate this quantum average in the basis of single-particle fermionic Jordan-Wigner Fock states. The result is (Silva 2008; Dorner et al. 2012): ) 0) 1 0 0) |0) | 1 )| 0) | 2 | 0)〉 ) ΔΕ cos2 Δ 1 ) sin2 Δ 1 0 S.T. Laurente Jr. and F.N.C. Paraan 57 In the previous equation, is the difference between the ground state energies of the post-quench and pre-quench Hamiltonians. Additionally, the angle is the difference between the post-quench and pre-quench Bogolyubov angles. These Bogolyubov angles satisfy Finally, the energy spectrum of elementary excitations is (7), where for odd N or for even N. The effect of N being odd or even is negligible in the thermodynamic limit . The excitation gap closes at the critical f ield value which signals a quantum phase transition between a ferromagnetic (⏐h i ⏐< 1) and paramagnetic (⏐h i ⏐>1) ground state. The average work done can be calculated from the f irst derivative of ln G, which yields We now take the thermodynamic limit , and f ind that the average work per spin becomes the def inite integral ∆n = θ1(qn) − θ0(qn) ∆E = E1 − E0 ) cos )2 sin2 qn 2πn/N qn 2π n ½)/N → ∞ ±1, 〈 〉 1 2 0 ) 1 ) cos 2Δ 1 0 → ∞ 〈 〉 ≡ lim →∞ 〈 〉⁄ 〈 〉 0 1 4 0 cos 0 cos )2 sin2 2 0 (9). The average work per spin is singular when the pre-quench Hamiltonian is quantum critical (h 0 = ±1). In the following section we prove that 〈w〉 is indeed non-analytic (not inf initely differentiable) at these quantum critical points. tan 2 ) sin cos (8). (6). Average Work Done in a Ground State Quench 58 AVERAGE WORK DONE The symmetry and periodicity of the trigonometric functions allow us to put the integral (9) in the form (10). After factoring and rearranging terms, the resulting integrals are recognized as the complete elliptic integrals (11), (12), with modulus We f ind thatκ. (13), where Also, from the known connection formulas for the complete elliptic integrals (Gradshteyn and Ryzhik 2007; Olver et al. 2010) (14), (15), (16), we can simplify our result to (17). A similar result for this compact formula (17) has been reported without derivation by Bayocboc Jr. and Paraan (2015). 〈 〉 0 1 0 1 2 sin2 0 1)2 4 0 sin2 /2 0 . ) 1 2 sin2 ) 1/2 /2 0 ) 1 2 sin2 )1/2 /2 0 〈 〉 0 1 2 0 | 0 1| 0 1 0 1 ) ) 2 4 0 0 1) 2⁄ 1. ) 1 0) 0), ) 1 1 0 2 0) 1 0 2 ) 0) , 1) 1 ) 1 2) ) 〈 〉 sgn 0 0 1) 0 1) S.T. Laurente Jr. and F.N.C. Paraan 59 SINGULARITIES AT CRITICALITY The exact expression (17) for reveals that the average work per spin is not analytic at (⏐h 0 ⏐= 1), since Ε(κ) is not inf initely differentiable at⏐κ⏐= 1. As seen in graphs of 〈w〉 at different pre-quench and post-quench f ields (Figure 1), quantum criticality is manifested as a sudden change of curvature of the average work done at the pre-quench critical f ields h 0 = ±1. In fact, we now prove that these points are actually inflection points by explicitly calculating the needed derivatives. 〈 〉 Figure 1. The average work done per spin 〈w〉 for different post-quench f ields h 1 generally have inflection points at the pre-quench critical f ields h 0 = ±1 . The f irst derivative of with respect to h 0 is exactly (18). This derivative is graphed in Figure 2 for several f ield quenches and we verify that it generally diverges at the critical points . The only exceptions are when the post-quench f ield is also at the same critical value as the post-quench f ield . We now investigate the divergence at critical h 0 using the asymptotic formulas for the elliptic integrals at unit modulus (Olver et al. 2010). The leading term of the asymptotic expansion of the exact derivative (18) about the critical f ield h 0 = 1 is (19). 〈 〉 〈 〉 0 sgn 0 0 1 0 1 ) 0 1) 0 1 ) 0 ±1 0 1 ±1 〈 〉 0 1 1 1) 2 ln 0 1 8 , 0 1 Average Work Done in a Ground State Quench 60 This result shows that the leading divergence in the derivative ∂〈w〉/∂h 0 is logarithmic in⏐h 0 –1⏐. This divergence vanishes when h 0 = h 1 = 1. Similarly, an expansion about h 0 = –1 yields These approximations about h 0 = ±1 are compared with the exact results in Figure 3 for the case when the post-quench Hamiltonian is also critical (h 1 = ±1). The logarithmic divergence is seen to be captured by the approximate expansions about h 0 = ±1 (dashed lines), and we f ind that the derivative remains continuous when h 0 = h 1 = ±1: (21). To f inally prove that the average work per spin has inflection points at critical pre- quench f ields, we need to verify that the second derivative changes sign at h 0 = ±1, as depicted in Figure 4. Let us def ine the complement . The second derivative becomes (22). Figure 2. The derivative ∂〈w〉/∂h 0 diverges at the pre-quench critical f ields h 0 = ±1, except when h 1 = h 0 . (20). 2〈 〉/ 0 2 2 〈 〉 0 2 sgn 0 0 2 0 ′ 2 0 3 0 2 1 2 1) 0 1) 0 ′ 2 0 1) 0 1 ) 0 ′ 2 ≡ 1 0 2 〈 〉 0 1 1 1) 2 ln 0 1 8 , 0 1 〈 〉 0 0 1 ±1 ± 1 S.T. Laurente Jr. and F.N.C. Paraan 61 Figure 3. The derivative (solid line) diverges logarithmically at the critical f ields except at h 0 = h 1 = ±1. The approximate expansions about h 0 = ±1 are also shown (dashed lines). 〈 〉/ 0 Figure 4. The second derivative changes sign at the pre-quench critical f ields h 0 = ±1, except when h 1 = h 0 . In the special case when both Hamiltonians are at the same critical f ield h 0 = h 1 = ±1 (black arrows), the second derivative of the average work per spin diverges logarithmically at h 0 = ±1. 2〈 〉/ 0 2 Again, asymptotic expansions about h 0 give the following leading order divergences when h 0 ≠ h 1 ≠1: (23). 2〈 〉 0 2 ~ 1 2 1 ∓ 1 0 ∓ 1 , 0 ±1) Average Work Done in a Ground State Quench 62 Since this derivative changes sign at h 0 = ±1, we conf irm that the average work per spin indeed has inflection points when the pre-quench f ield is at its critical value and h 1 ≠ h 0 . This time, in the special case that the post-quench field is critical h 1 = ±1, the leading divergence of the second derivative at h 0 = h 1 is logarithmic: (24). DISCUSSION AND CONCLUSION We have calculated the average work done in a zero temperature f ield quench of the quantum Ising model. In the thermodynamic limit, we presented an exact expression for the average work per spin in terms of complete elliptic integrals. We have shown that the average work per spin is not inf initely differentiable at the pre-quench f ields h 0 = ±1, where a quantum phase transition occurs. When the post-quench f ield h 1 ≠ ±1, the f irst derivative d o e s n o t e x i s t a n d diverges logarithmically at h 0 = ±1. In this case, the non-analyticity at the critical point takes the form of an inflection point. When the post-quench Hamiltonian is also critical, the non-analyticity at h 0 = h 1 = ±1 is weakened. That is, the f irst derivative at h 0 = h 1 exists and the logarithmic divergence f irst appears in the second derivative . It is interesting to note that complete elliptic integrals are encountered frequently in the study of the quantum Ising model. In fact, the derived expression for the average work done (10) is similar to that for the transverse magnetization of the quantum Ising model (Niemeijer 1967). Some exact solutions on a chain of spins 1/2. Physica 36:377-419). That is, if is the average magnetization of the ground state of the Ising model with f ield h 0 : then . Thus, the logarithmic divergence (19) of the f ield derivative at the pre-quench quantum critical point also mirrors the logarithmic divergence of the zero-temperature magnetic susceptibility of an unquenched Ising model at the critical f ield h = ±1, where the excitation gap vanishes and the quantum phase transition occurs. 〈 〉 〈 〉/ 0 →0 ⁄ 2〈 〉 0 2 1 ±1 ~ 5 2 1 ln 0 ∓ 1 8 , 0 ±1) 〈 〉/ 0 2〈 〉/ 0 2 lim →∞ 1 0)| | 0) 1 (25), 〈 〉 0 1) 2⁄ 〈 〉/ 0 S.T. Laurente Jr. and F.N.C. Paraan 63 ACKNOWLEDGEMENTS The authors are supported by the University of the Philippines Diliman Off ice of the Chancellor and the Off ice of the Vice-Chancellor for Research and Development through Project No. 141420 PhDIA. NOTE Some of the results in this paper were f irst reported in the research poster PB-48 at the 33rd Physics Congress of the Samahang Pisika ng Pilipinas and appeared in its Conference Handbook. REFERENCES Alba V, Heidrich-Meisner F. 2014. 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P a r a a n < f p a r a a n @ n i p . u p d . e d u . p h > i s a n A s s i s t a n t P r o f e s s o r a t t h e N a t i o n a l Institute of Physics, University of the Philippines Diliman. He is the Program Coordinator of the NIP Structure and Dynamics Research Program (www.nip.upd.edu.ph/sand).