Non-Equilibrium Quantum Spin Dynamics from Classical Stochastic Processes S. De Nicola,1 B. Doyon,2 and M. J. Bhaseen3 1IST Austria, Am Campus 1, 3400 Klosterneuburg, Austria 2Department of Mathematics, King’s College London, Strand, London WC2R 2LS, United Kingdom 3Department of Physics, King’s College London, Strand, London WC2R 2LS, United Kingdom Following on from our recent work, we investigate a stochastic approach to non-equilibrium quan- tum spin systems. We show how the method can be applied to a variety of physical observables and for different initial conditions. We provide exact formulae of broad applicability for the time- dependence of expectation values and correlation functions following a quantum quench in terms of averages over classical stochastic processes. We further explore the behavior of the classical stochastic variables in the presence of dynamical quantum phase transitions, including results for their distributions and correlation functions. We provide details on the numerical solution of the associated stochastic differential equations, and examine the growth of fluctuations in the classical description. We discuss the strengths and limitations of the current implementation of the stochastic approach and the potential for further development. I. INTRODUCTION formation on the stochastic approach itself and its nu- merical implementation. We also present new results on the dynamics of the classical stochastic variables, includ- The experimental realization of isolated quantum ing stochastic bounds on the Loschmidt rate function. many-body systems [1–6] has led to intense theoretical For other recent work exploring the connections between interest in their unitary time-evolution [7, 8]. The study quantum and classical dynamics see Refs [35–37]. of quantum quenches [9, 10] has provided fundamental in- The layout of this paper is as follows. In Section II we sights into their non-equilibrium behavior, including the recall the principal steps involved in the stochastic ap- absence of thermalization in low-dimensional integrable proach to quantum spin systems, adopting the notations systems [1, 11] and the role of the Generalized Gibbs En- of Refs [33, 34]. In Section III we show how quantum semble (GGE) [12–14]. This has stimulated the develop- observables can be computed in the stochastic formal- ment of new theoretical tools and methodologies, ranging ism providing results of general applicability for spin-1/2 from the quench action approach [15, 16] to recent appli- systems. In Section IV we illustrate the method by con- cations of hydrodynamics [17–21]. This has been comple- sidering quenches in the quantum Ising model, in one and mented by significant advances in numerical simulation two spatial dimensions. In Section V we investigate the techniques [22–27]. The theoretical prediction of dynam- relationship between DQPTs and the classical stochastic ical quantum phase transitions (DQPTs) [28, 29], which variables. In Sections VI and VII we discuss the strengths occur as a function of time, has been recently confirmed and limitations of the stochastic approach, exploring the using trapped ions [30]. These experiments provide a growth of fluctuations in the classical variables and the new set of tools for exploring the time-resolved dynamics computational cost of numerical simulations. We con- of quantum many-body systems using paradigmatic spin clude in Section VIII, summarizing our findings and in- Hamiltonians. dicating directions for future research. We also provide Recently, a theoretical approach to quantum spin sys- appendices on the technical details of the stochastic ap- tems has emerged, based on a mapping to classical proach and its numerical implementation. stochastic processes [31–34]. The procedure begins by decoupling the exchange interactions between spins using Hubbard–Stratonovich transformations. This yields an II. STOCHASTIC FORMALISM exact description in terms of independent quantum spins, where the effect of interactions is represented by Gaus- In this section we recall the principal steps involved arXiv:1909.13142v2 [cond-mat.stat-mech] 20 Dec 2019 sian distributed stochastic fields. Quantum expectation in the stochastic approach to quantum spin systems [31– values are then expressed as classical averages over these 34]. Following Refs [33, 34], we begin our discussion with stochastic fields. In recent work [34], we showed that a generic Heisenberg Hamiltonian this approach could be used to calculate the expectation values of time-dependent quantum observables, including Hˆ = abSˆaSˆb haSˆa, (1) − Jij i j − j j the experimentally measurable Loschmidt rate function ijab ja X X and the magnetization. We also verified that this ap- proach could handle both integrable and non-integrable where i, j indicate lattice sites and a,b label the spin models, including those in higher dimensions. Here, we components. The spin operators satisfy the su(2) com- ˆa ˆb abc ˆc extend our previous work in a number of directions, pro- mutation relations [Sj , S ] = iǫ δjkS , where a,b,c k k ∈ viding results for a broader range of observables under x,y,z , ǫabc is the antisymmetric symbol and we set different initial conditions. We also present more in- ~{ = 1.} The exchange interactions ab and the fields Jij 2 a 1 hj can, in general, be time-dependent. Away from equi- is the identity matrix, the noise action in Eq. (4) can librium, unitary dynamics under Hˆ is governed by the be recast in the diagonal form time-evolution operator tf 1 a a S[φ]= φ (t′)φ (t′) dt′, (6) tf i i t 2 Uˆ(t ,t )= T exp i dt Hˆ (t) , (2) ia Z i f i − X Zti a where φi are real-valued Gaussian white noise variables a a b where t and t denote the initial and final times, and T satisfying φ (t) = 0, φ (t)φ (t′) = δ(t t′)δij δab; see i f h i i h i j i − denotes time-ordering. In general, the time-evolution op- Appendix B. This yields a probabilistic interpretation of erator Uˆ(tf ,ti) is non-trivial, due to the quadratic spin Eq. (3) as an integral over Gaussian weighted stochastic a interactions in Hˆ , the non-commutativity of the spin op- paths φi (t) [31, 33]. The time-evolution operator can erators, and the time-ordering. However, some of these thus be written in the form difficulties can be circumvented in a two-step process. tf ′ ′ i Φa(t )Sˆadt ˆ Uˆ(t ,t )= Te ti ja j j , (7) First, the quadratic spin interactions in H can be decou- f i R P φ pled exactly using Hubbard–Stratonovich (HS) transfor- a a ab b mations. This leads to a physically appealing descrip- where Φj = hj + kb Ojkφk/√i and ... φ denotes tion in terms of independent quantum spins which are averaging with respect to the Gaussianh weighti given coupled via Gaussian distributed stochastic “magnetic” by Eq. (6). Equivalently,P Eq. (7) describes the time- fields [31, 33]. Second, the time-ordered exponential in evolution of individual decoupled spins moving under the a Eq. (2) can be recast as an ordinary exponential; the HS action of applied and stochastic “magnetic” fields, hj (t) decoupling renders the exponent linear in the su(2) gen- ˇa a √ ab b √ and hj (t) ϕj / i = kb Ojkφk(t)/ i, respectively. erators, allowing a simpler parameterization via group Although the≡ spins appear to be fully decoupled in the theory [32, 33]. This so-called disentanglement transfor- representation (7), the effectP of the interactions is en- mation [33, 34] can be regarded as a judicious parame- ˇa ab coded in the fields hj (t) via the matrix Ojk. Each spin terization of the time-evolution operator which takes ad- is governed by an effective stochastic Hamiltonian vantage of the Lie algebraic structure of the spin opera- tors. In Sections IIA and IIB we recall these two steps Hˆ s(t)= ha(t)+ hˇa(t) Sˆa. (8) j − j j j in turn, before summarizing the resulting stochastic dif- a ferential equations (SDEs) [33, 34]. In Section II C we X discuss the Ito form of these SDEs, which is useful for In general, this is non-Hermitian, as the stochastic fields ˇa numerical simulations. hj may be complex valued. Without loss of generality, in the remainder of this work we consider time-evolution over the interval [0,t] and set Uˆ(t) Uˆ(t, 0). A. Hubbard–Stratonovich Transformation ≡ B. Disentanglement Transformation As in Refs [31–34], the quadratic spin interactions can be decoupled via a HS transformation [38, 39] over aux- a The time-evolution operator defined by Eq. (7) is still iliary variables ϕj . Trotter slicing [40] the exponential in Eq. (2) and applying the HS transformation at each time non-trivial due to the time-ordering operation. However slice (Appendix A) one obtains the decoupled exponential is now linear in the spin oper- ators, and can therefore be simplified using group theory t ′ ′ S[ϕ]+i f Φa(t )Sˆa dt [31–33]. Specifically, one may rewrite the time-evolution Uˆ(t ,t )= T ϕ e− ti ja j j , (3) f i D R P operator acting at a given site j as Z ′ ′ − − i t Φa(t )Sˆadt ξ+(t)Sˆ+ ξz(t)Sˆz ξ (t)Sˆ where we refer to Te a 0 j j e j j e j j e j j , (9) P R ≡ tf 1 1 ab a b where the parameters ξa(t) are referred to as disentan- S[ϕ]= ( − ) ϕ (t′)ϕ (t′) dt′, (4) j 4 J ij i j gling variables [33]. This is also known as the Wei– ijab Zti X Norman–Kolokolov transformation [41, 42]. The rela- a noise action a a a √ tionship between the disentangling variables ξj (t) and as the .