arXiv:1909.13142v2 [cond-mat.stat-mech] 20 Dec 2019 ieetiiilcniin.W lopeetmr in- more present also under We observables of conditions. range broader initial pro- a directions, different of for we number results a Here, in viding work dimensions. previous ap- higher our this in extend non-integrable those that and including integrable verified models, both also handle We could proach magnetization. function the rate Loschmidt and measurable that experimentally expectation showed the including the observables, we calculate quantum time-dependent to [34], of used values work be recent could approach In this these over averages fields. classical as stochastic expectation expressed Gaus- Quantum then by are represented values fields. is stochastic interactions distributed of sian effect spins, the quantum independent where of an terms in yields classical description This exact to by begins transformations. mapping using spins Hubbard–Stratonovich procedure between a interactions The exchange the on decoupling [31–34]. based processes emerged, stochastic has tems spin paradigmatic a using Hamiltonians. dynamics provide systems time-resolved the many-body experiments exploring quantum These for of tools confirmed of recently [30]. set been new ions has time, trapped of which using function 29], a [28, as (DQPTs) occur transitions dynam- phase of quantum prediction ical simulation theoretical numerical The in [22–27]. techniques advances comple- significant been by has appli- This mented recent to [17–21]. 16] hydrodynamics of [15, cations approach action ranging quench methodologies, the and develop-from tools the theoretical En- stimulated new Gibbs of has ment Generalized This the [12–14]. of (GGE) role semble the integrable and 11] low-dimensional [1, in systems the thermalization including of behavior, absence non-equilibrium their in- fundamental study into provided The sights has 10] 8]. theoretical [9, quenches [7, intense quantum time-evolution of to unitary their led in has interest [1–6] systems many-body eety hoeia praht unu pnsys- spin quantum to approach theoretical a Recently, quantum isolated of realization experimental The o-qiiru unu pnDnmc rmCasclStoc Classical from Dynamics Spin Quantum Non-Equilibrium 2 eateto ahmtc,Kn’ olg odn Strand, London, College King’s Mathematics, of Department prahadteptnilfrfrhrdevelopment. further for th potential of the limitations and and approach strengths the examine discuss We and provide description. We equations, p differential functions. quantum stochastic furthe correlation dynamical associated We and of distributions presence their processes. the var in stochastic a variables classical function to stochastic over correlation applied formu and averages be exact values of can provide We expectation method of conditions. the dependence initial how different show for We and systems. spin tum 3 olwn nfo u eetwr,w netgt stochast a investigate we work, recent Lond our Strand, from London, on Following College King’s Physics, of Department .INTRODUCTION I. 1 S uti,A aps1 40Kotrebr,Austria Klosterneuburg, 3400 1, Campus Am Austria, IST .D Nicola, De S. 1 .Doyon, B. 2 pedcso h ehia eal ftesohsi ap- stochastic implementation. the numerical its of provide and details also proach technical We in- the research. on and future appendices findings for our con- summarizing directions We dicating VIII, the Section simulations. in and numerical variables clude of classical cost the in computational the fluctuations exploring approach, of stochastic growth strengths the the of discuss we limitations VII and and VI the stochastic Sections In investigate classical the we variables. and V DQPTs Section between In relationship and one con- dimensions. in by spatial model, method Ising two quantum the the illustrate in we quenches sidering IV Section In systems. 4.FloigRf 3,3] ebgnordsuso with discussion Hamiltonian our Heisenberg begin generic we [31– 34], a systems [33, spin Refs Following quantum to 34]. approach stochastic the in fRf 3,3] nScinIIw hwhwquantum formal- spin-1 how for stochastic show applicability the general we of in III results providing computed Section ism In be can 34]. observables ap- notations [33, the stochastic Refs adopting the systems, of in spin quantum involved to steps proach principal the recall [35–37]. Refs see dynamics function. between classical connections rate and the Loschmidt quantum exploring work the recent on other For bounds includ- variables, stochastic stochastic on classical nu-ing results the its of new dynamics and present the also itself We approach implementation. stochastic merical the on formation opnns h pnoeaosstsythe satisfy operators spin The components. where { ~ uainrltos[ relations mutation ,y z y, x, n .J Bhaseen J. M. and nti eto ercl h rnia tp involved steps principal the recall we section this In h aoto hsppri sflos nScinI we II Section In follows. as is paper this of layout The .Teecag interactions exchange The 1. = h rwho utain nteclassical the in fluctuations of growth the urn mlmnaino h stochastic the of implementation current e eal ntenmrclslto fthe of solution numerical the on details olwn unu unhi terms in quench quantum a following s } ,j i, xlr h eairo h classical the of behavior the explore r a fbodapiaiiyfrtetime- the for applicability broad of lae , aetastos nldn eut for results including transitions, hase I TCATCFORMALISM STOCHASTIC II. capoc onneulbimquan- non-equilibrium to approach ic odnW2 L,Uie Kingdom United 2LS, WC2R London ǫ abc niaeltiestsand sites lattice indicate nW2 L,Uie Kingdom United 2LS, WC2R on H ˆ steatsmercsmo n eset we and symbol antisymmetric the is = − eyo hsclobservables physical of iety ijab X S ˆ j a 3 , J S ˆ ij ab k b = ] S ˆ i a atcProcesses hastic S ˆ iǫ j b abc − X δ ja jk J ,b a, S ˆ h ij ab k c j a where , S ˆ ae h spin the label n h fields the and j a , su 2 com- (2) ,b c b, a, (1) / ∈ 2 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 . Here, we define Φj hj + ϕj / i a and further denote ≡ the variables Φj (t) can be made more explicit by differ- entiating Eq. (9) with respect to time. This yields [33] ϕ ϕa, (5) D ≡ D j j + + z + +2 Y iξ˙ =Φ +Φ ξ Φ−ξ , (10a) j j j j − j j where ϕa is the appropriately normalized integration z z + j iξ˙ =Φ 2Φ−ξ , (10b) measureD for each HS variable ϕa. Introducing the change j j − j j j z a ab b T 1 iξ˙− =Φ− exp ξ , (10c) of variables ϕ = O φ , where O − O/2= 1 and j j j i jb ij j J P 3 where the identity Uˆ(0) = 1 implies the initial condi- the SDEs, which are naturally suited for discrete time- a tions ξj (0) = 0 for all j, a. For completeness, we provide evolution. The Ito SDEs can be obtained by including a detailed derivation of these equations in Appendix C. an extra drift term. However, in the case of an interac- ab Alternative disentanglement transformations, based on tion matrix ij with vanishing diagonal elements, the different group parameterizations, have also been consid- additional ItoJ drift term vanishes identically and the Ito ered in the literature [31, 33]. and Stratonovich SDEs coincide; see Appendix D. The Equations (10) may be regarded as stochastic differ- time-dependence of a function f(ξ) corresponding to a a ential equations (SDEs) for the variables ξj , due to the physical observable ˆ(t) can be found via Ito calculus. presence of the (additive and multiplicative) Gaussian For a generic Ito SDEO written in the canonical form, a noise entering via Φj [33]. Applying the disentanglement dξa transformation (10) to the time-evolution operator (7) i = Aa(ξ)+ Bab(ξ)φb , (15) one obtains [33, 34] dt i ij j Xjb s Uˆ(t)= j Uˆ (t) φ, (11) one obtains h⊗ j i ∂f 1 ∂2f where we have defined on-site stochastic operators f˙ = (Aa + Babφb)+ BacBbc , ∂ξa i ij j 2 ∂ξa∂ξb ik jk − − ia i jb ijab i j ck ξ+(t)Sˆ+ z ˆz ξ (t)Sˆ Uˆ s(t) e j j eξj (t)Sj e j j . (12) X X X X j ≡ (16) as follows from Ito’s lemma [43]. In principle, it is pos- In general, this is a non-unitary operator, since the time- sible to analytically average these SDEs with respect to evolution of each spin is governed by the non-Hermitian the HS fields; in this approach, one obtains a system of Hamiltonian (8). Given a specific spin representation, ordinary differential equations (ODEs) [33]. However, as ˆ s Uj (t) can be written in matrix form. For example, for we discuss in Appendix E, this is formally equivalent to 1 ˆa a diagonalizing the Hamiltonian, whose matrix dimension spin- 2 , we may write S =σ ˆ /2 in terms of the Pauli matricesσ ˆa, where a x,y,z . This yields scales as (2N 2N ), where N is the total number of ∈{ } spins. Instead,O it× is more convenient to numerically per- z z z ξj /2 ξj /2 + ξj /2 + form the average in (14) over independent realizations of s e + e− ξj ξj− e− ξj Uˆ (t)= z z . (13) j ξj /2 ξj /2 the stochastic process. In this approach, the number of e− ξj− e− ! a stochastic variables ξj that one needs to simulate scales linearly with N. Moreover, the independent runs can be The product form of the evolution operator (11) makes readily parallelized. In Section III, we will provide exact it convenient for acting on spin states of interest; using stochastic formulae for a variety of quantum observables this, the quantum matrix elements of an operator ˆ(t) O ≡ that can be described in this way. We will return to a Uˆ †(t) ˆUˆ(t) can be expressed as the classical average of a O more detailed discussion of the numerical aspects in Sec- function f(ξ), over realizations of the stochastic process: tions VI and VII. ψ ˆ(t) ψ = f(ξ(t)) . (14) h F|O | Ii h iφ III. QUANTUM OBSERVABLES Here, the function f(ξ) depends on the disentangling variables ξ ξa , and is determined by the observ- j In order to illustrate the stochastic approach to non- ˆ ≡ { } able , and the chosen initial and final states, ψI and equilibrium quantum spin systems, we obtain below the O |ˆ i ψF . In writing (14), we consider operators with- classical formulae for a range of quantum observables. |outi explicit time-dependence: in the HeisenbergO picture their time-evolution is determined solely by Uˆ(t). In the Schr¨odinger picture, the matrix elements can be recast A. Loschmidt Amplitude as ψ (t) ˆ ψ (t) , where ψ(t) Uˆ(t) ψ(0) . In Sec- h F |O| I i | i ≡ | i tion III we will provide some explicit examples of the One of the simplest quantities to investigate in the quantum-classical correspondence (14), for different ob- stochastic formalism is the Loschmidt amplitude A(t). servables and for different initial and final states. This is defined as the amplitude for an initial state ψ(0) to return to itself after unitary evolution [28]: | i

C. Ito Equations of Motion A(t)= ψ(0) Uˆ(t) ψ(0) . (17) h | | i SDEs are defined by specifying a discretization scheme In general, A(t) is expected to decay exponentially with [43], with the most common choices being the Ito and the system size N. It is therefore convenient to define Stratonovich conventions. The SDEs (10) are initially in the Loschmidt rate function the Stratonovich form. However, for numerical simula- 1 λ(t) log A(t) 2. (18) tions, it is often convenient to work with the Ito form of ≡−N | | 4

This plays the role of a dynamical free energy density, In contrast to the Loschmidt amplitude (17), this involves since A(t) is analogous to a boundary partition function two time-evolution operators. This can be addressed [44] that is Wick-rotated to real time. This connection by two independent HS transformations over variables a ˜ ˜a led to the insightful prediction of dynamical quantum φ φi and φ φi , with associated disentangle- ≡ { } ≡ { a } ˜ ˜a ˜ phase transitions (DQPTs) occurring in λ(t) as a func- ment variables ξ ξi [φ] and ξ ξi [φ] . For sim- tion of time [28]. In the thermodynamic limit N , plicity, we illustrate≡ this { in} the case≡ where { the} observable → ∞ ˆ ˆz these transitions correspond to non-analyticities in λ(t), of interest is a product of Si operators at different and often occur on quenching across a quantum critical Osites. This class of operators includes the longitudinal point [28, 29]. The existence of DQPTs has been recently magnetization as well as correlation functions. For prod- confirmed in experiment using trapped ions [30]. This uct initial states, the argument of the classical average is experiment provides a realization of the quantum Ising factorized over the sites i. A given observable ˆ can then model with 6 to 10 spins, interacting via tunable dipo- be expressed in the stochastic language by multiplyingO a lar interactions. This allows access to the time-resolved set of on-site “building blocks”, given in Appendix F. In dynamics of an isolated quantum spin system. this framework, local expectation values are expressed as Here, we consider A(t) for the generic Hamiltonian (1). averages of functions of ξ and ξ˜, describing the forwards In principle, this may contain long range interactions as and backwards evolutions respectively. For example, the in the experiment [30], but this is not the primary thrust dynamics of the local magnetization for a system initial- of our investigation. For simplicity, we focus on initial ized in the state ψ(0) = is given by [34] | i |⇓i states of product form, ψ(0) = i ψ(0) i. Parameter- izing a generic superposition| i as ψ⊗(0)| =ia + b , | ii i|↑ii i|↓ii where and refer to spin-up and spin-down in the z ˜z∗ |↑i |↓i ξj +ξj ˆz 2 2 1 j 2 Si basis, with ai + bi = 1, one obtains ˆz −   + ˜+ + ˜+ | | | | Si (t) = e P (1 ξi ξi ∗) (1+ξj ξj ∗) . h i −2 − φ,φ˜ j=i ξz D Y6 E i 2 ξz + (21) A(t)= e− 2 ai e i + (aiξ− + bi)(a∗ξ + b∗) . | | i i i i The structure of (21) is relatively straightforward. It con- * i + Y   φ sists of an exponential factor like that in (19), together (19) + + with polynomial factors (1 ξ ξ˜ ∗) for each site, where ± j j In the special case of a fully-polarized initial state the minus sign is used for the chosen site i. As discussed |⇓i with all spins down, corresponding to ai = 0 and bi = 1, in Section III C, a similar structure also emerges in the one obtains the result given in our previous work [34]. evaluation of correlation functions. Analogous results for Sα(t) with α = x, y are given in Appendix F. The result (19) is more general and allows consideration h i i of spatially inhomogeneous initial states. In Section IV we will discuss the numerical evaluation of (19) in the context of the quantum Ising model, including domain wall initial conditions. For now, we gather the stochastic formulae describing local observables.

C. Equal-Time Correlation Functions B. One-Point Functions

The dynamics of a local observable ˆ is encoded in the Correlation functions of local operators can be com- time-dependent expectation value O puted in a similar manner to that described above. For example, the two-point function of the local magnetiza- z z ˆ(t) = ψ(0) Uˆ †(t) ˆUˆ(t) ψ(0) . (20) tion C (t) Sˆ (t)Sˆ (t) is given by hO i h | O | i ij ≡ h i j i

z ˜z∗ ξk+ξk 1 k −  2  + ˜+ + ˜+ + ˜+ Cij (t)= e P (1 ξi ξi ∗)(1 ξj ξj ∗) (1 + ξk ξk ∗) , (22) 4 − − φ,φ˜ k=i,j D Y6 E

ˆz for quenches starting in . The structure of (22) mirrors sign of the polynomial is negative for each factor of Si in that of (21), where now|⇓i there are two polynomial factors the correlation function. More generally, the expectation with minus signs, for the chosen sites i and j. This result value of a product of local operators starting from a prod- is readily generalized to arbitrary multi-point functions uct state can be decomposed into averages of products of of the local magnetization, starting in the state ; the the elementary “building blocks” referred to above; see |⇓i 5

Appendix F. In the case of an initial state , equa- formalism, by decoupling each of the time-evolution op- tions (21) and (22) can be equivalently decomposed|⇓i into erators. For example, the two-time correlation function ∗ (ξz +ξ˜z )/2 + + ˆz ˆz a “background” factor e− j j j (1+ξ ξ˜ ∗), for Cij (t,t′) Si (t)Sj (t′) can be written as j P j j ≡ h i all the sites that are not involved in the observable, to- gether with a multiplicativeQ factor for each inserted local operator. This structure is reminiscent of the form of correlation functions obtained from the algebraic Bethe z z C (t,t′)= ψ(0) Uˆ †(t)Sˆ Uˆ(t)Uˆ †(t′)Sˆ Uˆ(t′) ψ(0) . ansatz, see e.g. Ref. [45], although the present results ij h | i j | i apply to both integrable and non-integrable problems. (23)

D. Dynamical Correlation Functions

Dynamical correlation functions involving operators at Starting in the initial state ψ(0) = and using | i |⇓i different times can also be expressed in the stochastic Uˆ(t)Uˆ †(t′)= Uˆ(t t′), one obtains −

1 1 z ∗ z ′ z ′ z ′ (ξ ,l(t) +ξ ,l(t t )+ξ ,l(t )) + + + + ξ ,i(t t ) + Cij (t,t′)= e− 2 l 1 2 − 3 (ξ− (t t′)ξ (t′)+1) ξ (t t′)ξ (t)∗ 1 + ξ (t′)e 2 − ξ (t)∗ 4 P 2,i − 3,i 2,i − 1,i − 3,i 1,i ′ D + + h + + ξz (t t ) +  i (ξ− (t t′)ξ (t′) 1) ξ (t t′)ξ (t)∗ +1 + ξ (t′)e 2,j − ξ (t)∗ 2,j − 3,j − 2,j − 1,j 3,j 1,j × h   z ′ i + ∗ + + + + ξ2,k (t t ) + ξ1,k(t) ξ2−,k(t t′)ξ2,k(t t′)ξ3,k(t′)+ ξ2,k(t t′)+ ξ3,k(t′)e − + ξ2−,k(t t′)ξ3,k(t′)+1 . − − − − φ1,φ2,φ3 k=i,j Y6    E (24)

Here we have introduced three sets of disentangling vari- where J > 0 is the ferromagnetic nearest neighbor ex- ables ξ1,2,3 which are functionals of three independent change interaction and Γ is the transverse field. For sim- Gaussian white noise fields φ1,2,3. Although the expres- plicity, we consider periodic boundary conditions with ˆa ˆa sion (24) is rather non-trivial, it is general to dynamical SN+1 = S1 . In equilibrium, the model (25) exhibits a correlations of arbitrary spin-1/2 Heisenberg models (1), quantum phase transition at Γ = Γ J/2 between a c ≡ without reference to integrability or dimensionality. ferromagnetic (FM) phase for Γ < Γc and a paramag- netic (PM) phase for Γ > Γc. Out of equilibrium, the dynamics of the Hamiltonian (25) is encoded in the Ito E. Higher Dimensions SDEs

Γ 2 ˙+ + + √ A notable feature of the stochastic approach is that it iξj = (1 ξj )+ ξj Ojkφk/ i, (26a) applies to systems in arbitrary dimensions. Due to the − 2 − Xk on-site character of the stochastic time-evolution oper- + iξ˙z = Γξ + O φ /√i, (26b) ators Uˆ s(t), all of the formulae obtained above readily − j − j jk k j k generalize to arbitrary dimensions: the products simply X ˙ Γ z extend over all the lattice sites. In Section IV we will iξj− = exp ξj , (26c) provide an example of this in the context of the two- − 2 dimensional quantum Ising model. 1 where O is defined by O − O = 2δ , and we jk kl kiJkl lj ij take a symmetrized interaction matrix = J (δ + P Jij 2 ij+1 IV. QUANTUM ISING MODEL δij 1) [46]. Before embarking on a detailed examination of− (26), it is instructive to consider some limiting cases. In the non-interacting limit J = 0, one has Ojk = 0, and In order to illustrate how the stochastic method can (26) reduces to a set of deterministic equations which be applied in practice, we consider quantum quenches in can be solved exactly. As expected, these describe a the one-dimensional (1D) quantum Ising model [34]. The set of decoupled spins precessing in an external magnetic Hamiltonian is given by field Γ; see Appendix G. In the limit Γ = 0, the model N N (25) is purely classical. In this case ξ±(t) = 0 for all z z x j HˆI = J Sˆ Sˆ Γ Sˆ , (25) z − j j+1 − j t, while ξj (t) undergoes exactly solvable Brownian mo- j=1 j=1 X X tion; see Appendix G. For generic values of Γ and J, the 6

SDEs (26) can be solved numerically, as we highlighted ED in our previous work [34]. 3 Throughout this manuscript, we solve the SDEs using SDE the Euler scheme [43]. We also set J = 1 and use a discrete time-step ∆t = 10 5 in all of the figures. For 2 − ) t (

any non-zero ∆t, numerical solution algorithms for non- λ linear SDEs can give rise to diverging trajectories where the stochastic variables grow without bound [43, 47]; for 1 the Ising SDEs, this effect is most pronounced for large transverse fields Γ. Empirically, trajectories are found to 0 monotonically grow to numerical infinity when 0 1.5 t ξ˙+(t) ∆t ξ+(t) , (27) | i | ≥ | i | + FIG. 1. Loschmidt rate function λ(t) for the 1D quan- i.e. when the increment in ξi (t) in a given time-step + | | tum Ising model following a quantum quench from the fully- exceeds the value of ξi (t) . In the case of the Ising model, the increment| is given| by Eq. (26a). Since the polarized initial state |⇓i to the PM phase with Γ = 16Γc. The results obtained from the SDEs (filled circles) are in good fields φi are of order one and Γ∆t is typically small, + agreement with ED (solid line) for N = 9 spins. The SDE Eq. (27) can only be satisfied for large ξ . The in- 5 | i | results were obtained by averaging over 10 realizations of the crement is then dominated by the term proportional to stochastic process. The fraction of diverging trajectories at +2 ξi , and the requirement (27) translates into a diver- the stopping time is of order 1%. | | + + + gence condition ξi (t) > ξc , where ξc 2/Γ∆t. Di- verging trajectories| can| therefore be identified≡ by com- + + paring ξi (t) to ξc at each time t. With our choice of ∆t, to Γ = 16Γc, across the quantum phase transition at Γc. we retain between 99% and 100% of the total number of The results obtained from the numerical solution of the trajectories, depending on the chosen parameters. The SDEs in (26) are in good agreement with ED. They also stochastic averages are performed by retaining only the correctly reproduce the sharp peak in λ(t) correspond- non-diverging trajectories at a given time t. Whenever ing to a DQPT in the thermodynamic limit [28]. Going trajectories are excluded, we report their relative fraction beyond our previous work [34], it is also possible to con- in the associated figure caption. We estimate the magni- sider quenches from the PM phase to the FM phase. For tude of the fluctuations on our results via the standard example, for quenches starting in the PM ground state x error se = σ/√nB, where σ is the standard deviation for Γ = , where Sˆ =1/2 , the |⇒i ≡ ⊗i|→ii ∞ i |→ii |→ii obtained by splitting the data into nB = 5 batches of general formula (19) reduces to trajectories; we omit the bars when they are comparable N z to, or smaller than, the plot points. In order to illustrate 1 ξi z 2 + + ξi the general approach, we focus on relatively small sys- A(t)= e− ξi−ξi + ξi− + ξi + e +1 . 2 φ i tem sizes with N 10 spins. This aids comparison with D Y  E Exact Diagonalization≤ (ED) using the QuSpin package (29) [48] and reduces the computational cost, whilst exposing In Fig. 2 we show the time-evolution of λ(t) following a quench from Γ = (PM) to Γ = Γ /4 (FM), computed the main features. We also confine ourselves to times ∞ c t . 1/J, before stochastic fluctuations become impor- from Eq. (29). Again, we find very good agreement with tant. In Sections VI and VII we will examine the scaling ED. It is worth noting that, in contrast to the simple re- of the method with increasing N and discuss the eventual sult (28), the expression (29) features a sum of terms in- breakdown with increasing t. side the average. However, from a computational stand- point, this only involves a linear increase in the number of operations required. Furthermore, the averaging need A. Loschmidt Amplitude not be performed at each time-step: while for numeri- cal accuracy the SDEs are solved with a small time-step 5 (e.g. ∆t 10− ), observables may be computed on a In order to illustrate the general approach, we be- ≈ 3 coarser time interval (e.g. ∆t¯ 10− ). The main com- gin by considering the Loschmidt amplitude for differ- ≈ ent quantum quenches. For systems initialized in the putational cost of the method is associated with solving the SDEs, rather than performing the averages. There- fully-polarized state i i, corresponding to a FM ground state of the|⇓i Hamiltonian ≡ ⊗ |↓i (25) when Γ = 0, the fore, the presence of the additional terms in Eq. (29) general formula (19) reduces to does not significantly affect the computational cost: this applies to all the other examples considered in this Sec- N ξz(t) tion. Finally, we note that Eq. (29) can be evaluated A(t)= exp i , (28) − 2 φ from the same set of trajectories as used in Eq. (28). i D Y   E Thus, in contrast with other numerical techniques such as as reported in our previous work [34]. In Fig. 1 we show time-dependent Density Matrix Renormalization Group the time-evolution of λ(t) following a quench from Γ = 0 (tDMRG) approaches or ED, the same data can be used 7

0.6 tions: ED SDE ψ(0) = ... , (30) | i |↑i1 ⊗ |↑iM ⊗ |↓iM+1 · · · ⊗ |↓iN where 1 M

t ≤

( 0.3 λ M N z z ξi (t) ξj + 2 A(t)= e + ξj−ξj e− . (31) φ j=1 i=1 D Y   Y E 0 In Figs 3 and 4 we show the results for λ(t) for different 0 3.5 values of M, corresponding to a single spin flip and an t extended domain of inverted spins respectively. Once FIG. 2. Loschmidt rate function λ(t) for the 1D quantum again, the results are in good agreement with ED. Ising model following a quantum quench from the initial state |⇒i, corresponding to the paramagnetic ground state when B. Magnetization Dynamics Γ= ∞, to the FM phase with Γ = Γc/4. The results obtained from the SDEs (filled circles) are in good agreement with ED (solid line) for N = 5 spins. The SDE results were obtained A key observable for non-equilibrium quantum spin 5 by averaging over 10 realizations of the stochastic process. systems is the time-dependent magnetization (t) = N ˆz M i i/N where i(t) = Si (t) . Here we consider quantumM quenchesM from the initialh i state to different finalP values of Γ in the PM phase. As can|⇓i be seen in Fig. 5, the results obtained by performing the stochas- tic average in (21) are in very good agreement with ED; here we focus on small system sizes with N = 3 spins as to compute the time-evolution of different initial states. we need to average over two sets of disentangling vari- ables, ξi and ξ˜i. Again, we may consider different initial conditions, such as the inhomogeneous state (30). For As discussed in Section III A, the stochastic approach example, for an initial state where the spin at site i is can also handle spatially inhomogeneous initial states. pointing up and every other spin is pointing down, the For example, we may consider domain wall initial condi- time-dependent magnetization at site i is given by

z ˜z∗ ξj +ξj 1 j 2 z ˜z ˆz −   ξi + ξi ˜ ˜+ ˜ + ˜+ i(t)= Si (t) = e P (e + ξi−ξi )(e + ξi−ξi )∗) ξi−ξi−∗ (1 + ξj ξj ∗) , (32) M h i −2 − φ,φ˜ j=i D h i Y6 E where the disentangling variables ξ satisfy the Ising where O is the same as for the purely transverse field SDEs (26). This result can be obtained using the build- Ising model, as given in Section IV. In Fig. 6, we show re- ing blocks given in (F1) and (F2) of Appendix F. sults for (t) corresponding to quenches from the fully- A significant feature of the stochastic approach to non- polarizedM initial state to different values of Γ, with equilibrium quantum spin systems is that it is not re- h held fixed. Once again,|⇓i we find very good agreement stricted to integrable models. A simple way to break the with ED. It is interesting to note that the same formula integrability of the quantum Ising model (25) is through (21) governs the dynamics in both the integrable and the addition of a longitudinal magnetic field h, so that non-integrable cases; the Hamiltonian enters only via the ˆ ˆ ˆz a the Hamiltonian is given by H = HI + h j Sj . In the time-evolution of the disentangling variables ξi , not the stochastic formalism the dynamics of this non-integrable function being averaged. model is described by the Ito SDEs P As discussed in our previous work [34], the stochastic approach can also be used in higher dimensions. For sim- Γ 2 ˙+ + + + √ plicity, we focus on the two-dimensional (2D) quantum iξj = (1 ξj ) hξj + ξj Ojkφk/ i, (33a) − 2 − − Ising model with the Hamiltonian Xk ˙z + √ iξj = h Γξj + Ojkφk/ i, (33b) ˆ 2D ˆz ˆz ˆx − − − HI = J Si Sj Γ Si , (34) k − − X ij i h i Γ z X X iξ˙− = exp ξ , (33c) − j 2 j where i and j indicate sites on a square lattice. In equi- 8

2 ED 0.5 ED )

SDE t SDE (a) ( 0.0 M −0.5 ) t

( 1 λ 0.5 ED )

t SDE (b) ( 0.0 M −0.5 0 0 1.7 0.5 ED )

t t SDE (c) ( 0.0 M FIG. 3. Loschmidt rate function λ(t) for the 1D quantum −0.5 Ising model following a quantum quench from the spatially inhomogeneous initial state |↑↓↓↓↓i to the PM phase with Γ = 0 1 2 t 4Γc. The results obtained from the SDEs (filled circles) are in good agreement with ED (solid line) for N = 5 spins. The SDE results were obtained by averaging over 105 realizations FIG. 5. Time-evolution of the magnetization M(t) for the of the stochastic process. Less than 0.1% of the trajectories 1D quantum Ising model following quantum quenches from were found to be divergent at the stopping time. the fully-polarized initial state |⇓i to different values of the final transverse field. (a) Γ = 2Γc, (b) Γ=4Γc, (c) Γ=8Γc. The SDE results (filled circles) computed from (a) 2 × 105 (b) 3 × 105 and (c) 4 × 105 trajectories, are in good agreement with ED (solid line) for N = 3 spins. 3 ED SDE 0.5 ED 2 )

t SDE (a) ( 0.0 ) t ( M λ −0.5 1 0.5 ED )

t SDE (b) ( 0.0

0 M 0 1.3 −0.5 t 0.5 ED )

FIG. 4. Loschmidt rate function λ(t) for the 1D quantum t SDE (c) ( 0.0 Ising model following a quantum quench from the spatially M inhomogeneous initial state |↑↑↑↓↓i to the PM phase with Γ = −0.5 4Γ . The results obtained from the SDEs (filled circles) are c 0 1 2 in good agreement with ED (solid line) for N = 5 spins. The t SDE results were obtained by averaging over 105 realizations of the stochastic process. Less than 0.1% of the trajectories were found to be divergent at the stopping time. Larger error FIG. 6. Time-evolution of M(t) for the 1D quantum Ising bars are visible in the vicinity of the peak, due to the presence model with an integrability-breaking longitudinal field h = of enhanced stochastic fluctuations. 2J. We consider quantum quenches from the fully-polarized initial state |⇓i to (a) Γ = J, (b) Γ =2J, and (c) Γ = 4J. The SDE results (filled circles) computed from (a) 5 × 105 (b) 2 × 105 and (c) 6 × 105 trajectories are in good agreement with ED (solid line) for N = 3 spins. librium, this model exhibits a quantum phase transition 2D when Γ = Γc 1.523J [49, 50]. In Fig. 7 we show results for the magnetization≈ dynamics (t) following V. DYNAMICS OF THE DISENTANGLING a quantum quench from the fully-polarizedM initial state VARIABLES 2D to Γ 5.3Γc . Again, the results are in very good agreement|⇓i ≈ with ED. This highlights that the stochastic A notable feature of the stochastic approach to quan- ˆz formula for Si (t) in higher dimensions is the immediate tum spin systems is that it allows the derivation of ex- generalizationh of thei 1D result (21), where the products act stochastic formulae such as equations (19) and (21). are extended over all the lattice sites. The same holds In this framework, time-dependent quantum expectation true for other local observables. values are obtained by averaging explicit functions of the 9

4 N=7 N=25 N=50 0.5 ED φ i

SDE z

(a) χ h

Re 0 ) t ( 0 5 M

) 0 1 +

(b) χ 0 Re( -0.5 −5 5 0 1

) 0 1

t +

(c) χ 0

FIG. 7. Time-evolution of M(t) for a 2×3 site quantum Ising Im( model following a quantum quench from the fully-polarized −5 initial state |⇓i to Γ=8J. The SDE results (filled circles) 0 1 2 3 t obtained from 5×105 trajectories, are in very good agreement with ED (solid line). Less than 1% of the trajectories were found to be divergent at the stopping time. FIG. 8. Time-evolution of the disentangling variables in the 1D quantum Ising model following a quantum quench from the fully-polarized initial state |⇓i toΓ = 16Γc. (a) Dynamics z a of Re hχ (t)iφ for N = 7, 25, 50 spins showing maxima in the classical stochastic variables, ξi (t). It is therefore inter- esting to investigate to what extent the quantum dynam- vicinity of the Loschmidt peak times. The latter are obtained ics is reflected in these classical variables. by ED (dashed lines). Dynamics of the distribution of (b) Re χ+(t) and (c) Im χ+(t) for N = 7 spins showing signatures of the DQPTs. A. Distributions of the Classical Variables 6 tλ t + As we discussed in Section IV A, the stochastic formula χ for the Loschmidt amplitude has a particularly simple form for quantum quenches starting in the fully-polarized initial state . In this case A(t) can be written as [34] t 3 |⇓i N χz (t) A(t)= e− 2 , (35) h iφ where we define the site-averaged variables χa(t) 1 a ≡ N − i ξi (t). It is readily seen that the Loschmidt am- 0 plitude is directly determined by the statistical properties 1 4 8 12 16 of χzP(t). In particular, the functional form of (35) sug- Γ/Γc 1 2 gests that the peaks in λ(t) N − log A(t) occur in close proximity to (although≡− not necessarily| | coincident FIG. 9. Comparison of the time tλ of the first Loschmidt z peak obtained by ED (crosses) and the time tχ+ of the zeros with) the peaks in the distribution of χ (t), and its clas- + sical average χz(t) [34]. In Fig. 8(a) we show the time- of Imhχ (t)iφ (dots) for the 1D quantum Ising model with φ N = 7 sites. The data correspond to quantum quenches from evolution of theh latter,i which indeed exhibits maxima in the fully-polarized initial state |⇓i to different values of Γ. In the vicinity of the Loschmidt peaks, and has little depen- the limit of large Γ the results coincide, but for small Γ, the dence on system size. In addition, the turning points of results of ED differ from those given by the approximation z + + χ (t) φ coincide with the zeros of χ (t) φ due to the Imhχ (t)iφ = 0. Note that for Γ < Γc (not shown), there are h i z + h i + exact relation i χ˙ (t) φ =Γ χ (t) φ, which follows from no DQPTs for N →∞. However, zeros of Imhχ (t)i persist h i h i φ the Ising SDE in Eq. (26b) [34]. In general, it is impor- for Γ < Γc; these get pushed to later times as Γ decreases. tant to stress that the average of the exponential in (35) z is not the exponential of the average, N χ (t) φ/2. As z − h i such, the turning points of χ (t) φ are not in general the equations become deterministic and the average of located at the exact positionsh of thei Loschmidt peaks. the exponential in (35) is equal to the exponential of In Fig. 9 we show the comparison between the turning the average; away from this limit, this is not the case. z + points of χ (t) φ, or equivalently the zeros of χ (t) φ, Nonetheless, the exact formula (35) still applies, and its and the Loschmidth i peak times obtained via ED,h for dif-i predictions are in good agreement with ED. ferent quantum quenches. It is evident that these quanti- Signatures of the DQPTs can also be seen in the dis- ties are in excellent quantitative agreement for quenches tributions of the classical variables, which show marked with Γ Γc, but differ for Γ Γc. This can be under- features and enhanced broadening in their vicinity as stood from≫ the Ising SDEs in∼ (26). In the limit Γ shown in Figs 8(b) and (c). In particular, the distri- → ∞ 10

z bution of Re χ (t) is approximately Gaussian away from 0.035 the DQPTs, but is non-Gaussian in their proximity, as t=0.2 t=0.3 ) illustrated in Fig. 10(a). t t=0.4 (

The departures from Gaussianity can be quantified by D ) using the Kolmogorov–Smirnov (KS) test [51]. In this z 0 χ 0t 0.5 test, one considers the KS statistic D, which measures the (a) deviation of the observed distribution P (x) of a variable (Re P x from the best-fitting Gaussian distribution PG (x):

D max P (x) PG (x) . (36) ≡ x | − | 0 2 4 6 The aim of the test is to accept or reject the null hy- z pothesis that the observed data come from a Gaussian Re χ distribution, to a given statistical significance. The sta- 2 N = 7 tistical significance α is defined as the probability that ) N = 25 z the test rejects the null hypothesis when this is in fact χ N = 50 true, i.e. the probability that the test fails to recognize (b) (Re

a Gaussian distribution. The statistical significance α P determines a critical value Dc(α) for which the null hy- 0 pothesis is rejected with significance α when D>Dc(α). 2 3 4 5 For a sufficiently large number of samples , the limiting Re χz distribution of D is given by [51] N FIG. 10. (a) Time-evolution of the distribution of Re χz(t) ∞ r 1 2r2z2 for the 1D quantum Ising model with N = 7 spins. The data P (√ D>z)=2 ( 1) − e− . (37) N − correspond to a quantum quench from the fully-polarized ini- r=1 X tial state |⇓i with Γ =0 to Γ =16Γc, at times t = 0.2 (plus signs), t = 0.3 (crosses) and t = 0.4 (dots). We employ dif- The critical value Dc(α), for a given α and number of ferent normalizations for P (Re χz(t)) at different t for ease of samples , is then given by D (α) = z (α)/√ , where N c c N visualization. The distribution broadens on approaching the zc(α) is determined by solving P (√ D>zc)= α. Loschmidt peak at t = 0.39, and narrows afterwards. The dis- N z To analyze the distribution of Re χ (t), we evaluate tribution is approximately Gaussian away from the Loschmidt (36) with x = Re χz(t). The results of the KS test are peaks, as indicated by the Gaussian fits (solid lines), but is shown in the inset of Fig. 10(a). The null hypothesis is non-Gaussian in their vicinity. Inset: Time-evolution of the rejected at the α = 5% significance level in the shaded KS statistic D(t) on passing through the first Loschmidt peak. region surrounding the DQPT, indicating that the distri- We compare the value of D(t) to the critical value of Dc(α) bution of Re χz(t) is non-Gaussian in this region. This corresponding to the chosen significance of α = 5% (dashed- behavior persists for different system sizes, with the dis- dotted line). When D > Dc(α), the distribution can be re- tributions becoming narrower as N increases, as shown garded as non-Gaussian. This is observed in the shaded region near the DQPT, whose position is indicated by the dashed in Fig. 10(b). vertical line. (b) Variation of the distribution of Re χz(t) at t = 0.39 with increasing system size N. The distribution becomes more sharply peaked as N increases. B. Bounds

The stochastic approach also enables one to de- as confirmed in Fig. 11(b). As Γ , the three quan- z → ∞ rive bounds on the Loschmidt rate function, λ(t) tities λ(t), λb(t) and Re χ (t) φ all approach the non- 1 2 ≡ N − ln A(t) , where A(t) = f(χ(t)) and the func- h i − | | h iφ interacting result, given in Eq. (G10b) in Appendix G. tion f(χ(t)) depends on the initial conditions. Using the In this limit, as mentioned above, the SDEs (26) become fact that f(χ(t)) φ f(χ(t)) φ one immediately ob- purely deterministic and it is possible to replace the av- tains λ(t)|h λ (t)i where| ≤ h| |i ≥ b erage of the exponential in Eq. (35) with the exponential of the average. As such, λ(t) approaches Re χz(t) , in 2 h iφ λ (t) ln f(χ(t)) . (38) conformity with Fig. 8(a). b ≡−N h| |iφ This is confirmed in Fig. 11(a), where we consider quenches from the fully-polarized initial state , cor- C. Correlations of the Classical Variables Nχz(t)/2 |⇓i responding to f(χ(t)) = e− . Application of Jensen’s inequality [52] in this case also shows that The presence of DQPTs is also reflected in the cor-

N z relation functions of the disentangling variables. To see z 2 χ (t) φ Re χ (t) = ln e− 2 h i λ (t), (39) h iφ −N | |≥ b this, it is convenient to define the site-averaged connected 11

4 15 zz λED Re(C1 ) zz λSDE Im(C1 ) λb zz 1 0

(a) (a) C

−1 −15 z 4 Rehχ iφ 0 λb zz 2

(b) (b) C

zz Re(C2 ) zz Im(C2 ) −1 −60 0 1 2 3 0 1 2 3 t t

FIG. 11. (a) Time-dependent lower bound λb(t) (crosses) on FIG. 12. Time-dependent connected correlation functions of z the Loschmidt rate function following a quantum quench in the disentangling variables ξi following a quantum quench in the 1D quantum Ising model from Γ = 0 to Γ = 16Γc. The the 1D quantum Ising model from Γ = 0 to Γ = 16Γc. (a) zz data correspond to the solution of the Ising SDEs (filled cir- The first neighbor correlation function C1 (t) has a monoton- − z cles), ED (solid line) and Eq. (38) with f(χ(t)) = e Nχ (t)/2 ically decreasing real part and an oscillating imaginary part, (crosses) for an N = 7 site system. (b) The approximation with zeros occurring in the vicinity of the Loschmidt peaks z to the Loschmidt rate function Rehχ (t)iφ is also bounded (dashed lines, obtained from ED). (b) The second neighbor z zz by λb(t). For large values of Γ, λ(t), Rehχ (t)iφ and λb(t) correlation function C2 (t) has a vanishing imaginary part, coincide. but the real part decreases monotonically. The latter exhibits steeper gradients in the vicinity of the Loschmidt peaks. correlation function

N ab 1 a b a b n (t) ξi (t)ξi+n(t) φ ξi (t) ξi+n(t) φ ,

C ≡ N h i − h ih i ) i=1 3 ++ X
1

(40) (a) C where n indicates the separation between the two sites. Re( −3 zz As can be seen in Fig. 12(a), Re 1 (t) decreases smoothly ) 20 over time, but Im zz(t) exhibitsC oscillations within an ++ 1 1 increasing envelope.C In particular, Im zz(t) exhibits ze- (b) C C1 ros in the vicinity of the Loschmidt peaks. Likewise, the Im( 0 zz second-neighbor correlation function Re 2 (t) decreases ) 3 C zz ++ rapidly in the vicinity of the DQPTs, while Im 2 (t) 2 (c) C 0 remains zero for all t, as shown in Fig. 12(b); anC ana- zz Re( −3 lytical proof that Im Cn = 0 when n 2 is provided in Appendix H. Fig. 13 shows an analogous≥ analysis ) 3 ++ ++ 2 for (t). It can be seen that first neighbor correla- (d) C 0 Cn tion functions take small values everywhere, except in Im( −3 the vicinity of the DQPTs where Im ++(t) peaks; see C1 0 1 2 3 Fig. 13(a). Likewise, the second neighbor correlation t functions vanish on average for all times, but exhibit strong fluctuations in the vicinity of the Loschmidt peaks; see Fig. 13(b). The further neighbor correlation functions FIG. 13. Time-dependent connected correlation functions of + (not shown) are found to behave similarly to the second the disentangling variables ξi following a quantum quench neighbor case, due to the nearest neighbor form of the ij in the 1D quantum Ising model from Γ = 0 to Γ = 16Γc. matrix under consideration. As prescribed by Eq. (16),J (a-b) The first neighbor correlation functions are small ex- cept in the vicinity of the Loschmidt peaks (dashed lines), the Ito drift is proportional to k OikOjk =2 ij , which is only non-zero when j = i 1. As a result,J the when their imaginary part exhibits a sharp peak. (c-d) The first neighbor correlation functionsP± zz(t) and ++(t) are second neighbor correlation functions vanish away from the C1 C1 Loschmidt peaks, but show enhanced fluctuations as the peak qualitatively different from their further neighbor coun- times are approached. terparts. In Appendix H, we provide further information on the moments of the disentangling variables, explicitly 12 identifying a set of averages which vanish at all times. 5 4 Γ = 0 Γ = Γc/2 Γ = 16Γc 3 VI. FLUCTUATIONS 2 σ 1 5 10

As we have discussed above, the statistical proper- log ˜ N ties of the disentangling variables play a central role in 0 3 the stochastic approach to quantum spin systems. They −1 σ 2 Γ = 0 provide access to time-dependent quantum expectation −2 log ˜ 1 Γ = Γc values and exhibit notable signatures in the vicinity of −3 0 1 2 DQPTs. As we will discuss now, the fluctuations in the t disentangling variables also provide insights into the cur- rent limitations of the stochastic approach. From a nu- FIG. 14. Growth of fluctuations in the stochastic approach. merical perspective, the two main sources of error arise (a) Time-evolution of the normalized standard deviationσ ˜(t) from the non-zero discretization time-step ∆t, and the as defined in the text, for the Loschmidt amplitude A(t). We finite number of samples . The former is relatively be- N consider quantum quenches in the 1D quantum Ising model nign for short timescales, but eventually leads to diver- with N = 5 from the fully-polarized initial state |⇓i state to gences in the stochastic variables at late times [43, 47]. different values of Γ. The solid line shows the analytical result This effect is more pronounced in the presence of large for the classical case Γ = 0, corresponding to exponential transverse fields Γ, and can be mitigated by reducing the growth with t and N. For Γ > Γc, stronger fluctuations time-step ∆t. The latter is more important and arises become visible in the vicinity of the Loschmidt peaks, but from performing stochastic averages over a finite number the overall growth is consistent with the classical case. Inset: of samples . For a quantum observable ˆ(t) corre- growth of fluctuations with increasing N, for fixed Γ and t = 1. The results are consistent with exponential growth. sponding toN a stochastic function f(ξ(t)), ashO definedi by (14), the formally exact expression is approximated by appear in the vicinity of the Loschmidt peaks, but the 1 N ˆ(t) fr(t) S (t), (41) overall growth of fluctuations mirrors that in (42). Sim- hO i≈ ≡ N N r=1 ilar behavior is also observed for other observables and X for different initial conditions. In the case of local ob- where fr(t) is the value of f(t)= f(ξ(t)) for a given real- servables, the presence of two time-evolution operators ization r of the stochastic process. In the limit , N → ∞ in Eq. (20) translates to an extra factor of 2 in the ex- the central limit theorem implies that the sample aver- ponent, as shown in Appendix G for the magnetization. age S (t) is Gaussian distributed, even if the individ- N The exponential growth of fluctuations for large system ual fr(t) are not, provided that f(t) has finite variance. sizes and long times ultimately limits the stochastic ap- The resulting Gaussian distribution has mean f(t) , and h i proach in its current form. As the fluctuations inσ ˜(t) standard deviation σ (t)= σ(t)/√ , where σ(t) is the increase, an increasing number of runs is required for the N N standard deviation of f(t). The fluctuations in S (t) ob- sample mean S (t) to converge to f(t) . This is con- N tained by sampling the SDEs are therefore proportional sistent with ourN numerical observations,h asi illustrated in to σ(t); the value of σ(t) thus determines the number of Figs 15 and 16. The simulations typically breakdown at simulations required to achieve a given accuracy. a characteristic time tb 1/NJ, when the variance of In order to quantify the growth of fluctuations it is in- the spatially summed and∼ time-integrated HS fields is of structive to consider the Loschmidt amplitude A(t) given order unity; this can also be seen directly from Eq. (42). by (28), for quenches starting in the fully-polarized initial state . Since the Loschmidt amplitude is exponentially suppressed|⇓i with increasing system size, it is convenient to VII. COMPUTATIONAL COST consider the strength of the fluctuations relative to the mean, usingσ ˜(t) = σ(t)/ f(t) . In the classical limit where Γ = 0, one can show|h thati| f(t) = 1, so that A notable feature of the stochastic approach to quan- σ˜(t)= σ(t); see Appendix G. In this|h casei| tum spin systems is that the numerical solution of the SDEs is intrinsically parallelizable; the stochastic aver- JNt σ˜(t)= e 2 1, (42) ages are performed over independent trajectories and the − number of stochastic variables scales linearly in N, due which exhibits exponential growth with time t and sys- to the HS decoupling of the interactions. The simula- tem size N according toσ ˜(t) eJNt/2. In Fig. 14 we tion time also scales linearly with t and , and inversely confirm this dependence numerically∼ for quenches in the with ∆t. However, as t and N increase,N the exponential 1D quantum Ising model. A similar exponential growth growth in the fluctuations requires increasing ; this ne- N of fluctuations is also observed for Γ < Γc, as shown in cessitates much longer simulation times than suggested Fig. 14. In the regime Γ > Γc, enhanced fluctuations by the na¨ıve linear scaling. Eventually, the averages ob- 13

3 ED ED ED SDE SDE SDE 2

) 3 3 t ( λ ) )

1 t t ( ( λ λ

0 0 2 t

FIG. 15. Loschmidt rate function λ(t) for the same quantum 2 2 quench considered in Fig. 4, but extending the simulation 0.35 0.40 0.45 0.35 0.40 0.45 time. As can be seen from the vertical bars (light grey), for t & t t 1.3 strong fluctuations in the disentangling variables hamper (a) (b) the convergence of the stochastic averages (filled dots) to the results obtained by ED (solid line). 3 ED 3 SDE 2 ) t ( λ 1 2 ) t

( 0 λ 0t 1 1 tained from a given number of trajectories fail to con- verge to the required quantum expectation values, due 0 to the increasing variance of the stochastic variables; see 0 t 1 Figs 14, 15 and 16. In the case of the Loschmidt ampli- (c) 5 5 tude, each batch of = 10 simulations with ∆t = 10− takes approximatelyN 1 hour on 96 cores, per unit interval FIG. 16. (a) Close-up of the first Loschmidt peak for the 1D of time, and per lattice site, i.e. the data in Fig. 1 corre- quantum Ising model with N = 14 spins following a quantum spond to approximately 14 hours of simulation time. Lo- 6 quench from Γ = 0 to Γ = 16Γc. Using n = 3 × 10 inde- cal expectation values take a factor of two longer due to pendent trajectories we reproduce the ED result for λ(t). (b) the presence of two sets of disentangling variables. From Results for the same quench, but with n = 3 × 105 trajecto- a numerical perspective, this is clearly inferior to ED for ries. For this smaller number of simulations, λ(t) converges to small system sizes. However, for larger system sizes, the the ED result, except in the immediate vicinity of the peak. stochastic approach may offer some advantages as the (c) Loschmidt rate function for the quench in panel (a), but 6 number of stochastic variables scales linearly in N. In for N = 21 spins. For n = 5 × 10 simulations, λ(t) at the peak has not yet converged to the ED value. This is due to contrast to ED, one also avoids having to store an ex- z ponentially large matrix in memory. However, this ad- the enhanced fluctuations in the disentangling variables ξi in vantage is offset to some extent as the breakdown time the vicinity of the peaks, which grow with N; see Fig. 14. Near the peak, the sampling is insufficient to reproduce the t 1/NJ decreases with increasing system size due b ∼ ED result. However, in all other regions of the plot, including to the growth of stochastic fluctuations, i.e. there is times beyond the peak, the result obtained from the SDEs is a trade off between increasing the system size N and in good agreement with ED. This highlights that the method the timescale that can be addressed. We also observe is formally exact, but that sampling is important in order slower convergence in the regions where the fluctuations to achieve convergence. Inset: analogous results for a 5 × 5 are strongest; see Figs 14 and 16. This could perhaps quantum Ising model, corresponding to the upper limit for be mitigated through the use of enhanced sampling tech- comparison with ED. The system was initialized in the fully- polarized state |⇓i and time-evolved with Γ = 8J. The results niques. Nonetheless, in spite of these numerical and com- 7 putational challenges, the stochastic approach offers a for n = 4 × 10 are similar to those in panel (c): the SDE re- new set of tools for describing non-equilibrium quantum sults are in good agreement with ED before and after the peak, but the sampling is insufficient to resolve the peak. spin systems. This includes exact stochastic formulae with wide applicability, which hold in arbitrary dimen- sions and in the absence of integrability. The stochastic approach also provides direct links between quantum and classical dynamics, enabling the transfer of ideas between different domains of non-equilibrium science. 14

VIII. CONCLUSION Begg, John Chalker, Andrew Green, Vladimir Grit- sev, Lev Kantorovich and Austen Lamacraft. MJB is In this work we have investigated a stochastic approach very grateful to John Chalker for early discussions on to non-equilibrium quantum spin systems based on an ex- the Hubbard–Stratonovich and stochastic approaches to act mapping of quantum dynamics to classical SDEs. We quantum dynamics. BD is a Royal Society Leverhulme Trust Senior Research Fellow, ref. SRF R1 180103. have provided exact stochastic formulae for a variety of \ \ quantum observables, with broad applicability for spin- SDN acknowledges funding from the Institute of Sci- 1/2 systems. We have also outlined the general approach ence and Technology (IST) Austria, and from the Eu- to express other observables in this framework. We have ropean Union’s Horizon 2020 research and innovation illustrated the method in the context of the one- and programme under the Marie Sklodowska-Curie Grant two-dimensional quantum Ising model, highlighting the Agreement No. 754411. SDN also acknowledges fund- role of the classical stochastic variables and their rela- ing from the EPSRC Centre for Doctoral Training in tion to dynamical quantum phase transitions. We have Cross-Disciplinary Approaches to Non-Equilibrium Sys- also explored the growth of fluctuations in the stochas- tems (CANES) under grant EP/L015854/1. MJB, BD tic approach, discussing their scaling with time and sys- and SDN thank the Centre for Non-Equilibrium Sci- tem size, including details of the numerical aspects of the ence (CNES) and the Thomas Young Centre (TYC). We current implementation of the method. There are many are grateful to the UK Materials and Molecular Mod- directions for future research, including the development elling Hub for computational resources, which is partially of improved sampling methods as well as further explo- funded by EPSRC (EP/P020194/1). We acknowledge ration of the correspondence between the quantum and computer time on the Rosalind High Performance Com- classical dynamics. puter Cluster. We acknowledge helpful conversations with Samuel

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[40] H. F. Trotter, Proc. Amer. Math. Soc. 10, 545 (1959). [52] G. H. Hardy, J. E. Littlewood, and G. Polya, Inequalities [41] J. Wei and E. Norman, J. Math. Phys. 4, 575 (1963). (Cambridge University Press, 1934). [42] I. Kolokolov, Phys. Lett. A 114, 99 (1986). [53] K. Ito, Proc. Imp. Acad. 20, 519 (1944). [43] P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations (Springer, 1992). [44] A. LeClair, G. Mussardo, H. Saleur, and S. Skorik, Nucl. Phys. B 453, 581 (1995). Appendix A: Hubbard–Stratonovich Decoupling of [45] N. Kitanine, J. M. Maillet, N. A. Slavnov, and V. Terras, the Time-Evolution Operator (2005), arXiv:0505006. [46] For system sizes N that are multiples of 4, we add a In Section IIA we gave a brief overview of the constant diagonal shift to the interaction matrix J in Hubbard–Stratonovich decoupling of the time-evolution order to make it diagonalizable; see Appendix B. ′ i t Hˆ dt [47] M. Hutzenthaler, A. Jentzen, and P. E. Kloeden, Proc. operator Uˆ(t)= Te− 0 , where R R. Soc. A Math. Phys. Eng. Sci. 467, 1563 (2011). [48] P. Weinberg and M. Bukov, SciPost Phys. 2, 003 (2017). Hˆ = abSˆaSˆb haSˆa, (A1) [49] P. Pfeuty and R. J. Elliott, J. Phys. C Solid State Phys. − Jij i j − i i ia 4, 2370 (1971). Xijab X [50] M. S. L. du Croo de Jongh and J. M. J. van Leeuwen, Phys. Rev. B 57, 8494 (1998). is a generic Heisenberg model. Here we provide some of [51] F. James, Statistical Methods in Experimental Physics the technical details involved in this procedure. Trotter- (World Scientific, 2006). slicing the time-ordered exponential in Uˆ(t) one obtains

t n ab a b a a Uˆ(t)= T exp i dt′Hˆ (t′) = T lim exp i∆t (m∆t)Sˆ Sˆ + i∆t h (m∆t)Sˆ , (A2) − n Jij i j j j 0 →∞ m=1 ja  Z  Y  Xijab X  where ∆t t/n. Performing the Hubbard–Stratonovich transformation at each time slice yields ≡

− i∆t abSˆaSˆb a 1 ∆t ( 1)abϕaϕb+√i∆t ϕaSˆa e ijab Jij i j = (dϕ )e− 4 ijab J ij i j ja j j , (A3) P C i P P ia Z Y

a where is a normalization constant and ϕi are complex eigenvalues of , arranged in the same order as the scalarC fields chosen in such a way as to ensure conver- columns of Q, soJ that QT Q = D. Using these, Eq. (A4) gence of the integral in (A3); see the discussion following can be written as J Eq. (A8) below. In order to show that Eq. (A3) holds ˆ ˆ Sˆ2 i∆t αβ αβ SαSβ i∆t α λα α ˆa e J = e . (A6) for spin operators Si , it is convenient to introduce mul- P P ab ticomponent indices α = i,a so that e.g. ij αβ : where we have defined the operators Sˆ (QT ) Sˆ . { } J ≡ J α ≡ β αβ β ˆ ab ˆa ˆb ˆ ˆ For example, for the quantum Ising model (25) the Sα are i∆t ijab ij Si Sj i∆t αβ αβ SαSβ P e J e J . (A4) ˆz P ≡ P linear combinations of the Si operators at different sites. One can now factorize the infinitesimal exponentials in For simplicity, we assume that the matrix is sym- αβ Eq. (A6) over α using metric, as it can always be redefined so thatJ this is true. 2 2 Then, can be diagonalized as follows. We define the i∆t λαSˆ i∆tλαSˆ αβ e α α = e α , (A7) matrixJQ whose columns are the orthonormal eigenvec- P α (α) (β) Y tors e of , so that Qαβ = eα : J where we neglect terms of order (∆t)2 in the exponent. e(1) ... e(3N) For each factor in Eq. (A7), one obtains Q . (A5) ≡ ... 2 1 −1 2 i∆tλαSˆ ∆tλ ϕ¯ +√i∆tϕ¯αSˆα  ↓ ↓  e α = Cα d¯ϕαe− 4 α α (A8) This is an orthogonal matrix satisfying QQT = Z QT Q = 1. We also define the diagonal matrix D where C is a normalization constant. Since Sˆ com- ≡ α α diag(λ1,...,λ3N ), whose elements are the (real-valued) mutes with itself, the above Gaussian equality can be 16 proved by Taylor expansion of the integrand, where the Appendix C: Disentanglement Transformation integration overϕ ¯α is carried out along the real (imagi- nary) axis for all positive (negative) eigenvalues λα. Fi- As we discussed in Section II B, the stochastic time- T nally, by changing variables usingϕ ¯α = (Q )αβϕβ, ˆ s β evolution operator Uj (t) can be simplified by means of a the operator identity (A3) is verified. disentanglement transformation [31–33]: P − − s i t Φa(t)Sˆa ξ+(t)Sˆ+ ξz(t)Sˆz ξ (t)Sˆ Uˆ (t) Te 0 a j j = e j j e j j e j j , j ≡ R P Appendix B: Diagonalization of the Noise Action (C1)

Following the application of the Hubbard–Stratonovich where the explicit group parameterization eliminates the transformation, we define the noise action S[ϕ] as time-ordering operation. In order to obtain the evolution equations satisfied by the disentangling variables ξa [33], t j 1 1 ab a b one may differentiate (C1) with respect to time. Right- S[ϕ] ( − ) ϕ (t′)ϕ (t′)dt′. (B1) ≡ 4 J ij i j ˆ s 1 0 multiplying the result by (Uj )− one obtains Xijab Z a We want to perform a change of variables ϕ = s s 1 a a i (∂tUˆ )(Uˆ )− = i Φ Sˆ , (C2) Oabφb so that Eq. (B1) can be recast in the form j j j j jb ij j a X t P 1 a a or, equivalently, S[φ] φ (t′)φ (t′)dt′. (B2) ≡ 2 i i ia Z0 ˆ s X ∂Uj ∂ ξa (Uˆ s) 1 = i ΦaSˆa. (C3) For a symmetric interaction matrix ab, one can always t j a j − j j ij ∂ξj ! ab J a a construct a matrix Oij that diagonalizes (B1). Using the X X ˆa matrices Q and D introduced in Appendix A, the matrix For this equality to hold, the coefficients of Sj on each side of Eq. (C3) must be equal. Considering each spin O √2QD1/2 (B3) ≡ component a +, z, in turn, it may be verified that T 1 ∈{ −} satisfies O − O/2 = 1 and can thus be used to put J ∂Uˆ s the noise action in the desired form (B2). The matrix O j ˆ s 1 ˆ+ T + (Uj )− = Sj . (C4) also satisfies OO =2 , which is a useful relation when ∂ξj ! converting the SDEs betweenJ the Ito and Stratonovich conventions; see Appendix D. By writing O in terms of Similarly, its real and imaginary parts O , O , one also obtains R I ˆ s T T ∂Uj OROR OI OI = . Due to the definition (B3), where ˆ s 1 ˆz + ˆ+ − J z (Uj )− = Sj ξj Sj . (C5) the matrix Q is real valued and the entries of the diagonal ∂ξj ! − matrix D1/2 are either purely real or purely imaginary, O has either purely real or purely imaginary columns. In deriving this expression we invoke Hadamard’s lemma T T This implies that OI O = ORO = 0. We will use these R I Aˆ Aˆ 1 properties of O and O in Appendix G. The definition e Beˆ − = Bˆ + [A,ˆ Bˆ]+ [A,ˆ [A,ˆ Bˆ]] + ..., (C6) R I 2! of the matrix O is not unique and depends on the spe- cific ordering of the eigenvalues in (A5). This construc- and the commutation relations of su(2): [Sˆz, Sˆ+] = + z + z tion breaks down if the interaction matrix has vanishing Sˆ , [Sˆ , Sˆ−]= Sˆ−, [Sˆ , Sˆ−]=2Sˆ . Finally, eigenvalues and cannot be inverted. This is relevant to − ˆ s the quantum Ising model (25) for example. In this case, ∂Uj z ˆ s 1 ξj ˆ + ˆz +2 ˆ+ the interaction matrix is given by (Uj )− = e− Sj− +2ξj Sj ξj Sj . (C7) ∂ξj− ! −   ab J ij = δazδbz(δij+1 + δij 1) δazδbz ij . (B4) ˆa J 2 − ≡ J Equating the coefficients of each Sj one obtains

When the system size N is a multiple of 4, one of the + + ξz +2 + z iΦ = ξ˙ e− j ξ ξ˙− ξ ξ˙ , (C8a) eigenvalues of turns out to be zero. For spin-1/2 sys- j j − j j − j j J z tems this can be avoided by including a shift proportional z ˙z + ξj ˙ iΦj = ξj +2ξj e− ξj−, (C8b) to the identity operator in the Hamiltonian. Using the z z 2 N 1 ξj ˙ fact that i(Si ) = 4 , this may be achieved by adding iΦj− = e− ξj−. (C8c) a term Jsδij δazδbz to the interaction matrix. For Js = 1, P 6 ˙a becomes invertible. The corresponding time-evolution Rearranging for ξj yields the SDEs [33] operatorJ acquires a constant phase shift, which does not ˙+ + z + +2 affect the computation of physical observables. However, iξj =Φj +Φj ξj Φj−ξj , (C9a) T − − with this modification, ii = 0 and thus (OO )ii = 0. ˙z z + J 6 6 iξj =Φj 2Φ−ξ , (C9b) This leads to a change in the stochastic equations of mo- − − j j z tion, as discussed in Appendix D. iξ˙− =Φ− exp ξ . (C9c) − j j j 17

Appendix D: Ito and Stratonovich Conventions has no diagonal elements. However, as discussed in Ap- pendix B, for system sizes N that are multiples of 4 we In order to consistently define a stochastic differential add a diagonal constant shift to , in order to make it in- J + equation, it is necessary to specify a discretization con- vertible. In this case, the SDE for ξj takes the form (D5). vention [43]. These are distinguished by how the values This leads to different behavior for the classical variables, but does not affect the resulting physical observables. of a function f¯(tj ) defined at discrete times tj j∆t are assigned from the values of its continuous counterpart≡ f(t). Different discretization schemes are parameterized by a constant α as Appendix E: Analytical Averaging of the Equations of Motion

f¯(tj )= αf(tj )+(1 α)f(tj 1), 0 α 1. (D1) − − ≤ ≤ As we discussed in Section II C, in principle it is pos- ¯ The choice α = 0 gives the Ito convention f(tj ) = sible to analytically average the SDEs governing the dy- f(tj 1), while α = 1/2 gives the Stratonovich con- namics of physical observables. The expectation value of − ¯ vention. The latter corresponds to choosing f(tj ) as an observable ˆ following time-evolution from an initial the average of the values of f(t) at tj 1, tj . Since O − state ψ0 can be expressed as [f(tj )+ f(tj 1)]/2 f([tj + tj 1]/2), the Stratonovich | i convention is− also known≈ as the−mid-point prescription. ˆ(t) = f(t) ˜, (E1) SDEs in the Stratonovich convention satisfy the rules of hO i h iφ,φ ordinary calculus. However, when working with Ito SDEs where a specific calculus is required [43, 53]. If we interpret the disentangling equations (10) as SDEs, they are to be un- s s f(t)= ψ0 [Uˆ (ξ˜(t))]† ˆUˆ (ξ(t)) ψ0 . (E2) derstood as initially expressed in the Stratonovich con- h | O | i vention. This is the form which arises naturally in phys- Here, Uˆ s = Uˆ s and the two time-evolution operators ical applications involving well-defined continuous pro- ⊗i i depend on independent stochastic processes φ and φ˜ via cesses, i.e. noise with a finite correlation time, in the limit ˜ ˜ of the correlation time going to zero. However, equations ξ[φ] and ξ[φ]. The functional form of f, in terms of the ˜ in the Ito convention are typically mathematically and disentangling variables ξ and ξ, depends on the chosen computationally simpler to handle. It is therefore often observable and the initial state. The equation of mo- convenient to translate Stratonovich SDEs into the Ito tion of f is obtained from the Ito chain rule as given by ˙ form. In the Stratonovich form, the SDE for the disen- Eq. (16) in the main text. This can be written as f =Υf, a where we define the linear operator tangling variables ξi , collectively represented as a vector ξS, can be written as 2 a ab b ∂ 1 ac bc ∂ Υ (Ai + Bij φj ) a + Bik Bjk a b . dξS ≡ ∂ξ 2 ∂ξ ∂ξ ia i i j = AS (ξS,t)+ BS(ξS ,t)φ, (D2) X Xjb Xijab Xck dt (E3) where φ is a vector composed of the stochastic variables For notational economy, the indices a and b run over a +, ,z and over both the ξ, ξ˜ variables. The analyt- φj , AS is the drift vector and BS is a matrix of diffusion { − } ical expression for the average d (t)/dt = f˙ ˜ can coefficients. The corresponding SDE for the vector ξ in h O i h iφ,φ the Ito convention is given by be obtained by applying (E3) to the definition (E2) and averaging over the HS fields φ, φ˜: dξ = A(ξ,t)+ B(ξ,t)φ, (D3) dt ˙ ˆ s ˜ ˆ ˆ s f φ,φ˜ = ψ0 Υ[U (ξ)]† U (ξ) ψ0 h i h | O | i φ,φ˜ (E4) where D  s  s E + ψ0 [Uˆ (ξ˜)]† ˆ ΥUˆ (ξ) ψ0 . h | O | i φ,φ˜ 1 T T A = AS + (B ξ)B , (D4a) D   E 2 ∇ s Since Uˆ φ = Uˆ, we can simplify (E4) using B = BS. (D4b) h i d For the quantum Ising model (25), this modification only ΥUˆ s(ξ) = Uˆ s(ξ) = iHˆ U.ˆ (E5) + affects the Ito SDE (26a) for ξ , which becomes φ dt φ − j D E  

1 2 1 This can also be verified by directly evaluating ΥUˆ s(t) iξ˙+ = Γ(1 ξ+ )+ ξ+ O O +ξ+ O φ /√i. − j 2 − j 2 j jk jk j jk k and using the commutation relations of su(2). Similarly, k k X X (D5) + s d s In many cases, the extra term ξj k OjkOjk/2 vanishes, Υ[Uˆ (ξ˜)]† = [Uˆ (ξ˜)]† = iUˆ †H.ˆ (E6) φ˜ dt since OOT =2 and the interaction matrix typically  φ˜ J P J D E 18

Using these identities, the equation of motion (E4) for with the system size. Solving these ODEs is therefore ˆ(t) can be written as equivalent to diagonalizing the Hamiltonian. hO i ˆ d ˙ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ψ0 O ψ0 = f ˜ = i ψ0 HU † U U † UH ψ0 . Appendix F: Building Blocks for Local Observables h | dt | i h iφ,φ h | O − O | i  (E7) As discussed in Section III, expectation values of prod- This can be recognized as a matrix element of the Heisen- ucts of local operators, starting from a product state, berg equation of motion can be expressed in terms of stochastic averages over products of on-site “building blocks”. This follows from d ˆ(t)= i[H,ˆ ˆ(t)]. (E8) the fact that the time-evolution operator can be factor- dt s s O O ized over lattice sites as U(t) = U (t) φ where Uˆ (t) ˆ s h i ≡ Such matrix elements give rise to a set of coupled first or- iU (t). For example, if there are no spin operators in ⊗ i der ODEs, whose number in general grows exponentially the observable ˆ at site i we may use the building blocks O

s s ↑↑ Uˆ (ξ˜)†Uˆ (ξ) Bi ≡ ih↑| i i |↑ii z ˜z∗ ξi +ξi z ˜z∗ ˜z∗ z (F1a) 2 ξi +ξi ξi + ξi ˜ ˜+ ˜ + ˜+ = e− e + e ξi−ξi + e ξi−∗ξi + ξi−ξi−∗(1 + ξi ξi ∗) , z z∗ ξ +ξ˜ ∗ s  s i i ξ˜z + + +  ↑↓ Uˆ (ξ˜)†Uˆ (ξ) = e− 2 ξ˜−∗ + e i ξ + ξ˜−∗ξ ξ˜ ∗ , (F1b) Bi ≡ ih↑| i i |↓ii i i i i i ∗ ξz+ξ˜z s s i i  ξz + + +  ↓↑ Uˆ (ξ˜)†Uˆ (ξ) = e− 2 ξ− + e i ξ˜ ∗ + ξ−ξ ξ˜ ∗ , (F1c) Bi ≡ ih↓| i i |↑ii i i i i i ∗ ξz+ξ˜z s s i i  + +  ↓↓ Uˆ (ξ˜)†Uˆ (ξ) = e− 2 (1 + ξ ξ˜ ∗), (F1d) Bi ≡ ih↓| i i |↓ii i i

depending on the initial and final states. If the spin op- erator Sz is present in ˆ we may use one of the following: i O

z z∗ ξ +ξ˜ ∗ ∗ z s z s 1 i i ξz+ξ˜z ξ˜z + ξz + + + ↑↑ Uˆ (ξ˜)†Sˆ Uˆ (ξ) = e− 2 e i i + e i ξ−ξ + e i ξ˜−∗ξ˜ ∗ + ξ−ξ˜−∗( 1+ ξ ξ˜ ∗) , (F2a) Bi ≡ ih↑| i i i |↑ii 2 i i i i i i − i i ∗ ξz+ξ˜z ∗ z s z s 1 i i  ξ˜z + + +  ↑↓ Uˆ (ξ˜)†Sˆ Uˆ (ξ) = e− 2 ξ˜−∗ + e i ξ + ξ˜−∗ξ ξ˜ ∗ , (F2b) Bi ≡ ih↑| i i i |↓ii 2 − i i i i i ∗ ξz+ξ˜z z s z s 1 i i  ξz + + +  ↓↑ Uˆ (ξ˜)†Sˆ Uˆ (ξ) = e− 2 ξ− + e i ξ˜ ∗ + ξ−ξ ξ˜ ∗ , (F2c) Bi ≡ ih↓| i i i |↑ii 2 − i i i i i ∗ ξz+ξ˜z z s z s 1 i i  + +  ↓↓ Uˆ (ξ˜)†Sˆ Uˆ (ξ) = e− 2 ( 1+ ξ ξ˜ ∗). (F2d) Bi ≡ ih↓| i i i |↓ii 2 − i i

+ Similarly, if the observable ˆ contains Sˆ , Sˆ− we may use: O i i

∗ ξ˜z+ξz ∗ + s + s i i + ξ˜z ↑↑ Uˆ (ξ˜)†Sˆ Uˆ (ξ) = e− 2 ξ− (ξ˜−ξ˜ )∗ + e i (F3a) Bi ≡ ih↑| i i i |↑ii i i i z z∗ ξ˜ +ξ ∗ + s + s i i  + ξ˜z  ↑↓ Uˆ (ξ˜)†Sˆ Uˆ (ξ) = e− 2 (ξ˜−ξ˜ )∗ + e i , (F3b) Bi ≡ ih↑| i i i |↓ii i i ∗ ξ˜z ξz + i + i   ˆ s ˜ ˆ+ ˆ s 2 ˜+ i ↓↑ i Ui (ξ)†Si Ui (ξ) i = e− ξi−ξi ∗, (F3c) B ≡ h↓| |↑i ∗ ξ˜z+ξz + s + s i i + ↓↓ Uˆ (ξ˜)†Sˆ Uˆ (ξ) = e− 2 ξ˜ ∗. (F3d) Bi ≡ ih↓| i i i |↓ii i 19

∗ ξ˜z+ξz s s i i + ξz −↑↑ Uˆ (ξ˜)†Sˆ−Uˆ (ξ) = e− 2 ξ˜−∗ ξ−ξ + e i , (F4a) Bi ≡ ih↑| i i i |↑ii i i i ˜z z∗ ξi +ξi   ˆ s ˜ ˆ ˆ s 2 + ˜ i−↑↓ i Ui (ξ)†Si−Ui (ξ) i = e− ξi ξi−∗, (F4b) B ≡ h↑| |↓i ∗ ξ˜z+ξz s s i i + ξz −↓↑ Uˆ (ξ˜)†Sˆ−Uˆ (ξ) = e− 2 ξ−ξ + e i , (F4c) Bi ≡ ih↓| i i i |↑ii i i ∗ ξ˜z+ξz s s i i +  −↓↓ Uˆ (ξ˜)†Sˆ−Uˆ (ξ) = e− 2 ξ . (F4d) Bi ≡ ih↓| i i i |↓ii i

For example, starting in the fully-polarized initial state obtained in the classical limit with Γ = 0, and in the non- one readily obtains interacting limit with J = 0. We consider each below. |⇓i In the classical limit with Γ = 0, the equation of mo- + x 1 + 1 + tion for ξ (t) becomes linear. Due to the initial condition Sˆ (t) = Sˆ + Sˆ− = ( ↓↓ + −↓↓) ↓↓ ˜. j h i i 2h i i i 2h Bi Bi Bj iφ,φ + + j=i ξj (0) = 0 one obtains the trivial solution ξj (t) = 0. Sim- Y6 z (F5) ilarly, ξj−(t) = 0. The variable ξj (t) undergoes Brownian Using the above results one arrives at the exact formula motion and its time-evolution can be computed as

z ˜z∗ ξj +ξj t 1 j 2 z ˆx −   + ˜+ + ˜+ ξ (t)= i O dt′φ (t′)= √i O W (t). (G2) Si (t) = e P (ξi +ξi ∗) (1+ξj ξj ∗) . j jk k jk k h i 2 φ,φ˜ 0 j=i k Z k D Y6 E X X (F6) Similarly, The quantities Wk(t)areasetof N independent standard Wiener processes, which can be numerically generated as z ˜z∗ ξj +ξj i j ˆy −  2  + ˜+ + ˜+ √ Si (t) = e P (ξi ξi ∗) (1+ξj ξj ∗) . Wk(t)= t (0, 1), (G3) h i 2 − φ,φ˜ N j=i D Y6 E (F7) where (0, 1) is a random number extracted from a zero- ˆz mean,N unit-variance Gaussian distribution. In the classi- The result for Si (t) is given by (21) in the main text. h i cal limit with Γ = 0, there is no dynamics; this result can be recovered by substituting (G2) into the stochastic ex- Appendix G: Ising Stochastic Differential Equations pressions for observables. For example, for the Loschmidt amplitude (19) and the magnetization (obtained using the building blocks in Appendix F) one obtains As we discussed in Section IV, the Ito SDEs for the quantum Ising model are given by Eq. (26), which we A(t) =1, (G4) repeat here for convenience: | | Sˆz(t) = Sˆz(0) , (G5) h j i h j i Γ 2 iξ˙+ = (1 ξ+ )+ ξ+ O φ /√i, (G1a) − j 2 − j j jk k for any initial condition. For example, for quantum Xk quenches from the fully-polarized initial state , the ˙z + √ |⇓i iξj = Γξj + Ojkφk/ i, (G1b) Loschmidt amplitude and magnetization can be obtained − − + k by substituting (G2) and ξ = 0 into (28) and (21): X i Γ z iξ˙− = exp ξ . (G1c) − j 2 j (i + 1) A(t)= exp √NJW1(t) , (G6) 2 φ It is readily seen that the disentangling variable ξ+ plays D   E j √ a particularly important role for the quantum dynamics 1 NJ (t)= exp [(1 + i)W1(t)+(1 i)W˜ 1(t)] , in this parameterization: as we will discuss, ξ+ vanishes M −2 2 − ! φφ˜ j D E identically in the classical limit Γ = 0, and it is the only (G7) disentangling variable that is not dependent on the oth- ers, as follows from (G1a). Once ξ+ is known, ξz can be j j where W1(t) and W˜ 1(t) are independent Wiener processes obtained by integrating (G1b) with respect to time. In obtained as in Eq. (G3). Averaging the above equa- z turn, ξj− has a deterministic dependence on ξj , as follows tions with respect to the noises φ, one obtains the re- from (G1c). The non-linearity of the equation of motion sults (G4) and (G5). From (G2), one can also calculate + (G1a) for ξj , renders it non-trivial to solve. However, the variance of the stochastic functions fA and f cor- exact solutions to the full set of SDEs (G1) are readily responding to these observables, via A(t) f M and ≡ h Aiφ 20

(t) f φ,φ¯. One obtains In the above, we have introduced the shorthand nota- M ≡ h Mi tion ORiφ j (OR)ij φj , where (OR)ij Re(Oij ), and 2 NtJ ≡ ≡ σ (f )= e 2 1, (G8) similarly for OI . The identities we wish to show are A − P 1 most easily proved by introducing a convenient formal σ2(f )= eNtJ 1 . (G9) notation; this makes it simpler to analyze the system M 4 − of ODEs which arise from averaging the SDEs. In par-
In both cases, the variance grows exponentially with time ticular, let us represent the classical average of a given n m and the system size. The similar functional form of the monomial R I φ as a state ni,mi . To compute the h i i i | i Loschmidt amplitude (19) and the magnetization (21), time-evolution of this state, one applies the Ito chain ξz(t)/2 which both involve exponential factors of e− i i , rule (16); this requires differentiating with respect to Ri P leads to similar behavior for the fluctuations. An extra and Ii. Due to (H1), each differentiation with respect to factor of two is present in the exponent for the magneti- Ri (Ii) decreases ni (mi) by 1, and annihilates a state zation due to the presence of two Hubbard–Stratonovich where ni (mi) is equal to zero. This suggests that we transformations for local observables. The presence of can formally represent derivatives as annihilation opera- ∂ ∂ the exponential factors is suggestive of the exponential tors aˆRi ,a ˆIi satisfying ≡ ∂Ri ≡ ∂Ii growth of fluctuations even for non-zero Γ. This is ob- aˆ n,m = n n 1,m , (H2a) served numerically and is discussed in the main text. Ri| i i| i − ii In the non-interacting limit with J = 0 the equations aˆ n,m = m n ,m 1 . (H2b) Ii| i i| i i − i of motion for the disentangling variables become deter- Following the same line of reasoning, we can represent Ri ministic, as Ojk = 0. These can be solved analytically: and Ii themselves as creation operators satisfying

+ aˆRi† ni,mi = ni +1,mi , (H3a) ξj (t)= i tan(Γt/2), (G10a) | i | i z aˆIi† ni,mi = ni,mi +1 . (H3b) ξj (t)= 2 log cos(Γt/2), (G10b) | i | i − It can be seen that the operators satisfy bosonic commu- ξj−(t)= i tan(Γt/2). (G10c) tation relations [ˆa , aˆ† ′ ]= δ ′ where X,X′ R , I . X X XX ∈{ i i} As expected, (G10) parameterizes the precession of a sin- We can use the notations introduced above to com- gle spin in a magnetic field applied along the x-direction. pactly write the equation of motion for a general state n m This can be seen by writing the time-evolved state ni,mi = Ri Ii φ, obtained via the Ito chain rule, as ˆ | i h i ψ(t) = U(t) J=0 ψ(0) for product-state initial condi- d | i | | i 2 2 ni,mi = Hˆi ni,mi (H4) tions ψ(0) = j aj + bj with aj + bj = 1, dt| i − | i | i ⊗ |↑ij |↓ij | | | | using the values of ξ given in (G10). This yields   where we have defined an effective Hamiltonian Hˆi:

aj cos(Γt/2) ibj sin(Γt/2) ˆ Γ ψ(t) = − . (G11) Hi ΓˆaRi† aˆIi† aˆRi + 1 aˆRi† aˆRi† +ˆaIi† aˆIi† aˆIi | i iaj sin(Γt/2)+ bj cos(Γt/2) ≡ 2 − j −  O 1 T T   + (O O + O O ) (ˆa† aˆ† +ˆa† aˆ† )(ˆa aˆ +ˆa aˆ ). 2 I I R R ii Ri Ri Ii Ii Ri Ri Ii Ii Appendix H: Moments of the Disentangling (H5) Variables In deriving Eq. (H5), we used the Ito calculus property f(t)φ(t) φ = 0, and the properties of OR and OI given As we noted in Section VC, certain averages of the hin Appendixi B. In Eq. (H4), the time-evolution of a n m classical disentangling variables are identically zero for general classical average Ri (t)Ii (t) φ has been cast in all times. Here, we go further and demonstrate that a a form that is reminiscenth of a Euclideani Schr¨odinger + + set of monomials in Ri Re(ξi ) and Ii Im(ξi ) have equation, where the real-time variable t plays the role vanishing averages for all≡t. To see this we≡ note that the of an imaginary-time variable. The notations introduced coupled SDEs for Ri and Ii can be obtained by combining above allow us to obtain an infinite number of identities Eq. (G1a) with its complex conjugate: for monomials in Ri, Ii. As it can be seen from (H5), the Hamiltonian does not contain any term that raises or √2 lowers the index ni of a state ni,mi by an odd number. R˙ i(t) =ΓRiIi + [Ri(ORi OIi) Ii(ORi + OIi)]φ | i 2 − − Hence, states with odd and even ni belong to separate even and odd subspaces e and o. At t = 0, the initial + H H (H1a) condition ξi (0) = 0 translates into the initial conditions n ,m = δ δ . Since 0, 0 1 does not belong Γ i i ni0 mi0 φ I˙ (t)= (1 R2 + I2) |to thei odd subspace , all| statesi ≡ hni,m vanish i 2 − i i o i i o (H1b) identically at all times.H This finding| can bei∈ expressed H as √2 + [Ri(ORi + OIi)+ Ii(ORi OIi)]φ Rn(t)Im(t) =0, n odd. (H6) 2 − h i i iφ ∀ 21

This provides an analytical proof of the numerical obser- Hamiltonian becomes vation that Re ξ+(t) vanishes at all times, as we pre- h i iφ ˆ ˆ ˆ viously reported in Ref. [34]. Thus far, we have focused H2 = Hi + Hj on moments involving variables at a single site i. The 1 T T + (O O + O O ) (ˆa† aˆ† +ˆa† aˆ† )(ˆa aˆ +ˆa aˆ ) multi-site generalization is straightforward and is analo- 2 I I R R ij Ri Rj Ii Ij Ri Rj Ii Ij gous to the construction of a many-site Fock space from 1 T T + (O O + O O ) (ˆa† aˆ† aˆ† aˆ† )(ˆa aˆ aˆ aˆ ) a collection of single-site ones. Creation and annihila- 2 I I R R ij Ri Ij − Ii Rj Ri Ij − Ii Rj tion operators are defined analogously to the same-site + (ˆa† aˆ† aˆ† aˆ† )(ˆa aˆ +ˆa aˆ ) case; from their definition, it is clear that operators act- Jij Ri Rj − Ii Ij Ri Ij Ii Rj ing at different sites commute, since they are just mul- (ˆa† aˆ† +ˆa† aˆ† )(ˆa aˆ aˆ aˆ ). − Jij Ri Ij Ii Rj Ri Rj − Ii Ij tiplications or differentiations by independent variables. (H8) Of particular interest is the case of monomials involving two sites i = j, associated with two times ti, tj . States Eq. (H8) allows us to generalize the result (H6) for mono- 6 can be defined as mials involving two sites. Consider the total number of R factors in a given monomial, n = ni + nj . When i and ni mi nj mj j are not nearest neighbors, the last two terms in (H8) Ri (ti)Ii (ti)Rj (tj )Ij (tj ) φ h i ≡ (H7) vanish, and both the Hamiltonians (H5) and (H8) con- (ni,mi,ti)i; (nj ,mj,tj )j . serve the overall evenness or oddness of n. By the same argument as in the one-site case, states with odd n must E therefore vanish for all t , t . This finding can be used to i j Consider, for example, the evolution of the state with prove the vanishing of Im Czz(t) with n 2, as defined n ≥ respect to ti. For ti = tj, this is given by Eq. (H4) where in Eq. (40) and numerically studied in Section V. Using 6 zz the state is replaced by (H7). For tj = ti, the effective Ito calculus, we find that Cn (t) is given by

t t Czz(t)= Γ dt dt ξ+(t )ξ+ (t ) ξ+(t ) ξ+ (t ) . (H9) n − 1 2 h i 1 i+n 2 iφ − h i 1 iφh i+n 2 iφ i ZZ X 0 0  

+ By Eq. (H6), we know that ξi (t) is purely imaginary n 2 must be real valued. The operator description for all i, t, such that the secondh contributioni to Eq. (H9) of≥ the evolution of moments introduced in this Section is real-valued. The first contribution involves allows us to identify vanishing expectation values in a transparent way, without solving the SDEs (G1). The ξ+(t )ξ+ (t ) = R (t )R (t ) I (t )I (t ) h i 1 i+n 2 iφ h i 1 i+n 2 iφ − h i 1 i+n 2 iφ presence of identically vanishing moments suggests that + i R (t )I (t ) + i I (t )R (t ) . the Ising SDEs may contain a degree of redundancy, and h i 1 i+n 2 iφ h i 1 i+n 2 iφ (H10) that it may be possible to reduce them to a simpler form which automatically takes into account these vanishing It can be seen that the imaginary part of averages. The operator formalism employed to find the ξ+(t )ξ+ (t ) features only monomials with odd n, vanishing moments provides a rather general alternative h i 1 i+n 2 iφ which vanish for any t1, t2 by the previous argument. viewpoint, which may turn out to be a useful tool for zz Thus, the connected correlation function Cn (t) with future developments.