We set up a real-time path integral to study the evolution of quantum systems driven in real-time completely by the coupling of the system to the environment. For specifically chosen interactions, this can be interpreted as measurements being performed on the system. For a spin-1/2 system, in particular, when the measurement results are averaged over, the resulting sign problem completely

disappears, and the system can be simulated with an efficient cluster algorithm

Banerjee, Debasish

Hebenstreit, Florian

Jiang, Fu-Jiun

Kon, Mark

Wiese, Uwe-Jens

arXiv

Bern Open Repository and Information System (BORIS)

PoS(LATTICE 2015)281Real-time simulation of dissipation-driven quantumsystemsDebasish Banerjee∗NIC, DESY, Platanenallee 6, 15738 Zeuthen, GermanyE-mail: debasish.banerjee@desy.deFlorian Hebenstreit, Uwe-Jens Wiese †Albert Einstein Center, Institute for Theoretical Physics, Bern University, 3012 Bern, SwitzerlandE-mail: hebenstreit@itp.unibe.ch, wiese@itp.unibe.chFu-Jiun JiangDepartment of Physics, National Taiwan Normal University,88, Sec. 4, Ting-Chou Rd., Taipei 116, TaiwanE-mail: fjjiang@ntnu.edu.twMark KonDepartment of Mathematics, Boston University, Boston, Massachusetts, USAE-mail: MKON@bu.eduWe set up a real-time path integral to study the evolution of quantum systems driven in real-timecompletely by the coupling of the system to the environment. For specifically chosen interactions,this can be interpreted as measurements being performed on the system. For a spin-1/2 system, inparticular, when the measurement results are averaged over, the resulting sign problem completelydisappears, and the system can be simulated with an efficient cluster algorithm.The 33rd International Symposium on Lattice Field Theory14 -18 July 2015Kobe International Conference Center, Kobe, Japan*∗Speaker.†The research leading to these results has received funding from the Schweizerischer Nationalfonds and from theEuropean Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ ERC grantagreement 339220.c© Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). http://pos.sissa.it/source: https://doi.org/10.7892/boris.85291 | downloaded: 13.3.2017PoS(LATTICE 2015)281Real-time simulations Debasish Banerjee1. IntroductionThe generally established way to study the static properties of large quantum systems is usingMonte Carlo methods. However, the notorious ’sign problem’ may arise in cases where the proba-bility sampling over the various stochastically generated configurations have negative (or complex)weights. The particularly exciting branch of theoretical physics dealing with the real-time evolutionof large quantum systems, gets excluded from the reach of Monte Carlo methods since the config-urations contributing to the real-time path integral have complex weights. Moreover, diagonalizingthe Hamiltonian is impossible in practice, due to the exponential growth of the Hilbert space withthe system size. Matrix product states provide a basis for simulating the real-time evolution ofquantum systems, based on the density matrix renormalization group [1]. They are successful onlyfor gapped 1-D systems for moderate time-intervals [3, 4, 5, 6, 7, 8, 9]. Sign problems also occur inEuclidean time simulations of Quantum Chromodynamics at non-zero baryon chemical potentialand the fermionic Hubbard model away from half-filling. Frustrated systems and actions with non-zero topological terms again face the same problem. The different physical origins of the differentsign problems suggest that there might not be a unique solution applicable to all the cases [10].One reason why classical computers have problems to simulate quantum systems - especiallyin real-time - is because quantum entanglement is not easily representable, let alone computable,as classical information in conventional computers. This already led Feynman to propose the useof specially designed quantum devices to mimic quantum systems [11], very much along the linesof the usage of toy aeroplane models to study actual problems of aircraft manoeuvring in real-time flight situations. This so-called analog computing differs from a digital computer, which is amachine capable of computing an answer to a problem according to an algorithm. Ever since theexperimental realization of Bose-Einstein condensation [12, 13], quantum optics has progressedby leaps and bounds, and the degree of control available for ultracold atomic systems is trulyremarkable. The bosonic Hubbard model has been implemented with well-controlled ultracoldatoms in an optical lattice [14], and several aspects of this quantum simulation have been verifiedby comparison with accurate quantum Monte Carlo simulations [15]. Digital [16] and analog [17]quantum simulators are widely discussed in atomic and condensed matter physics [18, 19, 20, 21,22, 23], and more recently also in a particle physics context [24, 25, 26, 27, 28, 29, 30, 31, 32].However, quantum simulators are not always universally applicable, and even more impor-tantly they are not yet precision instruments. Therefore, the study of real-time evolution of quantumsystems using classical computers remains an open important challenge. Closed quantum systemstend to evolve into complicated entangled states under the action of the Hamiltonian in real timerendering their simulation difficult. A different approach is to consider open quantum systems, asthey undergo decoherence due to their continuous interaction with the environment.In this work, we have developed a method to simulate the dynamics of large quantum systems,driven entirely by the measurement process of the total spin (~Sx+~Sy)2 of pairs of spins 12 at nearestneighbors x and y. These measurements can also be interpreted as a dissipative coupling to theenvironment, especially when this interaction takes place stochastically. As we shall see, the systemis driven from a given equilibrium initial state to a new equilibrium final state for late real-times.This is the first example where the real-time evolution of a large strongly coupled quantum systemcan be studied over arbitrarily large periods of real time in any spatial dimension.2PoS(LATTICE 2015)281Real-time simulations Debasish Banerjee2. Formulation of the problemTo begin with, we outline the path integral representation for the real-time process drivenby measurements. Consider a general quantum system with a Hamiltonian, whose evolution inreal time from tk to tk+1 is described by the operator U(tk+1, tk) =U(tk, tk+1)†. An observable Okwith an eigenvalue ok is measured at time tk (k ∈ {1,2, . . . ,N}). The measurement projects thestate of the system to the subspace of the Hilbert space spanned by the eigenvectors of Ok witheigenvalue ok, and is represented by the Hermitean operator Pok . Starting from an initial densitymatrix ρ0 = ∑i pi|i〉〈i| (with 0 ≤ pi ≤ 1, ∑i pi = 1) at time t0, the probability to reach a final state| f 〉 at time t f , after a sequence of N measurements with results ok, is then given by [33]pρ0 f (o1,o2, . . . ,oN) = (2.1)∑i〈i|U(t0, t1)Po1U(t1, t2)Po2 . . .PoNU(tN , t f )| f 〉〈 f |U(t f , tN)PoN . . .Po2U(t2, t1)Po1U(t1, t0)|i〉pi.In general, matrix elements for both the time-evolution and the projection operators are complex,and cannot be simulated by Monte Carlo methods. However, classical measurements should dis-entangle the quantum system and reduce the sign problem. For simplicity, we consider systemscompletely driven by measurements, i. e. U(tk, tk+1) = 1. To construct the path integral, we insert∑nk |nk〉〈nk|= 1 into the first factor and independently ∑n′k |n′k〉〈n′k|= 1 into the second factor in eq.(2.1), at all times tk. After some rearranging, we arrive at the following expression for the real-timepath integral along the Keldysh contour leading from t0 to t f and back [34, 35]:pρ0 f (o1,o2, . . . ,oN) =∑ipi ∑n1,n′1. . . ∑nN ,n′NN∏k=0〈nkn′k|Pok ⊗P∗ok |nk+1n′k+1〉. (2.2)Here 〈nkn′k|Pok⊗P∗ok |nk+1n′k+1〉= 〈nk|Pok |nk+1〉〈n′k|Pok |n′k+1〉∗, 〈n0n′0|= 〈ii|, and |nN+1n′N+1〉= | f f 〉.Further, if the intermediate measurements are of no interest (as is usually the case for dissipation),we can sum over them to arrive at the following result:pρ0 f =∑o1∑o2. . .∑oNpρ0 f (o1,o2, . . . ,oN) =∑ipi ∑n1,n′1. . . ∑nN ,n′NN∏k=0〈nkn′k|P˜k|nk+1n′k+1〉, (2.3)A dissipative system continuously interacting with an environment can be described by the Lind-blad equation. For the system under consideration, the Lindblad operators [36, 37] Lok =√εγPok ,obey (1− εγN)1+∑k,ok L†okLok = 1, where γ determines the probability of measurements per unittime. The index k now labels the Lindblad operators at any fixed instant of time tk. The Lindbladequation is:∂tρ =1ε ∑k,ok(LokρL†ok −12L†okLokρ−12ρL†okLok)= γ∑k(∑okPokρPok −ρ). (2.4)The second equation is obtained in the continuous time limit, ε → 0. Based on the Lindbladequation, we can again arrive at a path integral expression similar to the one in eqn. (2.3).For illustration, lets consider two spins 12 , ~Sx and ~Sy. The total spin S eigenstates |SS3〉 (with3-component S3) are: |11〉 = | ↑↑〉, |10〉 = 1√2(| ↑↓〉+ | ↓↑〉), |1− 1〉 = | ↓↓〉, and |00〉 = 1√2(| ↑↓〉− | ↓↑〉). The projection operators corresponding to a measurement 1 or 0 of the total spin arethen given by P1 = |11〉〈11|+ |10〉〈10|+ |1−1〉〈1−1| and P0 = |00〉〈00|, such that3PoS(LATTICE 2015)281Real-time simulations Debasish BanerjeeP1 =1 0 0 00 1212 00 1212 00 0 0 1 , P0 =0 0 0 00 12 −12 00 −12 12 00 0 0 0 . (2.5)Note that the negative entries in P0 would give rise to the sign problem in a Monte Carlo simulation.However, when the measurement results are not distinguished, P˜= P1⊗P∗1 +P0⊗P∗0 all the matrixelements of the operator are non-negative. Furthermore, a very efficient loop-cluster algorithm hasbeen designed to simulate the system [39].3. Simulating a large quantum system in real-timeThe above example can be straightforwardly generalized to the case of a large quantum spinsystem in any dimension. We have implemented this process for a spin S = 12 system on a squarelattice of size L× L with periodic boundary conditions. The measurement process involves thenearest-neighbor spins, and is implemented in four steps. The first step measures all the neighbor-ing spins in the 1-direction, at x = (x1,x2) and y = (x1 + 1,x2) with even x1, simultaneously. Thesecond step looks at the spins at (x1,x2) and (x1,x2 + 1) with even x2. In a third and fourth mea-surement step, the total spins of pairs with odd x1 and x2 are measured. This is repeated an arbitrarynumber of times M, such that the total number of measurements is N = 4M. As a Lindblad process,the exact ordering of the measurement is irrelevant, and the measurements are stochastic. We usethe Hamiltonian of the Heisenberg anti-ferromagnet H = J∑〈xy〉~Sx ·~Sy, to prepare an initial densitymatrix ρ0 = exp(−βH). A cluster algorithm is used to update the whole system, including boththe Euclidean and real-time branches as explained in [39].At T = 0, the global SU(2) symmetry of the Heisenberg model is spontaneously broken, andthis gives a large signal in the staggered magnetization Ms=∑x(−1)x1+x2S3x of the initial state. Dis-crete measurements quickly destroy this order, and drive the system to a new steady state as shownin Fig. 1a. A Lindblad process also has the same effect, but since all the spin pairs are not affectedsimultaneously, the approach to the new steady state happens more slowly, and is dependent on thedissipative coupling strength. The Fourier modes, S˜(p) = ∑x S3x exp(ip1x1 + ip2x2), p = (p1, p2),show a similar behavior (Fig. 1c). The symmetry breaking is signalled by a large condensate signalat the Fourier mode (pi,pi), and in the nearby modes around this one. The uniform magnetization,~M = ∑x~Sx, on the other hand, is conserved by both the Hamiltonian and the measurement process.Therefore, while the mode (0,0) does not equilibrate at all, the nearby low-momentum modes ap-proach the final state very slowly, according to 〈|S˜(p)|2〉 → A(p)+B(p)exp(−t/τ(p)). For smallmomenta, the equilibration process is diffusive: 1/[γτ(p)] =C|pa|r, C = 1.26(8), r = 1.9(2) (Fig.1b).The dissipation process conserves not only the total spin, but also the lattice translation androtation symmetries. The final density matrix to which the system is driven, is therefore constrainedby these symmetries, and is proportional to the unit matrix in each of these sectors. This can beanalytically computed to give A(p) = L4/(L2−1), shown as the horizontal line in Fig. 1c. This isa stable fixed point of the full dynamics (Hamiltonian and the Lindbladian) at T = ∞, and acts as auniversal attractor for a large class of dissipative process [40].4PoS(LATTICE 2015)281Real-time simulations Debasish Banerjee 100 200 300 400 500 600 0  1  2  3  4  5ta) βJ=0.01βJ=0.1βJ=1.0βJ=10.0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0  0.2  0.4  0.6  0.8  1  1.2|p a|b)L=16aL=32a 100 200 300 400 500 600 0  1  2  3  4  5  6  7  8  9  10γ tc)(pi/8,pi/8)(pi/4,pi/4)(5pi/8,5pi/8)(3pi/4,3pi/4)(pi,pi)Figure 1: a) Real-time evolution of 〈M2s 〉 driven by discrete measurements, for βJ = 0.01, 0.1, 1, and10, for L = 16a. b) Inverse equilibration time 1/[γτ(p)] as a function of |p| for L = 16a, βJ = 40, andL = 32a, βJ = 80. c) Evolution of the Fourier modes 〈|S˜(p)|2〉 for the Heisenberg anti-ferromagnet drivenby a continuous Lindblad process. The initial ensemble is at a low temperature βJ = 5L/2a, where a is thelattice spacing.Another interesting issue is whether a phase transition occurs at a finite instant in real time.While the initial state breaks SU(2) symmetry spontaneously, the final state has a volume inde-pendent 〈M2s 〉/L2, indicating a restored SU(2) spin symmetry. This implies that the system passesthrough a (second-order) phase transition. However, since the Lindblad process takes the systemout of thermal equilibrium, this phase transition is not expected to fall into any of the standarddynamic universality classes [41]. To study this, we plot 〈M2s 〉/L4 as well as the Binder ratio〈M4s 〉/〈M2s 〉2 for βJ = 2L/3a in Fig. 2a and 2b. The finite volume curves do not intersect, but theirpoint of inflection moves to later and later real times with increasing spatial volume. The staggeredmagnetization densityMs, and the length scale ξ = c/(2piρs) (where c is the spin-wave velocityand ρs is the spin stiffness) are obtained as a function of time, by a fit to the equation〈Ms(t)2〉= Ms(t)2L433∑n=0cn(ξ (t)L)n, (3.1)which implicitly definesMs(t) and ξ (t).The constants c0 = 1, c1 = 5.7503(6), c2 = 16.31(2), c3 =5PoS(LATTICE 2015)281Real-time simulations Debasish Banerjee10-410-310-210-1 0  0.5  1  1.5  2  2.5  3  3.5  4γ ta)L=8aL=16aL=24aL=32aL=40aL=48aL=56aL=64a 1.8 2 2.2 2.4 2.6 2.8 3 0  0.5  1  1.5  2  2.5  3  3.5  4γ tb)L=8aL=16aL=24aL=32aL=40aL=48aL=56aL=64a 0.25 0.5 0.75 1 0  0.2  0.4  0.6  0.8  1  1.2  1.4γ tc) 0.8 1 1.2 1.4 1.6 1.8 2 2.2 0  0.2  0.4  0.6  0.8  1  1.2  1.4γ td)Figure 2: [Color online] a) 〈M2s 〉/L4 and b) Binder ratio 〈M4s 〉/〈M2s 〉2 as functions of time for L/a =12, . . . ,48, βJ = 8, . . . ,30. Evolution of c)Ms(t)/Ms(0) and d) ξ (t)/ξ (0).−84.8(2), accurately determined at t = 0, are assumed to be time-independent since they are relatedto the spatial geometry of the system. The exponential decay of the order parameter Ms(t) =Ms(0)exp(−t/τ), (withMs(0) = 0.30743(1)/a2 [42, 43]) suggests that it takes an infinite amountof time for the order to disappear completely and hence for the phase transition to be completed.4. Conclusion and OutlookWe have demonstrated that for certain measurement or dissipative processes, we can simulatethe real-time evolution of a large quantum system on classical computers. The processes underconsideration drive initial states (or density matrices) into a new state with only short-range cor-relations. However, the phase transition into a final disordered steady state without spontaneoussymmetry breaking is completed only in an infinite amount of time.Extending these investigations, we have also studied several other measurement processes andinitial states given by the ground states of different models [40]. All these cases show a diffusivebehavior of a conserved quantity (i.e., the staggered or the uniform magnetization). Further, thenon-equilibrium transport of magnetization in large open systems has also been studied with ourmethod [44]. A study of competing Lindblad processes, and the inclusion of a part of the Hamil-tonian along with the dissipation, as well as extension of this formalism to fermionic systems. arecurrently under investigation.6PoS(LATTICE 2015)281Real-time simulations Debasish BanerjeeReferences[1] S. R. White, Phys. Rev. Lett. 68 (1992) 2863.[2] U. Schollwöck, Rev. Mod. Phys. 77 (2005) 259.[3] G. Vidal, Phys. Rev. Lett. 91 (2003) 147902.[4] S. R. White and A. 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Real-time simulation of dissipation-driven quantum systems

We set up a real-time path integral to study the evolution of quantum systems driven in real-time completely by the coupling of the system to the environment. For specifically chosen interactions, this can be interpreted as measurements being performed on the system. For a spin-1/2 system, in particular, when the measurement results are averaged over, the resulting sign problem completely disappears, and the system can be simulated with an efficient cluster algorithm

Real-time simulation of dissipation-driven quantum systems

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