Relaxation of the Ising spin system coupled to a bosonic bath and the time dependentmean field equation

Mate Tibor Veszeli and Gabor VattayInstitute of Physics, Eotvos University, 1518 Budapest, Hungary

The Ising model doesn’t have a strictly defined dynamics, only a spectrum. There are differentways to equip it with a time dependence e.g. the Glauber or the Kawasaki dynamics, which are bothstochastic, but it means there is a master equation which can also describes their dynamics. Wepresent a Gluber-type master equation derived from the Redfield equation, where the spin systemis coupled to a bosonic bath. We derive a time dependent mean field equation which describes therelaxation of the spin system at finite temperature. Using the fully connected, uniform Ising modelthe relaxation time will be studied, and the critical behaviour around the critical temperature. Themaster equation shows the finite size effects, and the mean field equation the thermodynamic limit.

I. INTRODUCTION

Spin models are versatile, because they are simple, yet able to demonstrate fundamental phenomenons, like phasetransition [1–3]. Many complex physical models can be reduced to a simple Ising or Heisenberg model, like electronand nuclear spins [4], and even social situations [5]. It is also important in modern applied physics since one brach ofadiabatic quantum computers - like the D-Wave system [6] - are based on finding the global minimum of an artificialspin system [7, 8].

The Ising model is defined via its energy or in the quantum case, where it is often called Heisenberg model, viaits Hamiltonian operator. The former do not have a natural dynamics, and although the latter has one, i.e. theSchrodinger or the Heisenberg equation, it is not always what we want. For example if we want our system toconverge to the Boltzmann distribution, then the Schrodinger equation is not enough.

To describe such a system we must use the tools of open quantum systems [9, 10] like the Redfield [11] and theLindblad equation [12]. These equations have countless applications in quantum biology [13, 14], quantum optics [9],cold atomic gases [15], chemical physics [16] and besides it is also relevant in quantum computing [17–19].

Quantum dissipation and relaxation of spin systems in a bosonic bath and in magnetic field have been investigatedby many authors. [20–23]. The interaction between an adiabatic computer and its enviroment is meant to be small, sothe weak coupling Lindblad equation will be used, but of course there are improved methods to describe open quantumsystems, like slippage initial condition [24, 25], Nakajima-Zwanzig equation [26, 27] or the polaron transformation[28].

The structure of this paper is the following. In section II we present a Glauber-type master equation based on theRedfield equation. In section III we investigate the temperature dependence of the eigenvalues of the transition matrix,because they contain relevant informations on the time scales of the system, e.g. the relaxation time. We give anupper bound to the smallest nonzero eigenvalue, then in section IV the dynamics of the uniform, fully connected Isingmodel is investigated, and we show that the relaxation time diverges in the thermodynamic limit as the temperatureapproaches the critical temperature. In section V a time dependent mean field equation is derived from the masterequation, which will be tested in section VI using the uniform Ising model.

II. MASTER EQUATION OF QUANTUM ISING SYSTEM

In general if a system is connected to a bath, than its Hamiltonian operator is

Htot = H +HB +HI, (1)

where H acts only on the system of interest, HB only on the bath, and HI is the interaction between the twosubsystems, and it can be written as HI =

∑αAα⊗B, where Aα and Bα are system and bath operators respectively.

The dynamics of the total system is described by the von Neumann equation.

ρtot = −i[Htot, ρtot] (2)

If the interaction between the system and the bath is small, than after the Born and the Markov approximation aneffective equation can be derived to the density matrix of the system of interest (ρ := TrBρtot).

ρ(t) + i [H, ρ(t)] =∑α

(Aαρ(t)T †α −AαTαρ(t) + h.c.

), (3)

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1

(a) Ohmic case (b) Super-Ohmic case

FIG. 1: Fourier transform of the bath correlation functionβ1 < β2 < β3

where Tα =∑β

∫∞0

dtCαβ(t)AIβ(−t), AI

β(t) is in interaction picture, Cαβ(t) = 〈BIα(t)Bβ〉B is the bath correlation

function, and h.c means hermitian conjugate. This is the Redfield equation in weak-coupling limit [11]. After the socalled secular or rotating wave approximation one can get to equation

ρ+ i [H +HLS, ρ] =∑αβ

∑ω

γαβ(ω)(Aβ(ω)ρA†α(ω)− 1

2

{A†α(ω)Aβ(ω), ρ

} ), (4)

where Aα(ω) =∑ij |i〉〈i|Aα|j〉〈j|δω,εj−εi and |i〉 is the eigenvector of H with eigenvalue εi [9]. HLS is the Lamb shift

Hamiltonian, which is usually small, so we will neglect it, and γαβ(ω) is the Fourier transform of the bath correlationfunction.

γαβ(ω) :=

∫ ∞−∞

dteiωt〈B†α(t)Bβ(0)〉B (5)

There are two common bosonic bathes: the bath of phonons and the bath of photons. For phonons γohm(ω) ∼

e−|ω|ωc

ω

1− e−βω, which is called Ohmic case and for photons γsup(ω) ∼ ω3

1− e−βω, which is a super-Ohmic case. The

frequency ωc is the cutoff frequency. If we assume, that ωc is large compared to the energy distances of the system,

than e−ω|ωc| ≈ 1. Figure 1 shows the main features of the two γ functions. The main difference is that γohm is strictly

increasing and γohm(ω = 0) ∼ kBT , but γsup is non-monotonic, and γsup = 0. In the easiest case γαβ ∝ δαβ .The advantage of the weak-coupling limit is that a master equation can be derived to the diagonal elements of ρ.

Pi =∑j

MijPj ≡∑j

WijPj −∑j

WjiPi, (6)

where Pi = ρii, Wij =∑αβ γαβ (ωji) (Aα)ji(Aβ)ij and ωji = (εj − εi).

The system converges to the Boltzmann distribution if Wij satisfies the detailed balance condition i.e. Wij =Wji exp(−β(εi − εj)). Both the Ohmic and the super-Ohmic bath satisfy it, because

γ(−ω) = e−βωγ(ω) (7)

If the system of interest is the Ising model, then the Hamiltonian is

H = −∑ij

(i<j)

Jijσzi σ

zj −

∑i

hiσzi , (8)

where σzi is the Pauli z-matrix and the corresponding eigenvectors are

|S〉 ≡ |S1, S2, . . . , SN 〉 Si ∈ {±1} (9)

2

with eigenenergies

ES = −∑ij

(i<j)

JijSiSj −∑i

hiσzi (10)

The easiest way to couple the system to the bath is via a Pauli matrix i.e. Aα → σxi . Using σz in the interacion insteadof σx would not give any relevant dynamics, since the system and the interaction Hamiltonians would commute. Thepeculiarity of this system is that the populations decouple even without the secular approximation.

The σxi operator acting on |S〉 only flips the ith spin, so the WS S′ matrix element is

WS S′ =∑i

γ(ωS′S)(σxi )S′S(σxi )S S′ =

γ(ωS′S) | if the Hamming distance betweenS and S′ is 1

0 | otherwise.(11)

With Eq. (6) and (11) we have a dynamics for the Ising model.

PS =∑S′

MS S′PS′ , (12)

where MS S′ = WS S′ − δS S′∑S′′WS′′S′ is the transition matrix. This matrix is temperature dependent, and it has

at least one zero eigenvalue, which is the eigenvalue of the equilibrium distribution:

P eqS =

e−βES

Z(13)

For constant temperature the general solution of (12) is

PS(t) =∑S′

∑µ

e−λµtPRµ,SP

Lµ,S′PS(t = 0), (14)

where PRµ s are the right, and PR

µ s are the left eigenvectors of M with −λµ eigenvalues. All the λµs are nonnegative. Ifthe system is ergodic, then there is only one zero eigenvalue, and the other λs are positive. Let the smallest positivebe λmin and the largest be λmax. The relaxation time is tr = 1/λmin. This is the time scale in which all but theequilibrium mode dies out. The other relevant time scale is 1/λmax, which is the characteristic time of the fastestmode. If for example this spin system is a quantum computer, then the fastest mode is the more important, because ifthe computation is slower than this time scale, then the enviroment isn’t neglectable. In other words λmin is importantif we want the system to relax thermally, and λmax is important if we want to avoid any thermal influence.

III. TEMPERATURE DEPENDENCE OF THE EIGENVALUES

Both the smallest and the largest eigenvalue carry relevant information, and since M(β) is temperature dependentλmin(β) and λmax(β) are too.

At high temperature we can determine the temperature dependence of all λs by simply Taylor expanding γ(ω;β)for small β.

γ(ω;β) = ηωα

1− e−βω≈ ηω

α−1

β, (15)

where α = 1 in the Ohmic, and α = 3 in the super-Ohmic case. The transition matrix inherits this temperaturedependence: MS S′(β) ∼ β−1, and hence λ ∼ β−1.

In spite of the high temperature limit, where the elements of the dynamical matrix M diverges, in the low temper-ature limit they converge.

limβ→∞

γ(ω;β) =

{0 | ω ≤ 0ηωα | ω > 0

. (16)

It means all the eigenvalues also converge. As a consequence we can’t slow down arbitrary all the modes by reducingthe temperature. We have an upper limit in time for the quantum computing. Of course this calculation is valid onlyfor a time independent system, but the main features apply to more general cases.

3

Without external magnetic field (h = 0) at zero temperature the equilibrium Boltzmann distribution prefers onlythe two spin configurations with the lowest energies:

P eqS (T = 0) =

1

2(δS,Sg

+ δS,−Sg), (17)

where Sg and −Sg are the ground states. At zero temperature there is one more eigenvector with zero eigenvalue:

PRmin,S =

1

2(δS,Sg

− δS,−Sg) (18)

The question is how λmin(β) behaves at low temperature. We can give an upper bound. First let us introduce thefollowing symmetric matrix:

MS S′ = MS S′

√√√√P eqS′

P eqS

≡MS S′e−β

ES′−ES

2 (19)

This transformation doesn’t affect the eigenvalues, and the eigenvectors transform like

PµS =PRµ,S√P eqS

. (20)

Since M is symmetric its right and left eigenvectors are the same, and now the variational method applies to it:

λmin ≤ −∑S,S′

ΠSMS S′ΠS′ , (21)

where Π is an arbitrary vector with∑S Π2

S = 1, and it must be perpendicular to the equilibrium vector ( P eqS ≡

√P eqS

), because λmin is the second smallest eigenvalue of M . Let Π = 1√2(δS Sg

− δS−Sg). Then accordig to (21)

λmin ≤ −1

2(MSg,Sg

− MSg,−Sg− M−Sg,Sg

+ M−Sg,−Sg)

= −MSg,Sg= −MSg,Sg

=∑S

WS,Sg

=∑S

d(S,Sg)=1

γ(ωSg,S;β),

(22)

where d(S, Sg) is the Hamming distance, and the S 7→ −S symmetry was used. In the bosonic bath

λmin(β) ≤∑S

d(S,Sg)=1

η(∆ES

)α 1

eβ∆ES − 1, (23)

where ∆ES := ES − ESg> 0. At low temperature this is the sum of some e−β∆ES functions, so λmin(β) can be

estimated from above with an exponential function.Figure 2 shows λmin(β), λmax(β) and −MSg,Sg

(the upper bound) for a 4 × 4, ferromagnetic, 2D Ising model with

Ohmic bath. The dashed vertical line marks the critical temperature (βcJ = ln(1+√

2)2 ≈ 0.44). The left figure is

in log-log scale, where we can see, that at low temperature the eigenvalues has a β−1 temperature dependence, andλmax(β) converges, and the right figure with lin-log scale shows, that λmin(β) goes to zero exponentially.

IV. EIGENVALUES OF THE UNIFORM ISING MODEL

The M matrices are 2N × 2N large, therefore we can’t see how the eigenvalues behave at the thermodinamic limit.However the uniform, fully connected Ising model is so symmetric, that an effective equation can be derived, whichhas the same relaxation time as the original equation.

4

FIG. 2: Temperature dependence of λmin, λmax and MSg,Sg

2D, ferromagnetic, 4× 4 Ising model, Ohmic bath, J = 1, η = 1

The energy of the model is

ES = − JN

N∑i,j=1(i>j)

SiSj . (24)

The 1/N factor is to keep the energy extensive and J > 0. Given an S microstate, it consists of N↑ spins withSi = 1 and N↓ spins with Si = −1. The number of spins is constant, i.e. N↑ + N↓ = N = fix. The energy of such aconfiguration is

ES = − JN

[N↑(N↑ − 1)

2+N↓(N↓ − 1)

2−N↑N↓

], (25)

If N is fixed, then the energy is the function of only N↑. The symmetry of the system is that we can perturb thespins any way, the energy and the M matrix remains the same. If in the dynamic the initial condition also has thissymmetry, then the PS will inherit this property. The slowest mode propagates between the two deepest valley ofthe energy landscape, which are the ↑↑ . . . ↑ and ↓↓ . . . ↓. Assume that initially P↓↓...↓(t = 0) = 1, and we wantto determine relaxation time, where P↓↓...↓(tr) ≈ P↑↑...↑(tr). Since both the equations and the initial condition hasthe permutation symmetry all the probabilities, which has the same up spin has the same value, e.g. for 3 spinsP↑↓↓(t) = P↓↑↓(t) = P↓↓↑(t) ∀t. The probability can only flow between spin configurations if the Hamming distancebetween them is 1. Let us introduce the following probabilities:

PN↑ =∑′

S

PS =

(N

N↑

)P↑ . . . ↑︸ ︷︷ ︸

N↑

↓ . . . ↓︸ ︷︷ ︸N−N↑

, (26)

where the prime denotes that only such configurations count where there are N↑ up spin. We can give a closed set ofdifferential equations which only contain this new PN↑ probabilities.

PN↑ =

(N

N↑

)[N↑WN↑,N↑−1

PN↑−1(N

N↑−1

) + (N −N↑)WN↑,N↑+1

PN↑+1(N

N↑+1

)]−(N↑WN↑−1,N↑ + (N −N↑)WN↑+1,N↑

)PN↑

=(N −N↑ + 1)WN↑,N↑−1PN↑−1 + (N↑ + 1)WN↑,N↑+1PN↑+1

−(N↑WN↑−1,N↑ + (N −N↑)WN↑+1,N↑

)PN↑ ,

(27)

where WN↑,N↑+1 = γ(EN↑+1−EN↑) = γ(− 2N (2N↑ −N + 1)

). This master equation has only N+1 variables instead of

2N , thus easy to simulate for large systems. A comparison between the quantum and the thermal simulated annealingof the fully connected Ising model was investigated by Wauters et al. using a similar reduced master equation [29].Equation (27) has the form

PN↑ =

N∑N ′↑=0

M redN↑,N ′↑

PN ′↑ , (28)

5

(a) Temperature dependence of λmin fordifferent system sizes

(b) System size dependence of λmin below, aboveand at the critical temperature

FIG. 3: Fully connected, uniform Ising model, Smallest eigenvalue of Mred

J = 1, η = 1

FIG. 4: Critical behaviour of the fully connected Ising model above Tc.J = 1, η = 1

and we want to determine the lowest (nonzero) eigenvalue of M red, which is the same as the lowest (nonzero) eigenvalueof M . The matrix M red is sparse, because it is a tridiagonal matrix, i.e. only the main diagonal, the first diagonalbelow and above the main diagonal is nonzero. Figure 3a shows the temperature dependence of λmin for differentsystem sizes. As N increases we can see, that around the critical temperature (which is βcJ = 1) the behaviour ofthe system changes. At figure 3b we can see it better, that above the critical temperature (T > Tc) for large N valuesλmin converges, meaning for every system size there is a finite relaxation time. At the critical temperature (T = Tc),

it follows a power law (λmin ∝ N −0.5). Below the critical temperature (T < Tc) λmin goes to 0 for large N , but doesn’tfollow a power law. This behaviour is the famous critical slowing down phenomenon.

From the N → ∞ thermodinamic limit we can determine the dynamical critical exponent. Figure 4 showsλmin(T,N → ∞) as the function of the reduced temperature (T−Tc

Tc). This follows an easy power law, because

λmin ∝ T − Tc. In the next section we will see that this result can be obtained from the mean field approximation.

6

V. TIME DEPENDENT MEAN FIELD EQUATION

Since the primary interest is the magnetization (mi := 〈Si〉), we would like to derive a differential equation for it.Using the definition of mi and the master equation we get

mi =∑S

PSSi =∑S S′

WS S′PS′Si −∑S S′

WS′SPSS′i =

∑S′S

WS′SPS(S′i − Si) (29)

The WS′S matrix component is nonzero if the Hamming distance between S′ and S is one. Introducing

Λi(S, n) =

{Si | i 6= n−Si | i = n

(30)

we can rewrite the double sum in (29).

mi =∑S

N∑n=1

WΛ(S,n),S PS(Λi(S, n)− Si) =∑S

WΛ(S,i),S PS(−2Si) = −2〈WΛ(S,i),SSi〉 (31)

In the second step the (Λi(S, n) − Si) = −2Siδin identity was used. The nonzero elements of W are the function ofthe energy difference:

WΛ(S,i),S = γ(ES − EΛ(S,i)) = γ(− 2(∑

j

JijSj + hi

)Si

)≡ γ

(− 2hiSi

), (32)

where hi =∑j JijSj +hi, so this is still the function of the S random variable, but because Jii = 0 it is not a function

of Si. Since Si can be only 1 or −1 the γ(−2hiSi) as a function of Si must have the

γ(−2hiSi) ≡γ(−2hi) + γ(2hi)

2+γ(−2hi)− γ(2hi)

2Si (33)

form. Using (7) yields

γ(−2hiSi) = γ(2hi)

[e−2βhi + 1

2+

e−2βhi − 1

2Si

]= γ(2hi)

e−2βhi + 1

2

[1− tanh(βhi)Si

], (34)

then substituting back to (31) gives

mi = −⟨γ(2hi)

(e−2βhi + 1

)(Si − tanh(βhi)

)⟩. (35)

Equation (35) is similar to the Callen equation [30, 31] (〈Si〉 = 〈tanh(βhi)〉), where the averaging is outside thehyperbolic function. In order to get a closed equation to the expected values the average must move inside, andinstead of the Si random variables their mi expected values must be written.

mi = −γ(

2(ΣjJijmj + hi))(

1 + e−2β(∑j Jijmj+hi)

)(mi − tanh

(β(ΣjJijmj + hi)

))(36)

The right-hand side contains the self-consistent equation from the equilibrium statistical physics, hence if the equationof state is satisfied, then mi = 0.

Equation (36) contains both the real time and the temperature of the bath. The temperature can be also timedependent, and in that case what we could get is a thermal annealing, but if the temperature is constant we candetermine the relaxation time, and the dynamical critical exponent. If m(t) = meq + δm(t), where meq is theequilibrium solution and δm(t) is small, then the linearized equation of (36) is

δmi = −bi(meq)∑j

{(δij −

βJij

cosh2(β∑k Jikm

eqk + hi)

)δmj

}, (37)

where bi(meq) = γ

(2(ΣjJijm

eqj + hi)

) (1 + e−2β(

∑j Jijm

eqj +hi)

). Using the 1

cosh2(x)≡ 1 − tanh2(x) identity, and the

equation of state we get to

δmi = −bi(meq)∑j

{(δij − βJij

(1−

(meqj

)2))δmj

}. (38)

7

Equation (38) contains the inverse susceptibility of the mean field Ising model.

χ−1ij :=∂2FMFA

∂mi∂mj= −Jij +

Tδij1−m2

i

, (39)

where

FMFA(m,h, T ) = −1

2

∑ij

Jijmimj −∑i

himj + T∑i

[1 +mi

2ln

(1 +mi

2

)+

1−mi

2ln

(1−mi

2

)]. (40)

Substituting the inverse susceptibility back into (38) yields

δmi = −bi(meq)β(

1− (meqi )

2)∑

j

χ−1ij δmj ≡ −Γi∑j

χ−1ij δmj . (41)

This is a well known equation in the theory of dynamical critical phenomena [32], but it is usually derived fromthe m = −Γ∂mF

MFA phenomenological equation. Now we can see, how it is related to a master equation and thespin-boson model. If the system is symmetric in a sense, that all the spins behave the same, then (41) simplifies to

δm = −Γχ−1δm, (42)

where χ−1 =∑j χ

−1ij .

VI. TIME DEPENDENT MEAN FIELD EQUATION FOR THE UNIFORM ISING MODEL

As before in section IV the uniform Ising model will be studied, because in the equilibrium case in the thermodynamiclimit it gives back the exact results. According to (40) the mean field free energy is

FMFA(m,h, T )

N= −1

2Jm2 − hm+ T

(1 +m

2ln

(1 +m

2

)+

1−m2

ln

(1−m

2

)), (43)

and the time dependent mean field equation is

m = −γ(2(Jm+ h))(

1 + e−2β(Jm+h))

(m− tanh(β(Jm+ h))) . (44)

If h = 0 the critical temperature is Tc = J , and above this temperature the equilibrium solution is meq = 0. Theinverse susceptibility is

χ−1 = −J + T ≡ T − Tc, (45)

therefore

λmin = Γ(T − Tc), (46)

where Γ = 2γ(0;β)β = 2η in the Ohmic bath. Equation (46) is the same result that we have already seen in figure4. As in the equilibrium statistical physics the mean field approximation gives back the exact result for the uniformmodel in the thermodynamic limit.

At the critical temperature the inverse susceptibility is zero, the linear term vanishes, and we need the higher orderterms. Taylor expanding (44) at T = Tc around m = meq ≡ 0 up to third order gives.

δm = −2

3ηJδm3 (47)

which has the

δm(t) ∝ t− 12 (48)

solution for large ts, which means there isn’t a characteristic time.

8

FIG. 5: Free energy of the uniform Ising model with finite external magnetic field

Below the critical temperature the linearized equation is good again, only the meq changes. On the other handequation (44) can give different solution than the (28) master equation. If we want to compare these two equationsthe initial condition must be also the same which gives a restriction to the initial condition of (28). In the mean fieldapproximation the probability is a product of the one particle probabilities:

PMFAS =

∏i

1 +miSi2

, (49)

which means the initial probability of (28) is

PN↑(t = 0;m) =

(N

N↑

)(1 +m

2

)N↑ (1−m2

)N↓(50)

If h = 0, then the master equation must converge to the m = 0 solution, but the time dependent mean field equationfinds the global minimum of the free energy if initially m 6= 0. If h is finite and m is in the valley of the globalminimum (point A in figure 5), than in the N →∞ limit the solution of the master equation converges to the meanfield solution (figure 6a).

On the other hand if initially m is in the valley of the local minimum (point B in figure 5), then the solution of themean field equation converges to this local minimum, but the solutions of the master equation are totally different.The probabilities converge to the equilibrium

P eqN↑

(h,N) ∝(N

N↑

)e−βEN↑ (h,N) (51)

distribution, so they tend to approach meq for large N values in the t → ∞ limit, but as N grows, so does therelaxation time. In the thermodynamic limit the relaxation time diverges as in figure 3b. In the N → ∞ limit thesolution of the master equation converges to the solution of the mean field equation and none of them will approachthe global minimum, because the relaxation time will be infinit.

VII. CONCLUSION

In this work we have presented a Glauber-type master equation for the spin-boson model. Starting from the Redfieldequation the population decoupled even without the secular approximation. The most relevant dynamical propertiesare encoded in the eigenvalues of the transition matrix of the master equation. They are temperature dependent,and behave significantly different below, above and at the critical temperature as a function of the system size. Inthe case of the uniform, fully connected Ising model, in the thermodynamic limit, above the critical temperature therelaxation time follows a power law: tr ∝ (T − Tc)−1.

9

(a) Point A, Solutions when the initial m isclose to the global minimum

(b) Point B, Solutions when the initial m is closeto the local minimum

FIG. 6: Comparison of the time dependent mean field equation and the master equationJ = 1, η = 1, h = 0.02, T = 0.9, Ohmic bath

We have derived a time dependent mean field equation, which is an effective equation of the master equation,containing only the 〈Si〉 expected values. As every mean field theory it works best if the number of neighbours islarge, so the fully connected Ising model is the best candidate, and the numerical simulations show that in the N →∞limit the master equation gives back the same solutions as the mean field equation.

ACKNOWLEDGEMENT

This work was supported by NKFIH within the Quantum Technology National Excellence Program (Project No.2017-1.2.1-NKP-2017-00001) and within the Quantum Information National Laboratory of Hungary, by the ELTEInstitutional Excellence Program (TKP2020-IKA-05) financed by the Hungarian Ministry of Human Capacities, andInnovation Office (NKFIH) through Grant No. K134437.

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