Dynamics of Quantum Ising Systems

Abstract

The dynamics of quantum many-body systems have been the most important topic of the condensed matter and statistical physics in this century. Transverse Ising models have played a significant role in the study of the dynamics of quantum many-body systems. Chapter 7 exclusively discusses the dynamics of transverse Ising models. The first part mentions dynamical properties of transverse Ising models without explicit time dependence, where the tunneling dynamics of classical spin states are discussed. The second part, on the other hand, focuses on the dynamics of transverse Ising models with a time-dependent field. Theories of sudden as well as slow quantum quenches and quantum hysteresis are presented here. Also, the response to a pulsed transverse field is discussed. Appendices of this chapter include details on the mean field equation of motion, the Landau-Zener problem, and the microscopic equation of motion in the presence of an oscillatory field.

Appendix 7.A

7.1.1 7.A.1 Mean Field Equation of Motion

7.1.1.1 7.A.1.1 Some Analytic Solutions in the Linearised Limit

Linearisation of (7.2.113) can be carried out in three limits: namely, in the high temperature (T ⁻¹→0), the low tunnelling field amplitude (Γ ₀→0) and in the adiabatic (ω→0) limits.

In the first two cases, the linearised equations of motion take the forms:

$$ \tau \frac{dm^{x}}{dt} = -m^{x} + \frac{\varGamma(t)}{T} $$

(7.A.1)

and

$$ \tau \frac{dm^{z}}{dt} = - \biggl( 1- \frac{1}{T} \biggr) m^{z}, $$

(7.A.2)

in the T ⁻¹→0 limit, and

$$ \tau \frac{dm^{x}}{dt} = -m^{x} + \tanh \biggl( \frac{m^{z}}{T} \biggr) \frac{\varGamma(t)}{m^{z}} $$

(7.A.3)

and

$$ \tau\frac{d m^{z}}{dt} = -m^{z} + \tanh \biggl( \frac{m^{z}}{T} \biggr), $$

(7.A.4)

in the Γ ₀→0 limit. For T>1, m ^z decays to zero in the first case (7.A.1). The long time solution of the dynamical equation for m ^z in [2] can be obtained easily as the attractive fixed point of the map describing the discretised form of the differential equation: m ^z(t+τ)=tanh[m ^z(t)/T]. This gives m ^z=tanh(m ^z/T)≠0 for T<1 and m ^z=0 for T>1. Putting the solution for m ^z for T<1 into the dynamical equation for m ^x in (7.A.2), reduces it to the same form as (7.A.1) but with Γ(t)/T now replaced by Γ(t) in the second term. Defining now s≡ωt and λ=ωt, one can express the general solution of m ^x(t) as

$$ m^{x} (t) = J \cos(s) + K \sin(s) $$

(7.A.5)

with K=λJ=(Γ ₀/Γ)[λ/(1+λ ²)] for T>1 from (7.A.1) as well as from (7.A.2), and Γ ₀/T replaced by Γ ₀ for T<1 (from (7.A.2)). The loop area A _x =∮m ^x dΓ=2KΓ ₀ can then be expressed in the form given in (7.2.116) (containing the full Lorentzian scaling function g(λ)). The resulting exponents are α=2,β=1 and γ=0=δ in the T ⁻¹→0 limit (from (7.A.1) and (7.A.2) for T>1) and α=2, β=0=γ=δ in the Γ ₀→0 limit (from (7.A.2) for T<1).

In the adiabatic (ω ⁻¹≫τ, or small λ) limit, one can write , where and : m ₀=[tanh(|h|/T)](h/|h|). Collecting now the linear terms in λ, one gets

$$ \delta m^{x} = -\lambda\frac{dm_{0}^{x}}{ds} $$

(7.A.6)

and

$$ \delta m^{z} = \frac{\delta m^{z}}{T} \tanh \biggl( \frac{\varGamma}{T} \biggr). $$

(7.A.7)

For T>1, δm ^z=0 (of course $m_{0}^{z}= 0$). Using this to obtain $m_{0}^{x}$ for the right hand side of the other equation, one gets $\delta m^{x} = -\lambda\frac{d}{ds}\tanh[\varGamma(s)/T]$. Hence the loop area , where Γ=Γ ₀cos(s), can be expressed as

(7.A.8)

where y=Γ ₀/T. As I(y) goes to π and 4/y in the small and large y limits respectively, the loop area A _x can again be expressed in the form (7.2.116) with g(x)∼x in the x→0 limit, and with α=2, β=1, γ=0=δ in the Γ ₀≪T limit and with α=1, β=0=γ=δ in the Γ ₀≫T limit. Although these limiting values for the exponents are not observed (because of the inaccuracy of the Linearisation approximations in the range of the study), they provide useful bounds for the observed values, and the Lorentzian scaling function appears quite naturally here.

7.1.1.2 7.A.1.2 Approximate Analytic Form of Dynamic Phase Boundary

Using the mean field equation for the z-component of magnetisation (7.2.113)

$$ \tau \frac{dm^{z}}{dt} = -m^{z} + \tanh \biggl( \frac{|h|}{T} \biggr) \frac{m^{z}}{|h|} $$

(7.A.9)

one can estimate approximately the dynamic phase boundary. For small m ^z, |h|∼Γ(s), where s=ωt. Defining, λ=ωτ, the above equation can be approximated as

$$ \lambda \frac{dm^{z}}{dt} = -m^{z} + \tanh \biggl( \frac{\varGamma(s)}{T} \biggr) \frac{m^{z}}{\varGamma(s)}= \biggl[ \frac{1}{\varGamma(s)} \tanh \biggl( \frac{\varGamma(s)}{T} \biggr)-1 \biggr]m^{z} $$

(7.A.10)

or

$$ \frac{\lambda}{2\pi} \oint \frac{dm^{z}}{m^{z}}= \frac{1}{2\pi} \int _{0}^{2\pi} \biggl[ \frac{1}{\varGamma(s)} \tanh \biggl( \frac{\varGamma(s)}{T} \biggr) -1 \biggr] \,ds. $$

(7.A.11)

The right hand side of the above equation gives the logarithm of the factor by which m ^z grows over a cycle. So, the equation of the phase boundary can be written as

$$ \frac{1}{\varGamma_{0}} F \biggl( \frac{\varGamma_{0}}{T} \biggr) -1 = 0, $$

(7.A.12)

where

$$ F (y) = \frac{1}{2\pi} \int_{0}^{2\pi} \frac{\tanh(y \cos(s))}{ \cos(s)} \,ds; \quad y = \varGamma_{0}T. $$

(7.A.13)

For large y, ycos(s)∼y(s−π/2), and since the integrand in F(y) contributes significantly near s=π/2, we expand the tanh term as tanhx=x/(1+x ²)^1/2, and get

(7.A.14)

So the approximate analytic form of phase boundary is

$$ T = \frac{\pi\varGamma_{0}/2}{ \sinh(\pi\varGamma_{0}/2)}. $$

(7.A.15)

7.1.2 7.A.2 Landau-Zener Problem and Parabolic Cylinder Functions

Let us consider a time-dependent Hamiltonian of a two-level system,

$$ H_{\mathrm{LZ}} (t) = -\alpha t\sigma^3 + \sigma^1 = \left ( \begin{array}{c@{\quad}c} -\alpha t & 1 \\ 1 & \alpha t \end{array} \right ) , $$

(7.A.16)

where σ ¹ and σ ³ are Pauli matrices:

$$ \sigma^1 = \left ( \begin{array}{c@{\quad}c} 0 & 1 \\ 1 & 0 \end{array} \right ), \qquad \sigma^3 = \left ( \begin{array}{c@{\quad}c} 1 & 0 \\ 0 & -1 \end{array} \right ) . $$

(7.A.17)

Assume that the time t moves from t=−∞ to ∞. We express the basis vectors of the system as |0〉=(1 0)^T and |1〉=(0 1)^T. The eigenstates of this system are |0〉 and |1〉 when t=±∞, while those at any finite times are superpositions of |0〉 and |1〉. We assume that the initial state of the system is |Ψ(−∞)〉=|1〉. The time evolution is determined by the Schrödinger equation:

$$ i\frac{d}{dt}\bigl|\varPsi(t)\bigr\rangle= H_{\mathrm{LZ}}(t)\bigl|\varPsi(t)\bigr\rangle. $$

(7.A.18)

We expand |Ψ(t)〉 as

$$ \bigl|\varPsi(t)\bigr\rangle= \psi_0(t)|0\rangle+ \psi_1(t)|1 \rangle. $$

(7.A.19)

The Schrödinger equation yields

(7.A.20)

(7.A.21)

Our problem is to solve this set of equations under the initial condition, ψ ₀(−∞)=0 and |ψ ₁(−∞)|=1, up to an unessential phase factor. This problem is called the Landau-Zener problem and the solution is given by the parabolic cylinder functions [91, 242, 243, 388, 413, 443].

Eliminating ψ ₁ in (7.A.21) by using (7.A.20), one obtains an equation of ψ ₀(t) as

$$ \frac{d^2}{dt^2}\psi_0(t) + \bigl(1 - i\alpha+ \alpha^2 t^2 \bigr)\psi_0(t) = 0 . $$

(7.A.22)

Define a new variable z by

$$ t = \frac{e^{i\pi/4}}{(2\alpha)^{1/2}}z . $$

(7.A.23)

Then Eq. (7.A.22) is arranged as

$$ \frac{d^2}{dz^2}\psi_0\biggl(\frac{e^{i\pi/4}}{(2\alpha)^{1/2}}z\biggr) + \biggl(\frac{i}{2\alpha} + \frac{1}{2} - \frac{z^2}{4} \biggr) \psi_0\biggl(\frac{e^{i\pi/4}}{\sqrt{2\alpha}}z\biggr) = 0 . $$

(7.A.24)

Writing U(z)=ψ ₀(e ^iπ/4 z/(2α)^1/2), one obtains

$$ \frac{d^2}{dz^2}U(z) + \biggl(p + \frac{1}{2} - \frac{z^2}{4} \biggr)U(z) = 0 , \quad p = i/2\alpha. $$

(7.A.25)

The solutions of (7.A.25) are parabolic cylinder functions denoted by D _p (z). If D _p (z) is a solution, D _p (−z), D _−p−1(iz), and D _−p−1(−iz) are also solutions of the same equation. These four functions are linearly dependent. For instance, D _p (−z) and D _−p−1(−iz) are expressed by D _p (z) and D _−p−1(iz) as

(7.A.26)

(7.A.27)

D _p (z) has following asymptotic expansions for |z|≫1 and |z|≫|p|:

$$ D_p(z) \approx e^{-z^2/4}z^p \bigl(1 + O \bigl(z^{-2}\bigr) \bigr) \quad \biggl[\mbox{for } {|}\arg z| < \frac{3}{4}\pi\biggr] , $$

(7.A.28)

(7.A.29)

(7.A.30)

D _p (z) satisfies following recursion relations:

(7.A.31)

(7.A.32)

(7.A.33)

Now, the initial condition for U(z) is written as U(z→e ^±i3π/4×∞)→0, where upper and lower signs correspond to α>0 and α<0 respectively. To find the solution which meets this condition, one needs to look into the asymptotic values D _p (z), D _p (−z), D _−p−1(iz), and D _−p−1(−iz). Equations (7.A.28)–(7.A.30) are followed by

(7.A.34)

(7.A.35)

(7.A.36)

(7.A.37)

where note that p=i/2α is a pure-imaginary number. With this observation, one finds that D _−p−1(−iz) and D _−p−1(iz) are appropriate solutions for α>0 and α<0 respectively. Therefore

$$ \psi_0(t) = A D_{-p-1}(\mp iz) = A D_{-p-1} \bigl({\mp} ie^{-i\pi/4}(2\alpha)^{1/2}t\bigr) , $$

(7.A.38)

and

(7.A.39)

where a recursion relation, (7.A.33), is used in the last equality. The factor A is determined by the initial condition: |ψ ₁(−∞)|=1. An asymptotic expansion of D _−p (∓iz) for z→e ^±i3π/4×∞ yields

$$ D_{-p}(-iz) \approx e^{z^2/4}(\mp iz)^{-p} + O \bigl(z^{-1}\bigr) , $$

(7.A.40)

which is followed by

$$ \bigl|D_{-p}\bigl(-iz\to-ie^{i3\pi/4}\times\infty\bigr)\bigr| = \bigl| e^{\mp i p \pi/4} \bigr| = e^{\pm\pi/8\alpha} = e^{\pi/8|\alpha|} . $$

(7.A.41)

Therefore A is fixed as A=e ^−π/8|α|/(2α)^1/2 and the solution is obtained as

(7.A.42)

(7.A.43)

for α>0, and

(7.A.44)

(7.A.45)

for α<0.

Finally we investigate the limit of z→e ^∓iπ/4×∞ corresponding to t→∞. From asymptotic expansions (7.A.29) and (7.A.30), one has

(7.A.46)

$$ \bigl|D_{-p}\bigl(-iz\to e^{\pm i p 3\pi/4}\times\infty \bigr)\bigr|^2 = \bigl|e^{\pm i p 3\pi/4}\bigr|^2 = e^{\mp3\pi/4\alpha} , $$

(7.A.47)

where we have used an identity regarding the Gamma function: |Γ(1+iy)|²=2πy/(e ^πy−e ^−πy). Thus the Landau-Zener formula on the non-adiabatic transition probability is obtained as

$$ \bigl|\psi_1(t\to\infty)\bigr|^2 = e^{-\pi/|\alpha|} . $$

(7.A.48)

7.1.3 7.A.3 Microscopic Equation of Motion for Oscillatory Transverse Field

We start from the Liouville equation of motion (with Planck’s constant ħ=1)

$$ \frac{d \rho}{dt} = -i \bigl[ H_{\mathrm{tot}},\rho(t)\bigr] $$

(7.A.49)

where H _tot is the total Hamiltonian defined in (7.2.120). In the interaction picture the evolution is governed by

$$ i \frac{d \rho_{I}(t)}{dt} = \bigl[V_{I},\rho_{I}(t)\bigr] =V_{I}^{\times} (t) \rho_{I}(0) $$

(7.A.50)

where

$$ \rho_{I}(t) = \exp \bigl[i (H_{s}+H_{B})t \bigr] \rho(t) \exp \bigl[-i (H_{s}+H_{B}) t\bigr] $$

(7.A.51)

and

$$ V_{I} (t) = \exp \bigl[i(H_{s}+H_{B})\bigr] H_{I} \exp \bigl[-(H_{s}+H_{B})\bigr] . $$

(7.A.52)

$V_{I}^{\times}(t)$ is the Liouville operator associated with V _I (t). The solution of (7.A.50) can be formally written as

$$ \rho_{I} (t) = \exp _{T} \biggl[ {-}i \int _{0}^{t} V_{I}^{\times} \bigl(t'\bigr) \,dt' \biggr] \rho_{I} (0), $$

(7.A.53)

with exp _T denoting a time-ordered series with the operators for the latest time at the left. Note that at t=0, ρ _I (0)=ρ(0). Combining (7.A.51) and (7.A.53), we have

$$ \rho(t) = \exp \bigl[{-}i (H_{s}+H_{B})t\bigr] \exp_{T} \biggl[ {-}i \int_{0}^{t} V_{I}^{\times} \bigl(t'\bigr) \,dt' \biggr] \rho(0) \exp \bigl[i (H_{s}+H_{B})t\bigr]. $$

(7.A.54)

The rate equations for magnetisation m ^μ (μ=x,y,z) are obtained from

$$ \frac{dm^{\mu}}{dt} = \mathrm{Tr} \biggl[ \frac{d\rho(t)}{dt} S^{\mu} \biggr] $$

(7.A.55)

where

$$ m^{\mu} = \mathrm{Tr} \bigl(\rho(t) S^{\mu}\bigr) $$

(7.A.56)

and ρ(t) is given in (7.A.54). However, it is easier to work in terms of a reduced density matrix for the spin system alone:

$$ \rho_{s} (t) = \mathrm{Tr}_{b}\, \rho (t) $$

(7.A.57)

where Tr _b denotes a trace operation over the degrees of freedom of the heat bath. Thus, from (7.A.54)

(7.A.58)

The angular brackets 〈⋯〉 refer to an averaging over the bath degrees of freedom. It has been assumed that the density matrix can be factorised as

$$ \rho(0) \simeq \rho_{b} \otimes\rho_{s}, $$

(7.A.59)

and the cumulant expansion theorem has been used. The physical ground for writing (7.A.59) is that at t=0, the spin-system is assumed to be decoupled from the heat bath; it is at that instant that the perturbation H _I which couples the spin system to the bath is switched on. The subsequent time evolution of ρ _s (0) is what we are interested in. Assuming invariance under time translation, we can write

$$ \rho_{s}(t) = \mathrm{e}^{-i H_{s}t} \exp _{T} \biggl( -\int_{0}^{t} (t-\tau) \bigl\langle V_{I}^{\times} (\tau) V_{I}^{\times} (\tau) \bigr\rangle \,d\tau \biggr) \rho(0) \mathrm{e}^{iH_{s}t} . $$

(7.A.60)

There has been an additional assumption in writing the above equation, viz.,

$$ \bigl\langle V_{I}^{\times} \bigl(t'\bigr) \bigr\rangle= \bigl\langle V_{I}^{\times}(0) \bigr\rangle= 0. $$

(7.A.61)

This can always be ensured by an appropriate choice of the coupling term H _I . This assumption is necessitated by the physical requirement of the model that at a large enough time, the system should equilibrate to a situation governed by the Hamiltonian H _s alone. Using a short-time approximation for the bath correlation functions of the kind $\langle\hat{b} (\tau) \hat{b}(0)\rangle$, viz., that correlation functions die out after a time short compared to any other “times” of physical interest in H _s , the upper limit in the integrals in (7.A.60) can be extended to ∞. This enables us to further write the master equation as

$$ \frac{d\rho_{s}(t)}{dt} = -i\bigl[ H_{s},\rho_{s}(t)\bigr]- \mathrm{e}^{-iH_{s}t} \biggl[ \int_{0}^{\infty} d \tau \,\bigl\langle V_{I}^{\times} (\tau) V_{I}^{\times} (0) \bigr\rangle \biggr] \mathrm{e}^{iH_{s}t} \rho_{s}(t). $$

(7.A.62)

While using the above equation in the rotated frame, one must replace V by $\tilde{V}$, H _s by $\tilde{H}_{s}$, ρ(t) and $\tilde{\rho} (t)$ by etc. where, for example,

$$ \tilde{\rho} (t) = R_{y}^{-1} U_{z}^{-1} \rho(t) U_{z} R_{y}. $$

(7.A.63)

The rate equations for the components of magnetisation are obtained using the above formalism where we have

(7.A.64)

We then use $m^{\mu} = \mathrm{Tr}_{s}[\rho(t)S^{\mu}] = \mathrm{Tr}_{s}[ U_{z}R_{y} (t) \tilde{\rho}(t) R_{y}^{-1}U_{z}^{-1}S^{\mu} ]$ and (7.A.62) to get the rate equations (7.2.128)–(7.2.130) (see [22] for further details). The bath correlations which will appear in the calculations are not calculated but parametrised in terms of a phenomenological relaxation rate by making use of Kubo relations. In order to parametrise the bath correlations, we use the following Kubo relation

$$ \int_{-\infty}^{\infty} d\tau\, \mathrm{e}^{ih \tau} \bigl\langle \hat{b} (\tau) \hat{b} (0) \bigr\rangle= \mathrm{e}^{\beta h} \int _{-\infty}^{\infty} d\tau\, \mathrm{e}^{-ih\tau} \bigl \langle \hat{b} (\tau) \hat{b} (0) \bigr\rangle $$

(7.A.65)

so that we can write

$$ g^{2} \int_{-\infty}^{\infty} d\tau\, \mathrm{e}^{\pm ih \tau} \bigl\langle \hat{b}(\tau) \hat{b} (0) \bigr\rangle= \lambda \frac{\mathrm{e}^{\pm\beta h/2}}{ \mathrm{e}^{\beta h/2} + \mathrm{e}^{-\beta h/2}}, $$

(7.A.66)

where

(7.A.67)

is the phenomenological relaxation rate.