Dephasing enhanced transport
8.3 Transport properties with dephasing
This code uses the functions kron, NEye andUnvec, which are part of the Qulib library [45], a Mathematica library for Quantum Information tasks developed by Prof.
Landi. The core of the computation is done via the Mathematica bult-in method Lin-earSolve. Since the system does not exhibit any particular structure that can be ex-ploited, besides being sparse, the computational cost is much higher in comparison with the solution of the Lyapunov equation. For this reason, the maximum system size we were able to simulate is smaller when dephasing is present.
8.3 Transport properties with dephasing
In this section, we will study the effect of dephasing in the transport properties of the system. When the on-site potential is constant, or, without loss of generality, identically zero, the covariance matrix in the NESS can be found analitically. Using the same ansatz
(7.7), we obtain the following equation for the matrixDin the NESS::
i[H, D] +1
2{D,Γ}+ Γ∆(D) = ˜Γ. (8.20) An identical system is solved in Ref. [44], where the authors studied the effect of dephasing on a boundary-driven bosonic chain. According to that solution, the analytical expression for the current in the NESS is
J = 2γ(f1−fL)
4 +γ2+γΓ(L−1). (8.21)
Notice that whenΓis set to zero, the expression for the current without dephasing [(7.13)]
is recovered.
WhenL is large the constant terms in the denominator can be negleted, causing the current to scale as
J ∼ 2
Γ ∆f
L , L1, (8.22)
which is inversely proportinal toL. Therefore, in the presence dephasing the transport is diffusive and the current obeys Fourier’s law. Additionaly, Eq. (8.21) also shows that this happens foranynon-zeroΓ, as long asLis sufficiently large. This behaviour is illustrated in Fig.8.1. For all the values ofΓ, the corresponding curve reaches the diffusive scaling [(8.22)], indicated by the dashed line. However, the smaller the value of Γ, the larger is the size range in which the dephasing-induced diffusion sets in. Notice, in particular, the curve forΓ = 10−3, represented in orange. For small values ofL, the current is almost constant, similarly to the ballistic case, and the dephasing regime as achieved only for sizes of the order∼105.
Fig. 8.1 shows that that finite size effects play a significative role in the transport regime of the system. It also indicates that there exists a crossover size LΓ, which in-creases with Γ, above which the dephasing-induced diffusion dominates. Below this value, the transport regime is still influenced by the original Hamiltonian.
This characteristic length can be estimated using a simple scaling argument, similar to the one we used in section 7.2. Firstly, we notice that the coupling Γ determines a
10 100 1000 104 105 10-5
10-4 0.001 0.010 0.100 1
Figure 8.1: Scaling of the particle current withLwith zero on-site potential, for increasing values ofΓ. All the curves eventually reach the diffusive scaling∼L−1, indicated by the dashed line.
characteristic time for the dephasing effect, given by τΓ ∼ 1/Γ. This can be seen by inspecting the time evolution of the coherences in the single spin case with dephasing [(5.21)]. In this equation, the constantΓis the rate of relaxation. Then, this charateristic time may be compared with the time it takes for an excitation to traverse the whole chain, τ ∼ 1/Lα. IfτΓ τ, then the dephasing effect completely dominates, but ifτΓ τ the transport properties are still affected by the Hamiltonian. The crossover length is thereby obtained by equating these two characteristic times, which results in
LΓ ∼Γ−1/(ν+1), (8.23)
where we used the relation α = 1/(ν + 1). Notice that this derivation relies on the assumption that there is a single scaling coefficient.
To conclude this section, we will make a few remarks about Eq. (8.22), and precisely what it means to say that it follows Fourier’s law. The original law states that the heat current in a piece of material is proportional to the temperature gradient [(1.1)]. In our ongoing example of the metal bar, the temperature profile in the NESS is a simple linear interpolation betweenT1 andT2:
T(x) =T1 +
T2 −T1
L
x. (8.24)
Thus, by Eq. (1.1), the heat current in the NESS is given by J =κ∆T
L , (8.25)
where ∆T1 − T2. Therefore, for a fixed value of L, the current is proportional to the temperature difference. In Eq. (8.22), however, when L is fixed the current is instead proportional to the differencef1−fL. This is in agreement with the discussion we made in section6.1, where we introduced our choice of local master equations. As we mentioned, in this dissertation we avoid providing a definition of the temperature inside the chain, and refer to∆f simply as a “bias”.
Therefore, we say that Eq. (8.22) satisfies Fourier’s law by a simple analogy, with
∆f playing the role of ∆t. It is worth mentioning that, in the particular case of a free chain with dephasing, the authors of Ref. [44] have indeed tried to define an internal temperature , but we will not dive into the subtleties of this discussion.
The above statements can me made more precise in the spin chain framework. In this case, the analogous of Fourier’s law is Fick’s law of diffusion, which states that the spin current is proportional to the magnetization gradient:
J =−D∇ hσzii. (8.26)
Here,Dis thediffusion constant, which is the analogous of the conductivity in Fourier’s law. Furthermore, the two laws are, in fact, completely equivalent. This can be seen as follows. For largeL, the gradient of the magnetization is given by
∇ hσzii ≈ hσLzi − hσ1zi
L = hn1i − hnLi
L . (8.27)
Imposing the continuity equation on the boundaries [Eqs. (6.29) and (6.30)], we obtain γ(f1− hn1i) = γ(fL− hnLi), (8.28)
whencehn1i − hnLi=f1−fL. Therefore,
∇ hσzii ≈ ∆f
L , (8.29)
which shows that the two laws are equivalent.