Equilibrium dynamics of the sub-ohmic spin-boson model at finite temperature?

Ke Yang(楊珂) and Ning-Hua Tong(同寧華)

Department of Physics,Renmin University of China,Beijing 100072,China

Keywords: spin-boson model,full-density matrix renormalization group,quantum phase transition,dynamical correlation function,finite temperature

1. Introduction

The spin-boson model (SBM) is widely used in modeling a two-level system interacting with the environmental bath.[1–3]Using the SBM, one can easily build the decoherence[4]or damping[5]picture of the quantum impurity system due to influence of the bath. Thus, SBM is frequently applied in contexts such as damping in electric circuits,decoherence of quantum oscillations in qubits,[6–8]thermal conductance.[9]It has also been used in quantum information processing.[10]The coupling between the spin and bosonic bath is described by the spectral function J(ω)～αωs.In this paper,we focus on the sub-ohmic case with 0

Much less is known for the behavior ofC(ω)at finite temperature, however. Since all experiments are carried out at a finite temperature,the study of C(ω)at finite T is of more relevance to experiments. C(ω) contains information about the real time evolution of a two-level system subjected to the influence of a thermal bath.[15]One would expect that the thermal excitation of the SBM would significantly increase the damping of the two-level system and invalidate the Shiba relation in the low-frequency regime.[5]It is the purpose of this paper to present a quantitative study of C(ω)at finite T and zero bias.

The bosonic NRG is a powerful tool to study quantum impurity models including SBM. There are various ways to generate the equilibrium dynamical quantities such as C(ω)from NRG data, i.e., from the eigenstates and eigenenergies produced in the NRG iterative diagonalization. The patching scheme produces C(ω)by empirically combining the spectral functions generated from each energy shell of the SBM[31]and does not guarantee exact sum rule. The density-matrix NRG(DM NRG)combines the data of each energy shell using the reduced density matrix of the full system, such that the influence of the low energy state on the high frequency spectral function is well described.[32]Although DM NRG combines NRG data from all NRG iterations, it works through patching scheme. For finite temperature, one still needs to set the temperature according to the energy scale of a chosen shell.[33]A true multi-shell framework was built on the full density matrix and the complete basis sets introduced by Anders and Schiller.[34,35]The full-density matrix (FDM)NRG treats the density matrix exactly and guarantees the sum rule rigorously.[36]Recently, we developed the full excitation(FE) NRG method for calculating the equilibrium dynamical quantities.[37]It treats the density matrix exactly and employs the full excitations of NRG,i.e.,both intra-shell and inter-shell excitations are taken into account. This method guarantees both the sum rule and the positiveness of the diagonal spectral function. In this paper,we use FDM NRG to study finite temperature C(ω)for SBM,since FDM NRG is much faster than FE NRG and with suitable broadening, the problem of negative spectral function of FDM NRG method does not influence our conclusion.

2. Model and method

The Hamiltonian of the SBM reads

Usually the following power-law form of J(ω)with a cut-off ωc=1 is used:

Diagonalization of Hngives the eigenenergies and eigenstates of the nth energy shell. An iterative scheme is used to implement the diagonalization process for a given chain length N. We start from diagonalizing the Hamiltonian of a short chain whose Hilbert space dimension is small. Then we add a new bath site to the chain and build the Hamiltonian matrix on the product space of the diagonalized basis and the newly added site. The matrix is diagonalized again and the next site is added. To avoid the exponential enlargement of the Hilbert space dimension,a truncation of the energy spectrum is introduced after each diagonalization: only the lowest Mseigenstates are kept and used to form the new Hilbert space. The high energy discarded eigenstates are also stored for computing physical quantities later. This spectrum truncation introduces the NRG truncation error which diminishes in the limit Ms=∞.[38]For the bosonic NRG,the number of states of each bosonic bath site has also to be truncated.For the newly added site, we use the lowest Nboccupation number states to build the Hamiltonian matrix which has a linear size Ms×Nb. The infinitely large local bosonic Hilbert space is recovered in the limit Nb=∞.We therefore have three NRG parameters Λ,Ms,and Nbto control the NRG numerical error. The exact result are obtained only in the simultaneous limit Λ =1, Ms=∞,and Nb=∞. Conclusion from NRG study should be checked by extrapolating the numerical data to the above limit.

Fig.1. Schematic picture of NRG complete basis set for N =3. In this figure, we assume that H0 has two eigenstates, each bath site has Nb =4 bare states,and we keep Ms=4 lowest eigenstates after each diagonalization. In NRG algorithm,the coupling term ?Hn in Hn+1=Hn+?Hn is applied only to the lowest Ms=4 kept states(red solid lines)and splits their degeneracies(dashed arrows). ?Hn is not applied to the discarded states (dashed lines).Upon adding a new bath site, the discarded states only expand their degeneracies by a factor of 4 but keep their energies intact (solid right arrows).The three rectangular boxes contain the complete bases for N=1,2,and 3,respectively. Levels of different colors correspond to discarded states from different shells. In the last shell N=3,all eigenstates(red dashed lines and solid lines)for H3 are regarded to be discarded.

for the kept states (X = K) and the discarded states (X =D), one obtains the FDM expression for C(ω) which contains intra-shell excitations only. For details, we refer to Refs.[36,37]. In FDM NRG,the matrix elements of the density operator ρ = e?βHN/Tr(e?βHN) is treated exactly. That is, for a given temperature, the contribution from all eigenstates of ρ is taken into account according to the Boltzmann distribution. The excitation energies in the dynamical correlation function,however,are approximated by the intra-shell excitations only. The obtained FDM NRG method[33,36]has the advantage of conserving the sum rule and being computationally efficient,but the positiveness of diagonal spectral function is not guaranteed in certain situations.[37]We also compare the results from FE NRG[37]with those from FDM NRG and find no qualitative differences.

In both FDM and FE NRG methods, the delta peaks in the Lehmann representation of C(ω)are broadened with a log-Gaussian function,[31]

Here b is the broadening parameter.

3. Result

3.1. Existence of thermal peak at ω ～ωT

Fig.2. Correlation function C(ω) at different temperatures. The parameters are s = 0.7, ?= 0.01, ε = 0. Panels (a)–(c) correspond to α = 0.113872 ≈αc, 0.113, and 0.1, respectively. In each panel,T/?=10?18 ≈0(black solid line),10?8(red dashed line),10?6(green dash-dotted line), 10?4 (blue short-dashed line), and 10?2 (pink dashdot-dot line). NRG parameters are Nb=8,N=40,Ms=100,Λ =2.0,and b=1. The peak positions are marked by arrows.

In the following, we check the above observation of a peak in C(ω) at ωTto be not the artefact of NRG errors.Firstly,we study the influence of the discretization parameter Λ on the results. It is known that the discretization error could distort the NRG results for Λ >1. In Fig.3,we compare the C(ω)curve of T/?=10?5obtained at Λ =6.0,4.0,and 2.0 using Bulla’s discretization method, originally developed by Wilson.[39,40]For these calculations, we use broadening parameter b=1.0. We also plot two curves obtained at Λ =2.0 and Λ =6.0 using Zitko’s discretization method (upper two curves in Fig.3).

Fig.3. Correlation function C(ω)at T/?=10?5 obtained at different Λ values. The parameters are s=0.7, ?=0.01, ε =0, and α =0.1.NRG parameters are Nb =8, N =40, and Ms =100. Different colors refer to different Λ. The lowest three curves are obtained from Bulla’s discretization method and using b=1. The top two curves are from Zitko’s discretization method with Nz =10 and b=1/Nz. The arrows mark out the peak position.

Zitko’s discretization method[41]requires that the hybrid function after the discretization be equivalent to the original hybrid function. It is supposed to lead to much less discretization error than Bulla’s method.For the two curves from Zitko’s method,the z-average trick[42–44]is used to further reduce the effect of broadening. Specifically,the result is obtained by averaging Nz=10 curves, each obtained with slightly twisted boundaries on the energy axis in the discretization process and with a much smaller broadening parameter b=1/10.

In Fig.3, apart from the curve for Λ =6.0 with Bulla’s discretization, all curves have a peak at a common ωT. For Bulla’s discretization,with decreasing Λ,C(ω)shifts upwards and the peak at ωTgets more pronounced. The two curves from Zitko’s discretization method stay on the top of the figure and almost coincide with each other,implying that they represent the converged curve at Λ =1.0. The curve for Λ =6.0 has too much discretization error such that the thermal peak at ωTdoes not appear.

Fig.4. Correlation function C(ω)for(a)different Ms at Nb =16, and(b) different Nb at Ms =80. The parameters are s=0.7, ?=0.01,ε =0,α =0.1,T/?=10?5. The other NRG parameters are N =40,Λ =2,and b=1. The peak positions are marked by arrows.

3.2. C(ω)for various α and s values

Fig.5. Correlation function C(ω)at T/?=10?5 for various α values.Other parameters are s=0.7, ?=0.01 and ε =0. The NRG parameters are Nb =8, N =40, Ms =100,Λ =2, and b=1. From bottom to top in the low frequency regime, α =0.0, 10?4, 0.03, 0.1, 0.113,and 0.114 ≈αc,respectively. The peak positions are marked by arrows.The dashed lines show ωs and ω?s behavior.

Figure 6 shows the evolution of C(ω) curve with s increasing from 0.1 to 1.0,with fixed α =0.001 which is in the delocalized phase for all s values shown in the figure. A robust tunnelling peak at ω =?is always present through this process. The low frequency regime has cωsbehavior. Besides the change of slope in the log-log plot in the low frequency regime, the coefficient c decreases with increasing s. This is mainly because as s increases, αcincreases. This makes the state at α =0.001 further away from the critical point and the coupling strength is effectively reduced. In all these curves,there are thermal peaks at ωTwith an s-independent ωTvalue.The curves in ω<ωTdoes not show consistent pattern with s.

Fig.6. Correlation function C(ω) at T/?=10?5 for various s values. Other parameters are ?=0.01,ε =0,and α =0.001. The NRG parameters are Nb=8,N=40,Ms=100,Λ =2,and b=1.

3.3. C(ω)in ω hωT regime

Fig.7. The low-frequency part of C(ω)at T/?=10?5 for different Ms values. The parameters are s=0.7,?=0.01,ε =0,and α =0.1. The NRG parameters are Nb=8,N=40,Ms=100,Λ=2,and b=1.Inset:low frequency part of C(ω)at T/?=10?8 for different Λ values.

Fig.8. Comparison of C(ω)from FDM NRG and FE NRG.The parameters are s=0.7,?=0.01,ε =0,and α =0.1. The NRG parameters are Nb=8,N=40,Ms=100,Λ =2,and b=1. Peak positions are marked by arrows.

4. Conclusion

We have studied the σz?σzcorrelation function C(ω)at finite temperature for the SBM.A thermal peak is observed at the frequency ωT～T in the delocalized phase and zero bias.We find that ωTis controlled solely by temperature and it is independent of α and s. Above this frequency, C(ω) is almost identical to the zero-temperature curve. Below ωT,C(ω)significantly deviates from the zero-temperature curve and our NRG calculation gives irregular behavior. A definite conclusion for the behavior of C(ω) in this regime is yet to be obtained.

Acknowledgment

NHT acknowledges helpful discussions and share of the Green function equation of motion results of C(ω) from Zhiguo L¨u.