Introduction↩︎

The spin-boson model is a generic model for studying the decoherence of quantum two-level systems. Recently, it has attracted intense researchers’ attention due to rapid progress in fabrication of nanoscale devices, which have made it possible to study the dynamics of quantum systems strongly interacting with an environment. In the spin-boson model, the environment is considered a bosonic reservoir, the properties of which are characterized by a power law spectral density, \(f(\omega)=\alpha|\omega|^s\), where \(\alpha\) is a constant parameter.

Despite numerous studies on the subject, some aspects of the model remain unclear. It is well known that at zero temperature in the ohmic (\(s=1\)) and sub-ohmic (\(s<1\)) regime, the system undergoes a quantum phase transition from the localized (coherent) to the delocalized (incoherent) state [1], which is manifested by coherence breaking at the growth of the spin–boson coupling above a critical value. However, the only analytical description of this phenomenon as a Kosterlitz-Taules transition [2]–[4] is valid only in the ohmic regime at \(s=1\) and turns out to be inapplicable to the case of small exponents \(s<1/2\). Moreover, the results of numerous numerical calculations indicate that in the regime \(0<s<1\) this transition obeys the hyperscaling inherent to continuous phase transitions [5]–[9].

In this paper, we propose an alternative approach to this problem, based on the critical dynamics description of the spin-boson system presented in terms of the Majorana fermionization technique. For this purpose, we use the spin-fermionization by Majorana fermions and rely on the progress made in [10]–[14], where it was shown that correlations between transverse Majorana fermions can be described by an effective Gaussian action. This fact allows applying standard methods of phase transition theory to the study of the model under consideration. Using the method of renormalization group analysis, we consider the critical dynamics of the model and obtain analytical expressions for the dependence of the critical value of the spin-bosonic coupling constant and the \(1/\delta\) critical exponent on the spectral density exponent \(s\). The results allow us to claim that not only in the sub-ohmic but also in the ohmic regime the system exhibits the second order phase transition, which changes the previous notions following from the quantum classical mapping approach. In the final, we extend the application of the theory to the case of non-zero temperatures and consider the temperature dependence of the critical behavior of the system, which is straightforward in this approach.

The spin-boson model involves a single quantum \(1/2\)-spin \(\boldsymbol{S}\) interacting with the bosonic bath \(X\). We assume that only the \(x\)-th component of the spin interacts. Thus, the spin-boson model Hamiltonian has the general functional form: \[\begin{gather} \mathcal{H}=-B S_z-b S_x +\lambda S_xX+\mathcal{H}_X, \label{H1} \end{gather}\tag{1}\] where \(B\) and \(b\) are the transverse and longitudinal magnetic fields applied to the spin, respectively, \(\lambda\) is the bath-spin coupling constant and \(\mathcal{H}_X\) is the Hamiltonian of the bosonic bath, which governs the free dynamics of \(X\). The bath is characterized by the correlation function: \(\langle X X\rangle_{\omega} = f(\omega)\coth(\omega/2T)\) [15], [16].

First, we map the spin-1/2 operator onto fermionic degrees of freedom, which can be done, in particular, with the so-called Majorana fermions [10]–[14]. This mapping obeys the following correspondence principle: \[\begin{gather} S_{i}=-\frac{\displaystyle \mathrm{i}}{\displaystyle 2}\varepsilon_{ijk}\psi_{j}\psi_{k},\qquad i,\,j,\,k,=(x,\,y,\,z) \end{gather}\] where \(\psi\) is the Majorana spinor field, which has the following properties: \[\begin{gather} \psi^+=\psi,\quad \{\psi_{i}\psi_{j}\}=\psi_{i}\psi_{j}+\psi_{i}\psi_{j}=\delta_{ij},\\ \psi_i\psi_i=1/2, \quad \vec{\psi}\cdot\vec{\psi}=3/2. \end{gather}\]

The Hamiltonian (1 ) is written in the Majorana representation as follows \[\begin{gather} \mathcal{H}={\mathrm{i}}B \psi_x \psi_y + {\mathrm{i}}\left(b+{\lambda} X\right)\psi_y\psi_z +\mathcal{H}_{X}, \end{gather}\]

This representation allows employing the standard diagrammatic technique for calculations perturbative in the spin–boson interaction. Following the [10], [13], [14] we suppose that the Majorana field evolution is described by the Lagrangian \(\mathcal{L}=\psi_i\partial_t\psi_i+\mathcal{H}\). We investigate the critical behavior of the presented model in terms of the Shwinger–Keldysh technique [15], denoting by symbols \(\psi^+\) and \(\psi^-\) the Majorana field on the forward and backward branches of the Keldysh contour. By performing a Keldysh rotation, we rewrite the model in new fields \(\psi^{s}=(\psi^++\psi^-)/\sqrt{2}\), \(\psi^{d}=(\psi^+-\psi^-)/\sqrt{2}\) and represent the system’s action as follows: \[\begin{gather} \mathcal{S}=\mathrm{i}\int \mathrm{d}t\,\left[\vec{\psi}^{d}\partial_t\vec{\psi}^{s}+\vec{\psi}^{s}\partial_t\vec{\psi}^{d}+ {\mathrm{i}}B (\psi^{s}_x \psi^{d}_y+\psi^{s}_y \psi^{d}_x) \right.\\\left.{+\mathrm{i}}\left(b+{\lambda} X_x\right)(\psi^{d}_y\psi^{s}_z+\psi^{d}_z\psi^{s}_x)\right]. \label{I4} \end{gather}\tag{2}\] For future convenience, we use a Euclidean formulation, which can be obtained by “Wick rotation” \(t \to \mathrm{i}t\), and write the bare retarded and advanced propagators of the mass-less Majorana fermion field as \[\begin{gather} \langle\psi^{s}_i\psi^{d}_j\rangle^0_{\omega}=\frac{\displaystyle \delta_{ij}}{\displaystyle \mathrm{i}\omega+\epsilon},\quad \langle\psi^{d}_i\psi^{s}_j\rangle^0_{\omega}=\frac{\displaystyle \delta_{ij}}{\displaystyle -\mathrm{i}\omega+\epsilon}, \end{gather}\] where \(\epsilon \to 0\). In the presence of the \(B\) and \(b\) fields, the bare time correlation functions have the following form: \[\begin{gather} \langle\psi^{s}\psi^{d}\rangle_{\omega}= {\langle\psi^{d}\psi^{s}\rangle_{\omega}^*} =\\ \frac{\displaystyle 1}{\displaystyle \omega^2+B^2+b^2}\left( \begin{array}{ccc} \displaystyle\frac{\displaystyle \omega^2+b^2}{\displaystyle \mathrm{i}\omega+\epsilon} & \,\,\, \displaystyle -\mathrm{i}B & \displaystyle \frac{\displaystyle Bb}{\displaystyle \mathrm{i}\omega+\epsilon}\\[12pt] \mathrm{i}B & \quad -\mathrm{i}\omega & -\mathrm{i}b \\[12pt] \displaystyle \frac{\displaystyle Bb}{\displaystyle \mathrm{i}\omega+\epsilon} & \quad \mathrm{i}b & \quad \displaystyle\frac{\displaystyle \omega^2+B^2}{\displaystyle \mathrm{i}\omega+\epsilon} \end{array} \right), \end{gather}\]

The critical behavior of the system can be considered within the technique of critical dynamics [17] based on the hypothesis of dynamical scaling. The full diagrammatic set of the theory contains two propagators and six vertices (see Fig. 1). For simplicity, let us consider the case when the field \(B\) is relatively small, \(B\ll b\). We see from (2 ) that the model in this case contains only two types of vertices: the four vertices of second degree (\(c\), \(d\), \(e\), and \(f\) in Fig. 1); and two third degree ones’ (\(g\), and \(h\)).

In the fluctuation regime close to the phase transition critical point, the above vertices are renormalized. One-loop contributions to renormalization are shown in Fig. 2 and can be written in the following form: \[\begin{gather} \tag{3} Z_{b} \approx b-\frac{\displaystyle 2b}{\displaystyle 2!}\lambda^2\int\limits_{\omega'}^{\Lambda\omega'}\frac{\displaystyle \mathrm{d}\omega}{\displaystyle 2\pi}\frac{\displaystyle f(\omega)}{\displaystyle \omega^2},\\ \tag{4} Z_{\lambda}\approx \lambda -\frac{\displaystyle 6\lambda^3}{\displaystyle 3!}\int\limits_{\omega'}^{\Lambda\omega'}\frac{\displaystyle \mathrm{d}\omega}{\displaystyle 2\pi}\frac{\displaystyle f(\omega)}{\displaystyle \omega^2}. \end{gather}\]

First, we consider the quantum phase transition at \(T=0\). Usually, critical dynamics considers a system in the \(\omega'\to 0\) limit. However, this theoretical limit is practically unattainable when \(T=0\) since the bosonic bath described by harmonic oscillators system has finite zero-point energy, even at absolute zero. Thus, there is a natural lower frequency limit, \(\omega'\to\omega_0\neq 0\), corresponds to the lowest ground state [1], [18]. In this case, when \(f(\omega)=\alpha|\omega|^{s}\), and \(\alpha\) is the dimensionless parameter, the renormalization expressions (3 ), and (4 ) are proportional to the integral \[\begin{gather} I=\int\limits_{\omega_0}^{\Lambda\omega_0}\frac{\displaystyle \mathrm{d}\omega}{\displaystyle 2\pi}\frac{\displaystyle f(\omega)}{\displaystyle \omega^2}\approx\frac{\displaystyle \alpha{\omega_0}^{s-1}}{\displaystyle 2\pi}\ln\Lambda. \label{Int} \end{gather}\tag{5}\] Therefore, in the one-loop approximation we write the following renormalization equations (see [16]): \[\begin{gather} \label{R4} \frac{\displaystyle \mathrm{d}b}{\displaystyle \mathrm{d}\ln\left(\Lambda\right)} \approx b\left(1-\frac{\displaystyle \lambda^2\alpha}{\displaystyle 2\pi\omega_0^{1-s}}\right),\\ \frac{\displaystyle \mathrm{d}\lambda}{\displaystyle \mathrm{d}\ln\left(\Lambda\right)} \approx \lambda\left(\frac{\displaystyle 2-s}{\displaystyle 2}-\frac{\displaystyle \lambda^2 \alpha}{\displaystyle 2\pi\omega_0^{1-s}}\right). \end{gather}\tag{6}\] The flows of the renormalization group for the model are shown in Fig. 3. It can be seen that the theory contains two fixed points: the Gauss fixed point, \(\lambda =b=0\), and the Wilson–Fisher fixed point, \[\begin{gather} b=b^*=0,\quad\lambda=\lambda^*=\sqrt{\frac{\displaystyle (2-s)\pi\omega_0^{1-s}}{\displaystyle \alpha}}. \label{SS} \end{gather}\tag{7}\]

The above shows that in the ohmic fluctuation regime (\(s=1\)) the theory is logarithmic and the considered critical point corresponds to the second order phase transition. This result disagrees with the conclusion of the quantum–classical mapping, where this phase transition is considered equivalent to the Berezinskii–Kosterlitz–Thouless one in long-range ferromagnetic chains [1], [2], [6].

In the sub-ohmic fluctuation regime, when \(s\to 0\), the width of the coherent region depends on the cut-off frequency \(\omega_0\ll 1\). The infrared fixed point tends to \(\lambda^* =0\) (Fig. 4 b), showing the narrowing of the coherent region with increasing thermal fluctuations. Finally, at full dominance, \(s=0\), the coherent region disappears completely.

In the super-ohmic fluctuation regime, \(s>1\), the coherent region expands exponentially with \(s\) (Fig. 4 b). The width of the coherent region is also defined by the cutoff frequency, but at \(\omega_0\to 0\) the entire range of system parameters corresponds to the coherent state. In this limit, there is also no phase transition.

The critical exponent can be easily estimated in the one-loop approximation using the expression for the correlation function. Let us assume that the system is close to the critical point, so that the coherent time, \(\omega_c^{-1}\gg \omega_0^{-1}\), is relatively large, and \(\omega_c<b\), then, on the one hand, \[\begin{gather} \langle S_x\rangle\approx\int\limits^{\omega_c}_0\frac{\displaystyle \mathrm{d}\omega}{\displaystyle 2\pi}\langle\psi^s_z\psi^s_y\rangle_{\omega}= \int\limits^{\omega_c}_0\frac{\displaystyle \mathrm{d}\omega}{\displaystyle 2\pi}\frac{\displaystyle b}{\displaystyle \omega^2+b^2}\propto b^{-1}\omega_c. \end{gather}\] On the other hand, in the Wilson-Fisher fixed point \[\begin{gather} b=\lambda^2\int\limits_{0}^{\omega_c}\frac{\displaystyle \mathrm{d}\omega}{\displaystyle 2\pi}\frac{\displaystyle \alpha\,|\omega|^{s}b}{\displaystyle \omega^2+b^2}\propto b^{-1}\omega_c^{s+1}, \end{gather}\] therefore \(\omega_c\propto b^{{2}/({s+1})}\), and we can conclude that \(|\langle S_x\rangle |\propto b^{{(1-s)}/{(s+1)}}\). This results in the critical exponent of magnetization being \(1/\delta \approx {(1-s)}/{(s+1)}\), which is consistent with the results of the numerical calculations of the renormalization group [5], and with the numerical calculations using the method based on the product of the variational matrix states [7]–[9].

The obtained dependence of the coherence interval on the type of spectral function also allows establishing its dependence on the temperature of the thermal reservoir, using the results of the study of the classical-quantum crossover in a thermalized bosonic system. [19]–[22].

Consider the ohmic fluctuation regime \(s=1\). We assume that the boson system is thermalized and satisfies the fluctuation-dissipation theorem, so that the fluctuations obey the Bose-Einstein statistics. These fluctuations are described by the following correlation function [23]: \(\langle X_xX_x\rangle_{\omega} =2|\omega|\coth\left(\omega/2T\right)\), which depends on the temperature. Therefore, the temperature of the boson reservoir influences the critical behavior of the system we are considering. Previously, it was experimentally observed [19]–[22] and theoretically explained [24], [25] that at \(T\to 0\) the critical behavior significantly depends on the temperature. Indeed, according to the fluctuation theory of phase transitions the parameters of the critical behavior of our system (position of the critical point and critical exponents) are determined by the logarithmically divergent integral (5 ) corresponding to the loop of the correlation functions. The divergence of this integral depends both on the proximity of the system dimension to its critical dimension and on the fluctuation spectrum. At \(T\to 0\) the character of the system fluctuations changes from thermal, \(\langle X_xX_x\rangle_{\omega}\propto \omega^0\), to quantum \(\langle X_xX_x\rangle_{\omega}\propto \omega^1\). As a consequence, the nature of the divergence of the integral changes, causing the critical behavioral parameters observed to change [24], [25].

In the case of non-zero temperature, when \(b \to 0\), i.e. in the long-wavelength limit corresponding to the critical dynamics near the quantum critical point, the contribution of the correlation functions loop (5 ) is \[\begin{gather} I=\int\limits^{\Lambda \omega_0}_{\omega_0}\frac{\displaystyle \mathrm{d}\omega}{\displaystyle 2\pi}\frac{\displaystyle |\omega|\,cth\left({\beta\hbar\omega}/{2}\right)}{\displaystyle \omega^2}=\int\limits^{\Lambda x_0}_{x_0}\frac{\displaystyle \mathrm{d}x}{\displaystyle 2\pi}\frac{\displaystyle |x|\,cth\left(x\right)}{\displaystyle x^2}, \label{A0} \end{gather}\tag{8}\] where \(x_0=\omega_0/2T\approx 1\) corresponds to the classical-to-quantum crossover in a thermalized bosonic system [19]–[22].

Near \(x_0\) the integrable function can be represented as a power function [24], [25]. As a result, the loop contribution in (3 ) is \[\begin{gather} I\propto\int\limits^{\Lambda}_{1} \mathrm{d}x\,x^{s(x_0)-2}, \label{A2} \end{gather}\tag{9}\] where \(s(x_0) = 1-2x_0\,csch(2x_0)\) [16]. Thus, from (7 ) the temperature dependence of the the Wilson–Fisher fixed point is \[\begin{gather} \lambda^*(\beta)\approx\sqrt{\frac{\displaystyle \pi}{\displaystyle \alpha}\left(1+\frac{\omega_0}{T}\csch(\omega_0/T)\right)}\omega_0^{\omega_0\csch(\omega_0/T)/2T}. \end{gather}\] This means that at low temperatures the system is in the ohmic fluctuation regime, and the critical point does not depend on the value of the characteristic frequency \(\omega_0\): \(\lambda^*\to\sqrt{\pi/\alpha}\). At high temperatures, \(T\gg \omega_0\), the coherent region disappears (Fig. 4 a).

Our results confirm the suggestion made in [5] that some conclusions drawn in the past for effectively long-range Ising systems on the basis of the quantum classical mapping need to be reconsidered. The reason for this has long been suspected and lies in the peculiarity of the long-range interaction in imaginary time of the quantum model [5]. The quantum classical mapping method is worked out explicitly for the discrete one-dimensional Ising model with the inverse square law of interaction. At \(T\neq 0\) and an infinite number of discrete points, the distance between adjacent spins shrinks to zero, and consequently the interaction becomes infinitely large, leading to a discrepancy between the model and the real quantum system [26]. The quantum field approach we have outlined, based on the Majorana representation of 1/2-spin and the Schwinger-Keldysh technique, does not have this problem. We have shown that the quantum phase transition in the spin-boson model is the second order phase transition in the whole range of values of the exponent \(0<s\leq 1\), including the ohmic regime, which differs from the conclusions of the quantum classical mapping. However, unlike the latter, our approach is applicable in the whole ohmic–sub-ohmic range of exponent \(s\) values. Thus, the derived expression for the critical magnetization exponent, \(1/\delta=(1-s)/(1+s)\), argues for the validity of the hyperscaling law, which is in agreement with the results of numerical calculations [5], [7]–[9].

Another advantage of this approach is its applicability at non-zero temperatures and the possibility to directly determine the dependence of the critical transition point on the temperature of the bosonic thermal bath. The knowledge of such a dependence can be useful in practice, e.g. for estimating the decoherence region of qubits. The results of this work not only complement the already known ideas about dynamical phenomena in the spin-boson model, but also provide a deeper understanding of the nature of these phenomena based on the notion of critical dynamics of quantum phase transitions. Moreover, they give an additional impetus to the development of theoretical approaches based on the technique of representing 1/2-spin systems by Majorana spinors.

References↩︎