No spin-localization phase transition in the spin-boson model without local field
We explore the spin-boson model in a special case, i.e., with zero local field. In contrast to previous studies, we find no possibility for quantum phase transition (QPT) happening between the localized and delocalized phases, and the behavior of the model can be fully characterized by the even or odd parity as well as the parity breaking, instead of the QPT, owned by the ground state of the system. Our analytical treatment about the eigensolution of the ground state of the model presents for the first time a rigorous proof of no-degeneracy for the ground state of the model, which is independent of the bath type, the degrees of freedom of the bath and the calculation precision. We argue that the QPT mentioned previously appears due to incorrect employment of the ground state of the model and/or unreasonable treatment of the infrared divergence existing in the spectral functions for Ohmic and sub-Ohmic dissipations.
pacs:05.30.Rt, 05.70.Fh, 03.65.Yz, 05.10.-a
A two-level system coupled to an environment provides a unique test-bed for fundamental interests of quantum physics. Denoting the environment by a multi-mode harmonic oscillator, the spin-boson model (SBM) weiss (); leggett () presents a phenomenological description of the open quantum system, which plays an important role in quantum information science and condensed matter physics. Particularly, for the SBM at zero temperature, it has attracted intensive interests for the quantum phase transition (QPT) happening between localized and delocalized phases regarding the spin.
The standard SBM Hamiltonian in units of is given by leggett (),
where and are usual Pauli operators, and are, respectively, the local field (also called c-number bias leggett ()) and tunneling regarding the two levels of the spin. and are creation and annihilation operators of the bath modes with frequencies , and is the coupling between the spin and the bath modes. The effect of the harmonic oscillator environment is reflected by the spectral function for with the cutoff energy . In the infrared limit, i.e., 0, the power laws regarding are of particular importance. Considering the low-energy details of the spectrum, we have with and the dissipation strength . The exponent is responsible for different bath with super-Ohmic bath 1, Ohmic bath 1 and sub-Ohmic bath 1.
The local field introduces asymmetry in the model, which was considered to be less important than the tunneling and thereby sometimes neglected for convenience of treatments. For the Ohmic dissipation, it was mentioned hur1 (); Bu1 () that the SBM has a delocalized and a localized zero temperature phase, separated by a Kosterlitz-Thouless (KT) transition in the case of . In the delocalized phase, realized at small dissipation strength , the non-degenerate ground state behaves like a damped tunneling particle. In contrast, for large , the dissipation leads to a localization of the particle in one of the two eigenstates, implying a doubly degenerate ground state. On the other hand, intensive studies have recently been paid on the sub-Ohmic dissipation, which also demonstrated the QPT between localized and delocalized phases Bu1 (); hur1 (); kehrein (); vojta2 (); vojta1 (); hur (); winter (); Bu2 (); Bu3 (); and (); rmp1 (); cheng (); AA (); chin (); vojta3 (); wong (); florens (); zheng1 (); vojta4 (); zheng2 (). In particular, the QPT could also happen in the absence of the local field () vojta1 (); hur (); winter () from a non-degenerate ground state with zero magnetization below a critical coupling to a twofold-degenerate ground state with finite magnetization for a coupling larger than the critical value. The QPT has been found so far by different numerical studies, such as numerical renormalization group (NRG) technique vojta2 (); vojta1 (); hur (); Bu1 (); and (); rmp1 (); Bu2 (); Bu3 (); cheng (), quantum Monte Carlo winter (), the exact diagonalization AA (), the density matrix renormalization group approach wong (), and variational matrix product state approach vojta4 (). Analytical studies, such as unitary transformation method zheng1 (); zheng2 (), were also employed. The latest work by variational method also showed the appearance of the QPT between delocalization and localization chin () in the case of .
Concentrating on the SBM without the local field, we argue in the present work no possibility with the QPT happening between the localized and delocalized phases in above dissipative cases, opposite to previous results. The main behavior of the model could be characterized by the parity keeping or breaking, rather than the QPT, owned by the ground state in the variation of some characteristic parameters. The remarkable feature of our treatment is the analytical investigation, which is independent of the bath type, the degrees of freedom of the bath and the calculation precision. The key step of our treatment is the decomposition of the special SBM Hamiltonians in such case into two decoupled sub-Hamiltonians, and the key point is the focus on the parity of the ground state of the model. We show that the significant changes regarding the ground state of the model, if can be called ’QPT’, happen only when the parity breaks down, corresponding to appearance of the local field. Our investigation of the magnetization analytically indicates that the QPT discussed previously by different numerical studies is due to unreasonable treatment of the ground state and/or of the infrared divergence inherently existing in the spectral functions for Ohmic and sub-Ohmic dissipations.
We get started from the special SBM with zero local field. In such a case, we denote the Hamiltonian by with
and introduce a parity operator
to commute with , i.e., . Moreover, by introducing a unitary transformation
we may diagonalize to be two decoupled sub-Hamiltonians,
It implies that the eigensolution of consists of two independent eigensolutions of and . Since no local unitary transformation can change the eigenvalues, the ground state of should be the lower one of the ground states of and .
For the eigensolutions and , it is easy to prove that the eigenfunctions are, respectively, of even-parity and odd-parity with respect to the parity operator since and . Meanwhile, is actually the parity operator of after the unitary transformation has been performed, which means that also consists of two sub-Hamiltonians with the eigenfunctions of even-parity and odd-parity, respectively, with respect to the parity operator .
If QPT happens in the model of , there should be the possibility of degenerate ground states of the two decoupled sub-Hamiltonians and and also the possibility of releasing the degeneracy hur1 (); Bu1 (); vojta1 (); hur (); winter (). To check the possibilities, we may solve and collaboratively by expanding and using displaced coherent states
where are coefficients to be determined, are for different bosonic modes, is the product of displaced coherent states of different modes of the bosonic field with the displacement variables , , and the vacuum state of the bosons.
Expanding the tunneling terms in Eq. (5) by yields
where is given by TLiu ()
So the prefactor , relevant to and the tunneling terms, plays a crucial role in the SBM and should be non-zero. Otherwise the SBM in our case is physically meaningless due to neither local field nor tunneling involved. This point will be further emphasized later.
The solutions of and by Eq. (6) yield
Thus the lowest eigenenergies for the even- and odd-parity states are, respectively,
It is intuitive to reach degeneracy by setting , which is, however, physically meaningless in our case. For any non-vanishing value of , we show a rigorous proof in Appendix that there is no possibility of . That is to say, a physically meaningful solution of the SBM in our case requires no degeneracy for the ground states of and , which implies no QPT happening, in terms of the viewpoints in hur1 (); Bu1 (); kehrein (); vojta1 (); hur (); winter (), no matter how to change the characteristic parameters and which bath is considered.
Our results also mean that the ground state of is surely of a certain parity, and any employed state with mixed parity (e.g., the superposition of the ground states of and ) is definitely not the ground state of the model. In this sense, it is not a surprise that QPT was found in chin () using a presumed ground state which, due to mixture of odd- and even-parity, is actually not for the ground state of the model. In fact, our result could be obtained more simply by calculating the magnetization . Without losing generality, we assume the eigenfunction of with the two sub-eigensolutions correlated, i.e., , where are the ground states, respectively, of . So the magnetization of the Hamiltonian is
If or with , which corresponds to even-parity case or odd-parity case, the magnetization is absolutely zero, which implies no QPT happening. The finite value of occurs only in the case of the superposition of and , which, according to our discussion above, is not the ground state of the model.
Besides the incorrect employment of the state as the ground state chin (), the main reason for obtaining the QPT in previous literatures Bu1 (); hur1 (); kehrein (); vojta2 (); vojta1 (); hur (); winter (); Bu2 (); Bu3 (); and (); rmp1 (); cheng (); AA (); chin (); vojta3 (); wong (); florens (); zheng1 (); vojta4 (); zheng2 () is the unreasonable treatment of the infrared divergence existing in the spectral functions for Ohmic and sub-Ohmic dissipations, which induced the degeneracy of the ground states in the low frequency domain. To clarify this point, we demonstrate below the potential infrared divergence in the treatments using the bath mode with continuous and discretized spectra, respectively. The NRG is taken as an example to show that the prediction of the QPT in previous studies is completely due to divergent expansion in the infrared limit.
For treatment with the bath mode of continuous spectrum: We return to the prefactor , for which, using the spectral function , we have,
where and are defined above in the spectral function. is a small quantity regarding the frequency difference from . In the case of the infrared limit, i.e., , we have if , which leads to (See Eq. (8)) and thereby from Eq. (10). As a result, with calculation approaching the low frequency domain, the NRG gives a significant change from non-degeneracy to degeneracy, which was called QPT. In addition, we see from Eq. (7) that the tunneling terms are vanishing with the calculation approaching the infrared limit, which reduces to be
commuting with . This corresponds to the localization, sometimes also called the frozen spin hur (). It means that, once the system falls into a well-polarized state of the spin, the magnetization will remain unchanged in the dynamics. Consequently, it is the infrared divergence that wrongly predicts the QPT happening from the delocalization to the localization as the infrared limit is approached.
Eq. (12) presents the relationship between the prefactor and the dissipation strength , from which we see no possibility to have a finite value of as the critical number for the QPT. The degeneracy of the ground states occurs only in the case of , i.e., a physically meaningless case. On the other hand, considering Eq. (7), we may find that the infrared divergence also yields the parity operator to be zero. In fact, the correct understanding of the parity variation could be reached only after we have eliminated the influence from the infrared divergence. In such a case, the parity of the ground state of the model will explicitly break down as long as we introduce the non-zero local field into , which brings about a significant change in the magnetization as a function of . We prefer to call this phenomenon as parity breaking, instead of the QPT.
with a large number required by the method Bu1 (), . So in the case of and . As an example, we show in Fig. 1 the distribution of with respect to the mode-number for different . The divergence leads to in the case of Ohmic and sub-Ohmic dissipations, from which the QPT was predicted in previous studies.
Our work not only indicates for the first time that the parity keeping and breaking, rather the QPT, can fully characterize the behavior in the SBM, but also helps to understand a fact that the QPT had only been found by previous numerics in the sub-Ohmic and the Ohmic cases, instead of in the super-Ohmic case, which is because the infrared divergence never happens in the super-Ohmic case, e.g, for in the case of the continuous bath-mode spectrum. In fact, our investigation above has clearly shown that, whatever the bath-mode spectrum is, no QPT would happen if the local field is absent in the SBM.
In summary, we have presented analytical treatments to show reliably the impossibility of the QPT in the SBM without the local field. Since our treatments are independent of the bath type, the degrees of freedom of the bath and the calculation precision, we argue that the previous conclusions drawn for the QPT happening in the Ohmic and sub-Ohmic SBM need more serious reexamination. The parity discovered in the present work is strongly relevant to the configuration of the model and the parity breaking can significantly change the dynamics of the model. In addition, the SBM has wide application ranging from the electron transfer in biomolecules bio () to the entanglement in quantum information science weiss (); leggett (). Therefor, we believe that our study is of general interest and would be helpful for our deeper understanding of the weird phenomena in open quantum systems.
This work is supported by National Fundamental Research Program of
China (Grant No. 2012CB922102), by National Natural Science
Foundation of China under Grants No. 10974225 and No. 11004226, and
by funding from WIPM.
Appendix: Proof of the impossibility of degeneracy
The proof starts from Eq. (10). Excluding the trivial case , we have occurring only for (1) or (2) .
Employing the result
we first check the case (1). Since we require , the case implies , which is not satisfied unless . So the case (1) makes no sense and can be dropped.
For the case (2), if we have , which leads to , then we reach
where is the summation excluding the vacuum state of each bosonic mode. Since both sides of the above equation are of multivariate polynomials of , we may compare the terms of to determine if such an equation is satisfied. To this end, we further deduce the equation as
It is evident that the equation could not be satisfied since no constant term exists in the right side of the equator. This is contradictory to the above presumption . So we have proven the impossibility of degeneracy in the case of .
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