# Impurity-Induced Environmental Quantum Phase Transitions in the Quadratic-Coupling Spin-Boson Model

###### Abstract

We studied the zero temperature properties of the spin-boson model with quadratic spin-boson coupling. This model describes a superconducting qubit (spin) coupled to the circuit noise (bosons) at the optimal working point, and an impurity atom or quantum dot quadratically coupled to acoustic phonons. We show that quantum phase transitions can occur in the boson degrees of freedom, between a environment-stable state and a state with local environmental distortions. Using Wilson’s numerical renormalization group method and the exact solution at the non-tunnelling limit , we obtain the phase diagram that contains the first-order as well as continuous QPTs. The equilibrium state dynamical correlation functions of the spin and the bath are studied, showing the power-law -dependences at and away from the critical point. We discuss the relevance of these results to experiments.

###### pacs:

05.10.Cc, 64.70.Tg, 03.65.Yz, 05.30.Jp## I Introduction

The spin-boson model (SBM) is a frequently used paradigm to study the influence of the environmental noise on the quantum evolution of a two-level system Caldeira1 (); Leggett1 (); Weiss1 (). The noise-induced dissipation and dephasing is the central issue of a variety of research fields, ranging from the electron/energy transfer in bio-chemical systems Garg1 (); Xu1 (); Song1 (); Muehlbacher1 () to the endeavour of building a quantum computer Makhlin2 (); Shnirman1 (); Devoret1 (); You1 (); Novais1 (). Sufficiently strong coupling to the bosonic bath also induces a localize-delocalize quantum phase transition (QPT) in the two-level system Bulla1 (); Bulla1p (); Florens1 (); Zhou1 (); Zhang1 (); Lv1 (); Frenzel1 (); Vojta2 (). The non-trivial universality class of this QPT receives much attention in recent years Vojta1 (); Vojta1p (); Winter1 (); Alvermann1 (); Guo1 (). Experimental realization of the SBM have been proposed in various contexts, ranging from the mesoscopic metal ring to cold atom systems Tong1 (); Recati1 (); Orth1 (); Porras1 ().

The conventional SBM, composed of a quantum spin linearly coupled to the displacement operator of a group of harmonic oscillators, belongs to the quantum impurity problem where the properties of the impurity is determined by the (infinitely) many degrees of freedom in the bath, but the influence of the impurity on the bath is negligibly small. Recently, much attention is paid to the quantum two-level system quadratically coupled to a bosonic environment. Such a problem arises in various qubit systems, including the superconducting qubit Vion1 (); Bertet1 (); Ithier1 (); Yoshihara1 (); Kakuyanagi1 (), semiconductor quantum dot Petersson1 (), and the bismuth donors in silicon Wolfowicz1 (), where the linear qubit-environment coupling is tuned to zero to suppress the decoherence and the remaining leading order coupling is quadratic. Significant enhancement of the coherence time has been achieved at such an optimal working point. Theoretical studies have been carried out to understand the influence of the quadratic coupling on the dephasing of the qubit Makhlin1 (); Bergli1 (); Cywinski1 (); Balian1 (). In the quantum dot-based qubit systems, the quadratic electron-phonon coupling is derived to explain the anomalous temperature dependence of the absorption line shape Muljarov1 (); Borri1 (). For the quantum Brownian motion of a heavy particle quadratically coupled to the environment, Maghrebi et al. suggested that the quadratic couplings leads to the anomalous diffusion of the particle Maghrebi1 ().

As for the linear-coupling SBM, previous studies of the quadratic-coupling SBM focused mainly on the influence of the environment on the quantum two-level systems. However, in striking contrast to the linear-coupling case, the non-linearity of the spin-boson coupling makes the feedback effect of the impurity to the bath no longer negligible. Especially, the coupling quadratic in the coordinate of the environmental bosons will shift the boson energies and lead to new phenomenon such as the the softening of the environmental bosons. In this paper, we show that a strong quadratic coupling can even induce a QPT in the environment and qualitatively change the bath. At the QPT, the environmental bosons become unstable with inverted harmonic potentials. Considering higher order anharmonic potentials, this will lead to a local distortion in the environment as the new state. We find that the ground state phase diagram includes both the first-order and continuous QPTs. As a consequence of the QPT, the quantum dynamics of both the spin and the bath are severely changed, at and away from the QPTs. This makes such QPTs an indispensable ingredient to consider in the study of the above-stated systems.

The rest of the paper is organized as the following. In Section II, we describe the model and the methods used to study it. Section III is devoted to the main results, including the exact solution at and the NRG results for . Various related issues for the quadratic-coupling SBM are discussed in Section IV. The details of the exact solution at is presented in Appendix A. The NRG formalism is summarized in Appendix B. In Appendix C, we present quantitative comparison between the NRG data and the exact solution for .

## Ii Model and Methods

A general Hamiltonian describing a two-level system coupled to the environmental noise can be written as

(1) |

where is the local boson displacement operator. describes the local weight of the -th boson mode. The two-level system is represented by a spin with bias and tunnelling strength . It is coupled to the bosonic bath with mode energies in terms of and . In the weak coupling limit, the function can be expanded into Taylor series . The conventional SBM Hamiltonian is obtained by truncating the series at the linear order. At the optimal point of the superconducting qubit experiments Vion1 (); Bertet1 (); Ithier1 (); Yoshihara1 (); Kakuyanagi1 (); Petersson1 (); Wolfowicz1 (), the linear coupling is tuned to zero and the leading contribution to the coupling is quadratic in the boson coordination Makhlin1 (). Truncating the series at this order and absorbing the constant into , we obtain the Hamiltonian of the quadratic-coupling SBM as

(2) |

The effect of the bath on the spin is encoded into the bath spectral function which is defined as

(3) |

The coupling constant can be absorbed into , or equivalently, is set as unity in the numerical calculation below. We consider a power law form of in the low frequency limit with a hard cut-off at ,

(4) |

Here controls the strength of the spin-boson coupling. The exponent is determined by the distribution of energies and weights . is set as the energy unit. In this work, we focus mainly on the effect of sub-Ohmic bath with and will discuss the result for Ohmic () and super-Ohmic () baths in the end of section III.

Here we compare the symmetry of to that of the linear-coupling SBM . At , is invariant under the combined boson and spin transformation and . Previous studies disclosed that for the sub-Ohmic () and the Ohmic () baths, a strong coupling strength may induce a spontaneous breaking of this symmetry and the system enters the localized phase, in which the quantum system is trapped to one of the two states and the local bosons has a finite displacement Bulla1 (); Bulla1p (). The transition is the so-called delocalize-localize transition of the SBM.

The Hamiltonian of the quadratic-coupling SBM Eq.(2) is invariant under the parity transformation alone. In case the spin is in the state , the quadratic coupling contributes a negative boson energy proportional to which, when overcoming the energy of the low energy boson modes, may lead to an instability of the bosons. Physically, as the coupling strength increases, the harmonic potentials of the environmental particles are softened and the instability occurs at the potential inversion, accompanied with the divergence of particle numbers. A spontaneous breaking of the boson parity symmetry may occur at this transition. Under the constraints of higher order anharmonic potential of bosons beyond , this instability will lead to a local distortion in the environmental degrees of freedom. Even for a weak quadratic coupling strength, the feedback effect of the impurity to the bath can no longer be regarded as small and the bath is intrinsically non-Gaussian. New dynamical behaviour will thus emerge both in the bath and the impurity.

At the non-tunnelling point , the -component of the spin has no dynamics. The eigen-states of are in the form and . and are eigen-states of with energies and , respectively. are the corresponding boson states. In each spin sector, the quadratic boson Hamiltonian can be solved exactly. We use the equation of motion method for the double-time Green’s functions to obtain the exact properties of at . The derivation is summarized in Appendix A.

For general parameters, we study using the Wilson’s numerical renormalization group (NRG) method Wilson1 (); Bulla2 () adapted to the bosonic bath Bulla1 (); Bulla1p (). The Wilson chain Hamiltonian can be derived from an orthogonal transformation of the logarithmic-discretized bath. It is given as

(5) | |||||

Here are the on-site and hopping energies of the boson chain and is the logarithmic discretization parameter. The displacement operator in Eq.(2) is normalized as with . The local boson annihilation operator reads

(6) |

Here . The formalism used for NRG calculation is summarized in Appendix B. Thanks to the exponential decay of energy scales along the chain, the low energy eigen-energies and eigen-states of can be obtained reliably by iteratively diagonalizing and keeping the lowest states after each diagonalization. For each boson site to be added into the chain, we truncate its infinite dimensional Hilbert space into a -dimensional space on the occupation number basis. The accuracy of NRG result is controlled by three parameters: the logarithmic discretization parameter , the number of kept states , and the boson-state truncation parameter . In this work, the exact results at , , and are produced by extrapolating the NRG data obtained using , , and to the above limit.

## Iii Results

As analysed above, the quadratic-coupling SBM has an impurity-induced QPT in the environmental degrees of freedom, at which the boson parity symmetry is broken. An interesting feature of is that many of the universal properties of this non-trivial QPT are already well-described by the exactly soluble point . A finite quantum tunnelling introduces non-trivial dynamics of and modifies the phase diagram from the limit. Below, we first focus ourselves on the point , presenting the exact solution as well as the NRG results. Then, we use NRG to study the effect of finite quantum tunnelling .

### iii.1 Non-tunnelling point

The Hamiltonian at reads

(7) |

Here is the normalized boson displacement operator defined in Eq.(5). At this exact soluble limit, the dephasing properties were analysed in the context of the superconducting qubit at the optimal working point Makhlin1 () and the quantum dot qubit quadratically coupled to acoustic phonons Muljarov1 (). As confirmed by our NRG calculation below, the universal critical properties of the QPTs in are already well described by this limit.

#### iii.1.1 exact solution for

To detect the environmental QPT, we calculate the zero-temperature dynamical correlation function defined as

(8) |

with . We calculate the exact expression for and the energy difference between the two subspaces and , from which the exact ground state phase diagram can be extracted. Using the Green’s function equation of motion method, we obtain the exact expression for at as (see Appendix A for details),

(9) |

for . For , . The function is given as

(10) |

For the specific given in Eq.(4), and the above equation simplifies into

(11) |

Here is the hypergeometric function.

It is noted that describes the effective spectral function of bosons renormalized by the impurity-bath coupling. In the weak coupling limit , recovers the normalized bare spectral function. A finite quadratic coupling to the impurity exerts significant influence on . Especially, an environmental QPT occurs when a singularity develops in . Using the analytical continuation of from to , and considering , a continuous QPT is found at in the subspace and no QPT occurs for in the other subspace . The asymptotic behaviour of in the small frequency limit reads

(12) |

For , for and for . The peak position is given as

(13) |

It corresponds to the crossover energy scale between the boson-stable state and the quantum critical regime. As approaches from below, the peak position moves to zero frequency in a power law , giving the exact exponent .

The difference in the ground state energies of the two subspaces, , is obtained as

(14) |

in which

(15) |

and

(16) |

For a fixed coupling strength , for very large negative . increases with increasing until a level crossing occurs at , where the global ground state change from the subspace to . The spin-flip transition line is determined by . Taylor expanding with respect to , one obtains in the small limit

(17) |

As will be shown below, this transition from to is abrupt only at . For , due to the mixing of spin-up and spin-down sub-spaces, this spin flip is not a phase transition but a smooth change without singularity.

#### iii.1.2 NRG results for

In this section, we present the NRG results for and compare them with the exact solution. The purpose is to benchmark the NRG method and gain further insight into the phase diagram at . For simplicity, we present results only for a generic sub-Ohmic spectral function with . Unless specified otherwise, qualitatively similar results are obtained for other values. In our calculation we use and set as the energy unit.

Fig.1 shows the ground state phase diagram on the plane. Phase boundaries are obtained using NRG with and . There are three phases characterized by and or . In the small regime, the environmental bosons are in their stable state with . For a fixed in this regime, the ground state energies and changes with . When exceeds (up triangles), the two energies cross and it makes a spin flip from to . In the large regime, the boson parity symmetry is spontaneously broken and the order parameter . The environmental stable and the unstable phases are separated by a vertical continuous transition line (circles) at for larger , and by a first-order transition line (squares) for smaller . The three transition lines meet at a tri-critical point (, ) (solid dot in Fig.1).

The inset of Fig.1 compares the NRG results obtained using and (up triangles) with the exact line of solved from (solid line). The very good agreement is due to the cancellation of errors of and in the NRG calculation, since the error in the NRG ground state energy comes from its treatment of bosons, independent of the spin state.

The first-order QPT is a level crossing induced by the boson instability transition in the subspace . For and small , the subspace has higher energy than the subspace. As approaches the boson instability point , drops to abruptly while is unchanged, leading to a sharp level crossing between the two ground states.

We use and to denote the first-order and the continuous QPT points, respectively. The above scenario of the QPTs suggests for . Here is the critical point in the subspace and it is independent of . Indeed, NRG gives a vertical continuous QPT line at . It is slightly larger than due to the logarithmic discretization error at . For the first-order transition line, NRG gives an -dependent and it shifts to the left from to . As shown in Fig.13 of Appendix C, in the limit , the line of converges to the vertical line at (solid squares in Fig.1). Extrapolating the -converged lines to gives quantitative agreement with the exact value , as shown in Fig.12 of Appendix C.

In Fig.2, and are plotted as functions of for various values. For and which are larger than , a spin-flip transition and a continuous QPT occur successively as increases. For , , and which are smaller than , both quantities jump discontinuously at . It is noted that Fig.2 is a qualitative demonstration of the change of only. In the limit , as shown in Fig.11 and Fig.13 in Appendex C, diverges both at the continuous and the first-order QPTs, being consistent with the scenario of the QPTs presented above.

As a direct product from NRG, the flow of the excitation energy levels can help identify various fixed points in the phase diagram, which is not directly addressed by the exact solution Eqs.(9)-(16). As shown in Fig.3, we found three distinct fixed points for . The stable fixed point obtained for is identified as the free boson fixed point with and . For , the excitation energies flow towards a state with two-fold degeneracy. At this fixed point, the harmonic potential of bath particles is inverted and fluctuates between . In the large regime, the degeneracy will be split by the numerical error, breaking the boson parity symmetry and giving Note1 (). At , the excitation energies flow to an unstable critical fixed point and becomes nonzero continuously at this point.

Fig.4 further discloses the nature of the strong-coupling fixed point. Here we plot (), the eigen-energies directly obtained from diagonalizing the Wilson chain Hamiltonian , without subtracting the ground state energy . While the energies for is independent of , the strong-coupling fixed point energies in decreases with increasing . As shown in the inset, both and the boson number at the strong-coupling fixed point are linear functions of , diverging in the limit . As a result, the total NRG ground state energy tends to minus infinity in the limit . This supports that the strong-coupling fixed point is an unstable state of the environment.

To investigate the critical behaviour of the QPTs, the excitation energy flows are presented in Fig.5 for very close to and . In Fig.5(a), a typical critical behaviour is observed for , with the standard scaling form. The crossover energy scale is found to follow a power law, . The fitted exponent and from the two sides of agree well with the exact solution at . In Fig.5(b), near the first-order phase transition at , a level crossing in the energy flow is observed, accompanied with an abrupt jump from to .

### iii.2 Effects of finite quantum tunnelling

#### iii.2.1 the case of s=0.3

The quantum tunnelling introduces non-trivial dynamics of and modifies the phase diagram from the limit. Fig.6 shows the NRG phase diagram obtained for and . The boson-stable state () on the left side is separated from the boson-unstable phase () by a continuous or a first-order transition line, depending on the value of . The phase boundaries obtained using , , and are presented as different squares. Same as the case, the continuous transition (circles) and the spin-flip line (up triangles) are almost independent of , while the first-order transition line (squares) shifts to smaller with increasing , converging to an extrapolated line at (solid squares with eye-guiding line).

Several changes are induced by the finite quantum tunnelling. First, the phase boundaries has shifted quantitatively with respect to their position at . The line of is no longer vertical, especially near where the competition between and is strong. For , is independent of asymptotically. The jointing point (, ) of the continuous and the first-order QPT lines moves to larger direction, compared to case. Second, the spin-flip line is no longer accompanied with an abrupt jump of between and . As increases above for a given , changes smoothly from positive to negative values and no singularity is found in our NRG data. Therefore, the spin-flip line is not a phase transition line, due to the mixing of spin up and down subspaces by . Finally, the spin-flip line ends on the first-order phase boundary , instead of ending at the tri-ctitical point as in the case of . A small window of thus emerges below in which the first-order QPT is between the two states both with . In contrast, for the first-order QPT is always between the and the states.

In Fig.7, we show how the QPT transits from first order into continuous as increases above . The evolution of the curves with is shown in Fig.7(a). As shown in Fig.7(b) and (c), as approaches from below, the magnitude of the jumps in and at decrease to zero, leading to a continuous QPT. Note that the spin is polarized on both sides of the QPT. Similar to the case , NRG calculation shows that in the limit , the boson-unstable phase has , regardless of the order of QPT.

Besides the change of phase diagram, a finite also induces non-trivial dynamical correlation function. The coherence in the non-equilibrium evolution can be partly reflected in the equilibrium dynamical correlation function

(18) |

with . At , and it fulfils the sum rule . For a non-degenerate ground state , , where . At , there is no dynamics in the component and . For , the spin is no longer fully polarized in -direction and the weight of is partially transferred from to regime.

In Fig.8(a) and (b), and are presented for and . In Fig.8(a), the low frequency asymptotic behaviour of is shown to be the same as the case, i.e., and for and , respectively. approaches zero as . In Fig.8(b), is found to be characterized by a coherent peak around the effective Rabi frequency and a low frequency power law behaviour, for and at . For , the same crossover scale as in separates the (for ) and (for ) behaviour. A zero frequency peak is also present but not shown. Our NRG data for the sub-Ohmic regime support and (see Fig.10), being independent of and values. In contrast, the linear-coupling SBM has and . Here, the effect of the quadratic impurity-bath coupling is prominent: the existence of impurity influences the bath spectral function and it subsequently modifies the dynamics of the impurity itself.

Close to the first-order QPT at and , and are similar to the ones at and . At , both correlation functions change abruptly into an artefact produced by the finite used in NRG. In the case, the critical behaviour cannot be observed in and regime, because the lower subspace has no QPT. For , due to the mixing of two subspaces, quantum critical behaviour such as and can be observed in the intermediate frequency regime for the weak first-order QPT in and regime. The crossover scale decreases with increasing and reaches a finite value at the first-order QPT . As approaches from below, decreases to zero and the first order QPT transits into continuous one.

Fig.9 shows the dynamical correlation functions and at the critical point for a series of in the regime . Although is identical for different values, decreases with increasing and keeps the exponent intact. This is because as increases, decrease monotonically to , transferring the weight of from the regime to . The prominent Rabi peak corresponds to short-time coherent oscillations in the population of the non-equilibrium situation. The effective Rabi frequency increases with . Assuming an effective free spin Hamiltonian, we can write where contains both and the static mean field from the quadratic coupling . is the renormalized tunnelling strength. The estimated by assuming and using from NRG agrees well with the peak position in (vertical dashes in Fig.9). This shows that robust coherent spin evolution persists to the strongest coupling allowed before the environmental QPT occurs.

#### iii.2.2 other values

We carried out NRG study for other values and confirmed that the scenario of QPT established at applies to the whole sub-Ohmic regime , with important quantitative differences. For , the structure of the phase diagram is same as that of and NRG results agree well with the exact solution. For , we find that increases with and the first-order QPT line expands to larger values. At , the critical fluctuation of

(19) |

increases with . For larger , the ground state energy contains a term which changes more rapidly with the flipping of spin. This makes the continuous QPT more difficult to realize. Our NRG study for supports that for any finite , i.e., the transition is first-order for any and any . This behaviour is well understood at the extreme case where the infra-red divergence in makes the continuous QPT impossible. Here, the continuous versus first-order phase transition is an interesting problem on its own, giving its resemblance to the same problem in the crystal lattice Cowley1 (). A detailed study on this issue will be published elsewhere. At the other limit which is related to the noise in the quantum circuit, a finite induces a small but finite . diverges at and a QPT of the Kosterlitz-Thouless type occurs, as confirmed by the NRG calculation (not shown). This is similar to the situation of linear-coupling SBM Bulla1 ().

The above analysis also explains the observation that for larger , reliable NRG calculations require larger and are hence more difficult. For studying the continuous QPT at , insufficient could lead to artificial critical fixed point and produce incorrect exponents , and . For studying the first-order QPT at , it may make an artificial continuous QPT. Up to now, quantitatively accurate study of for is still a technical challenge for NRG. For the sub-Ohmic bath, however, we can get reliable results using the boson number truncation up to and a large logarithmic discretization parameter . For where the first-order QPT prevails, the critical exponents can still be extracted reliably from the intermediate frequency regime for and .

In Fig.10, we show the exponents and of . They are defined as for and for , with the same crossover scale of . Since they appear only at , there is no exact solution for them. The NRG data agree with the analytical expressions and within an error of . When extended to , such behaviour will lead to the breakdown of the sum rule of and prohibit the continuous QPT in the Ohmic- and super-Ohmic regime.

## Iv Discussion and Summary

In this section, we discuss several issues regarding to the impurity-induced environmental QPT that we studied in this paper.

First, we note that the unphysical result and in the boson-unstable state is a consequence of incompleteness of the present model. Whenever such QPT occurs in a realistic situation, the magnitude of boson displacement is confined by the higher order anharmonic terms of the boson energy beyond , leading to a local environmental distortion with finite as a new stable state. Such higher order terms will determine the properties of the new stable state of the environment but cannot remove the impurity-induced environmental QPT. The environmental instability shows up differently in real systems. For the superconducting flux qubit system Bertet1 (), corresponds to an additional bias current in the SQUID oscillator. In the experiment of quantum dot system Muljarov1 (); Borri1 (), however, the boson instability corresponds to a local distortion of the crystal lattice. In the optical spectra signal of an impurity center in crystals, the instability is detected by the anomalous temperature dependence of the zero-phonon line width due to the softening of bosonic modes close to the environmental QPT Hizhnyakov1 (). In the NRG calculation, the boson state truncation mimics such a higher order anharmonic effect accidentally. We find that although the existence of the QPT is robust under this constraint of Hilbert space, the critical exponents and may well be changed by it.

Second, we discuss the situation where both the linear- and the quadratic-coupling are present. In that case, the Hamiltonian reads

(20) |

For general parameters and , this Hamiltonian has a lower symmetry than both the linear-coupling SBM and the pure quadratic-coupling one. As a result, neither the delocalize-localize transition nor the environmental stable-unstable transition exists any more. Instead, similar to the situation of linear-coupling SBM under a finite bias , it is expect that the ground state smoothly interpolates between different limiting symmetry-broken states of purely linear- or quadratic-coupling Hamiltonians. The crossover lines separating these phases are determined by the relative strength of , , and the crossover energy scale to the quantum critical regimes in - and -only cases Zheng1 (); Hur1 (). However, both the bath and the spin dynamics will be severely influenced by the existence of the quadratic coupling terms.

At finite temperatures, the QPT observed in no longer exists. However, for parameters close to the critical point and , the quantum critical properties will appear in the temperatures regime , as demonstrated in the proposal of observing the impurity QPT in a mesoscopic metal ring system Tong1 (). This provides opportunities to experimentally observe the signatures of the environmental QPT discussed in this paper. Our conclusion about the environmental QPT can be straightforwardly extended to the single boson mode case. For the Hamiltonian of the circuit quantum electrodynamics Bertet1 (), the boson-instability occurs at for . Using the parameters of the experimental set up of Ref. Bertet1 (), we estimate that MHz. Given GHz, the actual ratio , much smaller the critical value. However, in the experiments of superconducting qubit, methods are available to both engineer the shape of Haeberlein1 () for low frequency continuous environment, and to enhance the spin-boson coupling to the ultra-strong regime for the case of discrete boson modes Niemczyk1 (); Diaz1 (); Baust1 (). Especially, the new technique of switchable coupling can boost the linear coupling from MHz level to GHz level, making it comparable to Peropadre1 (). The superconducting flux qubit Yoshihara1 (); Kakuyanagi1 () or the quantum dot Petersson1 () under the noise can also be tuned to the optimal working point . Considering that our results predict that the noise with quadratic spin-boson coupling gives a much smaller , we expect that these advances can make it feasible to detect the environmental QPT discussed in this work.

In summary, the quadratic-coupling spin-boson model is found to have an impurity-induced environmental QPT between the boson-stable and -unstable ground states. Using the exact solution at as well as the NRG calculation, we establish the ground state phase diagram which contains both continuous and first-order QPTs. The exact quantum critical behaviour is obtained. The non-trivial features of the spin as well as the bath dynamical correlation functions are revealed. Our results are relevant to various recent experimental set ups, including the superconducting flux qubit at the optimal working point Vion1 (); Bertet1 (); Ithier1 (), the quantum dot quadratically coupled to acoustic phonons Borri1 (), and the impurity atom embedded in a crystal Hizhnyakov1 (). The physical consequence of such QPTs and the feasibility of its experimental observation are discussed.

## V Acknowledgements

D.-C. Zheng and N.-H. Tong acknowledge helpful discussions with Y.-J. Yan. This work is supported by 973 Program of China (2012CB921704), NSFC grant (11374362), Fundamental Research Funds for the Central Universities, and the Research Funds of Renmin University of China 15XNLQ03.

## Appendix A Exact solution at

In this appendix, we derive the exact solution at . The Hamiltonian at reads

(21) |

Here and . For the spectral function specified in Eq.(4), .

To solve exactly, we employ the equation of motion (EOM) for the double-time Green’s function . At zero temperature , the dynamical correlation function is expressed in terms of as

(22) |

Here is an infinitesimal positive number. Note that is an even function of .

We start from the EOM of the following GF component,

(23) |

At , the commutators in the above equation read , . Using these expressions and their Hermitian conjugates, we obtain

(24) |

and

(25) | |||||

One can solve Eq.(A4) and (A5) to obtain

(26) | |||||

Multiplying on both sides of the above equation and summing over , we obtain a closed equation for . The solution of the equation reads