Entanglement entropy, decoherence, and quantum phase transitions of a dissipative two-level system

Entanglement entropy, decoherence, and quantum phase transitions of a dissipative two-level system

Karyn Le Hur
Abstract

The concept of entanglement entropy appears in multiple contexts, from black hole physics to quantum information theory, where it measures the entanglement of quantum states. We investigate the entanglement entropy in a simple model, the spin-boson model, which describes a qubit (two-level system) interacting with a collection of harmonic oscillators that models the environment responsible for decoherence and dissipation. The entanglement entropy allows to make a precise unification between entanglement of the spin with its environment, decoherence, and quantum phase transitions. We derive exact analytical results which are confirmed by Numerical Renormalization Group arguments both for an ohmic and a subohmic bosonic bath. Those demonstrate that the entanglement entropy obeys universal scalings. We make comparisons with entanglement properties in the quantum Ising model and in the Dicke model. We also emphasize the possibility of measuring this entanglement entropy using charge qubits subject to electromagnetic noise; such measurements would provide an empirical proof of the existence of entanglement entropy.

Department of Physics, Yale University, New Haven, CT 06520, USA

E-mail: karyn.lehur@yale.edu


Key Words:  Dissipative two-level systems; Spin-boson models; Entanglement entropy; Decoherence; Quantum Phase Transitions

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1 Introduction

Dissipation in quantum mechanics represents an important statistical mechanical problem having ramifications in systems as diverse as biological systems to limitations of quantum computation. A prototype model in this class is the Caldeira-Leggett model describing a quantum particle in a dissipative bath (environment) of harmonic oscillators [caldeira]. The spin-boson model can be seen as a variant of the Caldeira-Leggett model where the quantum system is a two-level system [Blume, Leggett]. Those impurity systems are interesting because they display both a localized (classical) and delocalized (quantum) phase for the spin [karyn, karyn2]. This class of models were intensively investigated to study the emergent quantum-classical transition [Matthias].

An environment generally induces some “uncertainty” in the spin direction which is responsible for quantum decoherence, i.e., for the rapid vanishing of the off-diagonal elements of the spin reduced density matrix. On the other hand, a finite coupling between the spin and its environment also induces entanglement between the spin and the environmental bosons, i.e., the wavefunction of the system cannot be written as a simple product state anymore. In this paper, we are interested in quantifying properly the entanglement properties between the spin and its dissipative environment. One good measure of entanglement for these well-defined bipartite systems is the von Neumann entanglement entropy [entropy, Bennett, Amico]. In fact, we anticipate a deep connection between decoherence of the spin and (entanglement) entropy which may be seen as follows: the decoherence irreversibly converts the averaged or environmentally traced over spin density matrix from a pure state to a reduced mixture. Combining exact analytical results and Numerical Renormalization Group techniques applied directly on the spin-boson model [karyn2, Bulla], we are prompted to flesh out the entanglement properties between the spin and its environment by computing the von Neumann entropy of the spin when integrating over the bath degrees of freedom. We expand our recent articles [karyn3, karyn4]. We will demonstrate that the resulting entanglement entropy constitutes an important order parameter (which exhibits universal scalings), not only of quantum phase transitions but also of crossovers. We will also confirm that loss of coherence of the two-level system (characterized either by the complete destruction of Rabi oscillations [Saleur] or by the strong suppression of persistent current in a ring [Buttiker, karyn3]) means a prominent enhancement in the entanglement of the system; this enhancement of entanglement corresponds to the actual quantum phase transition in the case of a subohmic bath and rather corresponds to the dynamical incoherent crossover from damped oscillatory to overdamped behavior in the case of an ohmic bath [Leggett]. For a subohmic bath, second order phase transitions can be interpreted as transitions where coherence is lost due to the emergence of a purely classical state at the transition. We also emphasize the possibility of measuring the entanglement entropy of the spin using charge qubits subject to tunable electromagnetic noise [Schon, Clarke].

2 Spin-Boson model, its relatives, Entropy

In this section, we provide useful definitions. In particular, we introduce the spin-boson Hamiltonian as well as useful mappings towards long-range Ising chains [Dyson] and Kondo models [Hewson], and we discuss more thoroughly the concept of entanglement entropy. We define properly the entanglement entropy of the spin with its environment in the context of the spin-boson model, and we discuss the different behaviors of in the delocalized and localized phase of the spin-boson model(s). The von Neumann entropy is strictly defined for a pure state at zero temperature; nevertheless, in Sec. 4.3 we will briefly comment on the quantum-classical crossover at finite temperature where the Boltzmann-Gibbs entropy proliferates [Lebowitz].

2.1. Spin-Boson model, Kondo physics, and Ising chains

The Hamiltonian for the spin-boson model with a level asymmetry takes the usual form [Leggett]:

(1)

where and are Pauli matrices and is the tunneling amplitude between the states with . is the Hamiltonian of an infinite number of harmonic oscillators with frequencies , which couple to the spin degree of freedom via the coupling constants . The heat bath is characterized by its spectral function:

(2)

where is a cutoff energy111To simplify notations, hereafter we fix the Planck constant . and the dimensionless parameter measures the strength of the dissipation. Similar models involving a two-level system coupled to a bath of harmonic oscillators are investigated in the context of stimulated photon emission. A known example is the Jaynes-Cummings model [Jaynes]. It should be noticed that our model has a dense spectrum of fundamental frequencies of the harmonic oscillators.

In the following, we will refer to the values as subohmic damping [Leggett, Matthias]. One standard approach is to integrate out the dissipative bath, leading to an effective action which is reminiscent of the classical spin chains with long-range correlations (here, in time):

(3)

with

(4)

at long times.

FIG. 1.    Phase diagram of the spin-boson model for the ohmic and subohmic case.

For a subohmic bath, correlations in time become sufficiently long-range that one may expect the localization of the spin for all values of , at least in the limit of . On the other hand, by increasing , one eventually expects a quantum phase transition towards a delocalized phase for the spins. In fact, the analogy with classical spin chains leads to the existence of second-order quantum transitions for between a localized phase at small and a delocalized phase at large [Dyson, Kosterlitz]; see the phase diagram of Fig. 1. It is relevant to mention that this transition has been seen by applying the Numerical Renormalization Group (NRG) method directly on the spin-boson model [Bulla, karyn4, Matthias] but this is not visible in the popular non-interacting blip approximation [Leggett].

In the localized regime, the tunnel splitting between the two levels renormalizes to zero, i.e., the spin gets trapped in one of the states or , whereas is strongly renormalized in the delocalized phase.

The special case of corresponds to the ohmic spin-boson model which is also equivalent to the anisotropic Kondo model; at small , here the system shows a Kosterlitz-Thouless (KT) type quantum phase transition222In the ohmic case, the phase transition is described by RG equations similar to those in the XY model in two dimensions. On the other hand, the phase transition in the Kondo model was found earlier than the KT phase transition [anderson]., separating the localized phase at from the delocalized phase at . To understand how the KT transition takes place as a function of the dissipation parameter , one may use the analogy with the anisotropic Kondo model, which describes the exchange interaction between conduction electrons and a magnetic impurity [anderson]:

(5)

where represents the kinetic energy of the electrons. The equivalence between the spin-boson model with ohmic dissipation and the Kondo model has been first realized by Chakravarty [chakravarty] and independently by Bray and Moore [bray]. One may identify:

(6)

where is the conduction electron density of states and the parameter is related to the phase shift caused by the Kondo term and is given by [Guinea]

(7)

The phase transition in the Kondo model (a purely quantum phase transition at the absolute zero) corresponds to the (bare) Kondo coupling changing sign. For (i.e., antiferromagnetic coupling of the Kondo spin to conduction electrons), the “fugacity” is a relevant operator. Physically, this means that the number of spin flips proliferates, and the Kondo spin forms a singlet state with the conduction electrons. In the opposite (ferromagnetic Kondo coupling) case, scales to zero. No spin flips remain, and the spin is “frozen” in time corresponding to the long range order of the Ising chain. Anderson, Yuval, and Hamman [anderson] succeeded in mapping the behavior of the Kondo model onto the solution of a long-range Ising spin chain. The tunneling events where the local moment spin flips due to scattering with conduction electrons, correspond to domain walls of the equivalent Ising chain. The problem is similar to ordinary tunneling of a two-level system. The physically new feature is the fact that conduction electrons represent a dissipative bath, and as result the tunneling events are not independent. Instead, they feature long-range interactions in time, and the equivalent Ising chain acquires long-range interactions.

The equivalence between the anisotropic Kondo model and the spin-boson model with ohmic damping can be formulated explicitly through bosonization [Guinea]. The delocalized region corresponds to the antiferromagnetic Kondo model, while the localized region corresponds to the ferromagnetic Kondo model where the spin is frozen in time. In the delocalized phase of the spin-boson model, the Kondo energy scale obeys333In the definition of the Kondo energy scale, we have introduced a high-energy cutoff for the conduction electrons which is of the order of the Fermi energy; the relation between the two cutoffs and will be fixed properly later; see Appendix C. [Bohr]

(8)

for values of not too close to the transition and close to the transition assumes the exponential form of the isotropic, antiferromagnetic Kondo model; . The critical line separating the localized and delocalized phase in the spin-boson model thus corresponds to or . Such results have been confirmed by applying the NRG approach directly to the spin-boson model [karyn2, Bulla]. In the delocalized phase of the spin-boson model, we expect a unique nondegenerate ground state and the expansion in the number of spin flips in the partition function does not lead to a dilute gas of flips for which reflects the basic difficulty that the exact ground state is orthogonal to that for a static spin ; consult Sec. 3.1.

2.2. Entanglement entropy

The problem of measuring entanglement in many-body systems is a lively field of research. Bipartite entanglement of pure states is conceptually well understood [Amico, Gil, Jordan]. A useful measure of many-body entanglement when the total system is in a pure state is the von Neumann entanglement entropy [entropy, Bennett, Amico]. This is obtained by focusing on bipartite systems where space can be divided into 2 regions, A and B. Beginning with the ground state pure density matrix, region B is traced over to define the reduced density matrix . From this, the von Neumann entanglement entropy

(9)

is obtained for the subsystem A of size . The rate at which grows with the spatial extent, r, of region is not only a fundamental measure of entanglement, it is also crucial to the functioning of the Density Matrix Renormalization Group [DMRG]. For systems with finite correlation lengths, it is generally expected that grows with the area of the boundary of region A [bombelli]. In the one-dimensional case, conformally invariant systems have where c is the “central charge” characterizing the conformal field theory [c, c2, Joel]; for a two-dimensional example, see Ref. [Fradkin]. Entanglement entropy has recently been shown to be a useful way of characterizing topological phases of many body theories [Kitaev, Fendley, Wen], which cannot be characterized by any standard order parameter. Entanglement entropy is also closely related to the thermodynamic entropy of black holes and to the “holographic principle” relating bulk to boundary field and string theories [Fendley, Ryu]. Entanglement entropy also brings a new light on quantum impurity systems; a spin can be entangled to a bath of conduction electrons [Affleck] or to a dissipative bosonic environment [Angela, Costi, karyn3, karyn4]. For a quantum impurity system (A is the spin):

(10)

where

(11)

For the spin-boson model, because is invariant under . Moreover, for the spin-boson model, we get the equalities:

(12)
(13)

where is the ground state energy of . Since and are related by a unitary transformation, they have the same ground state energy (up to an unimportant constant). The field couples directly to the spin in both models, so we have . A similar relationship does not hold between and , and thus does not (exactly) measure the entanglement between the Kondo impurity and the conduction band.

At , the qubit is decoupled from the environment; thus, and . With and , we check that for all values of and . Moreover, at large , we also must have because the qubit is localized in the state with and . In a similar way, in the localized phase, since the qubit is also localized in one classical state at arbitrarily small then 444This result is intuitive if one absorbs the coupling with the environment into the tunneling term [Buttiker]. In the localized phase, the tunneling term is renormalized (almost) to zero which then ensures a disentanglement (decoupling) with the environment.. Below, we want to understand how interpolates between all these limits. We argue that the entanglement entropy of the spin with its environment will allow us to make a unification between entanglement of the spin with its environment, decoherence, and quantum phase transitions. Entanglement near a quantum phase transition is of great current interest [Amico, Osborne, Osterloh].

It is also relevant to note the relation between entanglement and quantum decoherence of the spin. Quantum decoherence implies a rapid reduction of the off-diagonal elements of the spin density matrix, i.e., which results in and if vanishes at small (delocalized phase). We will show below that for the ohmic case, the quantum decoherence or maximal entanglement occurs at the Toulouse limit [Toulouse] whereas for the subohmic case this rather happens at the phase transition.

3 Ohmic case: Spin Observables and Entanglement

After these generalities, we consider the ohmic situation and a large level asymmetry where perturbation theory applies and a correspondence to the well-known theory for dissipative systems can be formulated [Nazarov]. We are also interested in extracting the spin observables.

3.1. Large level asymmetry and P(E) theory

FIG. 2.    Two-level system with a large level asymmetry .

For , the spin is completely localized in the state. Now, for finite , we want to evaluate the probability that this spin down electron flips up. In the absence of the environment, it is straightforward to obtain that . Below, we will incude the effect of the environment and see how this quantity gets modified. It should be noted that plays an important role because it is related to , through

(14)

For , one gets two classical states and the Hamiltonian for either of the states depends on the bosons:

(15)

We can easily handle this Hamiltonian when . The stationary wavefunctions are those with a fixed number of bosons in each mode corresponding to the energy . For , one may absorb the linear term in the redefinition

(16)

where . Since the bosons have been shifted, the vacua for the two states are not the same. More precisely, let us consider the vacuum

(17)

This is equivalent to

(18)

We then observe that is not the vaccum for the bosons linked to the state . This is rather a coherent state and thus

(19)

For convenience of notations, in this expression we have assumed that is fixed. To first order in , this implies that the ground state acquires corrections proportional to all possible states :

(20)

with

(21)

The probability that the spin flips up, then takes the form:

(22)

By carefully summing over all modes , one can make a formal link with the theory [Nazarov] of dissipative tunneling problems (see Appendix A):

(23)

If there is no dissipation, then , and thus this reduces to the known qubit result . In the case of Nyquist noise or ohmic dissipation where [Nazarov, karyn5], the delocalization probability rather evolves as

(24)

Note that is small at high and one reproduces . On the other hand, for , one observes that increases for smaller and eventually reaches the maximum value . This shows that the problem becomes highly non-perturbative at small for ; one indeed expects a delocalized phase for and a localized phase for . It is important to note that for small level asymmetries, the point is unstable, i.e., perturbation theory in already breaks down at small .

The result in Eq. (24) is in agreement with the expansion of the ground state energy to second order in (we consider the case where ) [Buttiker]:

(25)

where is the incomplete gamma function. In the regime of relatively large (but still small compared to the high-energy cutoff ), one may simplify

(26)

This immediately leads to:

Thus, can easily be derived. At relatively large values of the level asymmetry, one gets:

(28)

We check that . In a similar way, one can extract , resulting in:

(29)

It should be noted that the limit is always well-defined, since

(30)

In particular, this leads to:

(31)

in agreement with the exact results obtained from the non-interacting resonant level model at the Toulouse point [Toulouse]; see Appendix B. This implies the important equality between the two cutoffs of the theory:

(32)

4 Results for a vanishing level asymmetry

In the delocalized phase, one can use the Bethe Ansatz solution of the anisotropic Kondo model or of the equivalent interacting resonant level model; consult Ref. [Level, ponomarenko] and Appendix C. While general expressions are quite complicated [Buttiker], it is instructive to derive simple scaling forms for the observables in the limits and [karyn3]. We will also apply the NRG method which shows an excellent agreement with the Bethe Ansatz results; NRG parameters have been defined in our Ref. [karyn4].

4.1. Spin observables

For , we check that the Bethe Ansatz calculations reproduce the perturbative results of Sec. 3.1. For the sake of clarity, technical details are given (hidden) in Appendix C. For , we obtain,

(33)

where

(34)

Note that at small , in keeping with the Kondo Fermi liquid ground state. The local susceptibility of the spin converges to for 555For , , , , and . in accordance with the two-level description and diverges in the vicinity of the KT phase transition as a result of the exponential suppression of the Kondo energy scale. It is relevant to observe that the longitudinal spin magnetization only depends on the “fixed point” properties, i.e, this is a universal function of in the delocalized phase. Exactly at the KT transition, one can adapt the results of Anderson and Yuval [Anderson2] and the result is that jumps by the amount along the quantum critical line where .

This results in a non-universal jump at the KT transition [karyn2]. Remember that in contrast to the universal jump of the superfluid density in two-dimensional XY models, in the spin-boson model, the jump in the longitudinal magnetization is non-universal for finite . Finally, far in the localized phase, one rather predicts [Angela]. In Fig. 4.1. we present two curves of as a function of , which have been obtained with the Bethe Ansatz and with the NRG calculations, respectively. One can notice the very good agreement between the two approaches. By increasing the detuning , the abrupt jump in occurring at the quantum phase transition is progressively replaced by a smooth behavior; at finite there is a smooth crossover separating the delocalized and the localized regime.

The leading behavior of in the delocalized phase takes the form:

(35)

with

(36)

As , and , so we recover the exact result 666Exactly at , the (small) first term in Eq. (35) is not present; see Appendix C.. As we turn on the coupling to the environment, we introduce some uncertainty in the spin direction and decreases. It should be noted that does not only depend on fixed point properties; in the delocalized phase, still contains a perturbative part in !

For , the monotonic decrease of dominates. In contrast, for , the first term in Eq. (35) dominates and we have

(37)

In fact, this result can also be recovered using the perturbation theory of Sec. 3.1 (consult Eq. (29) for ). This result clearly emphasizes that the observable is continuous and small at the KT transition in the ohmic spin-boson model. This is also consistent with the work by Anderson and Yuval which predicts exactly at the phase transition [Anderson2].

The fact that becomes very small in the localized phase can be understood from the mapping between the anisotropic Kondo model and the Ising model with long-range correlations in time. Here, the partition function of the system can be expanded in powers of [Leggett, anderson, Anderson2]:

(38)

where the function describes the (long-range) interaction between kinks (spin flips) located at positions and , and .

The scaling procedure requires to modify the short-time cutoff . Two important features then appear by modifying : the dependence of can be included in a global renormalization of and configurations with a kink-antikink pair at distance between and have to be replaced by configurations where this pair is absent. The number of removed pairs is proportional to and the (free) energy variation is roughly :

(39)

This is equivalent to change :

(40)

The renormalization of in the ohmic case is [Leggett]:

(41)

This leads to

(42)

The ground state energy can be recovered by taking the limit . This is continuous at the quantum phase transition. Extending this result to the delocalized phase, we find that for this expression is effectively convergent at low energies, and essentially one recovers the result obtained by Bethe Ansatz or by perturbation theory:

(43)

This ensures that in the delocalized phase, for , is of the order of . This result is definitely emphasized in Figs. 4.1.. In fact, it should be noted that is always continuous at a quantum phase transition.

Finally, we can also check that evolves continuously close to . In the limit , one can take and use the identity to find , in agreement with the “non-interacting” resonant level description valid at the very specific point .

4.2. Entanglement entropy

Now, we are able to study thoroughly the entanglement between the spin and its environment, at least in the context of an ohmic bath; see Fig. 4.2..

First, in the delocalized phase, at , is entirely determined by . Thus, at small , grows linearly with and for , which ensures that , i.e., ; see Fig. 5. It should be remembered that in the ohmic case, the reduction of quantum coherence (superposition) of the spin rather occurs at and not at the quantum phase transition. Additionally, the maximal entanglement occurring at allows to explain the beginning of the “incoherent” dynamical behavior [Leggett]. More precisely, as , one has damped Rabi oscillations at low temperatures; the Rabi oscillation frequency and damping rate are given by and with [Leggett] . In contrast, for the Rabi oscillations disappear and the behavior becomes completely incoherent [Saleur]. When the entanglement with the environment becomes too important one cannot distinguish in which state, or , is the two-level system anymore!

The plateau of entanglement is reminiscent of the maximal entanglement between the spin and its cloud of electrons in the SU(2) Kondo model [Amico, Affleck].

Second, in the localized phase, we obtain and — and therefore — for infinitesimal . Since dissipation localizes the spin in the “down” state for , we do not expect the entropy to depend strongly on the external field; the perturbation theory in predicts to leading order for all [Angela]. As we approach the phase transition from the localized side, this behavior is replaced by in agreement with at the KT transition.

This implies a non-universal jump of the entanglement entropy at the KT phase transition. This non-universal jump is reminiscent of the non-universal jump in the longitudinal magnetization . It should be noted that at finite , the sharp non-analyticity at the quantum phase transition is replaced by a maximum which signifies the crossover regime [karyn3]. Finally, we like to emphasize that in the delocalized phase, the Kondo scale controls the entanglement between the spin and the bath. For , the disentanglement proceeds as whereas for , vanishes as , up to logarithmic corrections. This universal behavior stems from since evolves very smoothly with the level asymmetry ; see Appendix C. Although is defined at zero temperature, this universality is reminiscent of thermodynamic quantities [Zarand, Weiss].

To be more precise, for a small level asymmetry, we find the scaling,

(44)

where the coefficient is given by,

(45)

and

(46)

For a large level asymmetry this rather results in

(47)

where the pre-factor is given by

(48)

The fact that when is consistent with the fact that does not depend much on the level asymmetry in the localized phase. The evolutions of the coefficients and as a function of are shown in Fig. 6. At the Toulouse point [Toulouse], one can easily check that

(49)

in agreement with the non-interacting resonant level model of Appendix B. In Fig. 4.2., the full solutions of and obtained with Bethe Ansatz and with NRG have been used to plot versus for different .

4.3. Delocalized phase and Boltzmann-Gibbs entropy

There is a prominent entanglement entropy in the delocalized phase. Even though this von Neumann entanglement entropy is generally not a good measure of entanglement at finite temperature [Amico], it is certainly relevant to discuss the evolution of the entropy of the spin at finite temperature.

For temperatures much smaller than , the system is almost in a “pure” state since all the excited states only have a small weight . Thus, one does not expect a dramatic modification in the entanglement entropy of the system for temperatures smaller than . This can be seen by considering the entropy of formation of the system [Amico]. Entanglement at finite temperature has been evidenced in different materials [hit]. On the other hand, if the temperature becomes much larger than the Kondo energy scale, the entropy of the spin should be of purely “thermal” origin [Affleck2] and should converge to777The P(E) theory allows to prove this rigorously. For large temperatures, the (small) coupling does not affect much the spin observables since the P(E) function converges to that of the uncoupled two-level system (Appendix D). Zero-point fluctuations of the bath become clearly negligible in front of thermal excitations of the bath oscillators. (normalized to 1) where

(50)

here, is the Hamiltonian for the spin alone and the associated partition function. The main coupling with the reservoir becomes of thermal origin. Now, it should be noted that just above the coupling between the bath and the spin produces some corrections in the thermodynamic properties of the interacting resonant level model which produces a reduction of the thermal entropy of the spin (above ) [Zarand, Weiss]:

(51)

In this sense, the Kondo temperature corresponds to the energy scale at which the thermal entropy of the spin is strongly reduced and is replaced by a prominent entanglement entropy between the spin and its environment. It should be noted that the thermal entropy exhibits a similar power-law behavior as the von Neumann entropy at zero temperature.

In the localized phase, there is no entanglement entropy, and thus the spin is only governed by its thermal Boltzmann-Gibbs entropy.

5 Subohmic case: Critically entangled system

Below, we rather focus on the subohmic situation which exhibits a second-order quantum phase transition [Bulla, Matthias, karyn4, Subir] by analogy to classical spin chains [Dyson, Kosterlitz] (as a function of ). The case is of particular interest since it can be realized through a charge qubit (dot) subject to the electromagnetic noise of an transmission line [Matthias2]; the ohmic case corresponds to an transmission line [karyn, Buttiker]. In the limit of small , the theory of Appendix A suggests that the spin is localized for all . On the other hand, a quantum critical point [Matthias, karyn4] still exists and a delocalized phase for the spin occurs for where when . It should be mentioned that, at small , some critical exponents are different from those in the classical Ising model [Matthias, MatthiasIsing].

The order parameter for this delocalized-localized or quantum-classical transition is usually the longitudinal magnetization of the spin [Matthias] which now vanishes continuously at the phase transition,

(52)

where simple hyperscaling relations easily lead to (see Appendix D):

(53)

where the correlation length exponent obeys [Kosterlitz] for whereas at small , one finds[Matthias] through a small -expansion. For the subohmic spin-boson model, , which ensures that .

To better characterize those second-order quantum phase transitions, we are prompted to examine the entropy of entanglement shared between the spin and its environment. The subohmic model cannot be solved exactly. Then, we will resort to the “bosonic” NRG [Matthias, Bulla, Ingersent, karyn2, karyn4] and to hyperscaling relations which are applicable if the fixed point is not trivially Gaussian [Ingersent2]. Technical details are given in Appendix D. In fact, there is a simple way to conceive that the maximum of should coincide with the quantum phase transition for the subohmic situation. Starting from the delocalized phase, since the longitudinal magnetization at ,

(54)

where we have introduced the transverse susceptibility ; this expression is still valid at since for the subohmic case is zero at the quantum critical point. Since and are positive quantities, this implies that in the delocalized phase.

FIG. 7.    Susceptibility in the subohmic spin-boson model.

In the localized phase, is rather controlled by the finite longitudinal magnetization and by the susceptibility (see Fig. 5)888Here, we neglect the less relevant contribution from .:

(55)

is related to the rapid localization of the spin. Thus, in the localized phase. Eqs. (54) and (55) imply that the entanglement entropy is maximum at the phase transition. This shows that those second-order impurity quantum phase transitions are always accompanied by a maximum (cusp) in the entropy of entanglement. In general, such zero-temperature impurity critical points show a “fractional” entanglement entropy which depends on the dissipation strength through and thus is not universal. In fact, starting from the localized phase, by analogy to the ohmic case, we find where increases by decreasing .

On the other hand, unambiguously exhibits universal scalings even though the entanglement is two-sided, so that two numbers are necessary to specify (one for each of the two ways of approaching ). More precisely, near the qantum critical point , the transverse spin susceptibility obeys

(56)

where the exponent is defined as . For the subohmic spin-boson model, one finds for all , ensuring that does not diverge at the transition. Taking into account that is continuous at the transition, Eq. (56) thus implies that always rises linearly for means that we approach the quantum critical point from the delocalized (localized) region. It should be noted that the coefficients and can be different in the delocalized and in the localized phase and especially at [karyn4]. Through the NRG, we have also checked that , emphasizing that in the delocalized phase substantially increases at . For all , this strongly underlines the duality between the enhancement of entanglement and the strong reduction of the two-spin state quantum superposition near the phase transition.

In the localized phase, we obtain the scaling behavior:

(57)

here when , and we identify . For , . Using a small -expansion [Matthias] which predicts at small , we deduce that diverges at for , which is well verified with the NRG approach; see Fig. 7. Using Eqs. (52) and (53), then999For , one rather finds .

(58)

We observe that the decay of the von Neumann entropy in the localized phase is faster than linear for all and the behavior becomes strictly linear at . The entanglement entropy at is shown in Fig. 8. In the limit , we also check that becomes rapidly suppressed at which is a reminiscence of the KT transition (ohmic case, ).

Now, we shall discuss the scaling of with the longitudinal field. Using the hyperscaling relations, for all we find that (see Appendix D):

(59)

and

(60)

where is defined in a usual way as: . It is interesting to observe that one can always expand at small and to satisfy (the field favors a product state). It is also relevant to notice that in the delocalized phase, decreases as similar to the ohmic case, whereas in the localized phase by approaching the phase transition we rather find a linear decrease of with .

For the subohmic case, the entanglement entropy allows us to establish important connections between impurity entanglement, quantum decoherence (or strong reduction of the quantum superposition of the two spin states) when approaching the phase transition from the delocalized phase, and rapid disentanglement in the localized or classical phase for the spin (the spin is rapidly frozen in one classical state due to dissipation).

We note that the entanglement entropy of the spin is an interesting new order parameter of those second-order (impurity) quantum phase transitions even though the maximum of entanglement at the quantum critical point is not universal (but scalings are universal).

6 Comparison with other models

Here, we compare the entanglement entropy of the subohmic spin-boson model, which yields a parallel with long-range Ising spin chains and exhibits a second-order phase transition, with the entanglement entropy of the quantum Ising model[Osborne] and with the one of the Dicke model [Lambert, Amico].

6.1. One-dimensional Quantum Ising model

In fact, we argue that the cusp occurring in the subohmic situation of the spin-boson model is very typical of second-order phase transitions and of quantum Ising models. Remember that the spin-boson model can be mapped onto an effective action where the spin is subject to long-range correlations in time. Interchanging space and time, one thus expects that the the single-spin (single-site) entanglement properties in the quantum Ising model yields a very similar cusp at the phase transition.

As an example, let us consider the one-dimensional (1D) quantum Ising model which is described by the following Hamiltonian:

(61)

where is the Pauli matrix (a=x,y or z) at site , and describes the Ising coupling between spins whereas the transverse field has been set to . The model can be solved exactly using the Jordan-Wigner transformation [Subir] and the system exhibits a second-order quantum phase transition at separating a paramagnetic phase where from a ferromagnetic phase. In the ferromagnetic phase, the order parameter obeys (we use the fact that the dynamic critical exponent and [Subir]):

(62)

Here, as a reminiscence of the classical two-dimensional Ising model [Subir]. By analogy to the spin-boson model, one can verify that is continuous at the second-order quantum phase transition [Osborne]:

(63)

In the delocalized phase , we recover that whereas in the localized phase , we observe that is small . Similar to the spin-boson model, one can check that the transverse susceptibility does not diverge at the transition, which implies that in the delocalized phase, the single-spin (single-site) entanglement entropy rises linearly close to the quantum critical point [Osborne]. In the localized region, from Eq. (55), we also check that diverges at , meaning that goes very rapidly to zero in the localized phase. To summarize, the 1D quantum Ising model yields a single-site entanglement entropy which is very similar to that of the subohmic spin-boson model.

6.2. Dicke model

Now, we briefly discuss the entanglement properties [Lambert] in the one-mode superradiance (Dicke) model [Dicke] where collective and coherent behavior of pseudospins (atoms) is induced by coupling — with interaction — to a physically distinct single-boson subsystem. The Hamiltonian reads [Emary, Dusuel]:

(64)

where this form follows from the introduction of collective spin operators of length . in the thermodynamic limit , the system undergoes a quantum phase transition at a critical coupling , at which point the system changes from a large unexcited normal phase to a superradiant one in which both the field and atomic collection acquire macroscopic occupations [Emary]. In the thermodynamic limit, the problem reduces to a two-mode problem by using the Holstein-Primakoff transformation [Holstein] of the angular momentum operators , , and ; here, and are bosonic operators. In the normal phase , expanding the square roots directly this explicitly results in:

(65)

The problem is solvable going to the position-momentum representation for the two oscillators, and , with the momenta defined canonically. After diagonalizing the problem, one gets two independent (effective) oscillators, and the energies of the two independent oscillator modes read [Emary]:

(66)

Crucially, one can see that remains real only for ; thus remains valid only in the