Quantum phase transitions in the interacting boson model

# Quantum phase transitions in the interacting boson model

## Abstract

This review is focused on various properties of quantum phase transitions (QPTs) in the Interacting Boson Model (IBM) of nuclear structure. The model describes collective modes of motions in atomic nuclei at low energies in terms of a finite number of mutually interacting and bosons. Closely related approaches are applied in molecular physics. In the limit, the ground state is a boson condensate that exhibits shape-phase transitions between spherical (I), deformed prolate (II), and deformed oblate (III) forms when the interaction strengths are varied. Finite- precursors of such behavior are verified by robust variations of nuclear properties (nuclear masses, excitation energies, transition probabilities for low lying levels) across the chart of nuclides. Simultaneously, the model serves as a theoretical laboratory for studying diverse general features of QPTs in interacting many-body systems, which differ in many respects from lattice models of solid-state physics. We outline the most important fields of the present interest: (a) The coexistence of first- and second-order phase transitions supports studies related to the microscopic origin of the QPT phenomena. (b) The competing quantum phases are characterized by specific dynamical symmetries and novel symmetry related approaches are developed to describe also the transitional dynamical domains. (c) In some parameter regions, the QPT-like behavior can be ascribed also to individual excited states, which is linked to the thermodynamical and classical descriptions of the system. (d) The model and its phase structure can be extended in many directions: by separating proton and neutron excitations, considering odd-fermion degrees of freedom or different particle-hole configurations, by including other types of bosons, higher order interactions, and by imposing external rotation. All these aspects of IBM phase transitions are relevant in the interpretation of experimental data and important for a fundamental understanding of the QPT phenomenon.

Keywords: quantum phase transitions, interacting boson systems, novel theoretical approaches, shape transitions in nuclei

PACS codes: 05.30.Jp, 05.70.Fh, 21.60.Fw, 21.10.Re

Institute of Particle and Nuclear Physics, Faculty of Mathematics and Physics, Charles University, V Holešovičkách 2, 180 00 Prague, Czech Republic

Institute of Nuclear Physics, University of Cologne, Zülpicher Strasse 77, 50937 Cologne, Germany

## 1 Introduction

During the last decades, the classical notion of thermodynamical phase transitions [1] has been extended in several directions. One of these extensions is connected with finite systems, counting from thousands down to just few tens of particle constituents, see e.g. Refs. [2, 3, 4, 5]. The corresponding objects are Bose-Einstein condensates, atomic clusters, quantum dots and other mesoscopic systems. As the temperature or other external parameters are varied, different “phases”, i.e. specific structural configurations of these systems, suddenly emerge. Finiteness of the system unavoidably results in smoothening of all relevant phase transitional observables, but it makes sense to treat these effects as precursors of true phase transitions that would take place in an asymptotic regime. In particular, phase transitions are related to the scaling of some essential properties with the increasing size.

Another extension of standard phase transitions is to systems of interacting quantum objects at zero temperature. It turns out that when changing the interaction strength in some systems of this sort, one crosses a certain critical point of a nonanalytic change (in the infinite size limit) between ordered and disordered “phases”, which are represented by two distinct types of the ground state wave function. Since at zero temperature thermal fluctuations disappear, the only internal motion that can be responsible for the onset of disorder are quantum fluctuations. This situation is therefore referred to as the quantum phase transition (QPT). Quantum phase transitions were quickly recognized very relevant in a wide range of systems in condensed matter and many body physics, see e.g. Refs. [6, 7, 8, 9, 10, 11, 12].

Atomic nuclei are at an intersection of both above-outlined extensions of phase transitions. “Classical” phase transitions in nuclei are driven by intensive thermodynamical variables—temperature and/or rotational frequency—and take several well-known forms: (a) Transition from Fermi liquid to an ideal gas of nucleons is observed in multifragmentation phenomena induced by heavy-ion collisions [13, 14, 15]. (b) Experimental and theoretical studies (see e.g. [16, 17]) support the idea that nuclear matter exhibits a phase transition from the paired (superfluid) to an unpaired phase [18, 19]. (c) An analogous and maybe related effect is observed in the lowest rotational band of many nuclei as a sudden growth of the moment of inertia [20]. (d) Transitions between different quadrupole deformed shapes are anticipated to appear in a wide range of hot rotating nuclei [21].

Manifestations of “quantum” phase transitions in nuclei at zero temperature can be mostly observed as abrupt changes of nuclear shapes when crossing certain borders in the plane of neutron versus proton number, , see e.g. [9, 22, 23, 24]. Such transitions can be studied theoretically within some phenomenological models of nuclear structure, in which the variation of effective interaction strengths defines the relevant domain of application on the chart of nuclides, see e.g. Refs. [9, 25, 26, 27, 28].

This review is aimed at quantum phase transitions in the interacting boson model (IBM). Since its discovery by Arima and Iachello in 1975 [29, 30, 31], this model has played an important role in modeling collective motions of atomic nuclei [32, 33, 34, 35, 36, 37]. The success of the IBM is based on a simple and elegant algebraic formulation, and on a wide span of relevant dynamical regimes. Closely related algebraic approaches are applied in molecular physics [37, 38], hadronic physics [39, 40], and other areas [41, 42].

The IBM in its -boson “incarnation” exhibits both first- and second-order phase transitions between spherical, deformed-prolate and deformed-oblate shapes of the ground state [9, 43, 44, 45]. Although the nonanalytic nature of these transitions can be verified only in the unrealistic limit of the infinite boson number, (while in application to nuclei, coincides with the number of valence particle or hole pairs, so in all realistic cases), some finite-size precursors turn out to be very neat already at low boson numbers [23]. The shape transitional predictions of the IBM can be applied to low-energy spectroscopic data for numerous isotopic/isotonic chains.

Apart from describing actual data, the IBM can also be considered as a valuable theoretical tool for studying some general features of many-body quantum and mesoscopic systems. More than 30 years of the model history gradually disclosed that seemingly simple IBM hamiltonians encode a rich variety of complex physical phenomena, such as ground- and excited-state quantum phase transitions, effects of coexisting regular and chaotic motions, manifestations of various types of exact and approximate symmetries etc. The model offers a deeper insight into the roots of these phenomena. In this sense, it can be compared to the Hubbard model of solid-state physics, which was designed as a model describing rather specific systems, but in the course of time turned into a general testing ground for application of diverse theoretical concepts.

In attempts to understand the origin and consequences of quantum phase transitions in the IBM, considerable effort has been spent and a large number of results obtained—see the impressive (but certainly still incomplete) list of references [43117] below. Although these efforts are probably not yet completed, this might be about the right time to make a current summary.

In this review, the focus is set on both experimental and theoretical applications of the interacting boson model. With regard to the recent reviews of mostly experimental aspects of shape-phase transitions in nuclei [23, 24], more emphasis is put to the theoretical side of the problem. We start by describing some general features of quantum phase transitions in finite bosonic models and by pointing out some differences from the lattice models. A considerable part of the review deals with phase-transitional features of the simplest version of the model, the so-called IBM-1. It is presented as a solid reference frame for experimental studies, which at the same time serves as a theoretical workshop for analyzing the QPT underlying mechanisms. Applications in some of the more advanced versions of the IBM are illustrated afterwards. At the end, we will discuss recent generalization of the QPT type of behavior to excited states.

## 2 Quantum phase transitions in finite bosonic systems

Effective models of nuclear structure contain free parameters (interaction strengths) whose variations naturally induce changes of the ground-state wave function. In a finite system such changes are always of the “crossover” type, i.e., smooth in all observables. However, in some situations rather sharp transitions are encountered, which may signal true quantum phase transitions in the “thermodynamical limit”. To decide whether this is the case or not, one needs to identify a parameter, let us denote it , that measures the size of the system. Through this paper, the role of is played by the total number of bosons, . A necessary requirement for calling a given abrupt structural change of the ground state a “quantum phase transition” (more precisely, a finite- precursor of QPT) is that some of the related observables become discontinuous in the limit . Let us use this feature as a working definition of the quantum phase transition in the type of finite many-body systems that we study here.

Since in this review the focus is set on sharp transitions of nuclear shapes, the collective degrees of freedom are of primary interest. These can be represented by bosonic types of excitations, with individual bosons occupying a suitable finite-dimensional single-particle Hilbert space. The theoretical framework for this type of description is provided by the family of interacting boson models (Sec. 3), whose quantum phase transitions will be studied in the following sections. It turns out, however, that in relation to QPTs all finite bosonic models have some general properties in common. We therefore start our discussion by summarizing these properties and showing related examples in simpler IBM-like models.

### 2.1 Infinite-size limit of finite bosonic models

Consider a bosonic many-body hamiltonian with one- and two-body (and possibly higher) interaction terms conserving the total number of particles:

 H=E0+∑kϵkb†kbk+∑k,l,m,nνklmnb†kb†lbmbn+… (1)

Here and , respectively, create and annihilate a boson of the th type, while , and represent one- and two-body interaction strengths satisfying the condition

 νklmn=νlkmn=νklnm=νlknm=νnmlk=νnmkl=νmnlk=νmnkl,

which follows from the exchange symmetry and hermicity. is just an arbitrary energy shift. Let us note that (a part of) the subscript values may also be understood as denoting different components of the same boson species (e.g. different angular-momentum projections). Afterwards it will turn important that one of the bosons can be treated separately from the others, and for this boson we reserve the subscript . We therefore choose the numbering , where will be shown to coincide with the number of classical degrees of freedom associated with the system.

The total number of bosons, , is conserved by the hamiltonian (1). Therefore, we can consider realizations of the many-body system with different values of . However, because of the finiteness of the single-particle Hilbert space (its dimension is ), the increasing number of interacting particles makes the total energy grow faster than linearly with . Namely, the two-body part will grow proportional to , the three-body part (if any) as etc. Thus the energy is not an extensive quantity. To solve this problem, we have to attenuate individual terms by the respective factors, which means that we switch to a modified (extensive) hamiltonian with interaction strengths decreasing with . Expressing then the energy per particle, (an intensive quantity), one has:

 H=ε0+1N∑kϵkb†kbk+1N(N−1)∑k,l,m,nνklmnb†kb†lbmbn+… (2)

(we keep the adjustable energy shift, now denoted as ). Strictly speaking, this hamiltonian represents a different system than the original hamiltonian (1), although via an appropriate readjusting of the interaction strengths in Eq. (2) we can return to the system (1) for each individual value of .

Now, we can introduce self-adjoint coordinate and momentum operators

 qk=1√N(αkb†k+α∗kbk),pk=i√N(βkb†k−β∗kbk), (3)

or, equivalently,

 b†k=√NΓ(β∗kqk−iα∗kpk),bk=√NΓ(βkqk+iαkpk), (4)

with The commutator of and reads as:

 [qk,pl]=iΓNδkl. (5)

We want the commutator to be universal, independent of , so we take and such that represents a constant. For the sake of simplicity we may choose , hence .

The value in the commutator (5) plays the role of the Planck constant . We therefore conclude that in finite bosonic systems, in which the scaling from Eqs. (1) and (2) makes the coordinate-momentum representation (3) suitable, the limit represents just the classical limit. This is rather important finding since in the QPT systems with an infinite single-particle Hilbert space, where the scaling (2) is not employed, the limit represents the transition to the quantum field theory.

It is obvious that if Eq. (4) is substituted to Eq. (2), while replacing by for large , we obtain a coordinate-momentum representation of with no explicit dependence on . Using the commutation rule (5) and neglecting the contraction terms, the hamiltonian can be written in the following simple form:

 Hcl(q,p)=ε0+12∑kϵk(p2k+q2k) + 14∑k,l,m,nνklmn(pkplpmpn+qkqlqmqn) + 12∑k,l,m,nνklmn(plpnqkpm+plpmqkqn−pmpnqkpl).

Note that here we have used the above special choice of coefficients and . In fact, as the contractions were skipped, Eq. (2.1) represents the classical (i.e. ) limit of hamiltonian (2).

The classical hamiltonian (2.1) can be equivalently obtained using condensate states [8, 9, 43, 44, 118, 119, 120, 121, 122, 123]

 |N,c⟩∝[∑kckb†k]N|0⟩,ck=1√2(qk−ipk) (7)

where is the boson vacuum, and taking

 Hcl(q,p)≡⟨N,c|H|N,c⟩⟨N,c|N,c⟩=⟨H⟩|N,c⟩ (8)

in the limit . The condensate states (7) respect the conservation of the total boson number. Alternatively, one may use the Glauber coherent states [124]

 |⟨N⟩,d⟩∝exp[∑kdkb†k]|0⟩, (9)

which do not have a fixed , but allow to vary the average . For , both Glauber and condensate states yield the same results if taking .

It is easy to show that quantum fluctuations measured by the dispersion of the energy per particle in states (7) vanish for asymptotic :

 ⟨H2⟩|N,c⟩−⟨H⟩2|N,c⟩=O(N−1). (10)

This is a consequence of the attenuation of interaction terms in Eq. (2) with an increasing number of bosons. As both condensate and coherent states form overcomplete bases in the bosonic Hilbert space, the above result illustrates the fact that the asymptotic number of bosons indeed represents the classical limit of the present class of systems. In contrast, for an infinite lattice system with a finite range of interactions (not hindered by ) a quantity analogous to (10) behaves as .

There is an additional condition following from the conservation of the total boson number, namely , which is equivalent to the normalization . This constraint can be used to eliminate one of the degrees of freedom, for instance that connected with the boson. Indeed, the absolute value can be calculated from the other values with through the normalization condition, while the phase of can be chosen arbitrary, e.g. . We therefore obtain a system with degrees of freedom restricted by the condition . Furthermore, all coordinates with can be expressed relative to using the transformation , which is (for ) equivalent to setting in Eqs. (7) and (9). The phase space of the new coordinates and the corresponding momenta is unlimited. This procedure, however, implies the appearance of characteristic “form factors” in , namely the factors in the terms with -body interactions.

The above results can be viewed as a consequence of a more sophisticated group theoretical procedure, based on so-called coset spaces [8, 125]. The procedure starts by the identification of a certain subalgebra (called maximum stability subalgebra) of the spectrum generating algebra U(+1) associated with the bosonic system under study (the spectrum generating algebra is formed of bilinear products ). In the present case, the maximum stability subalgebra is taken as U()U(1), where the single separated degree of freedom is the one connected with the boson. The algebraic coherent states [125] associated with the factor algebra U(+1)/[U(f)U(1)] read as

 |N,z⟩∝exp[∑k>0(zkb†kb0+z∗kb†0bk)][b†0]N|0⟩. (11)

Using the Baker-Campbell-Hausdorf formula, one can prove that Eq. (11) transforms into the form (7) with and , where .

In the following, we implicitly use the above reduced set of normalized coordinates and the associated momenta, but skip tildes from the notation. We utilize shorthand symbols and . An important quantity for the analysis of the ground-state phase transitions is the potential energy. It is obtained from the hamiltonian by setting all momenta to zero, hence . Although the original hamiltonian (2.1) with degrees of freedom contains only quadratic and quartic terms in its potential , the elimination of creates also cubic terms. Moreover, the factors make the Taylor expansion of infinite. An explicit formula for the potential will be given below in relation to the interacting boson model (Subsec. 3.2). Here, the whole procedure was described just to elucidate the general method for obtaining that will be needed in the following.

### 2.2 Nonanalytic evolutions with control parameters

In the classical limit, , the ground state of hamiltonian (1) can be identified with the global minimum of the potential energy . Coordinates of the global minimum are and the ground-state energy reads as . Under “normal” conditions, and depend on the hamiltonian parameters and in a smooth, analytic way. However, in some cases—and these are of interest in this review—the changes of both and are nonanalytic for some values of the hamiltonian parameters. Such situations are of course well documented in the literature. There exist two standard approaches which make it possible to treat nonanalytic evolutions in a systematic way: the catastrophe theory and the Landau theory.

#### Catastrophe theory

The catastrophe theory, initiated by Thom in the mid 1970’s, has been broadly advertised among all kinds of scientists and technicians. In short, the theory deals with systems in which “continuous causes” can lead to “discontinuous effects”. Its application in quantum physics was pioneered by Gilmore [10], who showed that the catastrophe theory describes and classifies nonanalytic evolutions of the ground state properties of some many-body systems. The application of the catastrophe theory in the interacting boson model was first outlined by Feng, Gilmore, and Deans [44] and later elaborated by López-Moreno and Castaños [47].

The key idea opening the whole field is the concept of structural instability [126]. It can be easily explained on the potential of the quartic form , which will turn to be very relevant in the forthcoming analyses. But consider at first the harmonic-oscillator potential . Add a small perturbation to it, , with being an infinitesimally small number and a smooth function with all derivatives locally (around ) restricted by a common bound. Although the perturbation distorts the harmonic behavior close to the potential minimum, it is not difficult to see that there is no way how it could change the local topology of the problem, i.e., the number and ordering of the minima and maxima (if any) around . The harmonic potential is structurally stable. The same holds, less trivially, e.g. for a double-well potential , with the boundedness of now being imposed within a broader interval including all local extremes of .

On the other hand, the pure quartic oscillator, , is structurally unstable, since a small perturbation with does change the topology of the problem: the local minimum at becomes a maximum and emits two minima located symmetrically at . If we consider a family of potentials

 V(η;x)=(2η−1)x2+(1−η)x4 (12)

depending on parameter , a structurally unstable pure quartic potential is trapped at in between structurally stable potentials on both sides and .

Let us consider more general families of potentials with variables (denote their number by ) and adjustable parameters (there are of them). It turns out that for all possible forms of such families can be smoothly mapped (the transformation affecting the parameters as well as the variables) onto 13 “canonical” forms [126]. These forms classify all possible catastrophes in low dimension (). Example (12) belongs to the equivalence class called the cusp. Since ground-state phase transitions between spherical and deformed shapes in nuclei and other many-body systems are closely related to the cusp, we will introduce this special type of catastrophe.

The general cusp potential reads as

 Vcusp(a,b;x)=x4+ax2+bx, (13)

so it has and . In the parameter plane the potential has a single minimum except in the cusp-like region

 a<0,|b|≤43√6√(−a)3, (14)

where two local minima exist and one local maximum in between (a bistable form), see Fig. 1. Both minima are degenerate at , so that if changes from negative to positive values the minima swap and the system described by potential (13) undergoes a phase transition. In the region of parameter demarcated by condition (14) the two coexisting minima indicate a phase coexistence interval typical for first-order phase transitions. The limiting parameter values for the bistable form of the potential are called spinodal and antispinodal points.

We assume that the system driven through the phase coexistence region is ideally equilibrated, i.e., dwells at the bottom of the lowest potential minimum. Crossing the critical point (for ) then implies that the rate of change of the system’s energy flips from the value characteristic for one minimum to the value characteristic for the other minimum. The evolution of the ground-state energy has its first derivative discontinuous, which is indeed the defining condition of a first-order phase transition.

Consider now constant and decreasing from positive to negative values. In this case, the single potential minimum present at bifurcates at , forming two degenerate branches characterizing the situation at . The system can choose any of these branches as they are fully equivalent. If doing so, it turns out that the first derivative of the lowest energy varies in a continuous way this time, but the second derivative jumps. This is so-called second-order phase transition. Hence the cusp catastrophe accommodates both basic archetypes of phase transitional evolution in the systems which are of interest in this review.

#### Landau theory

Landau theory of phase transitions was formulated in the late 1930’s [127] as an attempt to develop a general method of analysis for various types of phase transitions in condensed matter physics (especially in crystals). It relies on two basic conditions, namely on (a) the assumption that the free energy is an analytic function of a quantity called order parameter, and on (b) the fact that the expression for the free energy must obey the symmetries of the system. Condition (a) is further strengthened by expressing the free energy as a Taylor series in the order parameter. It is now known that the Landau theory—being essentially the mean-field theory—fails in many systems. However, in the class of models we look at here it holds. The reason is the above-explained coincidence of the and limits in models with finite single-particle Hilbert spaces and properly scaled strengths of interactions.

Hamiltonian (1) has a number of external parameters. We will now assume a one-dimensional smooth path in the multidimensional parameter space, i.e., consider the hamiltonian parameters depending on a single real parameter . A phase-transitional evolution of the ground state (if any) shows up as a nonanalytic change of the ground-state energy at a certain critical point . Since the dependence of the potential on is smooth (as the path is smooth), the nonanalytic evolution of is always connected to a nonanalyticity in the trajectory of the potential minimum. We can write

 discontinuity of dkdηkE0⇔discontinuity of dk−1dηk−1qm⇔kthorder phase transition.

This results from the following sequence of expressions

 ddηE0=(∂∂ηV)qm+(∇V)qm0⋅ddηqm,  d2dη2E0=(∂2∂η2V)qm+(∂∂η∇V)qm⋅ddηqm… (15)

(where the dot represents the scalar product), which can be continued to an arbitrary order. Thus if jumps, so does (first-order transition), the jump of implies the same for (second-order transition) etc. The transition orders introduced here are consistent with the Ehrenfest classification of phase transitions. This classification is not applicable in general (since in real systems some derivatives may diverge), but it holds within the Landau theory.

Now we come to the essence of Landau theory [127, 128]. At zero temperature, the equilibrium free energy coincides with the ground-state energy and the role of thermodynamical variables is taken by the hamiltonian external parameters. The phase of the system can be characterized by a suitably chosen order parameter , which in the present context is a certain function of coordinates, . The free energy can be expressed as a function of , we denote it , and its equilibrium value, obtained by minimization, coincides with the ground-state energy: . Therefore, the above-derived properties of phase transitions of different orders hold also after the replacement .

As will turn out later, the order parameter relevant in our case is given by

 ξ=±√∑k>0q2k. (16)

It represents a radius in the -dimensional configuration space (the coordinate was eliminated in the way described in Subsec. 2.1) with the sign . Let us note that the existence of phases characterized by and is an important ingredient of the general Landau theory. The concrete meaning of the sign in Eq. (16) will be discussed in Subsec. 3.2. The above definition implies that for the system’s ground state is a condensate of bosons, see Eq. (7), while for the ground state is represented by a more complicated mixture of more types of bosons.

We will be mostly interested in transitions between the and phases. If there is a second-order transition of this kind, in its vicinity the order parameter takes arbitrarily small values. Therefore, the free energy can be expressed as a power expansion in ,

 VL(η;ξ)=V0(η)+A(η)ξ2+B(η)ξ3+C(η)ξ4+…, (17)

where the linear term was omitted because of symmetry constraints that apply in our case (see Subsec. 3.2) as well as in Landau’s original context. The condition for the second-order phase transition between and phases at reads as

 A(ηc)=0,B(ηc)=0, (18)

with . The phase is located on the side, while the phase on the side of . The evolution of on the side close to is , hence for a linear dependence on one has . This implies the critical exponent for the order parameter1 equal to . Note that this particular value is specific for the mean-field description of phase transitions of the above type [1] and therefore applies in the finite bosonic models studied here.

On the other hand, the first-order phase transition between and phases takes place if conditions

 VL(ηc;ξ)=VL(ηc;0),ddξVL(ηc;ξ)=0 (19)

are fulfilled for a certain value . Close to the second-order phase transition, where the terms of with and higher can be neglected and , one finds that a simultaneous solution of conditions (19) exists if

 B2(ηc)=4A(ηc)C(ηc). (20)

If higher than quartic terms in the potential (17) are neglected, it can be show that represents the only essential parameter of the potential with and . The remaining three “unimportant” parameters can be associated with an energy shift and two scale factors. Eq. (20) can therefore be written simply as . The phase is located on the side and the phase on the side. The spinodal and antispinodal points, demarcating the phase-coexistence region, are given by and . Let us stress that for an infinite-order potential (17) these numerical results may in general be valid only in a vicinity of the second-order phase transition (although for some infinite-order potentials, including those relevant here, they hold everywhere).

A closer inspection of Eq. (17) reveals that besides the first- and second-order transitions between and phases, there exists also a first-order transition between and phases. The critical point for this transition is given by

 A(ηc)<0,B(ηc)=0; (21)

at this point the equilibrium order parameter apparently changes the sign. The phase exists on the side and the phase on the side.

Therefore, the thermodynamical potential (17) describes a system with three phases: (phase I), (phase II), and (phase III). We see that the first-order phase transition between these phases appear on places determined by a single sharp constraint, either Eq. (20) or (21), while the second-order transition is limited to places determined by two constraints, Eq. (18). The second-order constraints simultaneously fulfill both first-order constraints. Therefore, in a general parameter space of dimension the first-order phase boundaries form two hypersurfaces of dimension and the second-order phase transition lies in their dimensional intersection. This phase structure will be illustrated by the interacting boson model.

There is a relation between the Landau potential (17) with terms up to and the cusp potential (13). Clearly, the potential at the second-order critical point coincides with the germ of the cusp catastrophe. The transition between the two forms can be achieved by a smooth transformation which does not affect the cusp topology. If setting in the Landau potential (the scale), the transformation involves just appropriate parameter-dependent shifts and in the cusp potential. The mapping between topologically equivalent forms (17) and (13) then reads as [103]:

 a=A−38B2,b=12B(14B2−A). (22)

We may therefore conclude that the phase structure sketched in the previous paragraph belongs to the cusp equivalence class.

### 2.3 Hamiltonians with a linear dependence on the control parameter

The classical analysis of the previous subsection needs to be connected with specific quantum signatures. To this end, consider a class of quantum models with a linear dependence on a single real control parameter . The hamiltonian reads as

 H(η)=H0+ηV, (23)

where and are mutually incompatible terms, . For the Landau analysis the linearity means that coefficients in Eq. (17) will be linear functions of , which alone would not be an important achievement. On the quantum level, however, the linearity represents a considerable advantage. In particular, the knowledge of the whole spectrum () at a single value of the control parameter, together with the complete set of instantaneous matrix elements , where is the th eigenstate of , determine all spectral observables for any value of the control parameter.

In the many-body case, the linear ansatz is naturally satisfied if the control parameter represents a weight factor at a certain interaction term of the hamiltonian. Otherwise it can be justified by the Taylor expansion of a general nonlinear hamiltonian around the point of interest. The dimensionless parameter may, in principle, vary within the unlimited domain , but in practice we are always interested in a certain restricted interval . Since we can redefine such that it coincides with and absorb the value in , the form (23) in its full generality can be studied using the constraint . An equivalent expression is then

 H(η)=(1−η)H(0)H0+ηH(1)H0+V. (24)

Quantum phase transitions are likely to appear if the limiting hamiltonians and correspond to two essentially different dynamical modes of the system, e.g. such represented by distinct dynamical symmetries (examples given below).

The evolution of the hamiltonian eigenvectors can be expressed as a unitary transformation in the Hilbert space that naturally conserves traces of all operators. This yields very simple predictions for bulk properties of the spectrum of hamiltonian (23), namely for the average energy and the spread of the spectrum measured by the statistical dispersion . While the spectrum average behaves linearly with , the dispersion is quadratic [98]:

 n¯¯¯¯E = TrH0+ηTrV, (25) n2(ΔE)2 = [nTrH20−Tr2H0]+2η[nTr(H0V)−TrH0TrV]+η2[nTrV2−Tr2V]. (26)

The minimum of Eq. (26) is at

 ηm=−nTr(H0V)−TrH0TrVnTrV2−Tr2V, (27)

where the proximity of energy levels induces rapid structural changes of the hamiltonian eigenfunctions , as can be seen from basic perturbation theory applied to , e.g. from the overlap formula

 |⟨ψi(η)|ψi(η+δη)⟩|2≈1−(δη)2∑j(≠i)|⟨ψi(η)|V|ψj(η)⟩|2[Ei(η)−Ej(η)]2 (28)

with the squared distance of levels in the denominator. On the other hand, for far away from the second term in Eq. (23) totally prevails and the corresponding wave functions approximately coincide with eigenfunctions of . The spectrum just linearly blows up.

The most interesting physics happens around the minimum of dispersion (26). This applies also to ground-state quantum phase transitions driven by . If such a transition exists in the given model, it most likely appears at a critical point that lies somewhere close to . We will assume that our choice of and in Eq. (23) was made so that both and (if any) are contained in the interval .

Let us have a closer look on situations when quantum phase transitions related to the structure of the ground state can typically take place. First, it is clear that since matrix elements of the hamiltonian in an arbitrary fixed basis vary with in a smooth (linear) way, any nonanalyticity of the eigenvalue and eigenvector evolutions can only occur if the Hilbert space dimension increases asymptotically, (which for the bosonic models considered here means ).

Elementary calculation yields the following expressions for the derivatives of the ground-state energy:

 ddηE0(η)=⟨ψ0(η)|V|ψ0(η)⟩,d2dη2E0(η)=−2∑i>0|⟨ψi(η)|V|ψ0(η)⟩|2Ei(η)−E0(η). (29)

Since the second derivative in Eq. (29) cannot be positive (), the ground-state average never increases. A common situation for the ground-state quantum phase transition to appear is when is a semi-positively definite operator or when some constrains do not allow the average to become negative. Then it is likely that the continuously decreasing average incidentally reaches zero at a certain point . If so, represents a critical point of the ground-state evolution. Indeed, at this point the first derivative of the ground-state energy in Eq. (29) gets fixed, for , and the second derivative jumps to zero, for . From Eq. (29) we know that everywhere on the right of the critical point there is for all (including ), and this yields for . In other words, the ground state becomes an eigenstate of with zero eigenvalue.

The situation described above constitutes a second-order ground-state phase transition from a “less symmetric”, , to a “more symmetric”, , phase. The notion of symmetry is invoked here in relation to the spontaneous symmetry breaking: the “less symmetric” form of the ground state usually brakes a certain symmetry that the hamiltonian itself maintains. The ground-state average can be used as a quantum order parameter and related to the classical (mean-field) order parameter introduced in Subsec. 2.2.2.

As will be discussed below, in case of the second-order QPT a nonanalytic (for ) change of this parameter is connected with a singular growth of the level density at and . Indeed, as follows from the Pechukas-Yukawa theory [129, 130], the evolution of levels with for a linear hamiltonian (23) is analogous to one-dimensional dynamics of a 2D Coulomb gas (however, with the product charges also subject to specific variations). To make any of the trajectories nonanalytic, one needs to produce an infinite local growth of “charge” at the corresponding place. The Pechukas-Yukawa approach and its implications for quantum phase transitions will be discussed in Subsec. 7.2.

For the second-order QPT the order parameter changes continuously, but with a discontinuous derivative. In contrast, the first-order QPT involves a discontinuous (for ) jump of the order parameter itself, i.e., the discontinuity of already the first derivative in Eq. (29) at . In terms of the Coulomb gas analogy, this needs locally an infinite “force”, which may be caused by a crossing of a pair of levels (or their sharp anticrossing, indistinguishable from a real crossing). Although such effects are usually accompanied by an infinite growth of the level density, similar to the second-order transition, this is not necessarily the case. Therefore, the mechanisms underlying the first- and second-order transitions (a sharp crossing or anticrossing of two levels and a local singularity of the level density, respectively) may be interrelated, but in general are different.

### 2.4 Simple example: Lipkin model

We would like to sketch here QPT properties of the model introduced by Lipkin, Meshkov and Glick in 1965 [131, 132, 133], which is often referred to as the Lipkin model. This will take us very close to the case of the interacting boson model that will be opened in the next section. One can find extensive literature investigating various properties of the Lipkin model, including its phase transitional behavior, see e.g. Refs. [7, 10, 27, 134, 135, 136, 137, 138].

The model is formulated in terms of pseudospin operators that form the SU(2) algebra. A simplified hamiltonian may be taken for example as follows,

 H′(ζ)=Jz−ζ1N(J++J−)2, (30)

where is a control parameter that should not yet be identified with from Eq. (24), see below, and is related to the spin quantum number denoting the chosen SU(2) representation. There exist various modifications of Eq. (30) with other quadratic combinations of the pseudospin operators in the form containing an additional parameter .

The usual interpretation of the SU(2) algebra involved in Eq. (30) is in terms of fermionic operators:

 Jz=−12Ω∑i=1a†i−ai−+12Ω∑i=1a†i+ai+,J±=Ω∑i=1a†i±ai∓. (31)

Here, and create and annihilate fermions on the lower single-particle level, while and do the same for the upper level. Both levels have capacity and the total number of fermions, , must satisfy the condition . The hamiltonian (30) and its generalized forms conserve the parity

 Π=(−)N+=(−)Jz+j. (32)

Alternatively, assuming , the Lipkin hamiltonian can be interpreted as describing interactions in an infinite array of spin- particles. As the strength of interactions between fermions on both levels—or between individual spins—increases with , the ground state is tempted to switch from the form with (all fermions on the lower level, or all spins down) to a “diamagnetic” form with both average occupation numbers and nonzero. Indeed, this happens at a certain critical point , which will be determined below for a slightly modified hamiltonian by the methods outlined in the previous subsections.

Using the Holstein-Primakoff mapping,

 Jz=b†b−j,J+=b†√2j−b†b,J−=√2j−b†b b, (33)

one can translate the hamiltonian into the bosonic form. However, this type of bosonic representation is not convenient for the present purposes since the total number of bosons is not conserved and since the boson interactions are of unlimited order (because of the square root). A simpler alternative is to employ the Swinger mapping,

 Jz=12(t†t−s†s),J+=t†s,J−=s†t, (34)

where and create and annihilate two types of bosons. Since the parity (32) can be expressed as , where is the -boson number operator, it is natural to consider to be a scalar and a pseudoscalar boson. The original pseudospin algebra of the model is now expressed in terms of the spectrum generating algebra U(2) of the system of and bosons with . We write hamiltonian (30) in a slightly modified form,

 H(η)=(1−η)[−1N(t†s+s†t)(t†s+s†t)]+ηnt=H′(ζ)−ζJz+(1−ζ)N2, (35)

where the control parameter was replaced by . The limits and are characterized by O(2) and U(1) dynamical symmetries, respectively, since the corresponding hamiltonians coincide with the Casimir invariants of the O(2) or U(1) subalgebras of U(2).

The U(1) case () represents a system of noninteracting bosons, with the - and -boson energies set to and , respectively. The term with introduces interactions between bosons of both types. If these interactions are too weak, the ground state for infinite boson numbers will be a pure -boson condensate. At a certain critical interaction strength, however, the ground state wave function flips into a mixed condensate of and bosons. The critical strength can be obtained from the variational analysis with trial states , as described in Subsec. 2.1, which leads to the following expression for the classical potential energy:

 V(η;q)=(5η−4)q2+ηq4(1+q2)2. (36)

Taking into account the Taylor expansion , we see that the potential (36) has the general Landau-like form (17) with the cubic term missing. Consequently, the nonanalytic evolution at (where the quadratic term in the numerator changes its sign) represents a second-order ground state phase transition. For , the potential in Eq. (36) has a minimum at , hence the ground state is indeed the pure condensate, as anticipated. For , there exist two degenerate minima at

 qm=±√5ε8−5ε, (37)

describing the mixed condensates. For , the minima converge to 0 as . The critical exponent for the “order parameter”  is therefore equal to .

The Lipkin hamiltonian can be generalized to get also the first-order phase transitions [44]. The only way to do this (while preserving the two-body character of the model) is to sacrifice the parity conservation. Indeed, as proposed in Ref. [95], modifying hamiltonian (35) to the form

 Hχ(η)=(1−η)[−1N(t†s+s†t+χt†t)(t†s+s†t+χt†t)]+ηnt, (38)

which contains parity-violating terms such as etc., one obtains the potential

 Vχ(η;q)=(5η−4)q2−4χ(1−η)q3+(η+ηχ2−χ2)q4(1+q2)2. (39)

Here, the cubic term is already present for and the ground state first-order phase transition takes place. It turns out that for potential (39) the condition (20) holds exactly regardless of the distance from the second-order critical point. The first-order phase transition therefore appears at

 ηχc=4+χ25+χ2. (40)

In the following section we will see that these results are very close to those obtained within the interacting boson model.

### 2.5 Historical and terminological remarks

As mentioned in Sec. 1, the term quantum phase transition comes from physics of infinite lattice systems of spin-like objects interacting via finite-range interactions. The order-disorder phase transitions in such systems at zero temperature are driven by external control parameters and can be related to zero point motions, i.e. purely quantum fluctuations, of the lattice constituents. The advent of this kind of physics was marked by a pioneering work of Hertz [6] in 1976. At present, the QPT field belongs to one of the most rapidly growing branches of condensed matter physics [11, 12].

On the other hand, the use of the QPT term in the context of models presently studied [7, 8, 9, 10] might seem slightly confusing, since—as we saw—the infinite- limit of such models is just the limit of classical physics. However, some of the models that belong to the category “finite” are indeed very close to those studied in condensed matter physics. This is most evident for the Lipkin model [131, 132, 133], which can be cast as a hamiltonian describing an infinite chain of spin- particles interacting by infinite-range interactions. A seemingly minor difference from the other lattice models—the infinite range of interactions—creates the necessity to damp the interaction constant with increasing and therefore leads to all consequences following from the convergence to the mean-field description with . Various forms of the interacting boson model [33], including the one with and bosons, is then just a natural extension of the Lipkin model from U(2) to higher spectrum generating algebras. As will be shown in the forthcoming sections, this extension results in a considerable enrichment of the phase structure, allowing also the first-order phase transitions to appear [8, 9, 43, 44].

The history of interaction-driven phase transitions, as one may alternatively call such phenomena, is however much longer (some historical remarks can be found in Ref. [83]). To our knowledge the first author who used the term “phase transition” in this context was Thouless in 1961 [25]. It was in connection with what is now known as a “collapse of the random phase approximation” (RPA) in nuclei: at some critical value of the hamiltonian control parameter the RPA phonon frequency (determining the elementary vibrational mode of the system) drops to zero and becomes imaginary beyond this point. This can be considered as a microscopic signature of a sudden structural change of the nuclear ground state.

The Lipkin [131, 132, 133] and related pseudospin models created a new wave of interest in similar phenomena, see e.g. [27]. The unified language for their description (based on coherent states and the catastrophe theory) was developed by Gilmore in the late 1970’s [7, 8]. Note that Gilmore proposed the term “ground-state energy phase transitions”, or simply ground-state phase transitions, which we sometimes adopt. The field continued growing in the 1980’s with the discovery of phase transitions in the -IBM by Dieperink, Scholten, and Iachello [9, 43, 44] and also in herefrom inspired so–called fermion dynamical symmetry model (FDSM) [139, 140, 141, 142, 143]. Since the phases in these models are defined by the equilibrium shape of a nucleus in its ground state, the related QPT-like phenomena are often called shape-phase transitions.

The recent increase of interest in this field comes back to the 1990’s [22, 47, 48, 144, 145]. The topic was reopened by several authors pursuing different goals, mainly an analysis of nuclear structure evolution [22, 48, 50, 51] and the application of so-called quasi dynamical symmetries [144, 145] and critical point symmetries [146, 147, 148, 149, 150]. Symmetry (regardless of its concrete incarnation) seems to be a unifying theme in a great majority of shape-phase transitional studies in nuclear physics. Whereas in the infinite lattice models the quantum phase transition separates ordered and disordered phases of the lattice, in nuclear-related models the transition is usually between two specific dynamical symmetries of the system, i.e. between two different types of order. In this respect, such transitions can be compared to structural phase transitions in solid-state physics.

In studies of quantum shape-phase transitions in nuclei, the interacting boson model, in its various forms, has attracted the major attention. Apart from a comparison with actual nuclear data [9, 46, 48], the IBM soon became an important testing ground for theoretical investigations of general concepts and methods, see e.g. Refs. [47, 52, 53]. Both these aspects are relevant since the IBM with its rich phase structure offers a basic example of a system whose features differ in many respects from the traditional QPT systems studied in the context of solid state physics. Therefore, the model may provide essential hints for deeper understanding of the QPT physics in general.

## 3 The interacting boson approximation

The Interacting Boson Model (IBM) was proposed in 1975 by Iachello and Arima to describe collective excitations of heavy or medium mass atomic nuclei [29, 30, 31]. This model combined ingredients of the two most successful paradigms used in nuclear physics at that time: the shell model and the geometrical (or collective) model [36]. The shell model considers the nucleus as an ensemble of weakly interacting fermions occupying single-particle orbits in the nuclear mean field. Despite a considerable truncation of the model Hilbert space achieved by activating only the nucleons on valence shells, calculations in heavy nuclei away from magic numbers were prohibitively complex. The geometric model attacked the nuclear many-body problem from the other side: Heavy nuclei can in some situations be considered as droplets of a quantum liquid, with elementary excitations identified with highly correlated collective vibrations and rotations. In even-even nuclei, the basic constituents of the model are quadrupole phonons which can be represented as bosons with spin and parity . The collective model has been successful in describing certain classes of nuclei away from closed shells.

The IBM is intermediate between these two complementary approaches in that it connects the bosonic behavior of the geometric model to the fermionic nature of the shell model. This is achieved using the pairing property of short-range residual interactions. Pairwise coupled nucleons (or holes, vacancies left by missing nucleons if the shell is more than half-filled) behave much like bosons. The energetically most likely combination of two identical nucleons coupled by a short-range force is the one with zero total angular momentum. This can be approximated by an boson, while the bifermion combination with angular momentum maps onto a boson.

The original version of the interacting boson model, nowadays abbreviated as IBM-1 [33], is applicable to even–even nuclei. The IBM-1 does not separate bosons connected with proton-proton and neutron-neutron pairs (this is done in an extended version of the model, the IBM-2 [33], which is suitable for the description of isovector collective excitations) and does not consider bosons connected with mixed proton-neutron pairs (these bosons, increasingly relevant as approaching the nuclei, are introduced in more advanced versions of the model, IBM-3 and IBM-4 [32]). The IBM-1 also does not consider single-nucleon excitations and their couplings with nucleon pairs (these are treated in the interacting boson-fermion model, the IBFM [35], which is applicable in odd nuclei). Some modified versions of the IBM also include bosons with other spins and parities, such as (), (), and () bosons [33]. Various IBM extensions and their quantum phase transitions will be discussed in Sec. 6, while here and in the following two sections we will focus on properties of the IBM-1.

### 3.1 Foundations and the algebraic structure

The interacting boson model, including its simplest version IBM-1, benefits from its transparent algebraic formulation. As indicated above, the model building blocks are and bosons that represent phenomenological images of and pairs of valence nucleons or holes and are also closely related to the basic quanta of nuclear collective excitations. Unitary transformations among the six states and , with , generate the Lie group U(6), which is identified with the spectrum generating (dynamical) group of the model. The 36 generators of the associated algebra can be written in the form , where and .

Although the separate boson numbers and are apparently not conserved by the dynamical U(6) algebra, the sum is. It means that collective states of an even-even nucleus with valence nucleons (or valence holes) are mapped to the IBM-1 Hilbert space of bosons. This is in contrast to the quadrupole phonon model, a bosonized version of the geometric model, where bosons directly represent quanta of collective excitations, so that their number varies within one nucleus [151]. For instance, in the latter model the ground state of a spherical nucleus is treated as the vacuum, a state with , while in the IBM the same state coincides with a condensate of bosons with given . In spite of these differences, both models can be connected and rooted in an underlying microscopic treatment [123, 152, 153, 154].

An -boson hamiltonian with one- and two-body interactions that conserves the total boson number and the total angular momentum has the following general form:

 H=E0+ϵdnd+∑l1l2l′1l′2lv(l)l1l2l′1l′2[[b†l1×b†l2](l)×[~bl′1×~bl′2](l)](0)0. (41)

The first term is a constant, which may be included to quantify the nuclear binding energy of the core. The second term represents the relevant one-body contributions (the -boson part can be eliminated using the relation ). The third part corresponds to two-body interaction, the coefficients being related to the interaction reduced matrix elements between normalized two-boson states with total angular momentum . We use the standard notation , it is and , and , where and , respectively, are rank