Excited-state quantum phase transitions in a two-fluid Lipkin model

Excited-state quantum phase transitions in a two-fluid Lipkin model J.E. Garc´ıa-Ramos1, P. P´erez-Fern´andez2, J.M Arias3 1Departamento de Ciencias Integradas, Universidad de Huelva, 21071 Huelva, Spain and Instituto Carlos I de F´ısica Te´orica y Computacional, Universidad de Granada, Fuentenueva s/n, 18071 Granada, Spain 2Departamento de F´ısica Aplicada III, Escuela T´ecnica Superior de Ingenier´ıa, Universidad de Sevilla, Sevilla, Spain 3Departamento de F´ısica At´omica, Molecular y Nuclear, Facultad de F´ısica, Universidad de Sevilla, Apartado 1065, 41080 Sevilla, Spain 1 arXiv:1705.02988v1 [nucl-th] 8 May 2017 Abstract Background: Composed systems have became of great interest in the framework of the ground state quantum phase transitions (QPTs) and many of their properties have been studied in detail. However, in these systems the study of the so called excited-state quantum phase transitions (ESQPTs) have not received so much attention. Purpose: A quantum analysis of the ESQPTs in the two-fluid Lipkin model is presented in this work. The study is performed through the Hamiltonian diagonalization for selected values of the control parameters in order to cover the most interesting regions of the system phase diagram. Method: A Hamiltonian that resembles the consistent-Q Hamiltonian of the interacting boson model (IBM) is diagonalized for selected values of the parameters and properties such as the density of states, the Peres lattices, the nearest-neighbor spacing distribution, and the participation ratio are analyzed. Results: An overview of the spectrum of the two-fluid Lipkin model for selected positions in the phase diagram has been obtained. The location of the excited-state quantum phase transition can be easily singled out with the Peres lattice, with the nearest-neighbor spacing distribution, with Poincar´e sections or with the participation ratio. Conclusions: This study completes the analysis of QPTs for the two-fluid Lipkin model, extending the previous study to excited states. The ESQPT signatures in composed systems behave in the same way as in single ones, although the evidences of their presence can be sometimes blurred. The Peres lattice turns out to be a convenient tool to look into the position of the ESQPT and to define the concept of phase in the excited states realm. PACS numbers: 21.60.Fw, 02.30.Oz, 05.70.Fh, 64.60.F- Keywords: Lipkin model, two-fluid system, excited-state quantum phase transitions 2 I. INTRODUCTION During almost twenty years, quantum phase transitions (QPTs) have been a hot topic in different areas of quantum many-body physics. On one hand, QPTs in Nuclear Physics have been deeply studied [1–3], from both theoretical and experimental points of view. On the other hand, other fields such as Molecular Physics [4, 5], Quantum Optics [6, 7] or Solid State Physics [8] put forward the QPTs studies. The well known thermodynamic phase transitions develop in systems with an infinite number of particles, i.e., in the thermodynamic limit, in this sense they are called classical phase transitions. QPTs are phenomena similar to classical phase transitions but differ in that QPTs take place at zero temperature. In a broad sense, QPTs appear in Hamiltonians that can be split into two parts, each of them presenting a different symmetry. In this situation a simple transitional Hamiltonian can be written as a function of one control parameter that governs the change in the system from one symmetry to the other, H(ξ) = ξ·H(symmetry1) + (1 −ξ)·H(symmetry2).(1) The phase of the system is characterized by a parameter, usually called order parameter, that is zero in one phase and different from zero in the other. A QPT is characterized by a sudden change in the value of the order parameter for a small variation around a particular value, ξc,of the control parameter, ξ. The value ξcwhere the QPT develops is known as the critical point and marks when the system undergoes a structural change from symmetry1to symmetry2. An appealing step forward in the QPT concept is its extension to composed systems, i.e., systems with different species of components. The simplest case is a quantum system with two of such species or fluids. One interesting case is the composed boson-fermion system [9–11], although here we will focus in a two fluid model in which the two species are bosons and are represented by creation and annihilation boson operators that fulfill the usual boson commutation relations. In this framework, it is worth to mention the very first studies in Nuclear Physics of two-fluid systems [12–14], conducted for the proton-neutron interacting boson model. In similar schemes, the bending dynamics of tetratomic molecules has also been studied with a two-fluid bosonic model where each fluid is associated with a bender [15, 16]. Other two-fluid systems, that still today act as landmarks, are the Dicke [17] and the Jaynes-Cumming [18] models for which the two fluids correspond to photons 3 and atoms. Finally, another simple composed model is the two-fluid Lipkin model [19], where the fluids may correspond to two species of atoms or to two vibrational modes. Dicke and Jaynes-Cumming models are algebraically connected with the two-fluid Lipkin model. Indeed, the dynamical algebra of the double Lipkin model is u(2) ⊗u(2) and one can go to Dicke and Jaynes-Cumming models through a contraction from the u(2) Lie algebra to the Heisenberg-Weyl one hw(1) [20]. The aim of this work is to extend the study of the two-fluid Lipkin model, whose QPTs and phase diagram were studied in detail in [21], to the excited states realm, in other words, to study the excited-state quantum phase transitions (ESQPTs) of the model. The term ESQPT was first coined in [22] and studied in detail in [23]. In [23] it is stated that “The infinite level density, moreover, propagates to higher excitation energy ... hence the concept of a continuation of the QPT to excited states”. Consequently, an ESQPT can be understood as the propagation of the QPT to excited states. Moreover, an ESQPT is deeply connected to the existence of a barrier in the potential energy surface and, in particular, with the height of the barrier. The presence of an ESQPT is marked by the presence of a singularity in the density of states, but it is also known to affect the structure of the states. In fact, these change, from being non-symmetric or deformed to symmetric or spherical (or viceversa), when crossing the ESQPT. Another relevant fact of the ESQPT is that it seems to be related to a change from a regular to a chaotic regime, although this point is still an open question. In this manuscript, the study of the phase diagram for the two-fluid Lipkin model is completed by extending the work of Ref. [21] to the excited states. The paper is organized as follows: in Section II the algebraic structure and the model Hamiltonian are outlined. In Section II B the main ingredients of its phase diagram are revised. In Section III the ESQPT concept is introduced and, in particular, the main tools used to study the onset of an ESQPT are commented. In Section IV the main outcome of this work is presented and we study in detail the onset of ESQPTs in some particular cases. Finally, Section V stands for the summary and conclusions. 4 II. THE LIPKIN MODEL AND ITS TWO-FLUID EXTENSION The Lipkin model was proposed in the 1960’s by Lipkin, Meshkov, and Glick [19] as a simple exactly solvable model to check the validity and limitations of different approximation methods used in Nuclear Physics. Since then, the model has been applied to other fields and many examples of its use can be found in the literature. A. Algebraic structure Using a boson representation, the Lipkin model is built in terms of scalar bosons that can occupy two non-degenerated energy levels labeled by sand t. In the case of a single fluid, the building blocks are the creation s†,t†, and annihilation s,t, boson operators. The four possible bilinear products of one creation and one annihilation boson operator generate the u(2) algebra. The next step to obtain the two-fluid Lipkin model is to combine two coupled Lipkin structures. In this model, there are two boson families identified by a subindex, s† 1,t† 1 and s† 2,t† 2, and the corresponding dynamical algebra will be u1(2) ⊗u2(2), whose generators are: s† isi,s† iti,t† isi, and t† iti, for i= 1,2. A detailed description of the u1(2)⊗u2(2) algebraic structure can be found in [24]. Here, we simply summarize some features that will be of interest along this work. Starting from the dynamical algebra u1(2) ⊗u2(2), the possible subalgebras chains are four. However, for us only two of them, in which there is an early coupling of the dynamical algebras into the direct-sum subalgebra u12(2) (or su12(2)), are relevant, u1(2) ⊗u2(2) ⊃u12(2) ⊃u12(1) ↓ ↓ ↓ N1⊗N2[h, h0]nt→basis |N1N2h nti ,(2) where the labels of the irreps verify the following branching rules: h+h0=N1+N2,h≥h0, 1/2(N1+N2+h0−h)≤nt≤1/2(N1+N2+h0−h), and u1(2) ⊗u2(2) ⊃su1(2) ⊗su2(2) ⊃su12(2) ⊃so12(2) ↓ ↓ ↓ ↓ N1⊗N2j1⊗j2j µ →basis |j1j2j µi ,(3) where ji=Ni/2, j= 1/2(N1+N2),1/2(N1+N2)−1, ..., 1/2|N1−N2|,−j≤µ≤j, and j= 1/2(h−h0). |...istands for the basis state in the corresponding dynamical symmetry. 5 The Hamiltonian used in this work is inspired in the consistent-Q-formalism of the interacting boson model [25]. The Hamiltonian can be written as, H=x(nt1+nt2)−1−x N1+N2 Q(y1,y2)·Q(y1,y2)(4) where nti=t† iti,(5) Q(y1,y2)= (Qy1 1+Qy2 2),(6) Qyi i=s† iti+t† isi+yit† iti.(7) The Hamiltonian (4) has three control parameters (x, y1, and y2). The model has two order parameters β1and β2associated to each fluid: βi= 0 values indicate the symmetric or spherical phase for fluid i, while values different from zero characterize non-symmetric or deformed phases. Due to the behaviour of the bosons under parity, the Hamiltonian (4) is, in general, non-parity conserving, except for y1=y2= 0. The Hamiltonian (4) is a mixture of dynamical symmetries of the problem, particularly u12(1) for x= 1, and so12(2) for x= 0 and y1=y2= 0. This form is specially suitable to study QPTs, because one can associate a symmetric (spherical) phase to the first term of the Hamiltonian and a non-symmetric (deformed) shape to the second term. Moreover, depending on the values of y1and y2, different kinds of deformation are generated. For us, it is specially important the case y1=y2, for which the dynamical algebra of the Hamiltonian will be u12(2) (su12(2)) and the states will belong to a single [h, h0] Young tableau (with a well defined value of j). In this case, one can separate the spectrum in families with given j-values. Note that although jhas the properties of an angular momentum, it is not an orbital angular momentum in the sense of Lin the interacting boson model or in the vibron model. It is, indeed, similar to the concept of F-spin in the proton-neutron interacting boson model [26]. B. The phase diagram The combination of numerical calculations with analytical results, as shown in [21], provides the phase diagram depicted in Fig. 1. In that reference an essential order parameter β= (β1+β2)/2 is defined and it is the parameter that characterizes the different phases: 6 FIG. 1: Phase diagram of the consistent-Q like two-fluid Lipkin model. In the diagram the different phases are represented: spherical and deformed, the first order QPT surfaces and the second order QPT line. Moreover the relevant control parameters (x,y= (y1+y2)/2, and y0= (y1−y2)/2) and dynamical symmetries also are shown. The marked points correspond to the cases studied in Section IV (see text). spherical means β= 0, while two different deformations can appear, β > 0 and β < 0. Note that the order parameter βis equivalent to βaas it appears in Ref.[21]. The Hamiltonian (4) can be more conveniently rewritten in terms of control parameters y= (y1+y2)/2, and y0= (y1−y2)/2. In Fig. 1 the phase diagram of the model in the space of coordinates x,y, and y0is presented. There, a first order phase transition surface (in red) separates the symmetric (spherical, β= 0) and non-symmetric (deformed) phases (the one shown here corresponds to β > 0). This first order phase transition region appears in many other models (quantum cusp, interacting boson model, Dicke and Jaynes-Cumming, among others) and essentially stems from the competition between single particle terms that lead to spherical shapes and two-body interactions that lead to deformed configurations. Therefore, this phase transition is related with the evolution and competition of the spherical and the deformed minima (there is a region of coexistence of minima and the critical point is defined 7 by the degeneration of both minima). This surface contains a line (black thick line in the figure) that runs from y=y0= 0 to y= 0, y0= 1 and corresponds to a second order phase transition. This line is, in fact, a triple point where three degenerated minima coexist (one spherical, and two deformed with different deformation). The point y= 0, y0= 1 shows a unique behaviour because presents a divergence [21] in the second derivative of the energy with respect to the control parameter. At this point, the spinodal and the antispinodal lines merge with the first order phase transition surface giving rise to a tricritical point. The phase diagram shown in Fig. 1 can be completed by extending it to negative values of yand y0. In particular the negative yvalues imply deformation β < 0. The vertical surface (in grey), separates deformed phases with different signs in the value of the order parameter β. This is a first order phase transition surface. This phase transition is connected with the existence of two deformed minima, β < 0 and β > 0, that eventually can become degenerated (the grey surface in the figure). Note that in this situation, the potential is symmetric under an appropriate interchange of the two shape variables (β1and β2, order parameters). At each side of the surface two deformed minima coexist but in the side shown in Fig. 1 the absolute minimum corresponds to β > 0 while at the other side the absolute minimum appears for β < 0. This situation is different to what happens at the phase transition surface separating spherical (β= 0) and deformed (β > 0) phases. In this case, the system transits from a single spherical minimum, β= 0, to a deformed minimum (β > 0) through a coexistence region. In the interacting boson model the first situation corresponds to the QPT between the SU(3) and the SU(3) limits [27], while the second corresponds to the QPT appearing when passing from the U(5) limit to the SU(3) −O(6) line [26]. III. EXCITED-STATE QUANTUM PHASE TRANSITIONS In many-body quantum systems the presence of a ground state QPT can give rise to an ESQPT when using the excitation energy as a control parameter [23]. Once in the deformed phase, keeping the control parameters fixed, one can go up in energy and look into a magnitude that marks the presence of a quantum phase transition in the excited states. This magnitude can be the density of states which is expected to have some kind of singularity when reaching the energy at which an ESQPT develops. In particular, this is the case for the one-fluid Lipkin model, where the QPT for the ground state seems to 8 “propagate” to the excited states. An ESQPT associated to second order QPT is defined as a singularity in the density of states or in one of its derivatives. The kind of singularity depends on the number of degrees of freedom of the system in the semiclassical limit [28, 29]. In particular, the Lipkin model, which has a single degree of freedom, presents a λsingularity in the density of states at the excitation energy corresponding to the ESQPT. The Dicke model, with two degrees of freedom, presents at the ESQPT critical energy a discontinuity in the first derivative of the density of states [28]. The sudden increase in the density of states is related with the presence of a maximum in the potential energy surface of the system. Starting from the bottom of the potential the states bunch up when reaching the maximum of the potential giving rise to an increase of the density of states. On the other hand, when the QPT is of first order, besides the previous behaviour, linked to the existence of a maximum, the presence of a new family of states related with the existence of an extra local minimum induces a finite increase in the density of states and, therefore, a discontinuity in the density of states. The energy at which the second family of states appears in the spectrum corresponds to the energy of the second minimum. To explain more in detail the connection between the presence of a maximum in the potential energy surface and the onset of an ESQPT, we present in Fig. 2 two calculations for large number of bosons, N= 1000, evaluating the density of states for the single Lipkin Hamiltonian. In panel (a) we perform a calculation for a Hamiltonian in the deformed phase. The Hamiltonian represents a situation in which the ground state is deformed but comes from the evolution of a spherical ground state through a second order phase transition, this means that there is a maximum at zero deformation (see inset of Fig. 2a). Note that there are, in fact, two degenerated minima that give rise to degenerated doublets below the ESQPT. The existence of the maximum induces many states bunching together in the spectrum, leading, therefore, to a sudden increase (a singularity in the thermodynamic limit) in the density of states at the energy of the maximum (this is seen at zero energy in panel (a)). In order to show the difference with the case of a first order phase transition, in panel (b) a calculation of the density of levels is presented for a Hamiltonian owning a deformed ground state that now comes from the evolution of a spherical ground state through a first order phase transition. This means that at the QPT there is coexistence of spherical and deformed minima. This fact has as a consequence that far from the QPT, in the deformed phase, the system presents a local deformed minimum too, being both deformed minima 9 -40 -20 0 20 40 60 Energy (arbitrary units) 0 0.5 1 1.5 2 2.5 3 Density of states j=70 j=65 j=60 j=55 j=50 a) -40 -20 0 20 40 60 Energy (arbitrary units) 0 50 100 Density of states b) FIG. 3: Density of states as a function of the excitation energy for a Hamiltonian with parameters y= 0, y0= 0, and x= 1/2 and a number of bosons N1=N2= 70. a) For selected values of j, b) for all the states without distinction in j. but taking into account what is observed in panel (a), one can notice that such a regularity is associated with the presence of jas an extra quantum number. The main feature of each set of points, corresponding to given j−values, is the sharp dip observed at energies around E≈0, with a shift in the dip position towards higher energies as jdecreases, in a similar way to Fig. 3a. Therefore, Fig. 4b is simply the superposition of different curves with sharp dips that move from E≈0 to E≈30. Both below and above the ESQPT, the Peres lattice shows an ordered pattern, pointing to a non-chaotic behaviour. Note that in this case, we do not analyze the NNSD because we have to separate in sets with the same value of jand, therefore, the number of states will be too low to calculate a reliable value of η(8). However, one can calculate the Poincar´e sections to study the regularity and chaos interplay. In panels (c), (d), (e), and (f) of Fig. 4 we depict the Poincar´e sections for energies E=−20, E= 0, E= 30, and E= 50, respectively. Because of the underlying U12(2) symmetry, all the cases correspond to a regular regime. In Fig. 5 we plot the value of the participation ratio as a function of the excitation energy. Once more, in panel (a) we separate in families with given j−values (different colors) and in panel (b) we plot all the states. In each family depicted in panel (a) one can see how the participation ratio starts increasing and reaches a maximum, then decreases with a 16 -40 -20 0 20 40 60 E (arbitrary units) 0 0.2 0.4 0.6 0.8 1 <nt1 >/N1 j=70 j=65 j=60 j=55 j=50 a) -40 -20 0 20 40 60 Energy (arbitrary units) 0 0.2 0.4 0.6 0.8 1 <nt1 >/N1 b) c) d) e) f) q c) e) d) f) 1 p 1 FIG. 4: Peres lattice for hnt1i N1versus the excitation energy for the same Hamiltonian and parameters as Fig. 3. a) For selected values of j, b) for all the states without distinction in j. Poincar´e sections for energies E=−10, E= 0, E= 20, and E= 50, in panels c), d), e), and f), respectively (energies marked in panel (b)). sharp minimum, then shows a new maximum, and finally ends with a minimum. This behaviour was already described in [34] and simply reflects the strong “localization” of the wave function at the ESQPT. Once more, the position of the minimum moves towards higher energies for decreasing j−values as it is shown in Figs. 3 and 4. Panel (b) corresponds to the superposition of the previously described curves which leads to the presence of certain parabolic curves somehow blurred. The set of well defined lines in the right bottom part of Fig. 5b corresponds to curves with small values of j. The second selected case of interest corresponds to parameters y= 1, y0=0(y1= 1, y2= 1), this is a line in the base of the phase diagram going from u12(1) to the deformed region crossing the line of first order phase transition which is located at xc≈4/5 (point B 17 -40 -20 0 20 40 60 E 0 0.02 0.04 0.06 0.08 0.1 0.12 Participation ratio j=70 j=65 J=60 j=55 j=50 a) -40 -20 0 20 40 60 Energy (arbitrary units) 0 0.02 0.04 0.06 0.08 0.1 0.12 Participation ratio b) FIG. 5: Normalized participation ratio as a function of the excitation energy for the same Hamiltonian and parameters as in Fig. 3. a) For selected values of j, b) for all the states without distinction in j. in Fig. 1). In particular we have selected the point x= 1/2. For this selection of the control parameters, the system ground state is deformed and since N1=N2, the indexes 1 and 2 can be interchanged, u12(2) is the dynamical algebra of the Hamiltonian and jis a good quantum number. Therefore, we are in a similar situation to the case previously discussed but now, besides the ESQPT, a set of states associated with the second (local) minimum of the potential energy surface will appear. We start with the analysis of the density of states. In Fig. 6, the density of states are plotted versus the excitation energy. In panel (a) each color line corresponds to a given j−value, while in panel (b) the total density of states is plotted independently of the j−values. In panel (a), in addition to the peak close to zero that is the precursor of the lambda divergence, a previous finite discontinuity is observed very close in energy (negative). In panel (b), a relatively broad peak is observed above zero energy, but nothing qualitatively different to Fig. 3b which corresponds to an ESQPT without a local minimum in the potential energy surface. In Fig. 7 we present the Peres lattice for hnt1i N1versus the excitation energy for given j−values (different colors) in panel (a), or for the whole set of states in panel (b). Here, narrow dips are observed in panel (a) around the energy of the ESQPT. The main difference 18 -150 -100 -50 050 E (arbitrary units) 0 0.5 1 1.5 2 Density of states j=70 j=65 j=60 j=55 j=50 a) -150 -100 -50 050 E (arbitrary units) 0 20 40 60 80 100 Density of states b) FIG. 6: Density of states as a function of the excitation energy for a Hamiltonian with parameters y= 1, y0= 0, and x= 1/2 and a number of bosons N1=N2= 70. a) For selected values of j, b) for all the states without distinction in j. with respect to the previous case is the appearance of a second family of states (see lowest part of Fig. 7 in both panels) related with the local minimum of the potential energy surface. Note that also in this case, we do not analyze the NNSD because we have to separate in sets with the same j−value and, therefore, the states will be too few to calculate the value of η. As in previous case, one can calculate the Poincar´e sections to study the regularity and chaos interplay. In panels (c), (d), (e), and (f) of Fig. 7 we depict the Poincar´e sections for energies E=−10, E= 0, E= 20, and E= 50, respectively. Because of the underlying U12(2) symmetry, all the cases correspond to a regular regime, too. In Fig. 8 we depict the participation ratio value as a function of the excitation energy, for selected particular j−values (different colors) in panel (a), and for the whole set of states in panel (b). In panel (a) each family of states has a narrow minimum around zero energy with broad maxima at right and left. Here, for each j−family a second set of states, related with the local maximum of the energy surface, appears just below the energy of the ESQPT, but it can be almost unnoticed. It is worth to mention that even in panel (b) the position of the ESQPT critical energy is clearly marked with a relatively narrow dip. Till now we have only studied cases where u12(2) was the dynamical algebra of the Hamiltonian and, therefore, they essentially correspond to the superposition of several one- 19 -150 -100 -50 050 E (arbitrary units) 0 0.2 0.4 0.6 0.8 <nt1 >/N1 j=70 j=65 j=60 j=55 j=50 a) -150 -100 -50 050 E (arbitrary units) 0 0.2 0.4 0.6 0.8 <nt1 >/N1 b) c) d) e) f) q c) e) d) f) 1 p 1 FIG. 7: Peres lattice for hnt1i N1versus the excitation energy for the same Hamiltonian and parameters than in Fig. 6. a) For selected values of j, b) for all the states without distinction in j. Poincar´e sections for energies E=−10, E= 0, E= 20, and E= 50, in panels c), d), e), and f), respectively (energies marked in panel (b)). fluid Lipkin systems with 2jnumber of bosons. Now we will move to the more interesting y16=y2cases, where jis no longer a good quantum number. The already studied cases have taught us how the patterns observed in Figs. 3b, 4b, 5b, 6b, 7b, and 8b are a consequence of the superposition of lines corresponding to different j−values, in other words, a consequence of the underlying u12(2) symmetry. Next case of interest corresponds to the parameter values y= 1/4, y0= 3/4 (y1= 1, y2=−1/2) which is a point in the line from spherical u12(1) to the deformed region but not in the base of the phase diagram, and consequently does not preserve the u12(2) symmetry. This line crosses the first order surface located around xc≈4/5. We have selected the point x= 1/2 which corresponds to a deformed ground state (point C in Fig. 1). In this case, as 20 -150 -100 -50 050 E (arbitrary units) 0 0.05 0.1 0.15 0.2 Participation ratio j=70 j=65 j=60 j=55 j=50 a) -150 -100 -50 050 E (arbitrary units) 0 0.05 0.1 0.15 0.2 Participation ratio b) FIG. 8: Participation ratio as a function of the excitation energy for the same Hamiltonian and parameters as in Fig. 6. a) For selected j−values, b) for all the states without distinction in j. already told, it is no longer possible to separate the states in terms of j. The potential energy surface of this Hamiltonian has a global deformed minimum and another local deformed one separated by a maximum. In Fig. 9a, the density of states is plotted as a function of the excitation energy. It can be observed that this quantity is not marking correctly the presence of the ESQPT at zero energy, as already learnt from the preceding calculations. It is worth noting that, although this case presents a second family of states associated to the local minimum starting at E≈ −40, nothing is observed in the density of states. In Fig. 9b the Peres lattice for hnt1i/N1is depicted as a function of the excitation energy. This quantity shows three clear regions, two on them at energies below the ESQPT critical energy (zero energy) and the third one above. The two lowest families of points correspond to states that are located in the well of the global deformed minimum (hnt1i/N1≈0.6) and in the well of the local one (hnt1i/N1≈0.2), also deformed. These two regions show a very regular pattern. The third region is above E≈0 and presents a relatively disordered structure, except in particular regions, as is 0.8<hnt1i/N1<1. The Peres lattice is clearly showing the existence of two different deformed phases, non-symmetric, with a more regular behaviour below zero energy and a symmetric phase with certain degree of chaoticity above the energy of the ESQPT. In panel (b) the value of η(red line), representing the NNSD, as a function of the energy is plotted too. This quantity shows a sudden decrease at the energy of the ESQPT, therefore pointing to a spectrum with a more regular behaviour 21 below the ESQPT and more chaotic above. However, even above the ESQPT, there is a regular region at E≈50 with hnt1i/N1≈0.7. This behaviour has been already observed in other models such as the Bose-Hubbard Hamiltonian [35]. Note that we do not reach the whole range of energies because we exclude the 10% of states with lowest and highest energies to calculate η. In Fig. 9c it is depicted the participation ratio versus the excitation energy. Here, one can note a rather different structure for energies below and above the critical energy of the ESQPT. For energies below zero, one can see two sets of inverted parabolas, with a minimum around zero. The two sets correspond to the two branches already seen in Fig. 9b. For energies above the ESQPT a single inverted parabola, though rather blurred, is observed. This region presents a behaviour similar to the one obtained with u12(2) conserving Hamiltonians. Finally, panels (d), (e), (f) and (g) correspond to Poincar´e sections with energies E=−20, E= 0, E= 30, and E= 50, respectively. These figures confirm the latter statements, i.e., that below the ESQPT energy a regular behaviour exits, as panel (d) confirms, at the ESQPT energy chaotic orbits star to appear (as shown in panel (e)), and above the ESQPT energy, regions with chaotic (see panel (f)) and partially regular motion coexist (see panel (g)). The three last cases correspond to Hamiltonians with y= 0, i.e., with y1=−y2, and, therefore, located on the gray vertical plane of the phase diagram. As we proved in [21], for values of y0<1 a second order QPT appears at xc= 4/5, for y0= 1 the QPT shows a divergence in d2E/dx2also at x= 4/5, but for y0>1 the QPT becomes of first order. Thus, next we try to disentangle whether or not there is a different qualitative behaviour between these three situations in terms of analyses of density of states, Peres lattice, Poincar´e section, and participation ratio. In Fig. 10 the results for the case of y0= 1/2 are presented, in Fig. 11 the case of y0= 1 is studied, and, finally, in Fig. 12 the case of y0= 3/2 is analyzed. In Fig. 10 we consider the case y= 0, y0= 1/2 and x= 1/2 (point D in Fig. 1). This corresponds to a deformed ground state in a line that changing xpresents a second order QPT at xc= 4/5. The energy surface has two deformed degenerate minima (at βand −β) separated by a spherical maximum. In panel (a) the value of the density of states is presented as a function of the excitation energy. A relatively broad peak is observed with a maximum well above the energy of the ESQPT. In panel (b) the Peres lattice for hnt1i/N1as a function of excitation energy is presented. Note that because of the presence of degenerated doublets, it is needed to include a tiny perturbation in the Hamiltonian for slightly break 22 -60 -40 -20 0 20 40 60 E (arbitrary units) 0 20 40 60 80 100 Density of states a) -60 -40 -20 0 20 40 60 E (arbitrary units) 0 0.2 0.4 0.6 0.8 1 <nt1 >/N1; η b) d) e) f) g) -60 -40 -20 0 20 40 60 E (arbitrary units) 0 0.05 0.1 0.15 0.2 0.25 Participation ratio c) q d) f) e) g) 1 p 1 FIG. 9: Different observables as a function of the excitation energy for a Hamiltonian with parameters y= 1/4, y0= 3/4, and x= 1/2 and a number of bosons N1=N2= 70: a) Density of states, b) Peres lattice (black points) for hnt1i N1and η−value (red solid curve), c) participation ratio, d), e), f), and g) Poincar´e sections for energies E=−20, E= 0, E= 30, and E= 50, respectively (energies marked in panel (b)). the degeneracy and avoid random linear combinations of states. Two branches of points are clearly seen for energies below the ESQPT, one centered around hnt1i/N1≈0.3 and the other around hnt1i/N1≈0.5. These two branches correspond to the two degenerated and symmetric minima for which hnt1iand hnt2iare interchanged. In both cases a clear ordered pattern is observed. Note that for each energy there exist two degenerated points, one in each branch. Above the ESQPT the order is lost and the points are located in a more or less random way around a straight line. As in previous cases, the ESQPT separates the regular and the chaotic zones. In this sense, the ESQPT separates two “shapes/phases” of the system. This is confirmed through the calculation of the NNSD η−value, which shows a sudden decrease in its value at zero energy, though with several oscillations, therefore having a more regular behaviour below the ESQPT and more chaotic above. However, even above 23 -40 -20 0 20 40 60 E (arbitrary units) 0 50 100 Density of states a) -40 -20 0 20 40 60 E (arbitrary units) 0 0.2 0.4 0.6 0.8 1 <nt1 >/N1; η b) d) e) f) g) -40 -20 0 20 40 60 E (arbitrary units) 0 0.05 0.1 0.15 0.2 Participation ratio c) q d) f) e) g) 1 p 1 FIG. 10: Same as in Fig. 9 but for y= 0, y0= 1/2, and x= 1/2. Panels d), e), f), and g) are Poincar´e sections for energies E=−10, E= 0, E= 20, and E= 50. the ESQPT, there is a more regular region at E≈50 with hnt1i/N1≈0.8. Panels (d), (e), (f) and (g), which correspond to Poincar´e sections with energies E=−10, E= 0, E= 20, and E= 50, respectively, confirm the above findings: below the ESQPT energy a regular behaviour exits (panel (d)), at the ESQPT energy chaotic orbits star to appear (as shown in panel (e)), and above the ESQPT energy, regions with chaotic (panel (f)) and partially regular motion coexist (see panel (g)). In panel (c) it is depicted the participation ratio versus the excitation energy. Below the ESQPT, the points (doubly degenerated) describe well separated inverted parabolas with a minimum at zero energy. For energies above the ESQPT, once more, the points define a very blurred parabola. Next case to be analyzed is y0= 1 which is probably the most exotic one because the associated QPT at xc= 4/5 presents a divergence in d2E/dx2(point E in Fig. 1). However, as we will explain, in fact, nothing special occurs in the spectrum. In Fig. 11a we depict the density of states and it looks like the one presented in Fig. 10a. In Fig. 11b we present the Peres lattice and, once more, we have two branches for energies below the ESQPT, one 24 -60 -40 -20 0 20 40 60 E (arbitrary units) 0 20 40 60 80 100 Density of states a) -60 -40 -20 0 20 40 60 E (arbitrary units) 0 0.2 0.4 0.6 0.8 1 <nt1 >/N1; η b) d) e) f) g) -60 -40 -20 0 20 40 60 E (arbitrary units) 0 0.05 0.1 0.15 0.2 Participation ratio c) d) f) e) g) 1 p q1 FIG. 11: Same as in Fig. 9 but for y= 0, y0= 1, and x= 1/2. Panels d), e), f), and g) are Poincar´e sections for energies E=−10, E= 0, E= 30, and E= 50. centered around hnt1i/N1≈0.2 and the other around hnt1i/N1≈0.6. Note that for each energy there exist two degenerated points. Above the ESQPT we obtain a cloud of points scattered around a straight line. Here too, the NNSD η−value is plotted and once more it presents a sudden lowering at the ESQPT energy, therefore pointing to a more regular spectrum below the ESQPT, while more chaotic above. Note that to calculate ηbelow the energy of the ESQPT, we only consider states in one of the branches, otherwise the results are wrong. The decreasing occurs in two steps, the first at zero energy and the second at E≈20. This fact is a consequence of the coexistence of intruder and regular states in the same energy region. In both regions several oscillations are observed. 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