Excited-state Quantum Phase Transitions in the Anharmonic Lipkin-Meshkov-Glick Model: Static Aspects

PHYSICAL REVIEW E 106, 044125 (2022) Excited-state quantum phase transitions in the anharmonic Lipkin-Meshkov-Glick model: Static aspects J. Gamito ,1J. Khalouf-Rivera ,2J. M. Arias ,1,3P. Pérez-Fernández ,3,4and F. Pérez-Bernal 2,3 1Departamento de Física Atómica, Molecular y Nuclear, Facultad de Física, Universidad de Sevilla, Apartado 1065, 41080 Sevilla, Spain 2Departamento de Ciencias Integradas y Centro de Estudios Avanzados en Física, Matemáticas y Computación, Universidad de Huelva, Huelva 21071, Spain 3Instituto Carlos I de Física Teórica y Computacional, Universidad de Granada, Fuentenueva s/n, 18071 Granada, Spain 4Departamento de Física Aplicada III, Escuela Técnica Superior de Ingeniería, Universidad de Sevilla, 41092 Sevilla, Spain (Received 10 March 2022; revised 20 July 2022; accepted 15 August 2022; published 17 October 2022) The basic Lipkin-Meshkov-Glick model displays a second-order ground-state quantum phase transition and an excited-state quantum phase transition (ESQPT). The inclusion of an anharmonic term in the Hamiltonian implies a second ESQPT of a different nature. We characterize this ESQPT using the mean field limit of the model. The alternative ESQPT, associated with the changes in the boundary of the finite Hilbert space of the system, can be properly described using the order parameter of the ground-state quantum phase transition, the energy gap between adjacent states, the participation ratio, and the quantum fidelity susceptibility. DOI: 10.1103/PhysRevE.106.044125 I. INTRODUCTION Ground-state quantum phase transitions (QPTs), zerotemperature transitions that are triggered by quantum fluctuations instead of thermal ones, have been in the limelight in recent years due to their deep implications in the understanding of many-body quantum systems [1]. In these transitions the ground state of the system undergoes abrupt and qualitative changes when one or several control parameters in the Hamiltonian straddle a critical value. Since the seminal works of Gilmore and co-workers [2–4], there have been numerous works characterizing QPTs in two-level quantum systems of different dimensionality used to model nuclear and molecular systems, such as the interacting boson model (IBM) or the vibron model (see Refs. [5–7] and references therein). A full classification of ground-state QPTs in twolevel models with different dimensionality can be found in Ref. [8]. The study of QPTs has been extended to the realm of excited states, introducing the concept of excited-state quantum phase transition (ESQPT) which implies a nonanalyticity in the density of states and the energy level flow which, in most cases, is associated with a ground-state QPT and the existence of a critical point in the energy functional obtained from the classical or mean field limit of the system [9–11]. The nature of the nonanalyticity in the energy level density in nondegenerate stationary points of systems with neffective degrees of freedom is such that the order of the derivative of the level density that is nonanalytic is n−1[11–15]. Excited-state quantum phase transitions have received a great deal of attention in many systems, fostered by detection of their precursors in molecular spectra [16–19], superconducting microwave billiards [20], and spinor condensates [21]. In the last case, some promising developments have been recently published [22,23]. Reference [24]isareview on different aspects of ESQPTs with an extensive reference list. A system where ESQPTs have been studied is the twodimensional limit of the vibron model (2DVM), a two-level model built upon the bilinear products of two Cartesian operators and a scalar bosonic operator [25,26]. This model has been used to reproduce bending spectra of molecules, as its two dynamical symmetries can be associated with the bending degrees of freedom for linear and bent molecular species, as well as the interesting situations that lie in between these two limiting cases [27,28]. The basic 2DVM Hamiltonian, including only a number and a pairing operator, presents a second-order ground-state QPT and an associated ESQPT that can be linked to the effect over excited energy levels of the barrier to linearity in nonrigid molecular systems [26]. In fact, most experimental ESQPT signatures that were identified in the bending degrees of freedom of nonrigid molecules are associated with this ESQPT [16,17,19]. The obtention of results of spectroscopic quality in the modeling of molecular spectra implies the explicit inclusion of anharmonic terms in the 2DVM Hamiltonian. In Ref. [29] it was shown that the inclusion of anharmonicity in the Hamiltonian results in a second ESQPT in the broken-symmetry phase, which is where the typical spectroscopic signatures of nonrigid molecules are found. More recently, in the pursuit of the description of the transition state in isomerization reactions [18], it has been found that the second ESQPT can also be present in the symmetric phase for sufficiently large values of the anharmonicity and it is not associated with the ground-state QPT but it stems from changes in the phase-space boundary of the 2DVM’s finite-dimensional Hilbert space [30]. A particular kind of such ESQPTs has been already studied in matter-radiation interaction models with two degrees of 2470-0045/2022/106(4)/044125(15) 044125-1 ©2022 American Physical Society J. GAMITO et al. PHYSICAL REVIEW E 106, 044125 (2022) freedom (Dicke and Tavis-Cummings models). In this case, the authors referred to these ESQPTs happening at the boundary of the model space as static ESQPTs, due to their lack of effect on the system dynamics. In this way, they can be distinguished from the more usual ESQPTs, which the authors refer to as dynamic ESQPTs [31]. More information on ESQPTs associated with the finiteness of the system Hilbert space can be found in Refs. [15,24]. Our aim is to extend the results obtained for the 2DVM [29,30] to the Lipkin-Meshkov-Glick (LMG) model. This model was introduced as a toy model for interacting fermions in nuclear physics, to help assess the performance of different approximations used in the study of nuclear structure [32–34]. It can be mapped to a set of spins with a long-range all-to-all interaction, and from their initial purpose it was later used in many different contexts. In particular, it was extensively used for the study of QPTs [35–45] and ESQPTs [46–55]. The general LMG model presents first-, second-, and third-order ground-state QPTs [43], something that has attracted the interest of researchers. In a basic formulation, with a Hamiltonian composed of two operators that can be mapped to a number and a pairing operator, the model has a second-order ground-state QPT and its associated ESQPT. A particular realization of the LMG model has a classical limit energy functional that is equal to classical limit of the IBM and it has been used as a tool to shed light upon QPTs and ESQPTs in this model [56–58]. It is also a toy model that can be applied in quantum computing [59–61]. The interest in the LMG model was further fostered by the achievement of different experimental realizations, using molecular magnets [62], optical cavities [63], Bose-Einstein condensates [64], nuclear magnetic resonance systems [65], trapped atoms [66–68], and cold atoms [69]. The purpose of the present article is to explore how the LMG model is modified, including in the model Hamiltonian the anharmonic two-body term ˆn(ˆn+1) with a negative control parameter α. We perform a mean field analysis of the anharmonic LMG model, characterize ground-state QPT in this case, and perform a study of the two ESQPTs that appear in the model making use of different quantities such as the energy gap between adjacent levels, the ground-state QPT order parameter, the participation ratio, and the quantum fidelity susceptibility. This paper will be followed by another one that focuses on the effect of the two ESQPTs on dynamical properties of the model [70]. The present paper is organized as follows. In Sec. II we introduce the anharmonic LMG model, review its algebraic structure, provide its Hamiltonian matrix elements, and study the model mean field or thermodynamic limit using the coherent state formalism. Section III includes a description of the different quantities used to characterize the anharmonic LMG ESQPTs and the results obtained for different anharmonicity and system-size values. We summarize in Sec. IV. II. MODEL The LMG model has recently been attracting increasing attention after the experimental realization of one-dimensional spin-1 2lattices with a variable interaction range [66,71]. Using Pauli spin matrices σi,β , with i=1,2,...,Nand β=x,y,z, the system Hamiltonian is ˆ H(a)=B N  i=1 σi,z+ N  i<j=1 K |i−j|aσi,xσj,x,(1) whereweassume¯h=1, Bis the amplitude of an external magnetic field, Kis the energy scale of the interactions between different spin sites, and the parameter acontrols the interaction range. For a=0 the interaction range is infinite and the Hamiltonian (1) is mapped to the LMG model [32–34] ˆ H=ˆ H(a=0) =(1 −ξ)(S+ˆ Sz)+2ξ SS2−ˆ S2 x,(2) where we introduce collective spin operators ˆ Sβ= 1 2N i=1σi,β for β=x,y,z, add a constant term 2BS −KS(2S−1), and define a single control parameter ξ, making 2B=1−ξand SK =−ξ. The control parameter is defined in the range ξ∈[0,1], driving the system from one phase to the other one. Indeed, from an algebraic point of view, the LMG Hamiltonian given by Eq. (2) presents a u(2) algebraic structure, with two limiting dynamical symmetries: u(2) ⊃u(1) and u(2) ⊃so(2). Each dynamical symmetry is associated with a different phase. For ξ=1, the Hamiltonian is diagonal in the basis associated with the so(2) subalgebra and this is known as the deformed or broken-symmetry phase; in contrast, for ξ=0 the Hamiltonian is diagonal in the u(1) subalgebra basis and it is in the normal or symmetric phase [72]. The LMG model Hamiltonian (2) experiences a second-order ground-state QPT at the critical value of the control parameter ξc=0.2[43]. Inspired by the work in Refs. [29,30], we include in the Hamiltonian (2) an anharmonic term, using a second-order operator on ˆ Sz, ˆ Hanh =(1 −ξ)(S+ˆ Sz)+α 2S(S+ˆ Sz)(S+ˆ Sz+1) +2ξ SS2−ˆ S2 x.(3) The new Hamiltonian still depends on the control parameter ξ, which drives the system between phases, but it also depends on a second control parameter α.Forα=0, we recover the original Hamiltonian (2), and for αvalues different from zero, the ξ=0 limit is transformed from a truncated onedimensional harmonic oscillator to an anharmonic oscillator. That is the reason why we refer to the Hamiltonian (3)asthe anharmonic LMG (ALMG) model. It is worth noting that the limit so(2) is no longer recovered for ξ=1, unless α=0. In Ref. [73], a preliminary study of an anharmonic LMG model using only operators diagonal in the u(1) basis was carried out. The Hilbert space for the Hamiltonian (1) has dimension 2N, but in the long-range interaction (3) there is a drastic reduction in the dimension of the Hilbert space due to the conservation of the total spin [ ˆ S2,ˆ H]=0. We focus on the sector that corresponds to the maximum angular momentum S=N/2, with a Hilbert space dimension N+1. In the ALMG model, as in the original LMG model, there are two bases available to carry out the calculations, one 044125-2 EXCITED-STATE QUANTUM PHASE TRANSITIONS IN … PHYSICAL REVIEW E 106, 044125 (2022) defined by the u(2) ⊃u(1) dynamical algebra, |S,Mz, where Mz=−S,...,0,...,Sis the projection of the total spin Sin the zdirection. The matrix elements of the Hamiltonian (3)in the u(1) basis are S,M z|ˆ Hanh|S,Mz=(1 −ξ)(S+Mz)+ξ 2S[4S2−(S−Mz)(S+Mz+1) −(S+Mz)(S−Mz+1)] +α 2S+1+2+1 SMz+M2 z sδM z,Mz −ξ 2S(S−Mz)(S−Mz−1)(S+Mz+2)(S+Mz+1)δM z,Mz+2.(4) As can be easily seen from the matrix elements in Eq. (4), only states with M z=Mzor M z=Mz±2 are connected; hence the ALMG Hamiltonian (3) conserves parity (−1)S+Mz and the Hamiltonian matrix is split into two blocks, one for positive or even parity and dimension S+1 and the other for negative or odd parity with dimension S. The second basis is associated with the u(2) ⊃so(2) symmetry |S,Mx, and in this case it is the projection of the spin in the xdirection, the second quantum label in the basis. The matrix elements in this case can be deduced from the previous ones once the system is rotated [72]. We depict in Fig. 1the correlation energy diagram for the ALMG Hamiltonian (3) with a system size N=2S=120 and α=0 and −0.6inFigs.1(a) and 1(b), respectively. In both cases, we plot the normalized excitation energy as a function of the control parameter ξ. Both positive- and negative-parity states are included in the figure; positiveparity energy levels are plotted with blue solid lines and odd-parity states with red dashed lines. Figure 1(a) shows clearly the ground-state QPT at ξc=0.2 and the separatrix, marked by a high density of states, which is the boundary between the two ESQPT phases. Excited states above the separatrix have a u(1) (symmetric) character, while those below it have an so(2) (broken symmetry) character. It can be appreciated in the figure how beyond the critical value of the control parameter ξand for states under the separatrix, the even- and odd-parity states are degenerated. From this figure it is already clear that the new ESQPT is not a static ESQPT using the notation introduced in Ref. [31]. In the present case, as it is confirmed in the semiclassical analysis in the next section, the nonanalyticity of the density of states is of the same kind in the two ESQPTs and both have noticeable effects on the system dynamics [70]. Figure 1(b) is the correlation energy diagram for a negative αvalue. As expected from the results obtained for the twodimensional limit of the vibron model [29,30], there is a new separatrix that crosses the one associated with the groundstate QPT and the degeneracy pattern is more complex. In order to better understand the second separatrix we study the classical limit of the model, making use of the spin coherent states formalism [74]. Classical limit of the model The classical limit of the Hamiltonian (3) can be obtained within a mean field analysis, studying the large-size limit of the model. We perform this study using spin coherent states [74]. From the usual definitions and using the notation introduced in the preceding section, the spin ladder operators are ˆ S±=ˆ Sx±iˆ Syand the spin coherent state is defined as |[S]μ= exp(μˆ S+) (1 +|μ|2)S|S,−S =1 (1 +|μ|2)S 2S  p=02S pμp|S,−S+p,(5) where μis a complex parameter that encompasses the classical position and momentum variables, |S,−S+pis a eigenstate of ˆ Sz,ˆ Sz|S,−S+p=(−S+p)|S,−S+p, and (ˆ S+)p|S,−S=p!(2S)! (2S−p)! |S,−S+p[74]. The expectation value of the operators ˆ Sz,ˆ S+, and ˆ S−with the spin coherent state (5)are [S]μ|ˆ Sz|[S]μ=S|μ|2−1 |μ|2+1, [S]μ|ˆ S+|[S]μ=S2μ∗ |μ|2+1, [S]μ|ˆ S−|[S]μ=S2μ |μ|2+1.(6) Thus, the classical limit of the ALMG model, Hcl(μ), can be obtained as the expectation value of the Hamiltonian (3) (per particle) with the coherent state in the large-system-size limit (S→∞) Hcl(μ)=[S]μ|ˆ Hanh|[S]μ 2S =(1 −ξ)|μ|2 1+|μ|2+α|μ|2 1+|μ|22 +ξ1−μ+μ∗ 1+|μ|22,(7) where the complex variable μcan be mapped into (q,p)(the canonical coordinate and momentum) by q=1 √2 μ+μ∗ 1+|μ|2,(8) p=−i √2 (μ−μ∗) 1+|μ|2.(9) 044125-3 J. GAMITO et al. PHYSICAL REVIEW E 106, 044125 (2022) (a) (b) FIG. 1. Correlation energy diagram depicting the normalized excitation energies for even (blue solid lines) and odd (red dashed lines) parity states as a function of the control parameter ξ∈[0,1] for a system with size N=120 and anharmonicity parameter (a) α=0 and (b) −0.6. Applying this transformation, the resulting classical Hamiltonian is Hcl(q,p)=1−ξ 2(p2+q2)+α 4(p2+q2)2 +ξq2(p2+q2−2) +ξ. (10) The resulting energy functional should provide the functional form of the separatrices that mark the critical ESQPT energies in Fig. 1, one in the α=0 case [Fig. 1(a)] and two for α<0 [Fig. 1(b)]. The separatrix starting at the critical value of the control parameter ξc=0.2 has been already characterized for α=0. We present results for both ESQPTs with a special focus on the anharmonicity-related critical line, exploring whether this line has the same nature and implications as the other and to what extent it has an impact in the system structure. The changes in the system dynamics associated with the introduction of the anharmonicity in the LMG model are explored in [70]. Assuming ξ∈[0,1] and α<0, the critical or stationary points of the Hamiltonian, where the first derivatives of Eq. (10) are zero, are ∂Hcl(q,p) ∂q=0 ∂Hcl(q,p) ∂p=0−→ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ q2 0=0,p2 0=0 q2 1=5ξ−1 4ξ+α,p2 1=0 q2 2=ξ−1−2α 2ξ,p2 2=1+2α+3ξ 2ξ q2 3=0,p2 3=ξ−1 α. (11) According to the classical limit, the origin is a stationary point and it corresponds to the system ground state in the control parameter range ξ∈[0,ξ c] where the critical point is ξc=0.2. For values of the control parameter ξ∈(ξc,1], the minimum energy is attained for the (q1,p1) coordinate and momentum values. In this way, the system ground-state energy as a function of the control parameters can be expressed as Egs(ξ,α)=ξ, ξ ⩽0.2 −1+ξ(10+4α−9ξ) 4(α+4ξ),ξ>0.2.(12) From this equation is clear that the Hamiltonian (3)inthe mean field limit has a critical point at ξc=0.2, where the second derivative of the ground-state energy with respect to the control parameter ξis discontinuous. This is in good agreement with the results obtained in [8] for a Hamiltonian including one- and two-body operators. Hence, the crossing of this critical point is marked by a second-order ground-state QPT. The energy gap vanishes and the ground state becomes parity degenerated. In addition to this and in a way similar to other systems with a transition between u(n) and so(n+1) dynamical symmetries [26], an ESQPT appears at an energy Hcl(q0,p0)=ξ. The critical excitation energy of the ESQPT defines the separatrix with a high density of states shown in Fig. 1. An analytical expression for this separatrix can be computed as f1(ξ,α)=Hcl(q0,p0)−Hcl(q1,p1)=(1 −5ξ)2 4(4ξ+α)(13) for ξ∈(ξc,1]. For negative αvalues there can be a second separatrix in this system, as shown in Fig. 1(b). When the condition α<(ξ−1)/2 is satisfied, the second ESQPT appears at a critical energy Hcl(q2,p2)=1+α. The excitation energy for this new ESQPT marks the second separatrix in Fig. 1(b) and can be computed as f2(ξ,α)=Hcl(q2,p2)−Hcl(q0,p0)=1+α−ξ, ξ ⩽ξc Hcl(q2,p2)−Hcl(q1,p1)=(1+2α+3ξ)2 4(α+4ξ),ξ>ξ c. (14) 044125-4 EXCITED-STATE QUANTUM PHASE TRANSITIONS IN … PHYSICAL REVIEW E 106, 044125 (2022) (a) ( b ) FIG. 2. Contour plots of the classical Hamiltonian (10) for systems with (a) α=0.0and(b)α=−0.6 and different values of the control parameter, from left to right: ξ=0.15, 0.3, 0.4, and 0.5. Each row shares the scale (right color axes). As previously mentioned, a similar ESQPT in the brokensymmetry phase of the two-dimensional limit of the vibron model was studied in Ref. [29] and it was recently shown to be present in the symmetric phase of the model too [30]. In the symmetric phase ξ∈[0,ξ c], pair of eigenstates with different parities are degenerate for energies above the f2(ξ,α) separatrix. In the broken-symmetry phase ξ∈(ξc,1], both separatrices can coexist and the pairs of eigenvalues with even and odd parities are degenerate below and above both of them and nondegenerate in between the two separatrices. This is shown clearly in Fig. 1(b). The critical point (q3,p3) marks the largest energy of the system, Hcl(q3,p3)=−1+(2+4α−ξ)ξ 4α, if the solution (q2,p2) exists. In fact, taking this into account and equating Hcl(q2,p2)=Hcl(q3,p3), the threshold value of αin order that the second ESQPT exists can be obtained as αth(ξ)=(ξ−1)/2. From this equation it is clear that the second ESQPT is present not only in the broken-symmetry phase, but also in the symmetric phase. This was recently discussed for a different model, the 2DVM [30], and it is a clear example of an ESQPT without an associated QPT, something that usually happens in systems with more than one control parameter for some trajectories in the parameter space [75–77]. In this case, the situation is somewhat different and the second ESQPT can be associated with changes in the boundaries in the finite Hilbert space of the system [15,24,31]. In Fig. 2we show the contour plot of the classical Hamiltonian (10) for two different values of the anharmonic interaction, one without the f2(ESQPT (14), with α=0.0 [Fig. 2(a)]) and another with a value of the control parameter smaller than the threshold, α=−0.6 [Fig. 2(b)]. In both cases different values of the control parameter ξhave been studied, from left to right ξ=0.15, 0.3, 0.4, and 0.5. The cases with α=0.0 present one minimum for ξ<0.2 and two minima in the broken-symmetry phase [Fig. 2(a)]. In the four examples the maximum values of the energy correspond to q2 2+p2 2=2. When the control parameter αtakes a value smaller than the threshold [Fig. 2(b)], two maxima appear for q3=0 and p3= 0, which correspond to the new system limit. In these cases, systems present a new ESQPT due to the system limit q2 2+p2 2=2. In order to check the threshold value αth(ξ), we have computed the maximum excitation energy for different values of α and system sizes and we have fit the maximum normalized excitation energy to a functional form Emax =Ecl +cN−bfor a constant ξ=ξc. The resulting values of Ecl for ξcand various values of αare a good estimate of the maximum excitation energy of the system in the classical limit. The difference between Ecl and the separatrix f2(ξc,α) is depicted in Fig. 3, FIG. 3. The olive green solid line is the difference between the maximum normalized excitation energy of the ALMG (see the text for details) and the separatrix (14)forξ=ξc=0.2. The horizontal blue dashed line marks the zero value. 044125-5 J. GAMITO et al. PHYSICAL REVIEW E 106, 044125 (2022) FIG. 4. Density of states ν(E) as a function of the normalized energy E/Nfor an ALMG Hamiltonian with α=−0.6and(a)ξ=0.15 and (b) ξ=0.6. The red solid line is the ν(E) density of states computed with Eq. (15). Also depicted is the density of states computed for the normalized eigenvalues of an ALMG Hamiltonian (3) with N=4000 (orange bars). where it is clear how the threshold value of αexists and is equal to the value αth(ξc)=−0.4. As can be clearly seen in Fig. 1, the existence of an ESQPT can be traced back to abrupt changes in the system available phase space which produces a maximum in the local density of states at the separatrices, for the critical energy of the transition [12,24,31,48]. From the classical Hamiltonian (10), we can evaluate the semiclassical approximation to the quantum density of states as a function of the system’s energy [78]. In the particular case of the LMG and ALMG models ν(E)=1 2π dqdpδ(E−Hcl(q,p)) =1 2π dφdjzδ(E−Hcl(φ, jz)),(15) where instead of the phase-space generalized coordinate and associated momentum we used a different pair of canonical variables φ∈[0,2π) and jz∈[−j,j] that facilitate the derivation of an analytical formula for the density of states of the ALMG model. Using the properties of the Dirac delta function (see the Appendix), we can obtain the analytical formula for the density of states ν(ε)=1 4π2π 0 ×dφ 1−ξ+α 22−(α+4ξcos2φ)2+2ξ+α 4−ξcos2φ−ε , (16) where εis the scaled energy value ε=E/N. In Fig. 4we plot the density of states ν(E) calculated with Eq. (16) (red line) versus the normalized energy for α=−0.6(thesameαvalue used in the correlation energy diagrams of Fig. 1) and two ξvalues: 0.15 [symmetric phase, Fig. 4(a)] and 0.6 [broken-symmetry phase, Fig. 4(b)]. For the sake of comparison, we also include the density of states computed from the eigenvalues of an ALMG Hamiltonian (3) for the same control parameters and a system size N=2000. It can be clearly appreciated how the density of states in Fig. 4(a) has a maximum at the f2(ξ=0.15,α = −0.6) +Egs(ξ=0.15,α =−0.6) =0.4 critical energy in the symmetric phase and in Fig. 4(b) at f1(ξ=0.6,α = −0.6) +Egs(ξ=0.6,α =−0.6) =0.4 (leftmost maximum) and f2(ξ=0.6,α =−0.6) +Egs(ξ=0.6,α =−0.6) =0.6 (rightmost maximum) critical energies in the brokensymmetry phase. The agreement between the results for Eq. (16) and the density of states computed from the system energies is excellent and it is worth noting how in all cases the peaks will transform into logarithmic divergences for an infinite system size, as expected [14]. This has deep implications in the system dynamics, as shown in [70]. In the next section we characterize the two ESQPTs and their associated separatrices using different quantities. III. RESULTS In this section we characterize the two ESQPTs that arise in the ALMG using four different quantities: the gap between adjacent positive-parity energy levels, the expectation value of the operator ˆn=S+ˆ Sz, the participation ratio, and the quantum fidelity susceptibility. 044125-6 EXCITED-STATE QUANTUM PHASE TRANSITIONS IN … PHYSICAL REVIEW E 106, 044125 (2022) FIG. 5. Gap between adjacent energy levels in the ALMG model as a function of the normalized excitation energy for (a) ξ=0.15 and (b) ξ=0.6; anharmonicity parameter α=0 (yellow), −0.3 (blue), −0.4 (green), and −0.6 (brown); and system size N=120 (dashed lines) and 1200 (solid lines). A. Energy gap The occurrence of an ESQPT is most often marked by a discontinuity in the density of states or one of its derivatives. In cases such as the LMG model (2), for a given value of the control parameter ξ>ξ c=0.2, the difference in energy between adjacent eigenstates is minimum at or close to the critical energy of the ESQPT, due to the high density of states at this excitation energy (see Fig. 1). Results are more complex once we introduce the anharmonic correction in the LMG model Hamiltonian (3). In Fig. 5we plot the difference between adjacent positive-parity energy levels as a function of the normalized excitation energy for system-size values N=2S=120 (dashed lines) and 1200 (solid lines), for control parameter ξequal to 0.15 [Fig. 5(a)] and 0.6 [Fig. 5(b)], and for α=0,−0.3,−0.4,−0.6. The results in Fig. 5(a), where the system is in the symmetric phase (ξ=0.15 <ξ c), can be explained by considering the second separatrix (14). The f2(ξ,α) line marks the critical energy value for the ESQPT associated with the anharmonic term in ξ>ξ cand, as explained above, there is a threshold αand the effects of the anharmonic term in this phase can be noticed only when the parameter α is beyond this threshold value [see the line for α=−0.4in Fig. 5(a)]. The crossing of the critical energy is marked by an abrupt minimum in the energy gap, as expected. When this study is extended to the broken-symmetry phase [Fig. 5(b)], there are two minima, one for each separatrix, for negative αvalues, except for α=−0.4, as this corresponds to the point associated with the crossing of f1(ξ,α) and f2(ξ,α)in Fig. 1(b). As expected, the ESQPT precursor is better defined for larger system size, though for N=2S=120 the results already clearly identify the ESQPT. The ALMG energy gap results are in good agreement with the 2DVM results obtained in the symmetric [30] and broken-symmetry [29] cases. B. Number operator The S+ˆ Szterm in Eqs. (2) and (3) is associated with the u(1) Casimir operator and it can be denoted by the system FIG. 6. Expectation value of the number operator ˆnin the system eigenstates for (a) ξ=0.15 and (b) ξ=0.6; anharmonicity parameter α=0(yellow),−0.3 (blue), −0.4 (green), and −0.6 (brown); and system size N=120 (dashed lines) and 1200 (solid lines). number operator ˆn. This Casimir operator provides a suitable realization of an order parameter for the ground-state QPT in the ALMG case [35–37,47,51]. The expectation value of this operator in the system eigenstates is a second ESQPT proxy in the Hamiltonian (3). We depict in Fig. 6the expectation value of ˆnas a function of the normalized system excitation energy for ξ=0.15 [Fig. 6(a)] and 0.6 [Fig. 6(b)] and different values of the anharmonicity parameter α(see the figure caption) for system size N=120 (dashed lines) and 1200 (solid lines). In all cases, the expectation value of ˆnis a well-defined minimum (maximum) for the eigenstates closer to the f1(ξ,α)[f2(ξ,α)] separatrix. This fact will be better understood once we introduce the inverse participation ratio to characterize both ESQPTs. It is worth emphasizing that in the symmetric case there is no minimum; the f1(ξ,α) line is only defined for ξ>ξ c. This is clearly shown in Fig. 6(a), where ξ=0.15 <ξ c.In addition, as in the energy gap case, this quantity has a peak only for values beyond the αthreshold value that corresponds to the second separatrix f2(ξ,α). C. Participation ratio The participation ratio (PR) provides the localization of a quantum state in a given basis [79]. For a state |ψ, expressed in a basis {|n}D−1 n=0as |ψ=D−1 n=0Cn|n, the PR is defined as P(ψ)=1 D−1 n=0|Cn|4,(17) where Dstands for the Hilbert space dimension, which in this case is D=N+1. This quantity is also often referred to as the inverse participation ratio [80] or the number of principal components [81]. In the case in which the state is maximally delocalized, Cn=1/√Dfor all nand P(ψ)=D; if the state is equal to one basis element (all Cn=0 but one that is equal to unity), the localization is maximal in the basis and P(ψ)=1. A series of works in algebraic models for systems in one-, two-, and three-dimensions having a u(n+1) dynamical algebra has shown that the eigenstates closer to the critical energy in a u(n)−so(n+1) second-order groundstate quantum phase transition are strongly localized in the 044125-7 J. GAMITO et al. PHYSICAL REVIEW E 106, 044125 (2022) (a) (b) (c) (d) FIG. 7. Normalized participation ratio as a function of the normalized excitation energy for an ALMG model with ξ=0.15, system size N=600, and different anharmonicity parameter values: (a) α=0 (yellow), (b) α=−0.3 (blue), (c) α=−0.4 (green), and (d) α=−0.6 (brown). In each case, the PR values for eigenstates of interest have been marked with black circles and the squared components of these eigenstates as a function of the corresponding u(1) basis state index are shown in the insets (see the text for details). u(n) basis [51,82,83]. In the ALMG case there is a strong localization in the u(1) basis of the states close to either one of the two separatrices. In particular, the eigenstates with energies close to the critical energy of the f1(ξ,α) separatrix are highly localized in the n=0 component (|S,Mz=−S)of the basis, something that was already checked in [51]forthe LMG model. However, for the localization of the eigenstates with energies close to the second separatrix f2(ξ,α), critical energy is due to a high component in the n=Nbasis state (|S,Mz=S). This is shown in Figs. 7and 8for systems in the symmetric phase (ξ=0.15) and in the broken-symmetry phase, respectively. In both cases, the system size is N=600 and results are depicted for (from top to bottom) α=0, −0.3, −0.4, and −0.6. In each panel some representative eigenstates have been chosen and the values of their squared components in the u(1) basis are shown in the corresponding insets. In Fig. 7the only αvalue beyond the threshold value is α=−0.6, shown in Fig. 7(d).InFigs.7(a)–7(c) the PR is minimum at the spectrum edges, where eigenstates are maximally localized in the |S,Mz=−Sand |S,Mz=Sstates, as expected in the symmetric phase where the u(1) dynamical symmetry provides a convenient approximation. However, in Fig. 7(d) (brown line), a well-defined PR minimum appears 044125-8 EXCITED-STATE QUANTUM PHASE TRANSITIONS IN … PHYSICAL REVIEW E 106, 044125 (2022) (a) (b) (c) (d) FIG. 8. Normalized participation ratio as a function of the normalized excitation energy for an ALMG model with ξ=0.6, system size N=600, and different anharmonicity parameter values: (a) α=0 (yellow), (b) α=−0.3 (blue), (c) α=−0.4 (green), and (d) α=−0.6 (brown). In each case, the PR values for eigenstates of interest have been marked with black circles and the squared components of these eigenstates as a function of the corresponding u(1) basis state index are shown in the insets (see the text for details). associated with the f2(ξ,α) separatrix and the ESQPT induced by the anharmonic term in Eq. (3). The eigenstate labeled as Cin this panel, i.e., the eigenstate with a minimum PR value which is not in the spectrum edges, is highly localized (low PR) and its squared components in the inset reveal that the localization takes place for the last basis state |S,Mz=S. The PR values in the broken-symmetry phase are shown in Fig. 8.Intheα=0 case [Fig. 8(a), yellow curve], the states at the edge of the spectrum have low values of the PR, as expected, and the ground state is not well localized in the n=0 basis state, as we have straddled the critical ground-state QPT and the u(1) dynamical symmetry is no longer a convenient approximation for the system eigenstates. There is a local PR minimum, labeled as C, associated with the f1(ξ,α) separatrix and the ESQPT that stems from the appearance of a maximum in the origin. When α=−0.3 [Fig. 8(b), blue curve], precursors from both ESQPTs can be observed. In this case the f1(ξ,α) separatrix is found at lower energies than the f2(ξ,α) separatrix and the states close to 044125-9