Tommaso Comparin

Fabio Mezzacapo

Martin Robert-De-Saint-Vincent

Tommaso Roscilde

Scalable spin squeezing from spontaneous breaking of a continuous symmetry

Spontaneous symmetry breaking (SSB) is a property of Hamiltonian equilibrium states which, in the thermodynamic limit, retain a finite average value of an order parameter even after a field coupled to it is adiabatically turned off. In the case of quantum spin models with continuous symmetry, we show that this adiabatic process is also accompanied by the suppression of the fluctuations of the symmetry generator -namely, the collective spin component along an axis of symmetry. In systems of S = 1/2 spins or qubits, the combination of the suppression of fluctuations along one direction and of the persistence of transverse magnetization leads to spin squeezing -a much sought-after property of quantum states, both for the purpose of entanglement detection as well as for metrological uses. Focusing on the case of XXZ models spontaneously breaking a U(1) (or even SU(2)) symmetry, we show that the adiabatically prepared states have nearly minimal spin uncertainty; that the minimum phase uncertainty that one can achieve with these states scales as N -3/4 with the number of spins N ; and that this scaling is attained after an adiabatic preparation time scaling linearly with N . Our findings open the door to the adiabatic preparation of strongly spin-squeezed states in a large variety of quantum many-body devices including e.g. optical lattice clocks.

Introduction. Many-body entanglement is at the heart of the fundamental complexity of quantum states [START_REF] Preskill | Quantum computing and the entanglement frontier[END_REF], and it is the basis mechanism by which a quantum manybody system relaxes to a stationary state after having been driven away from equilibrium [START_REF] Kaufman | [END_REF]. In this respect, the generation of many-body entangled quantum states is one of the main goals of a new generation of quantum devices, going from quantum simulators [3] to quantum computers [START_REF] Nielsen | Quantum Computation and Quantum Information[END_REF], whose common trait is the ability to perform coherent unitary evolutions of quantum many-body systems. Yet, certifying (let alone putting to use) manybody entanglement is a task which is restricted to a small class of quantum many-body states, most prominently those whose entanglement can be detected via the measurement of the lowest moments of the quantum-noise distribution [START_REF] Gühne | [END_REF]. In the context of ensembles of N qubits (S = 1/2 spins), characterized by the collective-spin operator J = N i=1 S i (where the S i 's are S = 1/2 quantum spin operators), one of the best example of such states is offered by spin-squeezed ones, whose entanglement is detected via the spin-squeezing parameter [6]

ξ 2 R = N min ⊥ Var(J ⊥ ) J 2 , (1)
where min ⊥ expresses the minimization of the variance on the collective-spin components perpendicular to the average spin orientation J . A state with ξ 2 R < 1/k (k ≥ 1) is entangled, with an entanglement depth (least number of entangled spins) of k + 1 [7,8]. The ξ 2 R parameter is also a fundamental figure of merit of the sensitivity of the state to rotations, expressing the reduction in phase-estimation error for a Ramsey interferometric protocol with respect to a factorized state [6]; and it offers the possibility to improve the efficiency of quantum devices such as atomic clocks [9][10][11][12] or quantum sensors [13][14][15] by using entanglement as a resource.

⇡ 1 ⌦ ⇠ 1/N ⌦ = 0 ⌦ J x J y J z J z ⇠ N 1/4
adiabatic squeezing FIG. 1: Adiabatic squeezing from spontaneous symmetry breaking in the XXZ model. Starting from a coherent spin state at Ω = ∞, an adiabatic reduction of the field Ω coupling to the order parameter leads to the appearance of scalable spin squeezing when Ω ∼ 1/N , due to the scaling of the uncertainty on the J z component, δJ z = Var(J z ) ∼ N 1/4 ; and to the absence of scaling of the order parameter m = J x /N , as a consequence of spontaneous symmetry breaking (SSB).

The red areas depict the uncertainty regions of the collective spin on a sphere of radius J 2 ∼ N . As a consequence the angular aperture of the uncertainty region along the z axis is δφ ≈ δJ z / J 2 ∼ N -3/4 , defining the sensitivity of the state to rotations around the y axis.

For the reasons listed above, devising many-body mechanisms that lead to the controlled preparation of spin-squeezed states [16] is a very significant endeavorand particularly so when the squeezing parameter can be parametrically reduced by increasing the number of resources. This situation leads to scalable squeezinggenerically ξ 2 R ∼ N -α (α > 0) -which allows one to surpass the standard quantum limit for the scaling of the phase-estimation error with the number of qubits [8]. The two main mechanisms that have been iden-arXiv:2202.08607v1 [quant-ph] 17 Feb 2022 tified and implemented so far in this direction are the preparation of spin-squeezed states via unitary evolutions with collective-spin interactions [17][18][19][20] and via the nondemolition measurement of a spin component [21,22]. Yet a third way to spin squeezing is offered by adiabatic preparation, which relies upon the identification of spin squeezing in the ground state of quantum spin Hamiltonians implemented e.g. by quantum simulation platforms. A clear example in this direction is offered by Ising quantum critical points, which exhibit scalable squeezing at and above the upper critical dimension (d ≥ 3 for shortrange interactions) [23].

hJ x i N ⇠ 2 R ⇠ ⌦ 1 / 2 ⇠ ⌦ 1 / 2 (a) (b) (c) (d
In this work we unveil another fundamental link between many-body physics of spin models and the generation of spin squeezing, namely the appearance of squeezing in the presence of spontaneous breaking of a continuous spin symmetry in the ground state. Without loss of generality, in the following we will be concerned with symmetry under U(1) rotations U z (φ) = e -iϕJ z generated by the collective spin component J z -this property is also present for SU(2)-symmetric Hamiltonians. On a finite-size system and in the absence of any symmetrybreaking field, the ground state of a U(1) symmetric Hamiltonian has Var(J z ) = 0, namely it exhibits socalled Dicke squeezing [8], lacking nonetheless a finite net magnetization J = 0. At the same time, the lowlying energy spectrum of that same Hamiltonian exhibits a so-called Anderson tower of states (ToS), whose energy collapses as 1/N onto that of the ground state [START_REF] Anderson | Basic Notions of Condensed Matter Physics[END_REF][START_REF] Tasaki | [END_REF][26][START_REF] Comparin | Universal spin squeezing from the tower of states of u(1)-symmetric spin hamiltonians[END_REF]. Hence a field Ω ∼ 1/N coupling to the order parameter in the x-y plane, e.g. -ΩJ x (without loss of generality), is sufficient to mix the ToS into a state exhibiting a net polarization m = J x /N = 0. The hallmark of spontaneous symmetry breaking (SSB) is then the persistence of a finite order parameter m in the limit N → ∞, in which the field is also parametrically set to zero. Here we investigate paradigmatic XXZ models with nearestneighbor interactions using finite-temperature and varia-tional quantum Monte Carlo simulations, as well as of spin-wave theory; and, in the presence of SSB in the ground state, we show that the state polarized by the minimal field Ω ∼ 1/N away from Dicke squeezing retains a strong asymmetry in the fluctuations of the collective spin components, exhibiting scalable (Wineland) spin squeezing with ξ 2 R ∼ N -1/2 . Such a state is shown to have minimal spin uncertainty, namely squeezing is its optimal metrological resource; and it can be prepared adiabatically, starting from a coherent spin state stabilized at Ω → ∞, and ramping down Ω to a value ∼ 1/N in a time scaling linearly with system size, τ ∼ N (see Fig. 1 for a summarizing sketch). This finding opens the possibility to squeeze the collective spin of quantum simulators of U(1)-(or SU(2)-)symmetric qubit Hamiltonians, with potential applications to quantum sensors [13,[START_REF] Facon | [END_REF] and atomic clocks [29,30].

Model and methods. We focus our attention on the S = 1/2 XXZ Hamiltonian:

H = - 1 2 ij J ij (S x i S x j +S y i S y j -∆S z i S z j )-Ω i i S x i (2)
where i, j are lattice sites on a d-dimensional hypercubic lattice of size N = L d with periodic boundary conditions. In the following we shall specialize our attention to nearest-neighbor (n.n.) interactions -J ij = J if i n.n. j. Moreover we choose J > 0 and -1 < ∆ ≤ 1, defining an XXZ model with ferromagnetic interactions in the plane and ferromagnetic or antiferromagnetic along the symmetry axis. Such a situation is realized e.g. in bosonic Mott insulators [31][32][33]. Under these assumptions, the above model is known to break a continuous symmetry in the ground state in d ≥ 2; and the field coupling to the order parameter is uniform, namely i = 1 ∀i. But most of our results are completely general, and apply to any model featuring SSB of a continuous symmetry -provided that the order parameter does not commute with the Hamiltonian (otherwise the ground state is simply a coherent spin state for all Ω, e.g. ⊗ N i=1 | → x i with | → x i the eigenstate of S x i with eigenvalue +1/2). In the case J < 0 (antiferromagnetic interactions in the xy plane) -as realized in fermionic Mott insulators [31,34,35] -the order parameter is the staggered magnetization, e.g. m = J x st /N = i i S x i /N ; the field coupling to the order parameter must therefore be staggered, i = (-1) i ; and the relevant rotations are generated by J y st = i i S y i . Yet, for n.n. interactions on a hypercubic lattice, the physics is equivalent to that of the bosonic insulators, as the two models are connected by a canonical transformation (rotation of π around the z axis for one of the two sublattices).

We have studied the ground-state physics of the XXZ Hamiltonian in d = 1, 2 and 3 making use of numerically exact quantum Monte Carlo (QMC) simulations, based on the Stochastic Series Expansion method [36]; as well as of spin-wave theory, valid in the presence of spontaneous symmetry breaking (d = 2, 3). Moreover we have investigated the (quasi-)adiabatic dynamics of preparation of the ground state starting from a large Ω by making use of time-dependent variational Monte Carlo (tVMC), based on pair-product (or spin-Jastrow) wavefunctions [START_REF] Comparin | Universal spin squeezing from the tower of states of u(1)-symmetric spin hamiltonians[END_REF]37] as well as of time-dependent spin-wave theory (see the Supplemental Material -SM -for an extended discussion of the methods [START_REF]) comparison between tVMC and LSW results for the evolution along quasi-adiabatic ramps[END_REF]).

Adiabatic squeezing from SSB. In the following we shall only show results for the case ∆ = 1 (Heisenberg model)analogous results for the case ∆ = 0 (XX model) are presented in the SM. Fig. 2 shows our QMC results for the 2d Heisenberg model calculated for different sizes N = L 2 at a temperature T = J /N chosen so as to effectively remove thermal effects at the energy scale of an applied field Ω ∼ J /N -as we shall see, this choice is rather conservative, because the field opens in fact a gap in the spectrum scaling as (J Ω) 1/2 . The QMC results are compared to linear spin-wave (LSW) theory -see SM for the details of the theory -in the thermodynamic limit, which is expected to be very accurate at large Ω, and to quantitatively capture some selected features in the limit Ω → 0 [START_REF] Manousakis | [END_REF]. The uniform magnetization J x /N , shown in Fig. 2(a), is indeed correctly predicted by LSW theory: the finite-size QMC data show that the LSW prediction is reproduced down to a field scaling ∼ 1/N , below which the finite-size gap between the states of the Anderson tower overcomes the field. Hence the SSB scenario, namely the persistence of a finite magnetization down to Ω = 0 when N → ∞, is clearly shown. Concomitantly, the suppression of the symmetry breaking field Ω leads to a strong suppression of the fluctuations of the U(1) symmetry generator J z ; LSW theory predicts that Var(J z ) ∼ Ω 1/2 when Ω → 0, a prediction which appears to be consistent with our finite-size QMC results for e.g. the 2d XX model (see Ref. [START_REF]) comparison between tVMC and LSW results for the evolution along quasi-adiabatic ramps[END_REF]) for fields down to Ω ≈ J /N ; while the 2d Heisenberg model shows significant beyond-LSW corrections, which interestingly appear to lead to a further reduction of the variance, namely to stronger squeezing [START_REF]) comparison between tVMC and LSW results for the evolution along quasi-adiabatic ramps[END_REF]. The combination of these two results implies naturally that the ξ 2 R parameter is smaller than unity for all values of Ω -in agreement with a recent theorem predicting ground-state squeezing in this model as soon as Ω < ∞ [40]; and which scales as Ω 1/2 (actually faster for the 2d Heisenberg model) down to fields ∼ J /N , namely as N -1/2 (or faster) for the lowest significant fields for each finite size N . The evidence of scalable spin squeezing resulting from SSB -namely from the absence of scaling (or persistence) of J x /Nis the main result of our work.

Another significant feature of the low-Ω state of XXZ models exhibiting SSB is that of being a state of minimal uncertainty for the collective spin, namely the collective spin components saturate the Heisenberg-Robertson inequality, Var(J y )Var(J z ) ≥ J x 2 /4. To discuss this aspect and its metrological implications, it is useful to introduce the quantum Fisher information (QFI) density [8] for the J y component, defined as

f (J y ) = 2 N nm (pn-pm) 2 pn+pm | m|J y |n | 2
, where |n and p n are the eigenstates and corresponding eigenvalues of the density matrix ρ. When the state in question is rotated around the y axis by the transformation U y (φ) = e -iφJ y , the QFI density expresses the minimal uncertainty on the angle φ, δφ ≥ (f Q N ) -1/2 , namely f Q = 1 implies a deviation of this uncertainty with respect to the standard quantum limit. The latter property, combined with the fact that 4Var(J y )/N is an upper bound to the QFI density, leads to the inequality chain:

ξ -2 R = J x 2 N Var(J z ) ≤ f Q (J y ) ≤ 4Var(J y ) N . (3)
If a state has minimal uncertainty, namely Var(J y )Var(J z ) J x 2 /4, the above inequality chain collapses to an equality, namely

ξ -2 R ≈ f Q (J y ) ≈ 4Var(J y )/N .
This collapse is clearly exhibited by our numerical data for all values of Ω > 0 and all systems sizes -see Fig. 2(d). In particular, among all the macroscopic observables built as a sum of local observables, J y is arguably the one with the largest QFI density [START_REF]) comparison between tVMC and LSW results for the evolution along quasi-adiabatic ramps[END_REF], so that the estimation of the rotation angle φ is the optimal phase-estimation protocol for the low-Ω states. The fact that ξ -2 R ≈ f Q implies that the measurement of the rotation of the average collective spin (corresponding to Ramsey interferometry) is the optimal measurement for this protocol, leading to a phase-estimation error δφ = ξ R / √ N ∼ N -3/4 . Finally, we would like to stress that the above results are not at all limited to the 2d Heisenberg model, but they are valid for all the XXZ models spontaneously breaking a U(1) (or SU [START_REF] Kaufman | [END_REF]) symmetry in the thermodynamic limit that we have explored. Fig. 3(a) shows the field dependence of the spin-squeezing parameters ξ 2 R for the Heisenberg model in d=1, 2, and 3. We observe that the scaling of ξ 2 R as Ω 1/2 is clearly exhibited in d = 3. On the other hand, for d = 1 (Heisenberg chain) SSB is not realized because of the critical strength of quantum fluctuations [START_REF] Auerbach | Interacting electrons and quantum magnetism[END_REF]: as a consequence, J x vanishes when Ω → 0, leading to the breakdown of the mechanism that underpins scalable spin squeezing in higher dimensions. Fig. 3(a) shows that J x 2 , vanishing as Ω 1/2 , leads to a squeezing parameter ξ 2 R that goes to a constant as Ω → 0. Similar results for the XX model (∆ = 0) are shown in the SM [START_REF]) comparison between tVMC and LSW results for the evolution along quasi-adiabatic ramps[END_REF].

Quasi-adiabatic ramps. The preparation of the ground state at low fields requires the initialization of the system in a coherent spin-state aligned with the Ω field with Ω J ; and the subsequent gradual reduction of the field along an adiabatic down-ramp -a protocol analog to that of adiabatic quantum computing [START_REF] Albash | [END_REF]. The adiabatic theorem mandates that the duration τ of an adiabatic ramp that prepares the system in the ground state at a final field Ω f should be τ

J > ∼ (∆E min /J ) -2 , where ∆E min = min Ω∈[Ω f ,∞] (E 1 (Ω) -E 0 (Ω))
is the minimal gap between the Ω-dependent ground-state energy (E 0 ) and the energy of the first excited state (E 1 ) over the field range of the ramp. This gap can be calculated by LSW theory [START_REF]) comparison between tVMC and LSW results for the evolution along quasi-adiabatic ramps[END_REF] -in good agreement with exact diagonalization on small system sizes [START_REF]) comparison between tVMC and LSW results for the evolution along quasi-adiabatic ramps[END_REF] -and for the Heisenberg model (∆ = 1) and Ω f J it is shown to be ∆E min /J ≈ (zΩ f /J ) 1/2 where z = 2d is the coordination number. This result implies that the adiabatic preparation of the ground state for the minimal field Ω f ∼ 1/N at size N takes a time τ /J > ∼ (zΩ f /J) -1 ∼ N . We complement the above general prediction from LSW theory with realistic calculations of quasi-adiabatic ramps based on tVMC -which show remarkable agree-ment with independent calculations based on timedependent LSW [START_REF]) comparison between tVMC and LSW results for the evolution along quasi-adiabatic ramps[END_REF], mutually corroborating their quantitative validity. We start the state evolution from the ground state at a large initial field value Ω i = 10J -obtained by minimization of the variational energy of the spin-Jastrow Ansatz [START_REF]) comparison between tVMC and LSW results for the evolution along quasi-adiabatic ramps[END_REF]; and then we ramp the field down to Ω f with the schedule Ω(t) = Ω i + F (t/τ )(Ω f -Ω i ), where F (x) = 1 2 e -1/x+2 θ(1/2x) + 1 -1 2 e -1/(1-x)+2 θ(x -1/2) for t ∈ [0, τ ], while Ω(t) = Ω f for t > τ . The function F (t) has the property of having vanishing derivatives at all orders at the two extremes of the [0, τ ] interval, so that it is continuous along with all of its derivatives when it is extended to t < 0 and t > τ by constant functions. Fig. 3(b-c) shows the tVMC results for evolution of the ξ 2 R parameter in the 2d Heisenberg model (L = 12) with two different final fields (Ω f /J = 10 -1 and 10 -2 ), and various ramp durations. Our main observation is that, even when the ramp fails to keep the system in its ground state down to Ω f , the squeezing parameter exceeds the adiabatic value only for t < ∼ τ , while it systematically evolves to lower values at immediately later times, and then oscillates around the adiabatic value. Therefore failure to follow a perfectly adiabatically ramp (which in Fig. 3 is observed for all considered ramp durations when Ω f = 10 -2 J ) does not per se imply a degradation of the amount of squeezing that can be produced in the system. A final comment concerns the possibly of imperfect preparation of the initial state of the quasi-adiabatic ramp: this would generically entail the presence of finite entropy in the initial state, persisting then in the evolved one. Fig. 3(d) tests the robustness of squeezing to the presence of finite entropy in the case of the equilibrium state of the 2d Heisenberg model. Not surprisingly, a finite entropy imposes a limit to the achievable squeezing; yet adiabatic spin squeezing can be obtained up to spin entropies S/N < ∼ 0.3k B . Conclusions. In this work we have demonstrated a fundamental mechanism for the equilibrium preparation of many-qubit entangled states featuring scalable spin squeezing, based on the adiabatic preparation of low-field magnetized ground states for Hamiltonians breaking a continuous (U(1) or SU(2)) symmetry in the thermodynamic limit. At variance with the existing schemes for spin squeezing using collective-spin interactions [17][18][19][20][21][22], here we offer a specific protocol for the production of scalable spin squeezing using short-range qubit Hamiltonians with continuous symmetry, whose implementation is common to nearly all quantum simulation platforms. Our results are immediately relevant for Mott insulators of bosonic ultracold atoms in optical lattices, realizing the XXZ model with SU(2) symmetry or U( 1) symmetry (easy-plane anisotropy) -see e.g. the two relevant cases of 7 Li [32] and of 87 Rb [33]); and to Mott insulators of fermionic atoms, realizing the Heisenberg antiferromagnet [34,35]. In the bosonic case the Ω field coupled to the order parameter is a uniform, coherent Rabi coupling between two internal states; while in the fermionic case the field coupling to the order parameter must be staggered, and it can be potentially created by Stark shifting a sublattice of a square or cubic lattice by a superlattice, therefore creating a Rabi-frequency difference between the two sublattices. This scheme opens the possibility to squeeze the spin state of optical-lattice clocks in the Mott insulating regime (e.g. based on 87 Sr [29,30]). Our protocol (with a uniform Rabi field Ω) is also relevant for superconducting circuits realizing e.g. the 2d XX Hamiltonian [START_REF] Mi | Information scrambling in computationally complex quantum circuits[END_REF]; for Rydberg atoms with resonant interactions [START_REF] Browaeys | [END_REF], realizing the dipolar XX model ∆ = 0, J ij ∼ |r ir j | α with α = 3; as well as for trapped ions, realizing the XX model with long-range interactions (0 < α < 3) [45]. Our findings pave the way for the controlled adiabatic preparation of scalable spin-squeezed states, with the double bonus of a solid entanglement certification via the measurement of the collective spin; and of the possibility to accelerate the size scaling of phase-estimation error compared to separable states.

Acknowledgements. We acknowledge useful discussions with M. Tarallo and G. Bertaina. We particularly thank B. Laburthe-Tolra and L. Vernac for their contributions to the conception of this project, and for their careful reading of our manuscript. This work is supported by ANR ("EELS" project) and by QuantERA ("MAQS" project). All numerical simulations have been performed on the PSMN cluster of the ENS of Lyon.

Supplemental Material

Scalable spin squeezing from spontaneous breaking of a continuous symmetry

SPIN-WAVE THEORY FOR THE XXZ MODEL

In this section we describe the well-known linear spinwave (LSW) theory as applied to the XXZ model in an applied field. We introduce the linearized Holstein-Primakoff transformation

S x i = 1 2 -n i (4)
S y i = 1 2 √ 1 -n i b i + b † i √ 1 -n i ≈ 1 2 b i + b † i S z i = 1 2i √ 1 -n i b i + b † i √ 1 -n i ≈ 1 2i b i -b † i in which n i = b † i b i and b i , b †
i are bosonic destruction and creation operators -where the linearization assumes that n i 1 for all quantum states of interest. Under this transformation the XXZ Hamiltonian takes the form of a quadratic bosonic Hamiltonian

H ≈ E 0 + 1 2 k 2A k b † k b k + B k (b k b -k + h.c.) (5)
where we have introduced the Fourier transformed bosonic operators

b k = 1 √ N i e -ik•ri b i ; (6)
the mean-field energy

E 0 = - ij J ij 4 - N Ω 2 ; ( 7
)
and the coefficients

A k = γ 0 + (∆ -1) γ k 2 + Ω B k = (∆ + 1) γ k 2 γ k = 1 N ij J ij e ik•(ri-rj ) . (8)
The above Hamiltonian is diagonalized via the Bogoliubov transformation

b k = u k β k -v k β † -k (9) into H = E 0 + k k β † k β k + const. ( 10
)
where

k = A 2 k -B 2 k and u k = 1 √ 2 A k k + 1 1/2 v k = sign(B k )(1 -u 2 k ) 1/2 . ( 11
)
10 -3 10 -2 10 -1 10 0 10 1 10 2 Ω/J

10 -1 10 0 10 1 10 2 ∆E/J ED, L = 4 LSW, L = 100
2 Ω/J FIG. 4: Energy gap of the 2d Heisenberg model as a function of the applied Ω field: the prediction of LSW theory for a large system (L = 100) is compared to that of exact diagonalization (ED) for a small system (L = 4), and to the expected lowfield behavior ∆E ≈ 2J Ω/J , valid in the thermodynamic limit.

The observables relevant for the calculation of the squeezing parameter are then

J x = N 2 - 1 2 k v 2 k = N - k A k 2 k (12)
Var(J z ) = N 4 1 + 2v 2 0 + u 0 v 0 = N 4 A 0 + B 0 A 0 -B 0 .
The comparison between the predictions of LSW theory and QMC is already shown in the main text, as well as in the next section. Fig. 4 shows the LSW prediction for the spectral gap k=0 , compared to the exact diagonalization (ED) for a N = 16 square lattice, in the case of the SU(2) model with ∆ = 1. The exact results were obtained via the QuSpin package [46,47]. We observe that the gap predicted by LSW theory is very accurate at large fields; the deviation from the ED results at lower field is due to the saturation of the ED gap to a value ∼ O(N -1 ) due to the finite size, while the LSW gap closes as Ω 1/2 when Ω → 0.

BEYOND-SPIN-WAVE BEHAVIOR OF SPIN SQUEEZING IN THE 2D HEISENBERG MODEL

Here we discuss in more details the field scaling of the variance of the squeezed spin component, Var(J z ), in the 2d Heisenberg model and in the 2d XX model. Fig. 5 shows the field dependence of Var(J z ) multiplied by a factor (Ω/J ) -λ . A λ exponent which leads to a fieldindependent product corresponds to the field-scaling exponent of Var(J z ) at low fields. While LSW theory predicts λ = 1/2, we clearly observe that this exponent is inadequate for the low-field behavior of the 2d Heisenberg model -see Fig. 5(a). Instead the heuristic choice λ = 2/3 seems to be more appropriate, and we retain this as the low-field scaling of Var(J z ), clearly faster than the LSW prediction. This picture is to be contrasted with that of the 2d XX model -see Fig. 5(b) -for which the low-field scaling of Var(J z ) is captured very well by the LSW prediction (within a ∼ 10% accuracy).

QMC RESULTS FOR THE XX MODEL

In this section we show our results for the XX model (∆ = 0), relevant e.g. for the physics of superconducting circuits [START_REF] Mi | Information scrambling in computationally complex quantum circuits[END_REF] as well of ultracold spinful atoms in optical lattices [32]. Fig. 6(a) shows the field scaling of the spin squeezing parameter for the models in d = 1, 2 and 3 dimensions. Similarly to the case of the Heisenberg model discussed in the main text, the scaling of the spin squeezing parameter as ξ 2 R ∼ Ω 1/2 is observed in d = 2, 3, while it is absent in d = 1 because of the lack of spontaneous symmetry breaking.

Fig. 6(b) also shows that for d = 2 the ground state at finite Ω is a state of minimal spin uncertainty, for which ξ -2 R ≈ 4Var(J y )/N . Similar result is also found in d = 3.

GROUND-STATE VARIANCE OF THE COLLECTIVE SPIN COMPONENTS

In Fig. 7 we compare the variance of the three spin components J x , J y and J z in the ground state of the 2d Heisenberg model in a finite Ω field. We clearly observe that Var(J y ) is the largest among the three, meaning that the highest sensitivity of the state to rotations is achieved when the rotation axis is the y axis. Also, among all operators of the form O = i o i (with o i a local operator associated to a finite neighborhood of site i) J y = i S y i is arguably the one that has the largest variance: this can be deduced from the fact that the variance is the integral of the correlation function Var(O) = ij o i o jo i o j , and the correlation function which has the slowest spatial decay, leading to the largest integral, is the one of the order parameter in the SSB mechanism, corresponding to any spin component in the xy plane in the limit Ω → 0.

ADIABATIC SQUEEZING AT FINITE ENTROPY

Entropy curves can be obtained via QMC using a simple scheme of linear interpolation on the energy data. A direct result of the QMC calculation is the average energy per spin e(T ) = H /N . From a sufficiently fine grid {T k } of temperature values, we can then reconstruct the e(T ) curve by linear interpolation between two successive temperatures e(T k ) and e(T k+1 ), and therefore obtain the specific heat at the mid-point temperature

R

and 4Var(J y )/N for the 2d XX model for various system sizes. The coincidence of the two quantities shows that the ground state of the system is a minimal-uncertainty quantum state for all the field values we explored.

T k+1/2 = (T k+1 + T k )/2 as c(T k+1/2 ) ≈ e(T k+1 ) -e(T k ) T k+1 -T k . (13)
In order to obtain the s(T ) = S(T )/(N k B ) curve via integration of the specific-heat data, we can proceed by a linear interpolation of the specific heat between midpoints

c(T ) ≈ c(T k+1/2 ) + c(T k+3/2 ) -c(T k+1/2 ) T k+3/2 -T k+1/2 (T -T k+1/2 ) ( 14
) valid for T k+1/2 ≤ T ≤ T k+3/2 . This then allows us to estimate the entropy increment between two temperatures of the grid {T k } as

∆s k = s(T k+1 ) -s(T k ) = δs k,< + δs k,> (15)
where

δs k,< = T k+1/2 T k dT c(T ) T ≈ c(T k-1/2 ) - c(T k+1/2 ) -c(T k-1/2 ) T k+1/2 -T k-1/2 log T k+1/2 T k + c(T k+1/2 ) -c(T k-1/2 ) T k+1/2 -T k-1/2 (T k+1/2 -T k ) (16)
V a r ( J y ) / N Var(J x )/N and

δs k,> = T k+1 T k+1/2 dT c(T ) T ≈ c(T k+1/2 ) - c(T k+3/2 ) -c(T k+1/2 ) T k+3/2 -T k+1/2 log T k+1 T k+1/2 + c(T k+3/2 ) -c(T k+1/2 ) T k+3/2 -T k+1/2 (T k+1 -T k+1/2 ) . (17)
We have verified that this entropy reconstruction scheme delivers a similar result as that of a high-order polynomial fit of the e(T ) curve, as used e.g. in Ref. [48]; yet it has the advantage that no fitting procedure is involved.

Fig. 8(a) shows the resulting entropy map for the 2d Heisenberg model as a function of temperature and applied field. We then combine the entropy map with the squeezing map at finite temperatures shown in Fig. 8(b), to reconstruct the squeezing dependence on field and entropy, reported in Fig. 3 of the main text.

TIME-DEPENDENT SPIN-WAVE THEORY VS. TIME-DEPENDENT VARIATIONAL MONTE CARLO

In order to corroborate the results coming from timedependent variational Monte Carlo (tVMC), shown in the main text, we have compared them with an independent calculation, based on time-dependent LSW theory. This comparison is interesting especially at high Ω fields, for which, as we have seen in the main text and in the results above, the predictions of (static) LSW theory are fully quantitative.

As in the main text, we consider an evolution which starts from the ground state at an initial field Ω i = 10J , and ends at a final value Ω f with schedule Ω(t) = Ω i + F (t/τ )(Ω f -Ω i ), where F (x) is reported in the main text. The time dependence of the field Ω(t) translates into a time-dependent A k (t) parameter entering in the linear bosonic Hamiltonian Eq. ( 5). The evolution of the Gaussian state of the bosonic variables within LSW is fully described by the two correlation functions G k = b † k b k and F k = b k b -k , which evolve according to the equations (descending from the Heisenberg equations for the b k operators):

dG k dt = -2 (1 + δ k,0 ) B k Im(F k ) ( 18
)
dF k dt = -i(1 + δ k,0 ) [2A k (t)F k + B k (1 + G k + G -k )]
and from which our main observables of interest can be deduced

J x = N 2 - k G k Var(J z ) = 1 4 [1 + G k + G -k + 2Re(F k )] . (19)
Fig. 9(a)-(b) shows the comparison between the results of tVMC and those of LSW theory, obtained for the 2d Heisenberg model with L = 12, for final fields Ω f = 0.1J and 0.01J and ramp durations τ J = 20, 40 (same as those shown in Fig. 3 in the main text). In particular we show the evolution of the spin squeezing parameter ξ 2 R and of the magnetization per spin J x /N . We observe that tVMC and LSW agree perfectly for most of the ramp duration, while they deviate upon approaching the end of the ramp (t = τ ) and in the subsequent time evolution, during which Ω is held fixed at its final value. This deviation is easily understood within an adiabatic picture, namely from the fact that (static) LSW theory overestimates the ξ 2 R parameter (as shown in the main text), namely it underestimates -10 log 10 (ξ 2 R ) shown in the figure; while the tVMC results for quasi-adiabatic ramps produce values of the ξ 2 R which are systematically closer to the (static) QMC estimate. From this comparison we conclude therefore that our tVMC results, while not numerically exact, are fully quantitative, as they reproduce the LSW results at large fields (t < τ ); and, for quasi-adiabatic ramps, they oscillate around a value compatible with the QMC prediction for t > τ .

FIG. 2 :

 2 FIG. 2: Adiabatic squeezing from SSB in the 2d Heisenberg model. (a) Field-induced magnetization J x /N for various lattice sizes N = L 2 ; (b) Variance of the collective spin component J z ; (c) resulting spin-squeezing parameter ξ 2 R ; (d) comparison between ξ -2R and 4Var(J y )/N . In all panels the solid black line indicates the prediction of linear spin-wave (LSW) theory, and the dotted line in panels (b)-(c) shows the Ω 1/2 scaling as a reference (multiplied by an arbitrary prefactor).

FIG. 3 :

 3 FIG. 3: Quasi-adiabatic preparation of the low-field state. (a) Comparison between the field dependence of the squeezing parameter ξ 2 R for the ground state of the Heisenberg model in d =1, 2, and 3. For each value of Ω, we use a system size N such that Ω ≥ J /N , at a temperature T /J = 1/N removing thermal effects. (b-c) The two panels show tVMC results for the evolution of the spin-squeezing parameter in the 2d Heisenberg model (L = 12) along two field ramps starting from Ωi/J = 10 and ending at (b) Ω f /J = 10 -1 and (c) 10 -2 -see text for the ramp protocol. Each panel shows three different ramps for different ramp durations τ . The dashed lines show the ground-state spin squeezing parameters -obtained by variational minimization of the energy with the spin-Jastrow Ansatz. (d) Spin squeezing parameter vs. applied field and entropy per spin in the 2d Heisenberg model, L = 24.

FIG. 5 :

 5 FIG.5: Variance of the squeezed collective-spin component Var(J z ) as a function of the applied field for (a) the 2d Heisenberg model and (b) the 2d XX model. The symbols are QMC data, and the solid line corresponds to LSW theory. The variance is multiplied by the factor (Ω/J ) λ so as to single out its power-law vanishing behavior as Ω → 0.

FIG. 6 :

 6 FIG. 6: Adiabatic squeezing in the d-dimensional XX model. (a) Spin squeezing parameter for various dimensions from simulations: ξ 2 R is found to scale to zero as √ Ω in the presence of spontaneous symmetry breaking (SSB) in the ground state for d = 2 and 3, while it does not vanish in its absence (d = 1). (b) Comparison between the inverse spin squeezing parameter ξ -2

FIG. 7 :

 7 FIG. 7: Variance of the three components of the collective spin in the ground state of the 2d Heisenberg model (QMC data for a L = 20 lattice). The dashed line marks the shotnoise limit Var(J α )/N = 1/4.

FIG. 8 : 2 RFIG. 9 :

 829 FIG. 8: Entropy and squeezing maps for the 2d Heisenberg model. (a) Entropy map in the (Ω/J , T /J ) plane; (b) squeezing map in the (Ω/J , T /J ) plane. All the QMC data shown here were obtained for a L = 24 lattice.


M Si

R Lozeva

H Naïdja

A Blanc

J.-M Daugas

F Didierjean

G Duchêne

U Köster

T Kurtukian-Nieto

F Le Blanc

P Mutti

M Ramdhane

W Urban

I Homm

F Ibrahim

L W Iskra

S Jazwari

J Jolie

A Korgul

P Koseoglou

Th Kröll

L Le- 774 Meur

J Ljungvall

A Lopez-Martens

I Matea

L Matthieu

K Miernik

J Nemer

S Oberstedt

W Paulsen

M Piersa

Y Popovitch

C Porzio

L Qi

D Ralet

P H Regan

D Rey- 777

Gadas Tello

K Rezynkina

V Sanchez

C Schmitt

P.-A Söder- 778

C Sürder

G Tocabens

V Vedia

D Verney

N Warr

B Wasilewska

M. S J Wiederhold

F Yavahchova

Zeiser

E A Stefanova

F Nowacki

T Rzaca- 782 Urban

J Wisniewski

I Ahmad

G De France

H Faust

J P Greene

G Simpson

A G Smith

T Soldner

G Häfner

M Lebois

N Jovancevic

D Thisse

D Etasse

R L Canavan

M Rudigier

J N Wilson

E Adamska

P Adsley

A Algora

M Babo

K Belvedere

J Benito

G Benzoni

A Blazhev

A Boso

S Bottoni

M Bunce

R Chakma

N Cieplicka-Orynczak

S M Collins

M L Cortés

P J Davies

C Delafosse

M Fallot

L M 792 Fraile

R.-B Gerst

D Gjestvang

V Guadilla

K Hauschild

C Henrich

L Le- 795 Meur

S Leoni

M Piersa- 797 Silkowska

P H Re- 798 Gan

D Reygadas Tello

V Sanchez-Tembleque

P.-A Söderström

V Ve

New -decay spectroscopy of the 137 Te nucleus

Background: Nuclear structure of the neutron-rich isotopes beyond 132 Sn is investigated.

Purpose:

The level scheme of 137 I is obtained after decay of 137 Te. Transitions in 136 I are detected after delayed neutron emission of 137 Te. The half life of 137 Te is measured.

Methods: -delayed -ray spectroscopy is employed for neutron-rich 137 Sb and 137 Te isotopes, produced at the ILL after neutron-induced fission to populate excited states in 137 I.

Results:

The new decay level scheme of 137 I is established. The half lives of 137 Sb and 137 Te are determined as 0.57(26) s and 2.46(5) s, respectively. The -delayed neutron emission probability P limiting value of 137 Te is deduced to be 2.17(66)%.

Conclusions:

The experimental results are an important input to the theoretical description of nuclei in the region, being well interpreted within LSSM calculations and provide essential information on the firstforbidden transitions beyond > 82 and > 50. 56 core and could lead to collective behaviour in 137 I, a quantity 57 of information on the shell evolution of nuclei is expected at 58 such extreme proton-neutron ratios. However, these I nuclei, 59 similarly to their Te isotones, have not shown a very collective 60 behaviour and a rather similar trend of excitation energies and 61 transition rates of their first excited states [8] with the increase 62 of . The three valence protons in the Iodines [7] were also re-63 garded as a cluster [9] to explain some features of these nuclei 64 [10]. It is a very interesting exploration field though only the 65 yrast states could be studied in both thermal neutron-induced 66 235 U-fission [11] and energetic neutron-induced 238 U-fission 67 [8], while many of the non-yrast states could not be populated 68 by this reaction. Based on the knowledge of the only previ-69 ous -decay work [12] with relatively scarce data on = 137 70 as compared to its = 135 isotope, only 5 excited states up 71 to excitation energy of 1169 keV with 9 -ray transitions in 72 137 I were associated to the 137 Te decay scheme [13]. There-73 fore, new results on 137 I are highly demanded. More recently, 74 benefiting from the development of experimental technology 75 with radioactive neutron-rich beams [14-16], the majority of 76 neutron-rich nuclei are investigated by spectroscopy of the fis-77 sion products directly. In this work we report on the decay of 78 137 Te, which is itself produced by the thermal neutron-induced 79 fission reaction on 235 U.

80 II. EXPERIMENT 81 The exotic neutron-rich 137 Sb isotopes are produced at the 82 ILL reactor, employing a thermal-neutron beam inducing fis-83 sion onto a 235 U target. The projectile fission fragments of 84

I. INTRODUCTION AND MOTIVATION

Close to the double shell closures in the nuclear chart with one of the good examples being the 132 50 Sn 82 doubly-magic nucleus, the nuclear shell model stays a major structural framework to understand such magic nuclei and their neighbours.

By investigating the nuclear configuration of states around the 132 Sn core, the extension of the magic core may be traced and the polarization effect of valence particles on the 132 Sn core may be studied in detail [1]. Valuable information on the nucleon-nucleon effective interaction and single-particle excitation energies may also be obtained. Furthermore, with increasing the neutron excess for these nuclei, a variety of new phenomena are predicted as the existence of neutron skin, vanishing of standard magic numbers or opening of new sub-shell gaps [2][START_REF] Shergur | and the ISOLDE Collaboration[END_REF][4][5]. These phenomena challenge recent competitive studies and boost the quest for new data to more and more neutron-rich species. In our previous review on 136 Te which has two valence protons and two valence neutrons outside the doubly-magic 132 Sn core, deviations of the observed transition rates from the ones predicted by the shell-model calculation are found [6]. It is of great interest to investigate the Iodine isotopic chain, where in our review in [7] of 136 Te decay, the excited levels of 136 I were established and compared to the shell model. In the present work new excited states also are identified in three valence protons systems beyond the magic number = 50. One of them, the 137 I nucleus, which has two valence neutrons outside the closed neutron shell = 82 is of particular interest since also its low-spin excitations have not been explored in detail, while being very important for the nuclear structure in the region. Intermediate-spin states are addressed in our review on 135-139 I nuclei [8].

The excess of valences particles could polarize the 132 Sn interest are selected and identified by the LOHENGRIN [17] separator for two separate data sets tuned on / setting for = 21 and = 25 to be able to disentangle more easily background from the radiation associated to the nucleus of interest. Further, the radioactive species are implanted onto our user detection system for a decay type of measurement. The detection setup consists of the LOENIE [18] -decay station of plastic detectors in 4 geometry, placed around a supported movable Al-coated Mylar tape, evacuating the radioactivity from the implanted ions.

The measurement cycle is adapted to the half life, T 1/2 , of the isotopes of interest. In this work, the time chopper is set to 3 s for injection, 3 s for measurement and 2 s for tape movement. This information is further used to extract the experimental T 1/2 value (see Sec. III A). Prompt and -delayed rays emitted from the implanted isotopes could be detected by two Clover detectors (with Anti-Compton (AC) shields, used as passive shielding) and a coaxial HPGe detector in a very close geometry. Data is collected by a standard triggerless VME electronics. Further experimental details, including information on the -ray detection efficiency are given also in Ref. [19]. As appropriate for the studied energy range, these were also cross-checked with the experimental relative efficiencies of the = 136 daughter nuclei [7].

For the = 137 study reported here, it is relevant to state, that isobars and long-lived contaminants as 137 I, and 137 Xe (and 142 Ba in = 25 case), produced directly in fission may also reach the experimental station, as the LOHENGRIN separator does not distinguish between isobars abundant quantities of the longer-lived isobars. Such activities are taken into account in the analysis as they are already well known and mostly contribute as a long-lived background.

III. DATA ANALYSIS AND RESULTS

A. Half life of 137 Te

Due to the very short half-life of the implanted 137 Sb ions and its exoticity with respect to the daughter products, the delayed spectra are dominated by these 137 Te, 137 I daughter/grand-daughters. Also, as explained above such = 137 contamination is present in the initially-implanted ions as they are produced stronger in the fission. The -ray energy after decay and the corresponding timing information, based on the beam chopping are used to construct the "time chopper-energy" matrix. This matrix allows to monitor the activity of -rays within the measurement cycle itself.

By gating on -delayed -rays in 137 I, we obtain the decay curve of 137 Te. To demonstrate this and in order to increase the statistics, we summed up the time distributions of already known -rays in this nucleus as 129 keV, 243 keV, 469 keV and 554 keV and fitted the half life using minimization procedure. As shown in Fig. 1, the contribution of the daughter 137 Te is dominating the time curve. While any 137 Sb contribution is minor (weakly produced and shorter) and may be neglected, the grand-daughter 137 I nuclei may contribute to the background and can be accounted either directly with the known T 1/2 in the Bateman equations fitting [20], or as a long- With the aim to expand and establish a new level scheme of 137 I from this new decay data, -, -(mutual) coincidences and intensity balances are used. The -coincidence relations are constructed between the Clover detectors and the coaxial HPGe detector. For all these, we also used a coincidence with the detected signal to suppress the prompt -ray background (similar information is given also in Ref. [19]).

In Fig. 4, we show a spectrum gated on the 243. present, in addition, their gated spectra, illustrated in Fig. 4.

The weakly seen 2047 keV -ray in Fig. 2, is marked as tentative. The new 227.1-keV transition, found in the 243.6 keV gated spectrum, shown in Fig. 4, is also in coincidence with the 229.1-keV line (in the 227 keV gated spectrum, see Fig. 6), and all transitions de-exciting the level 373.1 keV such as the 129.5-keV line). As its energy, in addition, fits the energy difference between the 600.6-keV and 373.1-keV levels, we place it as a new transition in the level scheme (see Fig. 7). The In order to obtain the 137 I feeding, its further decay to 137 to those in the 0-1 s of the Te decay, where only 3% follows decay (due to its 1/2 ). Analysing these portions showed that the -rays from Xe in this time range correspond to 20% of those in the entire decay and thus one may infer that 80% of the Xe actually comes from the Te decay and 20% is from direct production. Therefore, correcting for the above factors, the total error for the I g.s. feeding rises of up to 22% as reported in Table II. This reflects also the experimental log uncertainties of particular levels as a systematic error, while the statistical errors contribute to below or around the percent level in this measurement. It may be noted also that as several transitions in = 137 are in the energy range relevant for conversion, this is also investigated. While some of these transitions, as the 243.6 keV the multipolarity comes from previous measurements [11,28], for the others, the possible 1 and 2 assumptions can be made. However, while most of these transitions are weak in intensity and their proportion is found to be minor, for two relevant -rays of 129. Table II: Excited levels and transitions in the decay of 137 Te to 137 I. The initial level and its related information are given in the first four columns. The ray and its absolute intensity are provided in the fifth and sixth columns. The last two columns reveal the final level and its spin/parity. The superscript stands for a new transition, while marks also an assumption for uniqueness [START_REF] Emeric | Calculation program of log ft[END_REF]. The 620.5 keV state, observed in this work represents a single transition to the ground state. However, we could not detect any coincident transitions to represent feeders from above.

E (
Thus, its intensity should be mostly coming in direct feeding. Such scenario would not be entirely consistent with an earlier proposed spin/parity of 11/2 + and log value of 6.8 (1) evaluated in this work, therefore according to our data Δ < 2 possibility shall not be excluded. For example, if the presumed 11/2 + level would be a 9/2 + candidate, the 620.5-keV transition may have some mixing and this would not conflict several deviations from the theoretical angular correlation coefficients from the previous observation in Ref. [11]. Such scenario would include the margin in the intensity, observed in this experiment and respectively may be also suggested for the 620.5-keV level.

The 554.2-keV [13] state was set to 9/2 + based on the suggested mixed 1 + 2 multipolarity of the 554.2-keV transition, observed in Ref. [11], with respect to the 400.8-keV transition.

Both were placed in the level scheme in coincidence, that we indeed also observe in Fig. 4). The next candidate for the second (7/2 + ) state is the previously observed 713.5 keV state, de-exciting by three transitions, that we also observe, with the strongest one connecting to the 243.6 keV state with a spin/parity 5/2 + Ref. [11,12].

Its experimental I and log also suggest (7/2 + ) assignment.

However, such identification would be somewhat in variance with the SM prediction that expects the second 7/2 + level at 571 keV excitation energy, while the third 7/2 + state is calculated at 774 keV (see Sec. IV). Although they all are predicted with the same configuration, there is no clear reason why they would be compressed, except the possibility of unaccounted mixing in the configuration of these states. The third candidate for such spin/parity, according to our data, is at excitation energy of 1170.1 keV. This is very well probable and matches the findings of Ref. [12] to our data, where we observed three more transitions de-exiting this level (see Fig. 7). The level must have been among the strongest ones observed in that first -decay study and their proposed spins as 5/2 + and 7/2 + would be the favoured ones from the 7/2 -g.s. of the mother nucleus, thus very consistent.

The two levels at 830.2 keV and 853.2 keV have similar I and log values though lower than expected for spin/parity of 7/2 + . Besides, their interconnecting transitions to the lowerlying 5/2 + and 9/2 + states, make this 7/2 + possibility less adequate. Therefore, we propose both (5/2 + ,9/2 + ) as spin/parities assignments. This would agree with the SM results for 9/2 + and be underestimated for a 5/2 + , while the third 7/2 + and 3/2 + states, expected in the vicinity at 774 keV and 900 keV, require different experimental branching.

The next level which experimentally has strong I feeding and can be the (7/2 + ) state is located at 1170.1 keV, previously observed and matching the forth 7/2 + predicted at 1085 keV.

The state at 1451.9 keV which has similar characteristics may have the same origin, thus also proposed as a (7/2 + ) state.

Although the (5/2 + ) possibility cannot be completely excluded this would require more data to search for feeding branches that we could not observe. Interestingly, the five states we suggest with spin/parity 5/2 + or 7/2 + ( < 1.2 MeV), being the strongest fed, are communicated though left unassigned in Ref. [12].

In the data presented here, it can be seen that four similar new excited states at 830.2, 853.2, 1435.9 and 1451.9 keV are established. According to their possible de-excitation transition multipolarities, the spin/parities are consistent with the earlier proposed (9/2 + ) spin/parity, while due to the relatively low log value a (5/2 + ) possibility may be added or even (7/2 + ) for the later one.

Some spin-parity assignments suggested previously [11],

were based on the possible proton-neutron SM configuration (see Sec. IV). A possible decay of 137 Te to 137 I is expected to be primarily originated from the conversion of a neutron from in agreement with the suggested from systematics spin/parity 429 [12,13]. Three more levels with suggested spin/parity of 430 (7/2 + ) could be identified in the level scheme with log values 431 between 6.0 and 6.6.

432

Other levels candidates e.g. with a log 6.9(1) and a 433 smaller I of 2.4-2.8, possibly correspond to Δ = 1, 2 for 434 these, spin/parity of (3/2 + , 5/2 + ) would be more appropriate.

435

For about six levels with log ranging from 6.2 to 7. clei [32]. The 0 13/2 neutron and 2 1/2 proton orbital energies 474 are inferred from Ref. [33] and Ref. [34], respectively. The III.

486

For the states of interest, illustrated in Fig. 8, also the main

487

SM configurations are noted. The probable correspondence, 488 whenever possible, may also be seen (with the color code).

489

It is interesting to remark immediately that for these states, odd-odd to an odd-even system and finally to its structure.

506

As it can be seen from the current 137 Te. The new levels and transitions are in red (color online), tentative transitions are with dotted line. The Q( -) of 137 Te is taken from Ref. [25]. For the levels marked with (*), some assumption for uniqueness may be possible (see text, Table II).

While reporting on the first excites states in the 137 Te mother nucleus, in Ref. [28], some ℎ 9/2 ( 2 7/2 ) contributions are accounted e.g. for the yrast 9/2 -states, including the g.s.

together with the three valence give similar conclusions for this odd-decay.

A. The 7/2 -g.s. of 137 Te The g.s. spin/parity of the -decaying 137 Te nucleus can rather firmly be set to 7/2 -based on the experimental log values in the = 137 I daughter and very strong branch to its 7/2 + g.s. The direct g.s. to g.s. feeding is also relatively strong in this nucleus with about 47%. It behaves completely analogous to the = 135 nucleus with similar experimental strength [12], from where similar spin/parities can be concluded. The g.s. of 137 Te was suggested with quite some certitude in Ref. [11] based on the observation of a large part of its yrast scheme (up to spin/parity of (33/2 + )), multipolarity of several transitions connecting with the g.s. and SM calculations based on the Kuo-Herling (KH5082) [41] interaction, reasonably describing 137 I as the one proton coupling to the 136 Te nucleus.

The perfect match of this assignment for the 137 I ( = 53, = 84) isotope to systematics of heavier-Cs, La isotone nuclei is found in a follow-up fission work [42], where the entire excitation scheme could be built in a complete analogy to the 139 Cs ( = 55, = 84) isotone and in agreement with the data on other N=85 isotones [43]. These excitations are found also to be well described by the two-body CD-Bonn SM interaction using experimental single-particle energies from 133 Sb and 133 Sn nuclei [1]. The most strongly fed 7/2 + states from the direct feeding of the 7/2 -g.s. state are naturally in a good agreement with the = 7/2 assignments. Any other different spin/parity would not present these strong branches well visible in the level scheme (Fig. 7) and this certainly excludes the = 5/2 for the g.s. as e.g. the feeding to the 243.6 keV and 713.5

keV states is clearly different. The log values of the newly identified (7/2 + ) states are also of the order of 6 and the first excited after the g.s. seems very likely at 713.5 keV excitation energy, based on these experimentally established feeding, although slightly underestimated in energy by the theory (Fig. 8). Interestingly, in the earlier decay work, mostly the assigned as 7/2 + states are observed and where the next such candidate, previously set but unassigned at 1169.0 keV [12],

is confirmed here at 1170.0 keV. In this work, we propose one more candidate for 7/2 + at 1.4 MeV excitation energy (Fig. 8).

These states are purely based on the proton 3 7/2 configuration with the exception of the slightly competing 2 7/2 5/2 contribution for the 7/2 + 2 state and their theoretical correspondence is relatively good as it can be seen in the Figure.

As stated earlier, the non-observation of states containing GT strength, confirms that this channel is still blocked for 137 I nucleus and that the decay of the 137 Te is completely dominated by the first-forbidden transitions from the dominating

Figure 1 .

 1 Figure 1. Time distribution of 137 Te -decay events fitted with the resulting half-life.

140

  lived activity background with very similar outcomes. The 141 resulting -decay T 1/2 are reported in Table.I together with 142 the values from previous measurements[13]. The uncertainty 143 in the current results comes from the statistical error and the 144 fitting. The result on the 137 Sb comes from the limit of this 145 observation which is the 61.8 keV line (Fig.2), corresponding 146 to the deexictation of the excited state in 137 Te. We have to 147 note however, that it can be still polluted by some unidentified 148 origin as we could not observe any coincidence transitions, 149 e.g. those reported in Ref.[21], possibly due their much 150 lower intensity and the overall low production of 137 Sb. Thus, 151 the result can be taken as an experimental limit justifying an 152 agreement with previous observations. 153 154 B. 137 Te decay to excited states in 137 I 155 In the previousspectroscopic study of 137 I, 5 excited 156 levels and 9 transitions were established [13], far below the 157 value of 7053(9) keV (see Sec. III D). In the present work, 158 -delayed -ray spectrum of 137 I is obtained by applying -159 coincidence conditions to all Ge data. The -gated -ray 160 spectrum for different / settings on = 21 and = 25, as 161 taken in our measurement, is shown in Fig.2. One can clearly 162 see, as marked by their energies, that in both settings the 163 transitions from 137 I are well identified. The main background 164 comes from the 137 Xe grand-daughter nuclei, while for = 25, 165 also 142 Ba contributes to the background. In order to subtract 166 the background from long or short-lived nuclei, the analysis are 167 performed under the condition that the selected time window 168 corresponds to the chopper decay part (see Fig.3). Transitions 169 from the previous measurements, -decay of 137 I[12] and fission[8,11] are observed in this work.

Fig. 5 -

 5 (a) shows how the 241 known 243.6-keV transition behaves as an example, while the 242 1435.9 keV-transition is assigned as a new transitions regard-243 ing this criteria. Fig.5-(c,d) show how transitions from the 244 daughters 137 Xe and 137 Cs behave. Due to their long 1/2 the 245 statistics within the decay part remains basically the same for 246 these time projections. This criteria is used to cross-check all 247 the new transitions observed in singles or in gated spectra and 248 helps to reject wrongly suspected candidates. 249 To assign new transitions to the level scheme of 137 I we 250 used the following criteria: a) the same timing; b) mutual 251 coincidence relations with known or new transitions; c) the 252 same relative intensity in = 21 and = 25 settings; d) not 253 identified to the background or the contaminants. According 254 to these criteria, 17 excited levels could be established in this 255 work with a total of 32 -ray transitions following the decay 256 of 137 Te to 137 I. Among these, we observe for the first time 257 8 new and one tentative levels with 18 new and two tentative 258 transitions. Table. II summarizes the information about the 259 excited levels and transitions associated with this decay 260 to the 137 I nucleus. The -rays intensities are normalized 261 to the strongest 243.6-keV transition, and obtained using the 262 background subtracted full-energy peak areas from -gated 263 -ray spectrum selecting the time chopper decay part. The 264 intensity of each level is extracted from the feeding and 265decaying transitions of this particular state. Its log value 266 is calculated using the Q( -) = 7053(9) keV from the atomic 267 mass evaluation (AME2020[25]) and Ref.[START_REF] Emeric | Calculation program of log ft[END_REF]. 268 C. 137 Te feeding to the 137 I ground state 269

Figure 2 .

 2 Figure 2. The -gated -ray singles spectrum obtained following the -decay of 137 Te for both settings. Peaks belonging to 137 I are marked by their energies. The main background comes from 137 Xe grand-daughter nuclei and a 142 Ba contaminant for = 25.

Figure 4 .

 4 Figure 4. Coincidence -ray spectra gated on different transitions. All marked peaks are in mutual coincidence.

Figure 5 .

 5 Figure 5. Energy spectrum corresponding to the decay of the nucleus chopped at each second. Blue, red and green (color online) are for the first 1 , 2 and 3 second, respectively.

475

  N3LOP effective interaction[35] is used in this work. It was al-476 ready demonstrated in several articles to successfully describe 477 the spectroscopic properties and collectivity of nuclei in the 478 region beyond 132 Sn, quoting as examples the Refs. [7, 8, 35-479 37]. The diagonalization of the considered system is achieved 480 using the NATHAN shell-model code [38]. In this work, up 481 to several excited states for the spin/parity/energy range of in-482 terest are calculated and compared to the proposed spin/parity 483 assignments from the experiment in Fig. 8, the important 484 main configurations for the positive (+) parity states are listed 485 in Table

490

  the configuration is dominated by the proton configurations 491 and, especially, the presence of the 5/2 proton in the very 492 close vicinity of the 7/2 proton, inferred from the main oc-493 cupation of these orbitals. Almost no neutron excitations are 494 found to take place in the detailed analysis of the wave func-495 tions for the examined states, even for the highest in excitation 496 energy. Note that reviewing higher-lying states would not be 497 relevant to the observed experimental data. The details of the 498 main configurations for 137 I may be traced for the computed 499 states in Tab. III. This is certainly very different from the ob-500 served sequences in the 136 I neighbour, where various neutron 501 excitations could be seen [7], including some moderate 13/2 502 occupation (see e.g. Fig. 13). Such behaviour may be at-503 tributed to the proximity of the = 82 closed shell in the 504 former case, to the fast step with the extra neutrons from an

  505

Figure 8 .

 8 Figure 8. Experimental levels in 137 I compared to the theoretical shell-model (SM) calculations. The color code (online) indicates the main theoretical configurations and some of the possible experimental correspondences.

  g.s. This is in full agreement with the SM calculations that also do not expect such branching at low energy and the first possible transition between the 7/2 -( 137 Te) → 5/2 - ( 137 I) is expected at (5/2 -)=3.8 MeV with ( ) of about 0.007, which is not found experimentally. It would be further interesting to investigate whether this strength can again be unblocked e.g. due the expectations of more mixing and an evolving collectivity also in the heavier neighbours. V. SUMMARY 709 In this work, we performed new -decay spectroscopy of 710 the 137 Te nucleus to the daughter 137 I. We observed several 711 new transitions corresponding to first-forbidden branches to 712 excited states in 137 I. Due to the relatively weak ratio and 713 insufficient statistics, the transitions after -delayed neutron 714 emission were strongly hindered and only few indications of 715 states in 137 I could be seen. Therefore, such investigation 716 would require more experimental data. 717 From the investigations performed in this work, it is in-718 teresting to highlight the non-observation of GT strength for 719 this = 137 isotope, while a possible even-odd effect with 720 decrease of the GT strength at the expense of favoured first-

Table I

 I

	. -decay half-lives measured in this work compared with
	literature values
	nucleus	T 1/2 (s)	T 1/2 (s)
	137 Sb	0.57(26)	0.566(52) [22], 0.492(25) [22],0.507(27) [21]
	137 Te	2.46(5)	2.49(5) [12], 2.1(1) [23], 2.08(40) [24]

  the 554.2-keV transition is mostly an 1 type, mixed with some 2 multipolarity, the required Δ = 1 branch would connect similarly strong a 5/2 + to the 7/2 + ground state. Note that the SM calculations (see Sec. IV) predict, that the second 5/2 + state may be expected much below the first 9/2 + state and such a state would be populated directly and relatively strongly from the 7/2 -g.s. decay.

	350
	351	imental data, due to visible	intensity and relatively well
		fitting log values, the previously assigned 11/2 + and 13/2 +
	354
	355	As we detect the 554.2-keV transition as the second
	356	strongest, relative to that of the first excited state (also pre-
	357	viously identified with spin/parity of 5/2 + [12]), this gives
	358	quite some certitude for the spin of the originating state. It
		does well fit the spin/parity of 7/2 + for the g.s. and is in a very

Similarly, if the 400.8-keV 346 transition would originate from a 11/2 + instead of the 13/2 + , 347 suggested for the 955.0 keV level, some possible mixing due 348 some deviations in angular distribution, polarization data or 349 alternation of spins may need to be assumed. In this exper-352 states are now given possible alternative assignments in the 353 level scheme, shown in Fig. 7 and Table II. 359 good agreement with our shell-model (SM) expectations (see 360 Sec. IV) that places the first 9/2 + state at 554.2 keV excita-361 tion energy. Furthermore, looking at the I and log values 362 for the 554.2-keV level, some 5/2 + possibility cannot be com-363 pletely excluded. This tendency actually comes from the fact 364 that if

  Figure 6. Coincidence -ray spectra gated on the 247 keV and 569 keV transitions. All marked peaks are in mutual coincidence.

		Counts/keV	0 10 20 30	129.5	229.1	243.6	373.0	Gate 227 569.5	0 10 5	129.5	BG	227.1	243.6	BG	Gate 569 357.2 373.0
		10 -
		-20						5 -
			100		200	300	400	500 Energy/keV 600	100	150	200	250	300	350 Energy/keV 400 450
		137 I. Only one state at 87.3(7) keV with single transition to the
	460	g.s. is observed in this work. The resulting	amounts to
		2.17(66)%, compared to the latest literature value of 2.6(3)%
	463 464
		IV. SHELL-MODEL CALCULATIONS AND DISCUSSION
	469
	470	tire major shell 1 7/2 ,0ℎ 9/2 ,1 5/2 ,2 3/2 , 2 1/2 ,0 13/2 orbitals
		for neutrons and 0 7/2 ,1 5/2 ,1 3/2 ,2 1/2 , 0ℎ 11/2 orbitals for
	473
								443
								444	with 18 new transitions (with 2 tentative) are added to the
								445	revised previous knowledge. The tentative rays with 600 and
								446	2047 keV energy which are hardly distinguishable in -delayed
								447	-ray spectrum of 137 I are placed as they appear in gated
								448	spectrum and, moreover, fit the energy differences between
								449 450 451	already established levels.
								452					E.	branch
									Since the S of 137 Te is 2950(3) keV, within the Q -window
								458

1, we 436

propose the spin/parity of (5/2 + ) to be the most suitable can-437 didate. In addition, based on the branching, possible spins 438 and multipolarities it may be assumed that some levels are 439 candidates for Δ = 2 which suggests some uniqueness when 440 appropriate as marked in Table

II

.

441

The proposed level scheme from this work is illustrated 442 in Fig.

7

. Nine new levels (with one tentative among them) 453 of Q =7053(9) keV [25, 30], -delayed neutron emission 454 is possible for the decay of the 137 Te nucleus. To obtain 455 the -delayed neutron emission probability, , of 137 Te, we 456 consider the known decay of 136 Te [7]. The known -rays 457 in 136 I are regarded with the same criteria as for transitions in 461 and the evaluated database value of 2.91(

16

)% in Ref.

[31]

may 462 be regarded as lower limit for the knowledge of this quantity.

465

The experimental results are compared to theoretical calcu-466 lations from large-scale shell-model calculations (SM) calcu-467 lations. The 4ℎ -5 model space with the 132 Sn nucleus 468 as a core is used. The model-space, thus, consists of the en-471 protons. Single-particle energies for neutrons and protons are 472 taken from experimentally known data on 133 Sn and 133 Sb nu-

*

  Table, the purity of the twoFigure 7. The proposed level scheme of 137 I obtained from decay of

	85 I β─ log ft 0.2 7.3 (3/2+, / +) 9 2 52 Te 137 / 7 2 ─ 0.0 β ─	2047 2170.2 Q ─ =7053(9) keV	T 1 2 / =2.49(5) s	2290.6 2413.8
	0.1 7.7(3/2+, / +) 11 2	1833.7		2077.3
	0.8 7.1 (5/2+,9/2+) 2.5 6.6 ( 7/2+)	1155.4 1435.9 600 897.7 1078.6 1452.0	1435.9 1451.9
	2.9 6.6 0.3 7.6 (3/2+, / +) (7/2+) 9 2 0.2 7.7 (3/2+, / +) 11 2		569.5 797.6 925.9 1170.0 974.4	1170.1 1218.0 1399.0
	6. 2. 6.9 . 4 7 0 4 2. 6. 5 8 / +,1 9/2+ (3/2+, / +) 5 2 9 2 1/2+ 14 8 0 . 6. (7/2+) 1.5 7.0 (5/2+,9/2+) 1.5 7.0 (5/2+,9/2+) 0.3 7.6 11 2 / +,13/2+			554.2 227.1 357.2 600.9 620.5 340.9 469.7 713.2 229.1 276.2 830.6 139.7 299.0 609.6 400.8	554.2 600.6 620.5 713.5 830.2 853.2 955.0
	12 8 . 6. 2. 6.9 2 8	5/2+ (3/2+, / +) 5 2		517 518 519 520	proton excitations containing the 5/2 orbital shows clearly a competition with the 7/2 orbital for excited states, and is also the second important contribution (with about 20%) in 243.6 373.1 243.6 129.5 373.0
						the 7/2 + g.s. of 137 I.
	47 4 .	5.7	7 2+ /		523 524 525	mother 137 Te nucleus. The presence of this ℎ 9/2 component and its decay to a state with ℎ 11/2 component in the daughter 84 53 I 137 0
					526	is the main responsible transition for a GT strength at the
					527	beginning of this major shell. In the neighbouring 136 I case,
					528	several 1 + states originated from such main configuration are
					529	populated, with one very strong branch, as predicted by the SM
					530	at about 2 MeV excitation energy [7]. Similar range is recently
					531	suggested also for 138 I in Ref. [39]. Here, for = 137, the
					532	ℎ 9/2 component is really minor with a theoretically calculated
					533	proportion of only 2.5%. Note, that while present in = 136
					534	with about 10%, it is completely blocked e.g. for 135 I. It is
						worth to highlight, that for	= 140 of the same isotopic
					540

507

main configurations in the low-energy states of 137 I is not small 508 and, therefore, the other contributions appear with relatively 509 minor weight in the wave functions. We present these results 510 in accordance with the excited states that are candidates for the 511 experimentally-populated spin/parity. In the Table

,

we give 512 up to 10 excited states for the 3/2 + candidates for populated 513 states at intermediate energy, and up to 6 excited states, as 514 relevant for the spin/parities 5/2 + , 7/2 + , 9/2 + , 11/2 + and 13/2 + 515 (see Fig. 8). One may remark, that the neutron excitation to 516 the 3/2 orbital is applicable only in a very few cases. The 521 Another important point is the very small contribution of 522 the ℎ 9/2 orbital in the wave function of the 7/2 -g.s. of the 535 chain ( 140 I), this probability is again enhanced with respect to 536 = 137, based on the more mixed g.s. configuration on one 537 side and due to the lowered excitation energy of the GT states 538 themselves, on the other, as it can be seen from the results in 539

Ref.

[40]

. * *

  -and 5/2 -states, however they were not observed in the same study, while the position of these states is relevant in the situation of137 Te nucleus. These states could be seen only in the more recent -decay of 137 Sb → 137 Te

	) (3/2 ) ) (3/2, 11/2 (3/2, 9/2	2077 2290 2413					3/2 3/2 3/2	2054 2219 2339				f main configurations g d 7/2 2 7/2 2 3 f ν g π 7/2 2 7/2 5/2 ν π
							3/2 3/2	1843
							3/2									11/2
							3/2
							3/2							9/2		11/2
	(3/2, 9/2 ) (3/2, 11/2 )	1218 1399 (5/2, 9/2 )	(7/2 ) 1435 (7/2 )	1170 1451		3/2 3/2	1171 1404	5/2	1384	7/2 7/2	1085 1269	9/2 9/2	1286	11/2 11/2	13/2	1267
		(5/2, 9/2 )	853		11/2, 13/2	955	3/2			5/2 5/2 5/2	931 1037	7/2		9/2	1008	11/2	13/2	960
		(5/2, 9/2 )	830	(7/2 )	713							7/2		9/2	757
	(3/2, 5/2 )	600		9/2	9/2 ,11/2	620						7/2	571	9/2	567	11/2	588
					554		3/2	464
	(3/2, 5/2 )	373
		5/2	235				3/2	288	5/2 5/2	188
				7/2	0						7/2 7/2	0
			exp				137 53 84 I						SM
	3 7/2 configuration. These, as well as core vibrations coupled to them, were suggested to originate the 3/2 549 550 551 553 554 555	[21], where the first negative parity states, expected with such configurations are found at relatively high in energy and thus not among those populated in the experimental -decay results. branch ℎ 9/2 → ℎ 11/2 could be experimentally observed. As such contribution to the expected configurations is minor also
								556	in the examined 137 Te→ 137 I study here, and none of the excited
								557	states seem to present such a strong transition branching, we

552

It could, therefore, be concluded that at = 137 no allowed

  It is the position of the 3/2 + states that is well matching values of 24.2/16.4 and 6.3/6.4. In 137 I, two 657 5/2 + states with 5/2 proton are predicted among the first six 658 in about 1 MeV energy spacing, thus larger than in the lighter 659 neighbour. Indeed, two candidates can be traced among the 660 excited states around 850 keV and 1.4 MeV (see Fig. 7).Similar is the situation with the 9/2 + states, two of which 662 are based on the same proton excitation as the 5/2 + states, g.s.) are in competition (with the possible 5/2 + assignment for three of the states, or due to branching also to pairs of states with other possible (3/2 + and 11/2 + ) assignment, even if the most probable transition multipolarities are considered. Based on the enhanced branch to the first 7/2 + state and the predicted second SM state e.g. one may suggest for the 830-keV state more a probable 9/2 + than 5/2 + attribution, while for the higher lying ones such scenario may be relatively uncertain.

		611	cover except possible Δ = 2 (of the g.s.) also Δ = 1 and
		612	together with the spin/parity of 11/2 + , cover log values from
		613	6.8 to 7.7. For three of the observed states with Δ = 2 as
		614	most probable scenario, also uniqueness of the first-forbidden
		615	transitions may be assumed, reflecting the upper log value
			as given in Table II.
		617
		618	the predicted excitation energy by the SM calculations. It
		619	can be seen in Fig. 8, where the direct correspondence and the
		620	predicted configuration as part of the 2 7/2	3 7/2 multiplet may
		621	be traced. The competing 5/2 configuration is relevant only
		622	for states above 1.1 MeV and some of the higher-lying in energy
		623	experimental candidates that, nevertheless, may correspond to
		624	a theoretical state with such origin (see color code, online).
		625	However, some of the other propositions for these states as
		626	5/2 + , 9/2 + (or 11/2 + ) also predicted nearby would finally not
		627	be excluded.
			Of the 11/2 + states, only those with the lowest 2 7/2	3 7/2
		640
		641	recent data with similar conclusions [4, 7, 21]. Moreover, this
		581 642	state is reported also in the heavier 139 I at	= 208 keV and
		643	compared to the same SM prediction (	= 175 keV) with
		644	rather good agreement [5, 8]. It can be, therefore, concluded
		582	Both mother and daughter = 137 g.s. spin/parity assign-that no drastic change it the 5/2 orbital is present in these I
		583	ments are in a good agreement also with our SM calculations
		584	using the N3LOP effective interaction. They predict the 7/2 -
		585	state to be the 137 Te g.s.. It is already reported, that this pre-
		586	diction for lowest state with largest probability (>50%) as a
		587 651	single neutron state is in a full consistency with the Napoli SM
		588 652	interaction [21]. e.g. the 373.1 keV state, however it may also be assigned
		653	differently. The clear difference is based on the experimentally
		654	observed and respective log values that for the 5/2 + state
		655 589 590	It is worth to note that similar predictions were given also in our work amount to 12.8 and 6.2, respectively. It is in a in earlier works [28], where the 7/2 -assignment is compared very good agreement with the data on both 5/2 + states in 135 I,
		591	to systematics of the heavier isotones (up to	= 64) and
		592	concluded that it is best choice for this = 52 nucleus. Also, it
		593	is quoted in Ref. [28] that this would suite the non-observation
		594	(in their data) of candidates for the 3/2 -and 5/2 -states for this
		595 661	nucleus. These candidates could only be tentatively suggested
		596	from -decay to 137 Te to be at excitation energy of 61.8 keV
	identified in this work (see Fig.7 ). These states appear in	663	for the (5/2 -) state and at least 1136.6 keV for some possible while the rest belong to the 2 7/2 3 7/2 multiplet. It has to be
	610	602

597

(3/2 -) state

[21]

, despite the fact that this is somewhat in 598 variance with the SM theory with both N3LOP and Napoli 599 interactions, predicting the two states as first excited states at 600 around the g.s. for N3LOP and around 200 keV for the Napoli 601 SM interactions. 605 the neighbouring 135 I where the g.s. spin/parity is the same 606 as in the other odd-I isotopes, including = 137. These 607 states cover several possibilities in = 135 with log around 608 7.4-7.5 and together with the 11/2 + possibilities up to 8.1 609 [12]. The candidates in = 137, again similarly probable, 616 628 configuration (see

Table III) may be seen. Indeed, the low-629 est of these states being among the experimental possibilities 630 ( = 620 keV) has a remarkable theoretical ( = 588 keV) 631 correspondence. The experimental possibilities for the sec-632 ond and third 11/2 + states listed together with the first 13/2 + 633 or fifth 3/2 + possibilities, respectively, have also perfect SM 634 counterparts. 635 Concerning the 5/2 + states, we could classify with certitude 636 the first excited states of them, which is based as expected, on 637 the predominant 5/2 configuration (see Table III with more 638 than 40% probability). As it stays only at = 235 keV, the 639 orbital positioning may be inferred, and compared with other 645 isotopes, which is different than its behaviour of the Sb isotopes 646 [4, 21]. Interestingly, in the = 135 Iodine neighbour, the 647 second 5/2 + state is also observed in the vicinity of the first 648 (e.g. at about 270 keV excitation energy difference) [12]. This 649 is the predicted by the SM appearance in 137 I (see Fig. 8). We 650 have observed another candidate within the excitation energy, 656 with respective 664 underlined, however, that their wave-functions are much more 665 mixed than those of the 5/2 + states with relative contributions 666 of the order of 17%. Experimentally, only the first 9/2 + is 667 firmly assigned, while the other six candidates (Δ = 1 of the 668

The appearance of the possible (3/2 + ) states is uniquely forbidden decay seems to be present. Furthermore, the strong 722 effect of both 7/2 and 5/2 proton orbitals seems to com-