Semiclassical excited-state signatures of quantum phase transitionsin spin chains with variable-range interactions

# Semiclassical excited-state signatures of quantum phase transitions in spin chains with variable-range interactions

Manuel Gessner Physikalisches Institut, Albert-Ludwigs-Universität Freiburg, Hermann-Herder-Straße 3, 79104 Freiburg, Germany QSTAR (Quantum Science and Technology in Arcetri) and
LENS (European Laboratory for Non-Linear Spectroscopy), Largo Enrico Fermi 2, I-50125 Firenze, Italy
INRIM (Istituto Nazionale di Ricerca Metrologica), I-10135 Torino, Italy
Victor Manuel Bastidas Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany    Tobias Brandes Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany    Andreas Buchleitner Physikalisches Institut, Albert-Ludwigs-Universität Freiburg, Hermann-Herder-Straße 3, 79104 Freiburg, Germany
July 13, 2019
###### Abstract

We study the excitation spectrum of a family of transverse-field spin chain models with variable interaction range and arbitrary spin , which in the case of interpolates between the Lipkin-Meshkov-Glick and the Ising model. For any finite number of spins, a semiclassical energy manifold is derived in the large- limit employing bosonization methods, and its geometry is shown to determine not only the leading-order term but also the higher-order quantum fluctuations. Based on a multi-configurational mean-field ansatz, we obtain the semiclassical backbone of the quantum spectrum through the extremal points of a series of one-dimensional energy landscapes – each one exhibiting a bifurcation when the external magnetic field drops below a threshold value. The obtained spectra become exact in the limit of vanishing or very strong external, transverse magnetic fields. Further analysis of the higher-order corrections in enables us to analytically study the dispersion relations of spin-wave excitations around the semiclassical energy levels. Within the same model, we are able to investigate quantum bifurcations, which occur in the semiclassical () limit, and quantum phase transitions, which are observed in the thermodynamic () limit.

###### pacs:
05.30.Rt, 64.70.Tg, 03.65.Sq, 67.85.-d

## I Introduction

Phase transitions relate macroscopically observable, qualitative changes of the properties of a material to its microscopic structure and order. Thermally-driven phase transitions are usually described in terms of canonical ensembles, taking into account all the possible microstates of a system under appropriate boundary conditions sommerfeld (); landau (); reichl (). Since quantum phase transitions occur as a function of external control parameters at strictly zero temperature QPT1 (); Sachdev (); QPT2 (), when considering many-particle systems one is often tempted to restrict the theoretical treatment to a single quantum state – the ground state. In fact, according to a widely employed definition Sachdev (), any non-analytic behavior of the ground state energy under smooth changes of the external control parameter in an infinitely extended lattice is considered a quantum phase transition. To understand the origin of such a non-analyticity, however, one has to consider excited states: Different eigenstates may – due to their localization and/or symmetry properties – respond differently to changes of the external parameter, which can cause an excited state to cross the ground state from above. In an adiabatic picture, this naturally leads to non-analytic behavior of the ground state energy and, thus, evokes the quantum phase transition. In the presence of non-vanishing couplings between the eigenstates, one instead observes avoided crossings, which generate jumps of the second derivative of the ground state energy with respect to the control parameter, and for this reason in a many-particle context are called second-order quantum phase transitions.

Crossings and anti-crossings are however hardly specific to the ground state. Inasmuch as these are the expression of fundamental changes in the structural properties of the system – as in the above classical understanding of phase transitions – and not just the consequence of a perturbation-induced, local coupling between isolated pairs of states, one should in general expect dramatic structural changes throughout the entire spectrum, in the vicinity of a quantum phase transition. Indeed, quantum phase transitions are often accompanied Wu (); Brandes03 (); kolovsky2004 () by chaotic level statistics Haake (); Mehta (), which can lead to rich dynamics in the vicinity of the critical point EPL (). Moreover, level clusterings in the excited states cusp (); Heiss (); ESQPT (); ESQPT2 () or in quasienergy states of driven systems Victor () have been identified as analogs of quantum phase transitions. However, a compelling general connection between the rearrangement of excited-state levels and the ground-state quantum phase transition is still lacking.

In this work we develop semiclassical methods, based on variational approaches and bosonization techniques, to study the excitation spectrum of spin chain models with tunable interaction range undergoing a quantum phase transition. To be able to investigate both, the effect of the finite interaction range, and the interplay of thermodynamic and semiclassical limits, we introduce a model of interacting spins whose respective length naturally defines an effective Planck constant as . The semiclassical () and thermodynamic () limits of our model are fundamentally different—yet, non-analytic behavior can be observed in both cases. To distinguish between the two cases, we introduce the term quantum bifurcation, which describes non-analyticities of the ground state energy of infinitely-connected, semiclassical (e.g. mean-field) models; see also qb1 (); qb2 (); qb3 (). Such quantum bifurcations are encountered in the semiclassical limit of our model even if is finite, whereas for finite we observe a quantum phase transition in the thermodynamic limit.

The central element of our analysis is a multi-dimensional semiclassical energy landscape, whose geometry directly determines the -expansion of the Hamiltonian for large . In the first part of this paper, we study this multi-dimensional energy landscape obtained in the semiclassical limit for a finite number of spins. Imposing suitably chosen constraints, we obtain a series of one-dimensional sections of this energy landscape, whose extremal points reproduce key features of the full quantum spectrum, even in the most quantum case of . The obtained semiclassical spectrum is furthermore shown to converge to the exact quantum spectrum when the external magnetic field – the relevant control parameter – is either very large or very small. In the second part, employing a bosonized representation of the spin algebra, we study the quantum fluctuations contained in terms of higher order in . This allows us to identify the elementary spin-wave excitations of long-range interacting systems, and their dispersion relations. The disappearance of the excitation gap for the spin waves further predicts the exact critical point of the quantum bifurcation in a ring geometry with arbitrary and arbitrary interaction range. This critical point is shown to coincide with the bifurcation point of the corresponding semiclassical energy landscape. Increasing the number of spins, we observe the behavior of the spin waves close to the critical point as the system evolves from an effective semiclassical few-body system to an infinitely extended many-body system in the thermodynamic limit, , with tunable interaction range.

The model proposed and investigated in this paper includes, as special cases, several models that have been of recent experimental and theoretical interest. Examples include the long-range Ising model, which can be realized with strings of trapped ions for spin- systems PorrasCiracSpins (); Schaetz (); ExpSpinChains () and, recently, also for spin- Senko (), as well as the conventional Ising model with nearest-neighbor interactions, which may be studied with cold atoms in tilted optical lattices Sachdev2 (); Simon (); Meinert (). Models with algebraically decaying, long-range interactions are able to account for the finite interaction length of, e.g., dipolar Rydbergs (); PolarMolecules (); Alvarez () or Coulomb PorrasCiracSpins () interactions, and allow to assess the modified spreading behaviour of perturbations when compared ExpSpinChains (); Storch () to lattices with nearest-neighbor interactions LiebR (); Cheneau ().

## Ii The model

We consider a one-dimensional variable-range spin model

 H=−2SN∑i,j=1(i

where the spin-spin coupling reads . Furthermore, we have defined collective angular momentum operators with such that . In addition, for are Pauli matrices describing elementary spins at the -th site, in such a way that . In the course of this paper we will discuss both cases of open and periodic boundary conditions. Figure 1 depicts a sketch of the model and its interpretation in terms of collective angular momentum operators and elementary spins . In the special case of Cannas (); CannasRenor (); Cardy (); Dyson (); PorrasCiracSpins (); Schaetz (); ExpSpinChains (); Koffel (); EPL (); PhDGessner (), the Hamiltonian (1) reads

 H=−J0N∑i,j=1(i

As a function of , which determines the interaction range, the Hamiltonian (2) interpolates continuously between the infinite-range Lipkin-Meshkov-Glick model () LMG () and the one-dimensional Ising model with nearest-neighbor interactions () Ising (); Sachdev ().

The system’s properties are determined by the relative strength of the two competing interactions: The internal spin-spin interaction causes the spins to arrange their configuration depending on the -coordinates of neighboring spins, while the external field pushes the spins along the transverse -direction. The sign of determines whether the system arranges in ferromagnetic () or (anti-)ferromagnetic () order in the limit . The quantum phase transition occurs when the two potential energy terms proportional to and are of comparable order of magnitude, whereas the exact position of the critical point depends on .

For , the phase transition has been studied for the special cases of the Ising and Lipkin-Meshkov-Glick models – analytic solutions are available for both of them Lieb (); Pan (). For the system can be solved by Jordan-Wigner fermionization JordanWigner (), and exhibits a quantum phase transition from (anti-)ferromagnet to paramagnet at the critical field  Sachdev (). In the opposite limit , a Holstein-Primakoff bosonization HP () yields an efficient description of the system in orders of (since all spins can be combined into one large spin) Dusuel (), which is more practical than its exact solution LMGVidal (). Whenever , the spectrum is only bounded in the thermodynamic limit when is rescaled by Cannas (). The quantum bifurcation of the Lipkin-Meshkov-Glick model () occurs at with when Botet (); Zibold (), and at when Vidal (). Only few results are available for intermediate values of Koffel (). A phase transition in a classical long-range model for was shown to occur for the parameter range Dyson (). Further studies based on renormalization group techniques allowed to investigate the particular case Cardy () and to describe non-analyticities of the free energy as a function of CannasRenor ().

Conversely, in the semiclassical limit of Eq. (1), , the number of elementary spins at each lattice site becomes very large. As a consequence, each site is represented by a composite semiclassical spin, as depicted in Fig. 1, while different sites are coupled by a finite interaction range, determined by . The semiclassical limit allows for an exact mean-field analysis and produces sharp bifurcations of the energy landscapes, which directly imply non-analytic behavior of the quantum excitations for all values of . These phenomena are henceforth referred to as quantum bifurcations, to distinguish them from the quantum phase transition in systems that are infinitely extended along the interacting dimension. A discussion of quantum bifurcations will be provided in Sec. III.2, where the main features are illustrated with a simple special case of our model. A complete analysis of the quantum bifurcations in our model is then provided in Sec. V. The model additionally permits us to tune the number of composite spins to independently scan the transition to the thermodynamic limit, which is associated with the transition into an infinitely extended one-dimensional lattice whose interaction range is parametrized by .

## Iii Semiclassical expansion of the spin Hamiltonian

We begin by deriving a formal semiclassical expansion () of the Hamiltonian (1) for a finite number of spins. In this limit, the spectrum of each individual spin resembles that of a harmonic oscillator, which allows us to express the spin operators in terms of bosonic creation and annihilation operators. The associated Holstein-Primakoff transformation HP () then leads to a perturbative expansion in orders of .

### iii.1 General formalism

To obtain the semiclassical expansion, we restrict ourselves to the subspace of maximal angular momentum . In addition, it is convenient to introduce a local rotation operator of the -th spin as

 U(ϕ(i))=exp(iϕ(i)S(i)z). (3)

Based on these local spin rotations, we introduce the rotated Hamiltonian as in Ref. Dusuel ()

 H(ϕ)=U(ϕ)HU†(ϕ), (4)

which is given by

 H(ϕ) =−2J0SN∑i,j=1(i

In this expression, is a tensor product of the unitary operators defined in Eq. (3), and the local spin orientations are characterized by the vector . Since represents a unitary operation, the spectra of and coincide. We now invoke the Holstein-Primakoff representation of the angular momentum algebra HP ():

 S(i)y =S−a†iai, (6) S(i)x+iS(i)z =√2Sa†i ⎛⎝1−a†iai2S⎞⎠1/2, (7) S(i)x−iS(i)z =√2S⎛⎝1−a†iai2S⎞⎠1/2 ai, (8)

where is the total angular momentum at the -th site and are bosonic operators. In the case one can expand the Hamiltonian (III.1) as

 H(ϕ)=2SNE(ϕ)+√2SHL(ϕ)+HQ(ϕ)+O(1√2S), (9)

where the leading-order term in defines a semiclassical energy landscape

 E(ϕ)=−J0NN∑i,j=1(i

In the derivation of the energy landscape we considered the case of maximal angular momentum. However, in Section IV.1 we show how this assumption can be relaxed to obtain a generalized energy landscape by using a variational approach.

The partial derivatives of further determine the linear Hamiltonian , containing quantum corrections in linear order,

 HL(ϕ) =−NN∑i=1∂E(ϕ)∂ϕ(i)(a†i+ai). (11)

as well as the quadratic Hamiltonian,

 HQ(ϕ) =NN∑i,j=1(i

Thus, we find that the semiclassical energy landscape contains the complete information about the Hamiltonian for large , and determines, via its geometry, also the quantum corrections to the mean-field contribution. Of special interest are the stationary points of , which are defined as those spin configurations that satisfy the conditions

 ∂E(ϕ)∂ϕ(i)=−J0NN∑j=1(i≠j)cosϕ(i)sinϕ(j)|i−j|α+BNsinϕ(i)=0, (13)

for all . At these points, the linear Hamiltonian vanishes exactly, and the quantum fluctuations are described by , whose coefficients are given by

 ∂2E(ϕ)∂ϕ(i)2=J0NN∑j=1(i≠j)sinϕ(i)sinϕ(j)|i−j|α+BNcosϕ(i), (14)

and, for ,

 ∂2E(ϕ)∂ϕ(i)∂ϕ(j)=−J0Ncosϕ(i)cosϕ(j)|i−j|α. (15)

Using Eq. (13) in Eq. (14) simplifies the second derivative at a stationary point to

 ∂2E(ϕ)∂ϕ(i)2∣∣∣ϕ∈C=BN1cosϕ(i), (16)

provided that .

Section IV is dedicated to an analysis of the zero-order semiclassical energy landscape . In particular we will compare the semiclassical predictions for the energy spectrum based on a series of suitably defined one-dimensional sections of to the numerically obtained quantum spectra, with particular emphasis on the least classical case of . In this parameter regime far away from the semiclassical limit, the quantum phase transition can be observed in the thermodynamic limit when the external field approaches a critical value that depends on .

Later in Section V we study the quantum fluctuations in the large- limit, which in turn allows us to reveal quantum bifurcations for arbitrary and by analytical means. We conclude the present section by formally introducing the concept of quantum bifurcations, based on a simple illustrative example, and by discussing its relation to quantum phase transitions.

### iii.2 Quantum bifurcations in the semiclassical limit

In this section, we discuss the Hamiltonian (9) for in the particular case of just two lattice sites to illustrate the main features of quantum bifurcations. For a complete analysis of the quantum bifurcations in the model and additional details on the employed methods, we refer to Sec. V. Let us further assume an equal spin configuration , where is chosen such that the mean-field energy is minimized. We find a single minimum at when and two degenerate minima at when . As we will see in this section, this bifurcation of the classical mean-field energy landscape entails profound consequences for the quantum fluctuations of the higher-order terms in Eq. (9).

Due to Eq. (13), the linear Hamiltonian disappears at critical points of . The Hamiltonian (9) thus reduces to the mean-field energy and the quadratic corrections. Through their dependence on , both terms depend on the parameter . For , i.e., , we obtain the effective Hamiltonian

 H(ϕc)=−4SB−J0(a†1+a1)(a†2+a2)+2B(a†1a1+a†2a2). (17)

This Hamiltonian, in fact, can be identified with the effective Hamiltonian for the on-resonance Dicke model dicke () in the normal phase, where plays the role of atomic and mode resonances, reflects the atom-field coupling strength, and is the collective atomic spin Brandes03 (). A diagonalization of the Hamiltonian (17) leads to two collective bosonic modes with excitation energies . Correspondingly, is the gap between the ground state and the first excited state.

When , i.e., , the Hamiltonian reads

 H(ϕc) =−2S(J0+B2/J0)−B2/J0(a†1+a1)(a†2+a2) +2J0(a†1a1+a†2a2), (18)

Now, we obtain collective excitation energies of .

The above results indicate that when the energy gap above the ground state vanishes as , where is the bifurcation point. The energy gap directly defines a characteristic length scale which determines the spread of the ground state wave function Brandes03 (). Indeed, the ground state, which is a two-mode Gaussian state, gets strongly squeezed as the bifurcation point is approached Brandes03 (). Similarly, the ground state shows strong quantum correlations in the vicinity of the bifurcation point Brandes04 (). Furthermore, the divergent length scale and the closing gap can be associated with critical exponents, and finite-size scaling can be studied when is finite Brandes03 (); Dusuel (); LMGVidal (); Botet (); Brandes04 (). In the semiclassical limit, we find a sharp discontinuity of the ground state energy at the bifurcation point.

Evidently, these quantum signatures of the classical bifurcation stand in direct analogy to quantum phase transitions. The quantum phase transition, however, occurs in the thermodynamic limit, when the extension of the lattice of interacting, collective spins becomes infinite. In the semiclassical limit, which triggers the quantum bifurcation for any value of , we extend the zero-dimensional sub-lattice of elementary spins, which add up to form a collective spin as it is depicted in Fig. 1. According to Eqs. (6), (7) and (8), the number of elementary spins translates into the maximal occupation of the effective bosonic modes, and consequently, only in the semiclassical limit, these modes are unbounded and allow for a diverging spread of the ground state wavefunction. This way, the many-body character of the spin model is absorbed by a finite number of harmonic oscillator modes. The diverging length scale, however, despite being related to the ground-state correlations, does not identify a diverging spatial correlation length, since the elementary spins are arranged on a zero-dimensional lattice (see Fig. 1). This precisely identifies the difference between the semiclassical and thermodynamic limits, i.e., the quantum bifurcation and the quantum phase transition, respectively.

A further characteristic of the quantum bifurcation is that the mean-field description becomes exact in the classical limit . Moreover, the many-body aspect of the model for finite becomes irrelevant in the semiclassical limit: The quantum bifurcation does not explicitly depend on the substructure of the collective spins. The features close to the ground state are therefore reproduced by an effective -body system rather than an -body system. Yet, for an understanding of the excitation spectrum, this substructure is essential, as will become apparent in Sec. IV.

## Iv Semiclassical energy landscapes

The semiclassical energy landscape was derived as the leading-order contribution to a -expansion in a subspace of maximal angular momentum. We will show in the first part of this Section that this term can also be interpreted as the energy expectation value of a variational product ansatz of spin-coherent states in this particular subspace. Spin coherent states are formal analogues of coherent states of the harmonic oscillator Radcliffe (); Arecchi (); Zhang (); MandelWolf (). Their expectation values are characterized by Bloch vector coordinates, and they minimize the uncertainty relation with respect to certain angular momentum observables Arecchi (). These states are therefore often interpreted as semiclassical, and they allow to introduce an effective Planck constant Haake (); Zhang ().

Spin-coherent states are, however, not limited to the subspace of maximal angular momentum . We can therefore formulate a more general variational ansatz in terms of arbitrary spin-coherent states that will allow us to extend the definition of the semiclassical energy landscape (10) to include all subspaces of .

### iv.1 Variational approach based on spin coherent states

In this section we employ a product state ansatz for the trial wave function in terms of local spin coherent states

 |lμ,ϕμ⟩=N⨂i=1|l(i)μ,ϕ(i)μ⟩. (19)

The trial states (19) are characterized by the vectors and , which depend on the configuration labelled by . The variables and determine the respective length and orientation of the local spin coherent state which describes the spin at index as MandelWolf ()

 |l(i)μ,ϕ(i)μ⟩=12l(i)μl(i)μ∑m(i)z=−l(i)μ(2l(i)μl(i)μ+m(i)z)1/2(−ieiϕ(i)μ)l(i)μ+m(i)z|l(i)μ,m(i)z⟩. (20)

Here are the Dicke states dicke () of cooperation number , as defined by a total angular momentum of , and .

This ansatz, by construction, ensures that the local expectation values are restricted to the -plane. This choice is motivated by the fact that does not appear in the Hamiltonian (1), and therefore does not contribute to the energy. The expectation values are now conveniently represented by the Bloch vector coordinates as:

 ⟨l(i)μ,ϕ(i)μ|S(i)x|l(i)μ,ϕ(i)μ⟩ =l(i)μsinϕ(i)μ, (21) ⟨l(i)μ,ϕ(i)μ|S(i)y|l(i)μ,ϕ(i)μ⟩ =l(i)μcosϕ(i)μ, (22) ⟨l(i)μ,ϕ(i)μ|S(i)z|l(i)μ,ϕ(i)μ⟩ =0. (23)

Each of the spins is composed of spin- particles, leading to a total of elementary spins in the system. Using the trial states (19), together with (1,21,22), we obtain the average energy per elementary spin

 Eμ(lμ,ϕμ)=12SN⟨lμ,ϕμ|H|lμ,ϕμ⟩ =−J0S2NN∑i,j=1(i

This generalized semiclassical energy landscape indeed coincides with the energy landscape (10) when we restrict to the subspace where all of the composite spins have maximal angular momentum , i.e., when for all . In fact, the rotations introduced in Eq. (3) can be used to generate spin coherent states in this particular subspace as , where .

In the following we will show that, in the limiting cases and , the ansatz (19) allows to generate the exact spectra of (1).

### iv.2 (Anti-)ferromagnetic spectra at B=0

Let us first consider the spectrum in absence of an external field, . We define the local eigenstates by the eigenvalue equation , with . Introducing

 |S,mx⟩=N⨂i=1|S,m(i)x⟩, (25)

with , the Hamiltonian describing the internal spin-spin interaction

 Hin=H|B=0=−2J0SN∑i,j=1(i

which is obtained from (1) for , satisfies the eigenvalue equation

 Hin|S,mx⟩=−2J0SN∑i,j=1(i

To reproduce this exact spectrum (27) from the energy expectation value per spin (IV.1), we impose certain conditions on the spin coherent state ansatz (19). In particular, we assume that all the local spin orientations are represented by the same angle , while allowing individual spins to be inverted such that . Thus, we see from Eq. (IV.1) at that configurations with arbitrary sequences of positive or negative local angles or , lead, at the particular value of (see below), to the scaled energy expectation value

 Eμ(lμ,ϕμc)=−J0S2NN∑i,j=1(i

where we have defined via the equality . The possible values of Eq. (28) can now be determined by scanning over the full range of . Comparison to Eq. (27), while recalling that , demonstrates that the product state ansatz (19) of trial states is able to reproduce the exact spectrum at , i.e., the set of eigenvalues coincides with the set of all possible values of

 Eμ=2SNEμ, (29)

where represents the total energy, in contrast to the energy per elementary spin unit. Note that this observation is, in fact, independent of the coupling coefficients, which here are given by the algebraically decaying function .

Figure 2 illustrates two examples of spin configurations for , which differ by the number of inverted spins. The natural orientation of the spin-spin interaction along the -direction causes the spins to assume an orientation along the -axis in absence of the transverse field. Hence, the angle assumes the value , see Eqs. (21)-(23).

In total there are different configurations labeled by . In the case , this number indeed reflects the Hilbert space dimension. However, two symmetries lead to degeneracies of the :

• The invariance of the energy expectation value under a global sign flip, , originates in the -symmetry Sachdev (); Brandes03 () (-rotation around the -axis) of the Hamiltonian (1) and permits to restrict our analysis to configurations with at most half of the spins inverted.

• For open boundary conditions, which we impose in this Section, the energy of the chain remains invariant under a mirror reflection with respect to the center: .

In the following we discuss the quantity (28) where, for simplicity, we focus on the special case of . The analysis can be extended easily to larger spins. When , the length of the individual spin coherent states is fixed at , for all . Any given configuration is then fully determined by the orientations of the local spins. For this special case, we introduce the effective spin-spin coupling constant

 Jμ=J0NN∑i,j=1(i

which determines the energy spectrum at through [where we used in Eqs. (28,29)].

For arbitrary values of the interaction decay constant , the different possible configurations lead to rather irregular distributions of the energy eigenvalues. However, in certain extreme cases, when we recover Ising () or Lipkin-Meshkov-Glick interactions (), only few, strongly degenerate energy bands are obtained.

For example, in the limit of nearest-neighbor interactions, , we infer from (30)

 Eμ=−J0N−1∑i=1ϵ(i)μϵ(i+1)μ=−J0(N−1−2rμ), (31)

where counts the number of domain walls in the configuration . In a ferromagnet (anti-ferromagnet), these occur when two neighboring spins align in opposite (equal) directions Sachdev ().

Conversely, for an infinitely extended interaction range, , we have

 Eμ=−J0N∑i,j=1(i≠j)ϵ(i)μϵ(j)μ/2=−J02N[(N−2sμ)2−N], (32)

where denotes the number of inverted spins in .

The normalized counting function

 N(E)=2−N2N−1∑i=0θ(E−Ei) (33)

is obtained from the exact variational ansatz for , and is shown in Fig. 3 for different values of . We observe that, as a function of , the spectrum at interpolates smoothly between the two strongly degenerate cases of a quadratically spaced sequence of eigenvalues at and a harmonic spectrum, which is symmetric around zero at . For intermediate values of , the energy levels are broadly distributed between the two extreme values and , where

 Je=J0NN∑i,j=1(i

and

 Ja=J0NN∑i,j=1(i

are generated from Eq. (30) by means of an equal mean-field configuration of parallel spins, and an alternating mean-field configuration where the spin orientation of neighboring spins is inverted, respectively. These two configurations are depicted in Fig. 2. The latter describes a two-fold degenerate ground state configuration of an anti-ferromagnet as well as the highest excited states of a ferromagnet.

Let us briefly discuss the behavior of and in the thermodynamic limit . We rewrite

 Je =J0NN∑k=1N−kkα=J0N∑k=11kα+J0NN∑k=11kα−1. (36)

For , we have , where denotes Riemann’s zeta function Edwards (). In this case the second term approaches zero, since Cannas (). Hence, for , we have

 limN→∞Je=J0ζ(α), (37)

whereas for the sum diverges and the spectrum becomes unbounded. At the divergence is logarithmic in .

Employing an analogous rearrangement of terms, we find for ,

 limN→∞Ja=−J0η(α), (38)

where is Dirichlet’s eta function (Apostol (), which is related to Riemann’s zeta function by . Evaluating the sum explicitly at yields which coincides with the analytic continuation Cartan (), , allowing us to extend the above equality to all .

For both and coincide in magnitude in the thermodynamic limit, since , and, thus, the Ising spectrum is bounded between , which, in the considered limit of large , is consistent with Eq. (31). The convergence towards the thermodynamic limit is displayed in Fig. 4. To leading order, finite-size effects are caused by the prefactor of the second term of Eq. (36), and of the corresponding alternating expression for . Hence, to a good approximation, these finite-size effects are compensated by a factor , and small deviations can only be observed when is very small, as is shown in Fig. 4 b).

### iv.3 Paramagnetic spectrum at J0=0

After discussing the (anti-)ferromagnetic spectrum, we turn to the opposite limit of very strong external magnetic fields, by setting the spin-spin coupling to zero: . The Hamiltonian

 Hex=H|J0=0=−2BN∑i=1S(i)y, (39)

describing the interaction with the external field, is independent of . The spectrum of is easily found, e.g., by employing a treatment in complete analogy to the one shown in the beginning of the preceding Section: We introduce product states of local eigenstates of , characterized by a vector of eigenvalues . This leads to the eigenvalue equation

 Hex|S,my⟩=−2BN∑i=1m(i)y|S,my⟩. (40)

The resulting spectrum is harmonic and elementary excitations are given by spin flips against the magnetic field in -direction. Recalling Eqs. (21)–(23), we see that the previously introduced inversion of a spin, , which corresponds to a mirror reflection at the -axis, does not change the expectation value of the paramagnetic energy term. However, the configurations are able to account for such excitations via spin flips, defined by the operation , which describes a combined mirror reflection at the and -axes. We again describe the entire spin chain configuration in terms of a single angle , and introduce through to label the presence or absence of a spin flip at position .

Indeed, employing the spin coherent states (19), combined with the above constraints, generates the following energy expectation values [see Eq. (IV.1)] for and at :

 Eμ=−BSNN∑i=1l(i)μξ(i)μ, (41)

which, due to , reproduce the full spectrum, as given in Eq. (40). According to Eqs. (21)–(23) the angle reflects the polarization of the spins along the -direction of the external field.

Let us focus again on the special case of . We then can express the energy eigenvalues for as , where the effective magnetic fields are given by

 Bμ=(N−2kμ)B/N, (42)

and counts the number of flipped spins in the configuration characterized by . We recover the well-known equidistant energy levels of a paramagnetic chain. This resembles the spectrum at when . The difference between the two cases is that for , there can be between and inverted spins, which leads to energy bands, whereas for and there are between and domain walls, and, thus, only energy bands.

Notice that, due to the symmetry of the paramagnetic spectrum with respect to energy zero, a global change of the signs of all energy eigenvalues for arbitrary and always produces the spectrum of the chain for parameter values and , independently of and .

### iv.4 Semiclassical spectra from analytically determined extrema of one-dimensional energy landscapes: Multi-configurational mean-field approach

So far, we formulated a variational ansatz in terms of spin coherent states to reproduce the exact spectra when either or , for arbitrary and . Starting from a uniformly distributed spin arrangement , we employed combinations of spin inversions and spin flips to design excited-state configurations . Note that spin flips, , also change the (anti-)ferromagnetic energy expectation value, which can be compensated by an additional inversion, . In total, each spin can assume one of four different orientations, i.e., , (inverted), (flipped), and (inverted and flipped). Independently of the orientation, each of the spin coherent states also has a tunable length that can assume discrete values between the minimum value (if is integer) or (if is half-integer) and the maximum value .

The orientation of each individual spin is parametrized by a single angle , and, thus, the variational ansatz can be understood as a mean-field approach. Based on the above recipe, we obtain an entire family of mean-field descriptions (multi-configurational mean-field), labelled by the index , which represents a particular spin configuration. A spin configuration , is fully characterized by the two vectors and , which determine the local lengths and orientations of the spins, respectively. Suitable design of these configurations leads, via Eq. (19), to a series of single-parameter trial states that yield any arbitrary eigenvalue at and, independently, any arbitrary eigenvalue at . These two eigenvalue solutions will then be attained at different values of the parameter : As we saw in the previous Sections, the spin orientation in the paramagnetic phase is given by , while in the (anti-)ferromagnetic phase we have .

To continuously parametrize the energy spectrum for arbitrary and , we employ the configurations derived above to analyze the semiclassical energy landscape , introduced in Eq. (IV.1), as a function of and of the angle . For each , we obtain a one-dimensional semiclassical energy landscape

 Eμ(ϕ)=−Jμsin2ϕ−Bμcosϕ, (43)

with

 Jμ=J0S2NN∑i,j=1(i

and

 Bμ=BSNN∑i=1l(i)μξ(i)μ. (45)

The semiclassical energy landscape (43) is characterized by the effective magnetic field , which is proportional to , and the effective spin-spin coupling constant , proportional to . These two effective parameters determine the exact spectra in the extreme cases considered before. The resulting energy landscape is depicted as a function of in Fig. 5. The position of its extremal values shifts as a function of .

For and , we find the minimal energy

 (46)