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arxiv: 2604.12203 · v1 · submitted 2026-04-14 · ❄️ cond-mat.str-el · nucl-th

Spherical-tensor description of the Jahn--Teller--Hubbard molecule and local electron--phonon entanglement

Koichiro Takahashi , Shuichiro Ebata , Naotaka Yoshinaga , Shintaro Hoshino This is my paper

Pith reviewed 2026-05-10 16:11 UTC · model grok-4.3

classification ❄️ cond-mat.str-el nucl-th
keywords Jahn-Teller phononselectron-phonon entanglementcomposite quadrupoleMott insulatorA3C60spherical tensorangular momentum
0
0 comments X

The pith

In the Jahn-Teller-Hubbard model for A3C60, composite electron-phonon quadrupole operators do not couple to conventional quadrupoles or lattice displacements.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper models the Mott-insulating phase of A3C60 with a single-site multiorbital electron system coupled to anisotropic Jahn-Teller molecular vibrations. It applies the spherical-tensor formalism to the electron-phonon ground-state multiplet and finds that both the conventional electronic quadrupole moment and the associated lattice displacement vanish. The authors introduce composite two-body quadrupole operators built from electrons and phonons, then use quasispin selection rules to prove these operators remain decoupled from the standard ones. Analysis of the entanglement spectrum shows the ground state is a superposition of multi-phonon states carrying angular momenta L_ph=2 and L_ph=3 that arise from coupling to three-electron states with L=1 and L=2.

Core claim

The ground-state multiplet of the Jahn-Teller-Hubbard molecule exhibits vanishing conventional electronic quadrupole moments and lattice displacements. Composite two-body quadrupole operators involving electrons and phonons are introduced, and using quasispin selection rules, these are shown not to couple to the conventional operators, distinguishing them from standard quadrupolar degrees of freedom. The entanglement spectrum indicates the ground state is a superposition of multi-phonon states with L_ph=2 and 3 arising from coupling to three-electron states with L=1 and 2.

What carries the argument

Composite (two-body) quadrupole operators involving both electrons and phonons, analyzed within the spherical-tensor formalism

If this is right

  • Conventional electronic quadrupole moments and lattice displacements vanish in the ground-state multiplet.
  • Composite quadrupole operators remain decoupled from both conventional quadrupole and lattice-displacement operators.
  • The ground state consists of superpositions of multi-phonon states with L_ph=2 and L_ph=3 coupled to electron states with L=1 and L=2.
  • The composite operators exhibit parameter dependence that can be tracked numerically.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The decoupling may permit detection of hidden quadrupolar order in molecular solids without accompanying lattice distortion.
  • Similar composite operators could be constructed in other strongly coupled electron-phonon systems beyond single-site A3C60 models.
  • The angular-momentum characterization of the entanglement spectrum may extend to multi-site or dynamical-mean-field treatments.

Load-bearing premise

A single-site multiorbital electron model coupled to anisotropic molecular vibrations is sufficient to capture the localized-electron character of the Mott-insulating phase in A3C60.

What would settle it

A direct numerical diagonalization or experimental measurement that finds a non-zero expectation value for the conventional electronic quadrupole moment in the ground state would contradict the vanishing result.

Figures

Figures reproduced from arXiv: 2604.12203 by Koichiro Takahashi, Naotaka Yoshinaga, Shintaro Hoshino, Shuichiro Ebata.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic picture of the [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Lowest-energy levels [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The reduced matrix elements [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: (a) shows the g dependence of the entan￾glement entropy, calculated within a truncated Hilbert space with nd ≤ nc = 7. The entanglement entropy de￾pends on the magnetic quantum number |M| = 0, 1 and increases monotonically with increasing electron–phonon coupling. We also evaluate the entanglement entropy for the Ltot = 1 sector within the minimal truncation nd ≤ nc = 1, yielding SM ≃ −C 2 1 ln C 2 1 − C 2… view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Cutoff dependence of the reduced matrix elements [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗
read the original abstract

We investigate the localized-electron character of the Mott-insulating phase in A$_3$C$_{60}$ using a single-site multiorbital electron model coupled to anisotropic molecular vibrations (Jahn--Teller phonons). We apply the spherical-tensor formalism, a framework originally developed in nuclear physics, to analyze the electron--phonon-coupled ground-state multiplet. Focusing on multipole moments, we find that both the conventional electronic quadrupole moment and the lattice displacement associated with the molecular vibrations vanish, even though the degenerate ground-state multiplet implies the presence of quadrupolar degrees of freedom. By analyzing these degrees of freedom within the spherical-tensor framework, we introduce composite (two-body) quadrupole operators involving both electrons and phonons and study their parameter dependence numerically. Furthermore, using quasispin selection rules, we demonstrate that the composite quadrupole does not couple to either the conventional quadrupole or lattice-displacement operators, thereby distinguishing it fundamentally from standard quadrupolar degrees of freedom. In addition, we investigate the nature of the electron--phonon entanglement and characterize it from the viewpoint of angular momentum. Analysis of the entanglement spectrum reveals that the ground state consists of superpositions of multi-phonon states with angular momenta $L_{\rm ph}=2$ and $L_{\rm ph}=3$, formed through coupling to three-electron states with $L=1$ and $L=2$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper applies the spherical-tensor formalism, imported from nuclear physics, to a single-site Jahn-Teller-Hubbard model of three electrons coupled to anisotropic phonons. It reports that both the conventional electronic quadrupole moment and the lattice-displacement operator have vanishing expectation values in the degenerate ground-state multiplet. Composite two-body quadrupole operators are introduced; quasispin selection rules are invoked to show that these operators do not couple to the conventional quadrupole or displacement operators. The electron-phonon entanglement spectrum is analyzed in angular-momentum channels, revealing that the ground state is a superposition of multi-phonon states with L_ph=2 and L_ph=3 coupled to three-electron states with L=1 and L=2.

Significance. If the numerical and selection-rule results hold, the work supplies an analytically grounded distinction between standard and composite quadrupolar degrees of freedom in a Jahn-Teller molecule, together with a concrete angular-momentum characterization of local electron-phonon entanglement. This framework may prove useful for interpreting probes of quadrupolar fluctuations in the Mott phase of A3C60 and related multiorbital systems. The combination of exact selection rules with numerical parameter scans is a methodological strength.

major comments (2)
  1. [Numerical results and parameter-dependence section] The central claims that conventional quadrupole and displacement moments vanish and that the composite operators exhibit parameter-dependent decoupling rest on numerical diagonalizations, yet the manuscript provides no information on Hilbert-space dimension, basis truncation, convergence checks, or error estimates (see the sections describing the numerical parameter dependence and the entanglement-spectrum analysis).
  2. [Quasispin selection-rules subsection] The demonstration that the composite quadrupole does not couple to the conventional quadrupole or lattice-displacement operators is load-bearing for the claim of a fundamentally new quadrupolar degree of freedom. The specific quasispin quantum numbers assigned to each operator and the explicit matrix-element selection rules that enforce the decoupling should be tabulated or derived in an equation (currently only stated qualitatively).
minor comments (2)
  1. [Entanglement-spectrum paragraph] The abstract states that the ground state consists of superpositions with L_ph=2,3 and L=1,2; the main text should indicate whether these are the only contributing channels or whether higher-L components appear at finite JT coupling.
  2. [Formalism introduction] Notation for the spherical-tensor operators (e.g., the rank-2 composite quadrupole) should be introduced with an explicit definition before the selection-rule arguments are applied.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment of our work and for the constructive comments that help strengthen the manuscript. We address the two major points below and will revise the paper to incorporate the requested details and clarifications.

read point-by-point responses

  1. Referee: [Numerical results and parameter-dependence section] The central claims that conventional quadrupole and displacement moments vanish and that the composite operators exhibit parameter-dependent decoupling rest on numerical diagonalizations, yet the manuscript provides no information on Hilbert-space dimension, basis truncation, convergence checks, or error estimates (see the sections describing the numerical parameter dependence and the entanglement-spectrum analysis).

    Authors: We agree that additional documentation of the numerical procedure is necessary for full reproducibility and assessment of the results. The calculations are performed by exact diagonalization of the Hamiltonian in a truncated multi-phonon basis for the three-electron sector. In the revised manuscript we will insert a new paragraph (or subsection) in the numerical results section that specifies the Hilbert-space dimension of the electronic sector, the phonon occupation cutoff per mode, the convergence criterion used when increasing the cutoff, and quantitative error estimates on the reported expectation values and entanglement weights. These additions will directly support the claims of vanishing conventional moments and the observed parameter-dependent behavior. revision: yes

  2. Referee: [Quasispin selection-rules subsection] The demonstration that the composite quadrupole does not couple to the conventional quadrupole or lattice-displacement operators is load-bearing for the claim of a fundamentally new quadrupolar degree of freedom. The specific quasispin quantum numbers assigned to each operator and the explicit matrix-element selection rules that enforce the decoupling should be tabulated or derived in an equation (currently only stated qualitatively).

    Authors: We concur that making the quasispin selection rules fully explicit will strengthen the central claim of a distinct composite quadrupolar degree of freedom. In the revised version we will augment the quasispin selection-rules subsection with a compact table that lists the quasispin quantum numbers (including seniority and reduced matrix elements where relevant) for the conventional electronic quadrupole, the lattice-displacement operator, and each composite two-body operator. We will also add a short derivation, presented as a numbered equation, that shows how the selection rules on the quasispin quantum numbers force the relevant matrix elements to vanish. This will replace the current qualitative statement and render the decoupling argument self-contained. revision: yes

Circularity Check

0 steps flagged

Derivation is self-contained with no circular reductions

full rationale

The paper defines a single-site multiorbital Jahn-Teller-Hubbard Hamiltonian, imports the spherical-tensor formalism from nuclear physics as an external framework, introduces composite two-body quadrupole operators explicitly within the model, and applies quasispin selection rules (derived from the model's symmetries) to demonstrate non-coupling of the composite quadrupole to conventional electronic quadrupole and lattice-displacement operators. The entanglement spectrum analysis follows directly from the angular-momentum decomposition of the ground-state multiplet. All central claims are model-internal consequences of the defined Hamiltonian and symmetry considerations, with no reduction of predictions to fitted inputs, no load-bearing self-citations, and no ansatz smuggled via prior work by the same authors.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 1 invented entities

The analysis rests on a single-site truncation of the A3C60 problem and on the spherical-tensor algebra imported from nuclear physics; numerical diagonalization is performed for varying electron-phonon coupling strengths whose concrete values are not fixed by first principles.

free parameters (1)
  • Jahn-Teller electron-phonon coupling strength
    The ground-state multiplet and composite-operator expectation values depend on the strength of the anisotropic molecular vibrations, which is varied numerically but not derived from microscopic parameters.
axioms (2)
  • domain assumption Single-site approximation suffices to capture local electron-phonon entanglement in the Mott phase
    All results are obtained on an isolated molecule; inter-site hopping and long-range interactions are omitted by construction.
  • standard math Spherical-tensor formalism developed for nuclear physics applies directly to the molecular Jahn-Teller problem
    The paper imports the formalism without deriving its validity for the present electron-phonon Hilbert space.
invented entities (1)
  • composite (two-body) quadrupole operators no independent evidence
    purpose: To represent the hidden quadrupolar degrees of freedom that survive after conventional electronic and lattice quadrupoles average to zero
    These operators are defined within the paper to capture the degenerate multiplet structure; no independent experimental signature is proposed.

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Reference graph

Works this paper leans on

108 extracted references · 108 canonical work pages

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    The use of reduced matrix elements, however, allows one to factor out the angular- momentum coupling and thereby access quantities spe- cific to the electron or phonon sectors

    Reduced density matrix for electrons When the states are classified solely by the total angu- lar momentum, the electronic and phononic components are intrinsically entangled. The use of reduced matrix elements, however, allows one to factor out the angular- momentum coupling and thereby access quantities spe- cific to the electron or phonon sectors. The ...

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    Figure 5(c) shows thegdependence of the ex- pectation value⟨n d⟩and its standard deviation ∆n d =p ⟨n2 d⟩ − ⟨nd⟩2

    Reduced density matrix for phonons Electron–phonon entanglement induces phonon exci- tations in theL tot = 1 ground-state multiplet, lead- ing to an increasing weight of states with larger phonon number. Figure 5(c) shows thegdependence of the ex- pectation value⟨n d⟩and its standard deviation ∆n d =p ⟨n2 d⟩ − ⟨nd⟩2. For the cutoffn c = 1, these quantitie...

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    indicate the quantum numbers (L, S, K). With use of these tensors, one can define another triple tensor operators as T (L,S,K) ML,MS ,MK ≡ p 1, 1 2 , 1 2 ⊗p 1, 1 2 , 1 2 (L,S,K) ML,MS ,MK = X mm′σσ ′κκ′ ⟨1m1m′ |LML ⟩ 1 2 σ 1 2 σ′ |SMS 1 2 κ 1 2 κ′ |KM K ×p 1, 1 2 , 1 2 m,σ,κ p 1, 1 2 , 1 2 m′,σ′,κ′ .(103) For example, the angular momentum operator is con-...

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    Gell-Mann matrices and quadrupoles The Gell-Mann matrices are defined by ˆλ1 =   0 1 0 1 0 0 0 0 0   , ˆλ2 =   0−i 0 i 0 0 0 0 0   , ˆλ3 =   1 0 0 0−1 0 0 0 0   , ˆλ4 =   0 0 1 0 0 0 1 0 0   , ˆλ5 =   0 0−i 0 0 0 i 0 0   , ˆλ6 =   0 0 0 0 0 1 0 1 0   , ˆλ7 =   0 0 0 0 0−i 0 i 0   , ˆλ8 = r 1 3   1 0 0 0 1 0 0 0...

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    In fact,r (1) m =r q 4π 3 Y1m

    Spherical coordinate We also define the spherical coordinate by r(1) −1 = (x−iy)/ √ 2, r(1) 0 =z, r(1) 1 =−(x+ iy)/ √ 2, (A6) which satisfiesr (1)∗ m = (−1)mr(1) −m analogous to the spher- ical harmonicsY 1m. In fact,r (1) m =r q 4π 3 Y1m. In terms of the spherical coordinate, we can define the quadrupole in the form q(2) M = [r(1) ⊗r (1)](2) M (A7) = X m...

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    Explicit expressions of angular-momentum, spin, and quadrupole operators Usingp † mσ and ˜pmσ in Eqs. (37) and (39), one can construct any one-body spherical double tensor operator O(L,S) ML,MS with rank-L(0≤L≤2) for the orbital angular momentum and rank-S(0≤S≤1) for the spin angular momentum as O(L,S) ML,MS ≡O LS p† ⊗˜p (L,S) ML,MS ≡O LS X mm′ X σσ ′ ⟨1m...

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    The electron states are given explicitly in terms of the creation operatorsp † mσ

    Explicit tensor expressions of electron states in thep-shell Here, we consider the wave functions in tensor form. The electron states are given explicitly in terms of the creation operatorsp † mσ. The one-body state is given by 20 |pmσ⟩=p † mσ |0⟩with|0⟩being the closed-shell (or vac- uum in our model). For two-body states, one has |LMLSMS⟩ ≡ 1√ 2 p† ⊗p †...

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    We first define the Green’s function G(iωn) =− Z β 0 dτ⟨p(τ)p †⟩eiωnτ ,(D20) whereω n = (2n+ 1)πTis the fermionic Matsubara fre- quency

    Spectroscopic factor and overlap Let us also consider the single-particle excitation spec- trum as experimentally probed by the photoemission spectroscopy. We first define the Green’s function G(iωn) =− Z β 0 dτ⟨p(τ)p †⟩eiωnτ ,(D20) whereω n = (2n+ 1)πTis the fermionic Matsubara fre- quency. Then the spectrum is given at lowTby ρ(ε) =− 1 π ImG(ε+ i0 +) (D...

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    ⊗p i(L,S) h [p⊗p] (L2,S2) ⊗p i(L,S) U = 2 δL2,L′ 2 δS2,S′ 2 + 2F(L,S) L2,S2,L′ 2,S′ 2 ,(D24) 22 where F (L,S) L2,S2,L′ 2,S′ 2 ≡ q (2L2 + 1) (2L′ 2 + 1) (2S2 + 1) (2S′ 2 + 1) × ( 1L 2 1 L L ′ 2 1 )( 1 2 S2 1 2 S S ′ 2 1 2 ) ,(D25) and ( a b c d e f ) indicates a Wigner 6j-symbol. In the case withL ′ 2 =L 2, S′ 2 =S 2, the norm is given by N(L,S) L2,S2 ≡ h ...

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    ⊗p i(L,S) h [p⊗p] (L2,S2) ⊗p i(L,S) =O (L,S) L2,S2,L′ 2,S′ 2 N(L,S) L2,S2 N(L,S) L′ 2,S′ 2 −1/2 (D27) = √ 2(−1)L+S−3/2 N(L,S) L2,S2 (2L+ 1) (2S+ 1) −1/2 × h [p⊗p] (L′ 2,S′

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    ⊗p i(L,S) p† D[p⊗p] (L2,S2) . (D28) Here, for a spherical double tensorD (L,S) M,σ of orbital an- gular momentumL(third componentM) and spin an- gular momentumS(third componentσ), the double- reduced matrix element with respect toLandS, L2S2 D(L,S) DL1S1 has been defined as: D L2M2S2σ2 D(L,S) M,σ L1M1S1σ1 E = ⟨L1M1LM|L 2M2 ⟩√2L2+1 ⟨S1σ1Sσ|S 2σ2 ⟩√2S2+1 L2...

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    (D30) From Eq

    ⊗p i(L,S) p† D[p⊗p] (L2,S2) 2 . (D30) From Eq. (D29), we obtain S(L,S) L2,S2,L′ 2,S′ 2 = N(L,S) L2,S2 2 × h [p⊗p] (L′ 2,S′

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    (D31) This relation shows that the overlap matrix element is expressed in terms of the spectroscopic factor

    ⊗p i(L,S) h [p⊗p] (L2,S2) ⊗p i(L,S) 2 . (D31) This relation shows that the overlap matrix element is expressed in terms of the spectroscopic factor

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    Matrix elements for the states of three-body fermion system coupled tod-bosons At the end of this appendix, we show the matrix elements of the wave functions for the fermion–boson coupled states. Each term in the total Hamiltonian H=H ee +H p +H ep can now be rewritten in terms of the scalar product as Hee = J 2 (Q·Q),(D32) and Hp =ω 0(d† · ˜d) =ω 0nd,(D3...

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    quasispin operators Usingp † mσ and ˜pmσ in Eqs. (37) and (39), quasispin operators (K ±, Kz) are defined as follows: K+ ≡ r Ω 2 p† ⊗p † (0,0) ≡ r Ω 2 X m,m′,σ,σ′ ⟨1m1m′ |00⟩ 1 2 σ 1 2 σ′ |00 p† mσp† m′σ′, (E1) K− ≡(K +)† =− r Ω 2 [˜p⊗˜p](0,0), K z ≡ 1 2 (n−Ω), (E2) wherenis the number operator and Ω (= 3) is half the number of the maximally occupied elec...

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    For one-particle system, applyingK z top † mσ |0⟩ ≡ |K, K 0, m, σ⟩, one hasK z |K, K0, m, σ⟩= Kzp† mσ |0⟩= 1 2 (n−Ω)p † mσ |0⟩= (−1)p † mσ |0⟩

    Construction of states with definiteKandK 0 In the following we classify the electronic states accord- ing to quasispin. For one-particle system, applyingK z top † mσ |0⟩ ≡ |K, K 0, m, σ⟩, one hasK z |K, K0, m, σ⟩= Kzp† mσ |0⟩= 1 2 (n−Ω)p † mσ |0⟩= (−1)p † mσ |0⟩. Thus K0 =−1. Also, by applyingK 2, one obtains the quan- tum number K asK 2 |K, K0 =−1, α⟩= ...

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    (101) is satisfied,p † mσ is a spherical tensor T (1/2) 1/2 and ˜pmσ is a spherical tensorT (1/2) −1/2 in quasispin space, which is easily confirmed by Eq

    Quasispin tensors in terms of creation and annihilation operators Since Eq. (101) is satisfied,p † mσ is a spherical tensor T (1/2) 1/2 and ˜pmσ is a spherical tensorT (1/2) −1/2 in quasispin space, which is easily confirmed by Eq. (C1). As a triple tensor in the order ofL, SandK, we denotep (1,1/2,1/2) m,σ,1/2 ≡ p† mσ andp (1,1/2,1/2) m,σ,−1/2 ≡˜pmσ. The...

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    Specific cases of triple tensorsT (L,S,K) L0,S0,K0 For concreteness, we list several low-rank triple tensors forp-fermions below: T (0,0,1) 0,0,1 = h p(1,1/2,1/2) ⊗p (1,1/2,1/2) i(0,0,1) 0,0,1 = p† ⊗p † (0,0) = r 2 Ω K+,(E10) T (0,0,1) 0,0,−1 = h p(1,1/2,1/2) ⊗p (1,1/2,1/2) i(0,0,1) 0,0,−1 = [˜p⊗˜p](0,0) =− r 2 Ω K−,(E11) T (0,0,1) 0,0,0 = h p(1,1/2,1/2) ...

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    Suppose the one-body operatorQ (1) 0 hasK= 1 and K0 = 0 (e.g., a quadrupole operator)

    Wigner Eckart theorem in quasispin space We discuss concrete examples of the matrix elements. Suppose the one-body operatorQ (1) 0 hasK= 1 and K0 = 0 (e.g., a quadrupole operator). For diagonal ma- trix elements, the Wigner–Eckart theorem gives D K, Kz, α Q(1) 0 K, Kz, α E = ⟨K, Kz,1,0|K, K z ⟩√ 2K+ 1 D K, α Q(1) K K, α E .(E16) Here · · · Q(1) K · · · in...

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    Here we show its spe- cific value using quasispin formalism

    Calculation of reduced matrix element ⟨L= 2∥Q∥L= 1⟩using quasispin formalism Previously, we showed that the diagonal matrix ele- ments⟨L∥Q∥L⟩for any three-body state|L⟩vanish and also that off-diagonal matrix elements⟨L ′ ∥Q∥L⟩vanish except forL= 1 andL ′ = 2. Here we show its spe- cific value using quasispin formalism. We define a triple tensor, which is...

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    Similarly, theK= 0 operatorQ N M is constructed, us- ing orbital angular momentumL M defined in Eq

    Explicit construction of many-body operators having non-vanishing diagonal matrix elements In terms of quadrupole operatorQ (2,0,1) M,0,0 ≡ −2 p† ⊗˜p (2,0) M,0 =Q M, one can construct a two- body quadrupole operator withK= 2 andK 0 = 0 in a simple manner: QC M ≡ h Q(2,0,1) ⊗Q (2,0,1) i(2,0,2) M,0,0 = 1√ 6 h p† ⊗p † (2,0) ⊗[˜p⊗˜p](2,0) i(2,0) + 4√ 6 h p† ⊗...

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    Classification of many-body phonon states up ton d = 6

    Explicit construction of few-body systems in terms ofd-boson creation operators The group-theoretical analysis in the previous subsec- tion can be concretized by considering the explicit form of the wave functions as follows: (a) one-body system |nd = 1, L= 2, M⟩=d † M |0⟩,(F3) (b) two-body system |nd = 2, L= 0,2,4, M⟩= 1√ 2 d† ⊗d † (L) M |0⟩ = 1√ 2 X m,m...

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    In contrast, for|L= 0⟩and|L= 2⟩, such paired structures do not exist. We note that theL= 1 andL= 5 states are missing because their norms vanish, which is directly confirmed by substituting the value of the Wigner-6j symbol: O(1) 22 = 2 1 + 10 ( 2 2 2 1 2 2 )! = 0,(F13) O(5) 44 = 2 1 + 18 ( 2 4 2 5 4 2 )! = 0.(F14) We can also considerd-boson states forn ...

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    Quasispin ind-boson space In addition to the group structure ofU(5)⊃SO(5)⊃ SO(3), thed-boson system hasSU(1,1) structure. The generators ofSU(1,1) group are given in terms ofd- boson creation and annihilation operators as Kph + = 1 2 X m (−1)md† md† −m,(F16) Kph − = 1 2 X m′ (−1)m′ d−m′dm′,(F17) Kph z = 1 2 n+ 5 2 ,(F18) wheren= P m d† mdm is the number o...

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    Specific forms of double tensorsD (L,K) L0,K0 For concreteness, we list several low-rank double ten- sors for phonons below: D(0,1) 0,1 = 2√ 5 Kph + ,(F27) D(0,1) 0,−1 = 2√ 5 Kph − ,(F28) D(0,1) 0,0 = 2 5 √ 10Kph z ,(F29) D(1,0) M,0 = √ 2 h d† ⊗ ˜d i(1) M ,(F30) D(2,1) M,0 = √ 2 h d† ⊗ ˜d i(2) M .(F31)

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    Angular momentum and quadrupole operators The bosonic angular momentum operator (L=1) and a quadrupole operator (L=2) can be constructed in a manner similar to the fermions. The angular momentum in terms ofd-boson is given as L ph M = √ 10 h d† ⊗ ˜d i(1) M = √ 5D(1,0) M,0 .(F32) These satisfy the commutation relations of theSU(2) algebra: h L ph + ,L ph −...

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