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4c3d2a87f5b4165018478d41abfa0cedb0bd8e5d | subsection | 63 | 98 | Determinant expressions of the scalar products | In the tensor product of spin-\ell /2 representations we have& & S^{(\ell )}_n(\lbrace \mu _j \rbrace , \, \lbrace \lambda _{\alpha } \rbrace ; \lbrace \zeta _k \rbrace )
\\
& = &
{\frac{\prod _{\alpha =1}^{n} \prod _{j=1}^{n} \sinh (\mu _j - \lambda _{\alpha })}{\prod _{j >k} \sinh (\mu _k- \mu _j)
\prod _{\alpha < \b... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
"math.MP",
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2738b67f8e3210d5fafb272a7af21761de07eee1 | subsection | 64 | 98 | Norms of the Bethe states in the higher-spin case | For two sets of n parameters, \mu _1, \ldots , \mu _n
and \lambda _1, \ldots , \lambda _n,
we define matrix elements H_ab byH_{ab}(\lbrace \mu _j \rbrace , \lbrace \lambda _{\alpha }; \lbrace \xi _k \rbrace )
= {\frac{\sinh \eta }{\sinh (\lambda _a - \mu _b)}}
\left( {\frac{a(\mu _b)}{d(\mu _b)}}
\prod _{k=1; \ne a}^{n... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
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b06aa2361830198cb57b61cbf34e87037903ca48 | subsection | 65 | 98 | Norms of the Bethe states in the higher-spin case | Then we have\lim _{\mu _j \rightarrow \lambda _j}
det H(\lbrace \lambda _{\alpha } \rbrace , \lbrace \mu _{j} \rbrace ; \lbrace \xi _k \rbrace )
= \sinh ^n \eta \prod _{\beta =1}^{n} \prod _{m=1; m \ne \beta }^{n}
\sinh (\lambda _m - \lambda _{\beta } - \eta ) \cdot det \Phi ^{^{\prime }}
(\lbrace \lambda _{\alpha } \r... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
"math.MP",
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616a5d2109f1f77db93168b65cc9df044c114075 | subsection | 66 | 98 | Norms of the Bethe states in the higher-spin case | Gaudin's formula for the square of he norm of the Bethe state
is given by{\mbox{$N$}}_n(\lbrace \lambda _{\alpha } \rbrace ; \lbrace \xi _j \rbrace )
& = & \langle 0 | \prod _{j=1}^{n} C(\lambda _j) \prod _{j=1}^{n}B(\lambda _k)|
0 \rangle \\
& = & \sinh ^n \eta \prod _{\alpha , \beta =1; \alpha \ne \beta }^{n}
b^{-1}(... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
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5c49b64fa367acf0e1ba32289cc8265191c270a0 | subsection | 67 | 98 | Formulas of the quantum inverse scattering problem | Let us briefly review the derivation of the fundamental lemma of the
quantum-inverse scattering problem for the spin-1/2 XXZ spin chain
. It will thus becomes clear how
the pseudo-diagonalization of B
and C are important.Proposition 36 Let us denote by p_q sequence p_q=(1, 2, \ldots , n).
Recall the notation
\widetilde... | {
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{
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"raw": "J.M. Maillet and J. Sanchez de Santos, Drinfel'd twists and algebraic Bethe ansatz, ed. M. Semenov-Tian-Shansky, Amer. Math. Soc. Transl. 201 Ser. 2, (Providence, R.I.: Ameri. Math. Soc., 2000) pp. ... | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
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2f3f0241b0dd4d55fc25d6aa20a3024404fee9e7 | subsection | 68 | 98 | Formulas of the quantum inverse scattering problem | We have\widetilde{ R}_{0, p_q} = \widetilde{ R}_{0, \sigma (p_q)} \, ,
\quad {\rm for} \, \, \sigma \in {\cal S}_n \, .We shall show (REF ) in Appendix B.The following lemma plays a central role
in the quantum inverse-scattering problem .Lemma 37 For arbitrary inhomogeneous parameters
\xi _1, \xi _2, \ldots , \xi _L we... | {
"cite_spans": [
{
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"raw": "N. Kitanine, J.M. Maillet and V. Terras, Form factors of the XXZ Heisenberg spin-1/2 finite chain, Nucl. Phys. B 554 [FS] (1999) 647–678",
"source_ref_id": "efe99452be2366a35304b0c64d11b12e7d2... | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
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f196a4b8c2d19ad17a0b668fce873d69ff8c05e3 | subsection | 69 | 98 | Formulas of the quantum inverse scattering problem | From the expression
of F^{-1}_{i \cdots L \, 1 \cdots i-1} F_{1 \cdots L}
we now have{\rm tr}_0(x_0 R_{0, 1 \cdots L}) = \prod _{\alpha =1}^{i-1}
\left( (A+D)(\xi _{\alpha }) \right)^{-1}
\cdot x_i \cdot \prod _{\alpha =1}^{i} (A+D)(\xi _{\alpha }) \, . | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
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2837542fe507fbee403393dd7cd66ce4b94b7d05 | subsection | 70 | 98 | Quantum inverse-scattering problem
for the higher-spin operators | Let us consider monodromy matrix T^{+}_{0, 1 \cdots \ell N_{s}}.
Here we recall L=\ell N_{s}. | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
"math.MP",
"math.QA",
"nlin.SI",
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9ea8d924f6874684205e98d80f4a31501f6a0356 | subsection | 71 | 98 | Quantum inverse-scattering problem
for the higher-spin operators | For simplicity,
we shall suppress the superscript `+'
for A, B, C and D operators through this subsection.We recall the following:
\Delta ^{(n-1)}(K) = K^{\otimes n} and\Delta ^{(n-1)}(X^{+}) & = & \sum _{j=1}^{n}
K^{\otimes (j-1)} \otimes X^{+}_j \otimes I^{\otimes (n-j)} \, , \\
\Delta ^{(n-1)}(X^{-}) & = & \sum _{j=... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
"math.MP",
"math.QA",
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ffe860ade3fd6e13a60ceabb36152f0d372bed11 | subsection | 72 | 98 | Quantum inverse-scattering problem
for the higher-spin operators | \\ | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
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"Tetsuo Deguchi",
"Chihiro Matsui"
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"cond-mat.stat-mech",
"hep-th",
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ac9e1599edfa3eb2f43883b6a429808acadb0bf0 | subsection | 73 | 98 | Useful formulas in the higher-spin case | Let us denote by X^{\pm (\ell )} the matrix representations
of generators X^{\pm } in the spin-\ell /2 representation of
U_q(sl_2). Here we recall that the matrix representations of
X^{\pm (\ell )} are obtained by calculating
the action of \Delta ^{(\ell -1)}(X^{\pm })
on the basis \lbrace || \ell , n \rangle \rbrace .... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
"math.MP",
"math.QA",
"nlin.SI",
"quant-ph"
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257fd6fc38faf15ec8d4d6d136b3925cf23e5480 | subsection | 74 | 98 | Useful formulas in the higher-spin case | Multiplying projection operators to them,
we obtain the following formulas:P^{(\ell )}_{1 \cdots \ell } \, \sigma _{1}^{-} \, P^{(\ell )}_{1 \cdots \ell }
& = & {\frac{1}{[\ell ]_q}} \, X^{- (\ell +)} \\
P^{(\ell )}_{1 \cdots \ell } \, \sigma _{\ell }^{+} \, P^{(\ell )}_{1 \cdots \ell }
& = & {\frac{1}{[\ell ]_q}} \, X... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
"math.MP",
"math.QA",
"nlin.SI",
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77ef8750b96ef52dca245073f2083b5b60dcb32c | subsection | 75 | 98 | Useful formulas in the higher-spin case | This reduces the calculational task very much.In the derivation of (REF ), we first note\chi _{1 \cdots L} \sigma _{(i-1)\ell +1}^{-} \chi _{1 \cdots L}^{-1}
\exp ( - \xi _{(i-1)\ell +1}) \, ,and then we show the following:\sigma _{(i-1)\ell +1}^{-}
= \prod _{\alpha =1}^{(i-1)\ell } (A^{+}+ D^{+})(\xi _{\alpha }) \,
B^... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
"math.MP",
"math.QA",
"nlin.SI",
"quant-ph"
] | 2,008 | en | Physics | [
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3ac5f4e2ceee947adeb42a5fae6e1abd4e4f0ed4 | subsection | 76 | 98 | Useful formulas in the higher-spin case | We can show the following:|| \ell , m \rangle \langle \ell , n ||
& = &
\left[
\begin{array}{c}
\ell \\
m
\end{array}
\right]_q \,
q^{m(m+1)/2 - n(n-1)/2 + n \ell - (i_1 + \cdots + i_m + j_1 + \cdots + j_n)} \\
& & \times \,
P_{1 \cdots \ell }^{(\ell )}
\left( \prod _{k=1}^{m} e_{i_k}^{21}
\cdot \prod _{p=1;p \ne i_k, ... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
"math.MP",
"math.QA",
"nlin.SI",
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41bb97da7efa359e95e03dac00c2a686e0f4e4d6 | subsection | 77 | 98 | Useful formulas in the higher-spin case | We now define E_i^{mn (\ell +)} by the
unit matrices acting on the ith component
of the tensor product (V^{(\ell )})^{\otimes N_s}.
Explicitly, we haveE_{i}^{mn \, (\ell +)} = (I^{(\ell )})^{\otimes (i-1)} \otimes E^{mn}
\otimes (I^{(\ell )})^{\otimes (N_s-i)}where I^{(\ell )} denotes the (\ell +1) \times (\ell +1) ide... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
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"hep-th",
"math-ph",
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1dfcaebbbcc60ebe914e5dd4bc5ba00eff881f91 | subsection | 78 | 98 | Useful formulas in the higher-spin case | For m > n we haveE_{i}^{mn \, (\ell +)}
& = & \left[
\begin{array}{c}
\ell \\
m
\end{array}
\right]_q \,
q^{n(\ell -n)} \,
P^{(\ell )}_{1 \cdots L}
\,
\prod _{\alpha =1}^{(i-1)\ell } (A+D)(\xi _{\alpha })
\prod _{k=1}^{n} D(\xi _{(i-1)\ell +k} )
\prod _{k=n+1}^{m} B(\xi _{(i-1)\ell +k})
\\
& & \quad \times \, \prod _{k... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
"math.MP",
"math.QA",
"nlin.SI",
"quant-ph"
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c88fea304ba747255f721da15f269cfa6e1b660b | subsection | 79 | 98 | Useful formulas in the higher-spin case | It is easy to show the following:\sigma _{i_1}^{-} \cdots \sigma _{i_m}^{-} | 0 \rangle \langle 0 | \sigma _{j_1}^{+} \cdots \sigma _{j_n}^{+}
=
e_{i_1}^{21} \cdots e_{i_n}^{21}
\prod _{p=1; p \ne i_k, j_q}^{\ell } e_{p}^{11}
e_{i_1}^{12} \cdots e_{i_n}^{12}Then, making use of expressions
(REF ) and (REF ),
we obtain (... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
"math.MP",
"math.QA",
"nlin.SI",
"quant-ph"
] | 2,008 | en | Physics | [
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59dfa5a3a0967925a594854e95f98819afb27a77 | subsection | 80 | 98 | Form factors for higher-spin operators | Making use of the fundamental lemma of the quantum inverse-scattering
problem, lemma REF , together with the useful formulas
given in §7.3 such as
(REF ) and (), and (REF ), (REF )
and (REF ),
we can systematically calculate form factors for the higher-spin cases.
Here we note that the form factors associated with gene... | {
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"end": 404,
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"raw": "O.A. Castro-Alvaredo and J.M. Maillet, Form factors of integrable Heisenberg (higher) spin chains, J. Phys. A: Math. Theor. 40 (2007) 7451–7471.",
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4213d145615c60704a5f8707cb9995e57930659a | subsection | 81 | 98 | Form factors for higher-spin operators | Putting (REF ) into (REF ) and making use of the fact
that projector P_{1 \cdots L}^{(\ell )} commutes with
the matrix elements of R_{0, 1 \cdots L}^{+},
we have& & \langle 0 | \prod _{j=1}^{n+1} C^{+}(\mu _j) \cdot X_i^{- (\ell +)} \prod _{k=1}^{n} B^{+}(\lambda _k) | 0 \rangle \\
& = & [\ell ]_q \, \langle 0 | \prod ... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
"math.MP",
"math.QA",
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20d1df724ee1fc7051f1f82428142f3442bacb2a | subsection | 82 | 98 | Form factors for higher-spin operators | Then, we obtain the following expression.& & F_n^{- (\ell )}(i; \, \lbrace \mu _j \rbrace , \lbrace \lambda _k \rbrace )
= [\ell ]_q \, {\frac{\phi _{(i-1)\ell }(\lbrace \mu _j \rbrace _{n+1})}{\phi _{(i-1)\ell +1}(\lbrace \lambda \rbrace _n)}} \\
& & \qquad \times S_{n+1}\left(\lbrace \mu _j \rbrace _{n+1},
\lbrace \x... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
"math.MP",
"math.QA",
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c09559c050903216dbca0390a9114c8e7f655032 | subsection | 83 | 98 | Derivation of symmetry relations for | Lemma 8.1 Let p be a sequence of n integers, 1, 2, \ldots , n.
For any \sigma _A, \sigma _B \in {\cal S}_n we have(\sigma _A \sigma _B) \, p = \sigma _B (\sigma _A p) \, .Let us denote p_{\sigma _A i} by q_i for i=1, 2, \ldots , n.
We thus have\sigma _B (q_1, \ldots , q_n) & = & (q_{\sigma _B 1}, \ldots , q_{\sigma _B ... | {
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quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
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6e0c0b6038417ae94f2fa2338a2dba26098c120e | subsection | 84 | 98 | Derivation of symmetry relations for | For \sigma _c=(1 2 \cdots n) we haveR^{\sigma _c}_{p} = R_{p_1, p_2 \cdots p_n} \, .Noting
(1 2 \cdots n) = (1 2)(2 3) \cdots (n-1 \, n)
= s_1 s_2 \cdots s_{n-1}.
we haveR_p^{\sigma _c} & = & R_p^{s_1 s_2 \cdots s_{n-1}}
=R_{s_1 p}^{s_2 \cdots s_{n-1}} R_p^{s_1} \\
& = & R_{s_2 (s_1 p)}^{s_3 \cdots s_{n-1}} R_{s_1 p}^{... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
"math.MP",
"math.QA",
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4af0b03dfad7e9f00e2e3039e428ebb40c966f04 | subsection | 85 | 98 | Derivation of symmetry relations for | We define R_{j,k} byR_{j, k} = \sum _{a} id_1 \otimes \cdots \otimes r_j^{(a,1)}
\otimes \cdots \otimes r_k^{(a,2)} \otimes \cdots \otimes id_n
\quad \in {\cal A}^{\otimes n}
\, .If R_{j,k} satisfy the inversion relations and the Yang-Baxter equations:R_{12}R_{21} & = & id \, , \\
R_{12}R_{13}R_{23} & = & R_{23} R_{13}... | {
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quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
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e0b27896aaef0151895aeeb7713c244e37d38ad3 | subsection | 86 | 98 | Derivation of symmetry relations for | Assuming a < b we have& & R_{\sigma _A j, \, \sigma _A (j+1)} \, \,
\sigma _A \circ \Delta ^{(n-1)}(x)
\\
& = &R_{a, b} \,
\sum x_1^{(\sigma _A^{-1} 1)} \otimes x_2^{(\sigma _A^{-1} 2)} \otimes \cdots \otimes x^{(\sigma _A^{-1} a)}_a \otimes \cdots \otimes x^{(\sigma _A^{-1} b)}_b \otimes \cdots \otimes x^{(\sigma _A^{... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
"math.MP",
"math.QA",
"nlin.SI",
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6621063e46bd55de657b0f36874a29c6a12a5a30 | subsection | 87 | 98 | Derivation of symmetry relations for | Let us consider m auxiliary spaces with suffices
a(1), a(2), \ldots , a(m), respectively.
We denote the monodromy matrix
T_{a(j)}(\lambda _{a(j)}; \xi _1, \ldots , \xi _L)
simply by T_{a(j)}. We denote by \Delta ^{(m-1)}(T)
the following operator:\Delta ^{(m-1)}(T) = T_{a(1)} T_{a(2)} \cdots T_{a(m)}Let \sigma an eleme... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
"math.MP",
"math.QA",
"nlin.SI",
"quant-ph"
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67aeaf59ee67feef81b5c76016d37e529cdbbef5 | subsection | 88 | 98 | Symmetric-group action on products of | Lemma 9.1 ()
(i) Cocycle conditions hold for n \le L.R_{2 \cdots n-1, \, n} R_{1, \, 2 \cdots n} =
R_{1, \, 2 \cdots n-1} R_{1 2 \cdots n-1, \, n}(ii) The unitarity relations hold for n \le L.R_{1, \, 2 \cdots n} R_{2 3 \cdots n, \, 1} = I^{\otimes L}Cocycle conditions (REF ) are derived
from the Yang-Baxter equations.... | {
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quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
"math.MP",
"math.QA",
"nlin.SI",
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ec291da91f88973aaa7b2163134eb56afdddfb46 | subsection | 89 | 98 | Symmetric-group action on products of | We haveR_{p}^{\sigma } R_{0, \, p} & = & R_{0, \, \sigma (p)} R_{p}^{\sigma }
\\
R_{p}^{\sigma } R_{p, \, 0} & = & R_{\sigma (p), \, 0} R_{p}^{\sigma }Expressing permutation \sigma as a product of generators s_j,
and applying (REF ) many times, we can show the symmetry relations.We define the action of \sigma on the F-... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
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"math.QA",
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b5e594ca35d02bafdfa99193746cc0a909455df7 | subsection | 90 | 98 | Symmetric-group action on products of | We haveF_{i \cdots L \, 1 \cdots i-1} = F_{\sigma _c^{i-1}(p_q)}
= F_{1 \cdots L} R_{p_q}^{\sigma _c^{i-1}}and hence we haveF^{-1}_{i \cdots L \, 1 \cdots i-1} F_{1 \cdots L} & = &
\left(F_{1 \cdots L}
R_{p_q}^{\sigma _c^{i-1}} \right)^{-1}
F_{1 \cdots L} \\
& = & R_{p_q}^{\sigma _c^{i-1}} \,
F_{1 \cdots L}^{-1} F_{1 \... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
"math.MP",
"math.QA",
"nlin.SI",
"quant-ph"
] | 2,008 | en | Physics | [
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f4b9a4b78f36cbc9bb2638e466077a5dfaac47ea | subsection | 91 | 98 | Formulas of the | Lemma 10.1 For two integers \ell and n satisfying
0 \le n \le \ell we have\sum _{1 \le i_1 < \cdots < i_n \le \ell } q^{2 i_1 + \cdots + 2 i_n}
=
q^{n (\ell +1)} \,
\left[
\begin{array}{c}
\ell \\
n
\end{array}
\right]_{q}We can show by induction on \ell the q-binomial expansion as follows.\prod _{k=0}^{\ell -1} \left(... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
"math.MP",
"math.QA",
"nlin.SI",
"quant-ph"
] | 2,008 | en | Physics | [
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90ae234a213e0e256257d427ad363b0100f895d8 | subsection | 92 | 98 | Formulas of the | Similarly, we can show
().Let us review some points of the diagonalization process
of the A and D operators .Lemma 11.1 Operators A and D are upper- and lower-triangular matrices,
respectively. Moreover,
the eigenvalues of operators A and D are given by{\rm diag} \left( D_{1 2 \cdots n}(\lambda _0) \right)
& = &
\bigot... | {
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quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
"math.MP",
"math.QA",
"nlin.SI",
"quant-ph"
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3180bfff9c8230d803db3588f4677e58bcabcb4f | subsection | 93 | 98 | Formulas of the | From R_{0, 1 \cdots n} = R_{0, 2 \cdots n} R_{0, 1} we haveD_{1 2 \cdots n } & = & C_{2 \cdots n} B_1 + D_{2 \cdots n} D_1
\\
& = &
\left(
\begin{array}{cc}
b_{01} D_{2 \cdots n}(\lambda _0) & 0 \\
c_{01} C_{2 \cdots n}(\lambda _0) & D_{2 \cdots n}(\lambda _0)
\end{array}
\right)_{[0]} \, .We thus calculateF_{1 \cdots ... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
"math.MP",
"math.QA",
"nlin.SI",
"quant-ph"
] | 2,008 | en | Physics | [
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14fcd60b2b8b5896e19505d4790525abca5cf623 | subsection | 94 | 98 | Formulas of the | Similarly, we can diagonalize A, A^{\dagger } ]
and D^{\dagger }.Lemma 11.3 The diagonalized form of F_{0, 1 \cdots n}
{\bar{F}}^{\dagger }_{0, 1 \cdots n} is given by the following:F_{1 \cdots n} \left(F_{0, 1 \cdots n}
{\bar{F}}^{\dagger }_{0, 1 \cdots n} \right)
F_{1 \cdots n}^{-1} = {\hat{\delta }}^{-1}_{0, 1 \cdot... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
"math.MP",
"math.QA",
"nlin.SI",
"quant-ph"
] | 2,008 | en | Physics | [
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4c4f30856d9de284f7eefec002c96866fbb9ec48 | subsection | 95 | 98 | Formulas of the | We have{\bar{F}}^{\dagger }_{0 1 \cdots n} \widehat{\delta }_{0 1 \cdots n}
= {\bar{F}}^{\dagger }_{0 1 \cdots n} \left(
{\bar{F}}^{\dagger }_{1 \cdots n}
\widehat{\delta }_{1 \cdots n} \right)
\widehat{\delta }_{0, 1 \cdots n}
= {\bar{F}}^{\dagger }_{0 1 \cdots n}
{F}^{-1}_{1 \cdots n}
\widehat{\delta }_{0, 1 \cdots n... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
"math.MP",
"math.QA",
"nlin.SI",
"quant-ph"
] | 2,008 | en | Physics | [
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71e931aa288c82ce1c8b18f8195e102b3a8db64e | subsection | 96 | 98 | Formulas of the | We first show{\bar{F}}_{1, 2 \cdots n}^{\dagger }
= {\cal C}_{1 2 \cdots n}
\left(e_1^{11} + e_1^{22}R_{1, 2 \cdots n} \right)^{\dagger }
{\cal C}_{1 2 \cdots n} = F_{n \cdots 2, 1}^{t_1 \cdots t_n} \, .Making use of the induction assumption we showF_{1 2 \cdots n}^{\dagger } & = &
(F_{2 \cdots n} F_{1, 2 \cdots n})^{\... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
"math.MP",
"math.QA",
"nlin.SI",
"quant-ph"
] | 2,008 | en | Physics | [
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25cc809ade886ed2b51a85b77565edbc1ebe73b1 | subsection | 97 | 98 | Lemmas for Diagonalizing the | Lemma 12.1 Let X^{+} be the generator of the quantum group U_q(sl_2) and
X_i^{+} the matrix representation of X^{+}
acting on the ith site. We have{\widetilde{\Delta }}_{1 \cdots n}(X^{+})
= \left(X_n^{+} + e_n^{11}
{\widetilde{A^{+}} }^{\dagger }_{1 \cdots n-1}(\xi _n)
{\widetilde{\Delta }}_{1 \cdots n-1}(X^{+})
+ e_n... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
"math.MP",
"math.QA",
"nlin.SI",
"quant-ph"
] | 2,008 | en | Physics | [
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97b5032c9f697f76c109180c90dbbed290c7c1c4 | abstract | 0 | 1 | Abstract | We report on high-field magnetic properties of the silver vanadium phosphate
Ag(2)VOP(2)O(7). This compound has a layered crystal structure, but the
specific topology of the V-P-O framework gives rise to a one-dimensional spin
system, a frustrated alternating chain. Low-field magnetization measurements
and band structu... | {
"cite_spans": []
} | 10.1088/1742-6596/145/1/012067 | 0807.1849 | Magnetic interactions and high-field properties of Ag(2)VOP(2)O(7):
frustrated alternating chain close to the dimer limit | [
"Alexander A. Tsirlin",
"Ramesh Nath",
"Franziska Weickert",
"Yurii Skourski",
"Christoph Geibel",
"Helge Rosner"
] | [
"cond-mat.str-el",
"cond-mat.mtrl-sci"
] | 2,008 | en | Physics | [
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5b2cfdf82ac43fb2c19552071bfb7c08e574989e | abstract | 0 | 18 | Abstract | Symmetry adapted bases in quantum chemistry and bases adapted to quantum
information share a common characteristics: both of them are constructed from
subspaces of the representation space of the group SO(3) or its double group
(i.e., spinor group) SU(2). We exploit this fact for generating spin bases of
relevance for ... | {
"cite_spans": []
} | 10.1135/cccc20081281 | 0807.1850 | Generalized spin bases for quantum chemistry and quantum information | [
"M. Kibler"
] | [
"quant-ph"
] | 2,008 | en | Physics | [
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b70a8f21906ecb163a380cb46bf87851aa798004 | subsection | 1 | 18 | Introduction | The notion of symmetry adapted functions (or vectors) in physical chemistry
and solid state physics goes back to the fifties^{1}. The use of bases
consisting of such functions allows to simplify the calculation of matrix
elements of operators and to factorize the secular equation. Symmetry adaptation
generally requires... | {
"cite_spans": []
} | 10.1135/cccc20081281 | 0807.1850 | Generalized spin bases for quantum chemistry and quantum information | [
"M. Kibler"
] | [
"quant-ph"
] | 2,008 | en | Physics | [
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1ad02a4727aa2d2d48c66e3161e4784ffbbb50e7 | subsection | 2 | 18 | AN ALTERNATIVE TO THE | Let us consider a generalized angular momentum. We note j^2 its square and j_z
its z-component. The common eigenvectors of j^2 and j_z are denoted as
| j , m \rangle . We know that^{11}j^2 |j , m \rangle = j(j+1) |j , m \rangle , \quad j_z |j , m \rangle = m |j , m \ranglein a system of units where the rationalized Pla... | {
"cite_spans": []
} | 10.1135/cccc20081281 | 0807.1850 | Generalized spin bases for quantum chemistry and quantum information | [
"M. Kibler"
] | [
"quant-ph"
] | 2,008 | en | Physics | [
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79079dd354500299ecd4e17d1031449115ee965f | subsection | 3 | 18 | AN ALTERNATIVE TO THE | (REF ), we can check that the action of v_{ra} on the state
| j , m \rangle is given byv_{ra} |j , m \rangle = \left( 1 - \delta _{m,j} \right) q^{(j-m)a}
|j , m+1 \rangle + \delta _{m,j}
{e}^{{i} 2 \pi j r}
|j , -j \rangle .Furthermore, the matrix V_{ra} of the operator v_{ra} on the basis b_s readsV_{ra} =
\pmatrix {... | {
"cite_spans": []
} | 10.1135/cccc20081281 | 0807.1850 | Generalized spin bases for quantum chemistry and quantum information | [
"M. Kibler"
] | [
"quant-ph"
] | 2,008 | en | Physics | [
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125339fae50cd3ac4225fb7657e7c2eac1aada9c | subsection | 4 | 18 | AN ALTERNATIVE TO THE | In the particular case where 2j+1 is a prime integer,
the overlap between the bases B_{ra} and B_{rb} is such that^{13}| \langle j \alpha ; r a | j \beta ; r b \rangle | =
\delta _{\alpha , \beta } \delta _{a , b} + \frac{1}{\sqrt{2j+1}} (1 - \delta _{a , b})a property of considerable importance in quantum information.... | {
"cite_spans": []
} | 10.1135/cccc20081281 | 0807.1850 | Generalized spin bases for quantum chemistry and quantum information | [
"M. Kibler"
] | [
"quant-ph"
] | 2,008 | en | Physics | [
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2bfe7115b11a4eb1b76c9d516c924089ec5b2307 | subsection | 5 | 18 | A FORMULATION FOR | The parameter r is of interest for group-theoretical analyses but turns out to be of no concern
here. Therefore, we shall restrict ourselves in the following to the case r=0. In addition, we shall
adopt the notationk = j - m, \quad | k \rangle = | j , m \rangle , \quad | a \alpha \rangle = | j \alpha ; 0 a \rangle , \q... | {
"cite_spans": []
} | 10.1135/cccc20081281 | 0807.1850 | Generalized spin bases for quantum chemistry and quantum information | [
"M. Kibler"
] | [
"quant-ph"
] | 2,008 | en | Physics | [
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64a13136ec30757dbd824161644f3cb4fcef67fb | subsection | 6 | 18 | The case | In this case, relevant for a spin j = 1/2 or for a qubit, we have q = -1 and a, \alpha \in \lbrace 0 , 1 \rbrace . The matrices of the
operators v_{0a} areV_{00} =
\pmatrix {
0 &1 \cr 1 &0 \cr }, \quad V_{01} =
\pmatrix {
0 &-1 \cr 1 &0 \cr }.We note in passing a connection (to be generalized below) with the Pauli matr... | {
"cite_spans": []
} | 10.1135/cccc20081281 | 0807.1850 | Generalized spin bases for quantum chemistry and quantum information | [
"M. Kibler"
] | [
"quant-ph"
] | 2,008 | en | Physics | [
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abd454005770106211fe389fde08c5ec13a7c34b | subsection | 7 | 18 | The case | This leads toB_{2} &:& \alpha \rightarrow \pmatrix {
1 \cr 0 \cr }, \quad \beta \rightarrow \pmatrix {
0 \cr 1 \cr } \\
B_{00} &:& | 0 0 \rangle \rightarrow \frac{1}{\sqrt{2}} \pmatrix {
1 \cr 1 \cr }, \quad | 0 1 \rangle \rightarrow - \frac{1}{\sqrt{2}} \pmatrix {
1 \cr -1 \cr } \\
B_{01} &:& | 1 0 \rangle \rightarrow... | {
"cite_spans": []
} | 10.1135/cccc20081281 | 0807.1850 | Generalized spin bases for quantum chemistry and quantum information | [
"M. Kibler"
] | [
"quant-ph"
] | 2,008 | en | Physics | [
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4408a4b278f81d3af57013e8d8db93f1e011f613 | subsection | 8 | 18 | The case | The
matrices of the operators v_{0a} areV_{00} =
\pmatrix {
0 &1 &0 \cr 0 &0 &1 \cr 1 &0 &0 \cr }, \quad \pmatrix {
0 &q &0 \cr 0 &0 &q^2 \cr 1 &0 &0 \cr }, \quad \pmatrix {
0 &q^2 &0 \cr 0 &0 &q \cr 1 &0 &0 \cr }.The bases B_{3}, B_{00} and B_{01} B_{02} areB_{3}: & & | 0 \rangle , \ | 1 \rangle , \ | 2 \rangle \\
B_{... | {
"cite_spans": []
} | 10.1135/cccc20081281 | 0807.1850 | Generalized spin bases for quantum chemistry and quantum information | [
"M. Kibler"
] | [
"quant-ph"
] | 2,008 | en | Physics | [
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af1bc3df717fa2e20a320967d2df8187fd9d56ad | subsection | 9 | 18 | The case | In terms of colum vectors, we haveB_{3} &:& | 0 \rangle \rightarrow \pmatrix {
1 \cr 0 \cr 0 \cr }, \quad | 1 \rangle \rightarrow \pmatrix {
0 \cr 1 \cr 0 \cr }, \quad | 2 \rangle \rightarrow \pmatrix {
0 \cr 0 \cr 1 \cr } \\
B_{00} &:& | 0 0 \rangle \rightarrow \frac{1}{\sqrt{3}} \pmatrix {
1 \cr 1 \cr 1 \cr }, \quad ... | {
"cite_spans": []
} | 10.1135/cccc20081281 | 0807.1850 | Generalized spin bases for quantum chemistry and quantum information | [
"M. Kibler"
] | [
"quant-ph"
] | 2,008 | en | Physics | [
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0326bdec1288450ff8333483078632103b7a53d0 | subsection | 10 | 18 | The case | This can be achieved by replacing the space {\cal E}(4)
spanned by \lbrace | 3/2 , m \rangle : m = 3/2, 1/2, -1/2, -3/2 \rbrace by the tensor product space
{\cal E}(2) \otimes {\cal E}(2) spanned by the basis\lbrace \alpha \otimes \alpha , \alpha \otimes \beta , \beta \otimes \alpha , \beta \otimes \beta \rbrace .The s... | {
"cite_spans": []
} | 10.1135/cccc20081281 | 0807.1850 | Generalized spin bases for quantum chemistry and quantum information | [
"M. Kibler"
] | [
"quant-ph"
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ffa88c7d715940df593734dc6416e1dfa911a728 | subsection | 11 | 18 | The case | As a result, we have the d+1 = 5 following mutually unbiased bases where
\lambda = (1-i)/2 and \mu = (1+i)/2.The canonical basis:\alpha \otimes \alpha , \quad \alpha \otimes \beta , \quad \beta \otimes \alpha , \quad \beta \otimes \betaor in column vectors\pmatrix {
1 \cr 0 \cr 0 \cr 0 \cr }, \quad \pmatrix {
0 \cr 1 \... | {
"cite_spans": []
} | 10.1135/cccc20081281 | 0807.1850 | Generalized spin bases for quantum chemistry and quantum information | [
"M. Kibler"
] | [
"quant-ph"
] | 2,008 | en | Physics | [
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5975fe92aa66e1c709f1236f3fd556f7eb1e5aa8 | subsection | 12 | 18 | GENERALIZED PAULI MATRICES | From the operators v_{0a}, it is possible to define two basic operators x and z which can be
used for generating generalized Pauli matrices. Let us putx = v_{00}, \quad z = v_{00}^{\dagger } v_{01}.The action of x and z on the space {\cal E}(2j+1) is given byx |j , m \rangle = \left( 1 - \delta _{m,j} \right) |j , m+1 ... | {
"cite_spans": []
} | 10.1135/cccc20081281 | 0807.1850 | Generalized spin bases for quantum chemistry and quantum information | [
"M. Kibler"
] | [
"quant-ph"
] | 2,008 | en | Physics | [
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dd14ea5a5f5b4724548c4f51f92961aa15df3eb9 | subsection | 13 | 18 | GENERALIZED PAULI MATRICES | Additionally, the commutator
[u_{ab} , u_{a^{\prime }b^{\prime }}]_- and the
anti-commutator [u_{ab} , u_{a^{\prime }b^{\prime }}]_+ of u_{ab} and u_{a^{\prime }b^{\prime }} are given by[u_{ab} , u_{a^{\prime }b^{\prime }}]_{\mp } = \left( q^{-ba^{\prime }} \mp q^{-ab^{\prime }} \right) u_{a^{\prime \prime } b^{\prime ... | {
"cite_spans": []
} | 10.1135/cccc20081281 | 0807.1850 | Generalized spin bases for quantum chemistry and quantum information | [
"M. Kibler"
] | [
"quant-ph"
] | 2,008 | en | Physics | [
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91e830bb0435a61d4000149d3b037952d0195535 | subsection | 14 | 18 | Exemple 1 | In the case j = 1/2 \Leftrightarrow d = 2 (\Rightarrow q = -1), the matrices of the 4
operators u_{ab} with a, b = 0,1 areI = X^0 Z^0 =
\pmatrix {
1 &0 \cr 0 &1 \cr }, \quad X = X^1 Z^0 =
\pmatrix {
0 &1 \cr 1 &0 \cr }Z = X^0 Z^1 =
\pmatrix {
1 &0 \cr 0 &-1 \cr }, \quad Y = X^1 Z^1 =
\pmatrix {
0 &-1 \cr 1 &0 \cr }.In ... | {
"cite_spans": []
} | 10.1135/cccc20081281 | 0807.1850 | Generalized spin bases for quantum chemistry and quantum information | [
"M. Kibler"
] | [
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] | 2,008 | en | Physics | [
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3e6e4a893999255f2c63364cbfb177f7c38a7fd0 | subsection | 15 | 18 | Exemple 2 | In the case j = 1 \Leftrightarrow d = 3 (\Rightarrow q = \exp (2 \pi i/3)), the matrices of the 9
operators u_{ab} with a, b = 0,1,2, viz.,X^0 Z^0 = I \quad X^1 Z^0 = X \quad X^2 Z^0 = X^2 \quad X^0 Z^1 = Z \quad X^0 Z^2 = Z^2X^1 Z^1 = X Z \quad X^2 Z^2 \quad X^2 Z^1 = X^2 Z \quad X^1 Z^2 = X Z^2areI =
\pmatrix {
1 &0 ... | {
"cite_spans": []
} | 10.1135/cccc20081281 | 0807.1850 | Generalized spin bases for quantum chemistry and quantum information | [
"M. Kibler"
] | [
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4cae3afa1fa5b037b60b915f4230977b9054b766 | subsection | 16 | 18 | CONCLUDING REMARKS | The various bases described in the present paper are of central importance in quantum information and
quantum computation. They also play an important role for quantum (chemical and physical) systems with
cyclic symmetry. By way of illustration, we would like to mention two examples.Let us consider a ring shape molecul... | {
"cite_spans": []
} | 10.1135/cccc20081281 | 0807.1850 | Generalized spin bases for quantum chemistry and quantum information | [
"M. Kibler"
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e16509acd16f7ac436a5a8b67f585d72cb112978 | subsection | 17 | 18 | CONCLUDING REMARKS | A similar decomposition holds of SU(d) in the case where d = p^e, with p prime
integer and e positive integer^{22}. However, in this case we need to replae {\cal E}(d) by
{\cal E}(p)^{\otimes e}.A second group-theoretical remark concern a finite group known as the Pauli group or the finite
Heisenberg-Weyl group^{17, 18... | {
"cite_spans": []
} | 10.1135/cccc20081281 | 0807.1850 | Generalized spin bases for quantum chemistry and quantum information | [
"M. Kibler"
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2b904bc4b827c2db4d894df59b48c792544ca875 | abstract | 0 | 21 | Abstract | In this paper we classify a linear family of Lie brackets on the space of
rectangular matrices $Mat(n\times m,\K)$ and we give an analogue of the Ado's
Theorem. We give also a similar classification on the algebra of the square
matrices $Mat(n, \K)$ and as a consequence, we prove that we can't built a
faithful represen... | {
"cite_spans": []
} | 0807.1851 | Linear family of Lie brackets on the space of matrices $Mat(n\times
m,\K)$ and Ado's Theorem | [
"Bechir Dali"
] | [
"math.RT"
] | 2,008 | en | Mathematics | [
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203d76e45a9e5df658393380a9da6aa685e854aa | subsection | 1 | 21 | Introduction | We begin by setting some notations which will be used throughout
the paper. Let \mathbb {K} be a field with characteristic p=0,
Mat(n\times m,\mathbb {K}) be the linear space of n\times m rectangular
matrices with coefficients in \mathbb {K} and Mat(n,\mathbb {K}) is the
associative algebra of square matrices with coef... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 2856,
"openalex_id": "",
"raw": "Yanovski A. B., Linear Bundles of Lie Brackets and their Applications, J. Math Phys. 41 (2000) 7869-7882.",
"source_ref_id": "84deae5611d3952ae95d044b9203258247dcd715",
"start": 2630
... | 0807.1851 | Linear family of Lie brackets on the space of matrices $Mat(n\times
m,\K)$ and Ado's Theorem | [
"Bechir Dali"
] | [
"math.RT"
] | 2,008 | en | Mathematics | [
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589bd812d3c78f79abcfda447dd5cbced2da8034 | subsection | 2 | 21 | Introduction | This induces a new Lie algebra
structure, defined by the bracket[u,~v]_w=u.w.v-v.w.u \quad \quad u,~v\in \mathcal {A}\quad \quad (1).Thus we obtain a family of Lie brackets, labelled by the element
w. It is readily seen that we actually have a linear space of
Lie brackets, since the sum of two such brackets is also a L... | {
"cite_spans": []
} | 0807.1851 | Linear family of Lie brackets on the space of matrices $Mat(n\times
m,\K)$ and Ado's Theorem | [
"Bechir Dali"
] | [
"math.RT"
] | 2,008 | en | Mathematics | [
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03be5772ee8f830ab3eaad04bbe05c24ef3aa183 | subsection | 3 | 21 | Preliminaries | Let n, m be integers in \mathbb {N}^*=\mathbb {N}\setminus \lbrace
0\rbrace and let Mat(n\times m,\mathbb {K}) be the linear vector space of
n\times m rectangular matrices. We denote its canonical basis
(E_{i,j})_{1\le i\le n,1\le j\le m} withE_{i,j}=(\delta _{p,i}\delta _{q,j})_{1\le p\le n,1\le q\le m},where \delta ... | {
"cite_spans": []
} | 0807.1851 | Linear family of Lie brackets on the space of matrices $Mat(n\times
m,\K)$ and Ado's Theorem | [
"Bechir Dali"
] | [
"math.RT"
] | 2,008 | en | Mathematics | [
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d61253199e07881c4a3db3c95472fde382e06ba4 | subsection | 4 | 21 | Preliminaries | Note that
J_{n,n}=I_n the identity matrix.Consider now the linear space Mat(n\times m,\mathbb {K}) and for J\in Mat(m\times n,\mathbb {K}) put[A,B]_J=AJB-BJA,\quad A,B\in Mat(n\times m,\mathbb {K}),then we haveLemma 1.5 (i) [A,B]_{J+\alpha J^{\prime }}=[A,B]_J+\alpha [A,B]_{J^{\prime }}, \forall \alpha \in \mathbb {K},... | {
"cite_spans": []
} | 0807.1851 | Linear family of Lie brackets on the space of matrices $Mat(n\times
m,\K)$ and Ado's Theorem | [
"Bechir Dali"
] | [
"math.RT"
] | 2,008 | en | Mathematics | [
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ad27ee8aa25d13a375e44cc0a16ddfc11b858ba0 | subsection | 5 | 21 | Preliminaries | Recall that J_{n,n}=I_n, the
identity matrix of Mat(n,\mathbb {K}).For easy of notations, we shall denote the Lie algebra
\bigl (Mat(n\times m,\mathbb {K}),[~,~]_{J_{m,n,r}}\bigr ) by
\mathfrak {gl}(n,m,r,\mathbb {K}) and in the case when m=n, we simply
denotes the lie algebra \bigl (Mat(n,\mathbb {K}),[~,~]_{J_{n,r}}\... | {
"cite_spans": []
} | 0807.1851 | Linear family of Lie brackets on the space of matrices $Mat(n\times
m,\K)$ and Ado's Theorem | [
"Bechir Dali"
] | [
"math.RT"
] | 2,008 | en | Mathematics | [
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b3193904425b4fc2c5981100042c5d7ea1582f54 | subsection | 6 | 21 | The case of | In this section we will deal with the linear vector space of
square matrices Mat(n,\mathbb {K}). It is well known that it has a
structure of associative algebra and thus a Lie algebra with the
commutator[A,B]=AB-BA, \quad A, B\in Mat(n,\mathbb {K}).The commutator of two matrices E_{i,j} and E_{k,\ell } of the
canonical... | {
"cite_spans": []
} | 0807.1851 | Linear family of Lie brackets on the space of matrices $Mat(n\times
m,\K)$ and Ado's Theorem | [
"Bechir Dali"
] | [
"math.RT"
] | 2,008 | en | Mathematics | [
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e22dfb3b4c15999d306c05e92088b897ad2d1a88 | subsection | 7 | 21 | The case of | Now for each X\in End(V_1), consider\aligned \pi _X:\ \mathfrak {n} &\longrightarrow \mathfrak {n} \\
(A,B,C)& \longmapsto (-AX,XB,0).Then we easily verify that X\longmapsto \pi _X is a Lie algebra
homomorphism from \mathfrak {gl}(n,\mathbb {K}) into {\partial }
(\mathfrak {n}) the Lie algebra of the derivations of \ma... | {
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"star... | 0807.1851 | Linear family of Lie brackets on the space of matrices $Mat(n\times
m,\K)$ and Ado's Theorem | [
"Bechir Dali"
] | [
"math.RT"
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3fb7933d652f6b071b3d0b630281974fe5589ef7 | subsection | 8 | 21 | The case of | Since End(V_1)\subset \mathfrak {g} then we must have
[Z, X]=0, for any Z\in \mathfrak {z} and X\in End(V_1).Put Z=\displaystyle {\begin{pmatrix}
Z_1&Z_3\\
Z_2&Z_4
\end{pmatrix}, X=\begin{pmatrix}
X&0\\
0&0
\end{pmatrix}} then[Z,X]_{J_{n,r}}=\begin{pmatrix}
[Z_1,X]&-XZ_3\\
Z_2X&0
\end{pmatrix}=0,and thus
Z=\begin{pmatr... | {
"cite_spans": []
} | 0807.1851 | Linear family of Lie brackets on the space of matrices $Mat(n\times
m,\K)$ and Ado's Theorem | [
"Bechir Dali"
] | [
"math.RT"
] | 2,008 | en | Mathematics | [
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5728ad189d1526afe55691a506b5eec5a7d27cd9 | subsection | 9 | 21 | The case of | Let us denote it by \mathcal {Z}_{J}, then we have the followingProposition 2.3
Put r=rank(J), if r<n then \dim \mathcal {Z}_{J}=(n-r)^2
while if r=n, then \dim \mathcal {Z}_{J}=1.Case 1: rank(J)=nIn this case, the mapping \varphi : A\longmapsto JA is a Lie
algebra isomorphism from \left(Mat(n,\mathbb {K}),[~,~]_J\rig... | {
"cite_spans": []
} | 0807.1851 | Linear family of Lie brackets on the space of matrices $Mat(n\times
m,\K)$ and Ado's Theorem | [
"Bechir Dali"
] | [
"math.RT"
] | 2,008 | en | Mathematics | [
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1bbb9d170784c9616fedb92bd30c6e283aab83c7 | subsection | 10 | 21 | The case of | The Lie algebras
\left(Mat(n,\mathbb {K}),[~,~]_J\right) and \left(Mat(n,\mathbb {K}),[~,~]_{J^{\prime }}\right) are isomorphic
if and only if the matrices J and J^{\prime } are equivalent.Let J, J^{\prime }\in Mat(n,\mathbb {K}). If J and J^{\prime } are equivalent then from Proposition
REF the
corresponding Lie algeb... | {
"cite_spans": []
} | 0807.1851 | Linear family of Lie brackets on the space of matrices $Mat(n\times
m,\K)$ and Ado's Theorem | [
"Bechir Dali"
] | [
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] | 2,008 | en | Mathematics | [
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0cb95626483ec02bbcb9e72d3c3bbf52576753ab | subsection | 11 | 21 | The case of | If such mapping \rho exists, then we must have\begin{aligned}\rho \bigl ([X_i,Y_i]_{J_{n+2,n+1}}\bigr )&=[\rho (X_i),\rho (Y_i)]\\
\rho (Z)&= [\rho (X_i),\rho (Y_i)],
\end{aligned}but since \rho is an
injective Lie algebra morphism, then \rho (Z)=\lambda I_r with
\lambda \ne 0 wick is impossible because I_r is traceles... | {
"cite_spans": []
} | 0807.1851 | Linear family of Lie brackets on the space of matrices $Mat(n\times
m,\K)$ and Ado's Theorem | [
"Bechir Dali"
] | [
"math.RT"
] | 2,008 | en | Mathematics | [
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c429622edba8b14fe4af71c02001c43a53bbad99 | subsection | 12 | 21 | Example | Let \mathfrak {H}=span\lbrace Z,Y,X\rbrace with the only non vanishing bracket is
[X,Y]=Z then the classical representation of \mathfrak {H} isX=\begin{pmatrix}
0&1&0\\
0&0&0\\
0&0&0
\end{pmatrix},\quad Y=\begin{pmatrix}
0&0&0\\
0&0&1\\
0&0&0
\end{pmatrix},\quad Z=\begin{pmatrix}
0&0&1\\
0&0&0\\
0&0&0
\end{pmatrix}.Fro... | {
"cite_spans": []
} | 0807.1851 | Linear family of Lie brackets on the space of matrices $Mat(n\times
m,\K)$ and Ado's Theorem | [
"Bechir Dali"
] | [
"math.RT"
] | 2,008 | en | Mathematics | [
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801fee19b27427e6e24517fa754e78c0cbafcff0 | subsection | 13 | 21 | Example | Then we have the followingLemma 4.2 (, ) The map (A,B)\longmapsto [A,B]_J is a two-coboundary for the
adjoint representation of \mathfrak {gl}(n,\mathbb {K}).We can easily check that[A,B]_J=ad_A\alpha (B)-ad_B\alpha (A)-\alpha ([A,B])=(d\alpha )(A,B).where\alpha (X)=\frac{1}{2}(XJ+JX).Let t\in [0,1] and J=J_{n,r} (with... | {
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},... | 0807.1851 | Linear family of Lie brackets on the space of matrices $Mat(n\times
m,\K)$ and Ado's Theorem | [
"Bechir Dali"
] | [
"math.RT"
] | 2,008 | en | Mathematics | [
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cd427f3abd92e05233913c7e39722086d896916f | subsection | 14 | 21 | Example | The mapping\Psi _t:\left(Mat(n,\mathbb {K}),[~,~]_{(1-t)I+tJ}\right)\longrightarrow \mathfrak {gl}(n,\mathbb {K}),\quad \left(
\begin{array}{cc}
X_1& X_2 \\
X_3& X_4 \\
\end{array}
\right)\longmapsto \left(
\begin{array}{cc}
X_1& (1-t)X_2 \\
X_3& (1-t)X_4 \\
\end{array}
\right)is for any t\in [0,1[ invertible and
[X,Y... | {
"cite_spans": []
} | 0807.1851 | Linear family of Lie brackets on the space of matrices $Mat(n\times
m,\K)$ and Ado's Theorem | [
"Bechir Dali"
] | [
"math.RT"
] | 2,008 | en | Mathematics | [
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442d52f0e1349d9c9b0554cda133427c721be81d | subsection | 15 | 21 | Example | Then we have the following
subalgebras inclusions\begin{pmatrix}
\mathfrak {g}&\mathfrak {h}\\
0&0
\end{pmatrix}\subset \begin{pmatrix}
\mathfrak {g}&\mathfrak {h}\\
0&\mathfrak {h}"
\end{pmatrix}\subset \mathfrak {gl}(n,r,\mathbb {K})and\begin{pmatrix}
\mathfrak {g}&0\\
\mathfrak {h}^{\prime }&0
\end{pmatrix}\subset \... | {
"cite_spans": []
} | 0807.1851 | Linear family of Lie brackets on the space of matrices $Mat(n\times
m,\K)$ and Ado's Theorem | [
"Bechir Dali"
] | [
"math.RT"
] | 2,008 | en | Mathematics | [
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0f8c9471abccecde0d2ac4f13bf8160d21ca013b | subsection | 16 | 21 | The case of | In this section we will
deal with the linear space of strict rectangular matrices
Mat(n\times m,\mathbb {K}) which is not an associative algebra. Fix
J\in Mat(m\times n, \mathbb {K}) and put[A,B]_J=AJB-BJA,\quad A, B\in Mat(n\times m,\mathbb {K}).Recall that we have denoted the matrix
\displaystyle {\begin{pmatrix}
I_r... | {
"cite_spans": []
} | 0807.1851 | Linear family of Lie brackets on the space of matrices $Mat(n\times
m,\K)$ and Ado's Theorem | [
"Bechir Dali"
] | [
"math.RT"
] | 2,008 | en | Mathematics | [
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b47ef7d51932b636a402c03688f3bf1da386ea6f | subsection | 17 | 21 | The case of | Then the Lie algebras
\left(Mat(n\times m,\mathbb {K}),[~,~]_J\right) and \left(Mat(n\times m,\mathbb {K}),[~,~]_{J^{\prime }}\right)
are isomorphic if and only if J and J^{\prime } are equivalent in
Mat(m\times n,\mathbb {K}).Let J and J^{\prime } two equivalent matrices in Mat(m\times n, \mathbb {K})
then from Propos... | {
"cite_spans": []
} | 0807.1851 | Linear family of Lie brackets on the space of matrices $Mat(n\times
m,\K)$ and Ado's Theorem | [
"Bechir Dali"
] | [
"math.RT"
] | 2,008 | en | Mathematics | [
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fd6b7958c5531ccad8aeaf2be23468f74db73520 | subsection | 18 | 21 | Examples | (a) Let V=\mathbb {R}^2, we identify V with Mat(2\times 1,\mathbb {R}), we choose a basis (e_1,e_2)
in V with e_1=\begin{pmatrix}
1 \\
0
\end{pmatrix}, e_2=\begin{pmatrix}
0 \\
1
\end{pmatrix}. Then we can check that [e_2,e_1]_J=e_1 with J=(1,0). Then the
two-dimensional affine Lie algebra can be viewed as (Hom(\mathbb... | {
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m,\K)$ and Ado's Theorem | [
"Bechir Dali"
] | [
"math.RT"
] | 2,008 | en | Mathematics | [
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a0c65d5b5985cfe6743747d83a94ac292cba3131 | subsection | 19 | 21 | Contractions and Extensions of Lie algebras | Let n, r\in \mathbb {N} with n\ge r, n\ge 2 and put J=J_{n,r},
then we have the followingProposition 4.1 The Lie algebra \left(Mat(n,\mathbb {K}),[~,~]_{J_{n,r}}\right) is a contraction of
\mathfrak {gl}(n,\mathbb {K}).Let (E_{i,j}) the canonical basis of Mat(n,\mathbb {K}), and defineg=Span\lbrace E^{\prime }_{i,j}, \... | {
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} | 0807.1851 | Linear family of Lie brackets on the space of matrices $Mat(n\times
m,\K)$ and Ado's Theorem | [
"Bechir Dali"
] | [
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7caf069bd756e919c300a46fff7a5b643183384c | subsection | 20 | 21 | Contractions and Extensions of Lie algebras | Then we have[E^{\prime }_{i,j},E^{\prime }_{k,l}]= \left\lbrace \aligned \delta _{j,k}E^{\prime }_{i,l}-\delta _{l,i}E^{\prime }_{k,j}&\qquad \hbox{ if }
i,j,k,l\le r,\\
\delta _{j,k}E^{\prime }_{i,l}&\qquad \hbox{ if } i,j,
k\le r \hbox{ and } l>r,\\
-\delta _{l,i}E^{\prime }_{k,j} &\qquad \hbox{ if } i,j,l\le r\quad ... | {
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} | 0807.1851 | Linear family of Lie brackets on the space of matrices $Mat(n\times
m,\K)$ and Ado's Theorem | [
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b3cff51561651cee5a89d13e0428b4b0f228dd4a | abstract | 0 | 32 | Abstract | The elastic and capillary interactions between a pair of colloidal particles
trapped on top of a nematic film are studied theoretically for large
separations $d$. The elastic interaction is repulsive and of quadrupolar type,
varying as $d^{-5}$. For macroscopically thick films, the capillary interaction
is likewise rep... | {
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} | 10.1140/epje/i2008-10360-1 | 0807.1852 | Effective interactions of colloids on nematic films | [
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21d946831ab1eca99db9fa86c7d8044f9a6e39d3 | subsection | 1 | 32 | Introduction | The interactions of colloidal particles trapped at fluid interfaces have been found to differ
significantly from the corresponding interactions in bulk solvents. This has been studied
mostly for electrically charged particles trapped at interfaces with water. On
one hand, the presence of the interface gives rise to dir... | {
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859f9b4443bc6170fbbb50148e1af4efa6a44bf0 | subsection | 2 | 32 | Introduction | We will show that mechanical isolation
of the system “nematic film – colloid – air" can be violated through a subtle interplay
between the finite thickness of the film and the anchoring conditions at the colloids
and at the nematic interfaces with the substrate and with the air, respectively.
However, for
experimental ... | {
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245f42f19c38d06f8d715e5ffee0e9203a102461 | subsection | 3 | 32 | Coarse–grained model | In view of the mesoscopic length scales involved we describe the bulk part of the
nematic free energy associated with the director deformations
in terms of the Frank free energy expression within the one–coupling approximation
{\cal F}^{\rm b}_{\rm ne} &=& \int _{V_{\rm ne}} d^3 r \; f^{\rm b} (\mathbf {r}) \; \\
&=& ... | {
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e4b2aafb9414e1f42b319aad413e7304ccfda998 | subsection | 4 | 32 | Coarse–grained model | Thus, in the “strong anchoring” limit which we shall consider, the effect of the boundary terms is so strong that
as a first approximation it amounts to
fixing the angle between the director and the surface normal.
We shall adopt W_1<0 (normal alignment at the nematic–air interface) and W_2>0
(parallel alignment at the... | {
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492068bf48da15d08467056bb19194f2c84c0076 | subsection | 5 | 32 | Macroscopically thick nematic film | First we consider the limiting case h \rightarrow \infty (i.e., very thick nematic films (see
Fig. REF )).This was
implicitly assumed also by the authors of Ref. in discussing Fig. 2
therein. Due to
the small values of the elastic coupling constant, K \ll \gamma \, R,
and
of the anchoring energy, |W_i| \ll \gamma , th... | {
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b05d7e74fa7c7833500b43549b4c2d6d5065721c | subsection | 6 | 32 | Macroscopically thick nematic film | Furthermore
we have used the relation
\nabla \cdot {\mathbf {\Pi }}=0 in volumina V_1 and V_2 which is valid because
the reference configuration is taken
to be in force equilibrium. This also implies that the isotropic pressures above the
interface (p_{\rm air}) and below it (p) are equal
and that the director configur... | {
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97b293a6788930d32c669f90b8b81d5de03c8f02 | subsection | 7 | 32 | Macroscopically thick nematic film | Note that
to leading (quadratic) order in \varepsilon _\pi ,\varepsilon _F
the free energy change of the nematic due to the shifted interface
and due to a change in the director configuration with respect
to the reference configuration is captured
by the term \propto \int \pi _{zz}\,u.
(The analogous textbook argument ... | {
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33d9dc4cc8b68d2196652ee0e19809045cea863c | subsection | 8 | 32 | Asymptotic director configuration and elastic force between colloids | In Refs. , it has been shown that a colloidal drop
immersed in the bulk of a liquid crystal
is accompanied by a single counterdefect such that the total topological charge is zero
(here, the volume occupied by the colloid contains a topological charge which
may be represented by a virtual defect inside the colloid) and... | {
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292c884094c254c0a1188bdfdf98ee00e3fccf73 | subsection | 9 | 32 | Asymptotic director configuration and elastic force between colloids | Analyzing the multipole ansatz (with \mathbf {r}=(r_1, r_2, r_3))n_i &=& q_i\,\frac{1}{r} + \sum _{\alpha =1}^3 P_{i\alpha }\,\frac{r_\alpha }{r^3} +
\sum _{\alpha ,\beta =1}^3 Q_{i\alpha \beta }\,\frac{r_\alpha \,r_\beta }{r^5}
+ \dotsit follows that rotational covariance requires q_i=0,
P_{i\alpha } = P\,\delta _{i\... | {
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2c42be64392519d50f096a75ff61d582721c904c | subsection | 10 | 32 | Asymptotic director configuration and elastic force between colloids | For the present configuration (distance
vector perpendicular to the asymptotic director) the elastic
potential is repulsive and varies asV_{\rm el} \propto \frac{K\,Q^2}{d^5} \propto \gamma \rho _0^2\,
\varepsilon _F\left(\frac{\rho _0}{d}\right)^5\;.We have used that the dimensionless force parameter \varepsilon _F is... | {
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} | 10.1140/epje/i2008-10360-1 | 0807.1852 | Effective interactions of colloids on nematic films | [
"M. Oettel",
"A. Dominguez",
"M. Tasinkevych",
"S. Dietrich"
] | [
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] | 2,008 | en | Physics | [
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d9501ab2680ce1dcfd3a30d90ced5d846378663e | subsection | 11 | 32 | Asymptotic behavior of the stress on the interface and meniscus–induced
effective potential between colloids | The asymptotic behavior of the stress tensor component \pi _{zz} at the interface
follows from inserting
Eq. (REF ) into Eq. (REF ):\left. \pi _{zz}\right|_{\rm interface} &=&
\left.\frac{K}{2} \sum _{i=1}^2 \left(n_{i,z}^2-n_{i,r_1}^2-n_{i,r_2}^2
\right)\right|_{z=0}
\\
&\stackrel{r\rightarrow \infty }{\longrightarro... | {
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"M. Oettel",
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"M. Tasinkevych",
"S. Dietrich"
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61add42f88e16b26f7ecd440043c45b5326a9b64 | subsection | 12 | 32 | Asymptotic behavior of the stress on the interface and meniscus–induced
effective potential between colloids | The interfacial stress \hat{\pi }_{zz} may be decomposed generally as\hat{\pi }_{zz}(\rho )& = &\pi _{zz}(|\rho -\rho _1|) + \pi _{zz}(|\rho -\rho _2|)
+ 2\,\pi _{zz, {\rm m}}(\rho )
\\
& \equiv & \pi _{zz,1}+\pi _{zz,2}+2\,\pi _{zz, {\rm m}} \;.Here, \pi _{zz, i} denotes the stress around colloid i which pertains to ... | {
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... | 10.1140/epje/i2008-10360-1 | 0807.1852 | Effective interactions of colloids on nematic films | [
"M. Oettel",
"A. Dominguez",
"M. Tasinkevych",
"S. Dietrich"
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e0fd088d877134d8c8f4f9f00a657a354ed7f669 | subsection | 13 | 32 | Asymptotic behavior of the stress on the interface and meniscus–induced
effective potential between colloids | This term has two peaks around the colloid centers and
therefore close to the colloids it can be approximated by\pi _{zz,{\rm m}} \approx \frac{K \,Q^2}{2\,d^4} \sum _{i=1}^2 (-1)^i
\frac{ \mathbf {e}_d \cdot (\rho -\rho _i)}{|\rho -\rho _i|^5}where \mathbf {e}_d = (\rho _2-\rho _1)/d.
As discussed in Ref. , the qualit... | {
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"M. Oettel",
"A. Dominguez",
"M. Tasinkevych",
"S. Dietrich"
] | [
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01c16e554cf7f729b41d49b25e746bbad5c1e7d3 | subsection | 14 | 32 | Asymptotic behavior of the stress on the interface and meniscus–induced
effective potential between colloids | Below we shall investigate whether a net force on the system “colloid and interface"
may appear if the
thickness of the nematic phase is finite, as it is the case in the actual experiment. | {
"cite_spans": []
} | 10.1140/epje/i2008-10360-1 | 0807.1852 | Effective interactions of colloids on nematic films | [
"M. Oettel",
"A. Dominguez",
"M. Tasinkevych",
"S. Dietrich"
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5b09d1db10a53ed33c6ef44eba58576e71940ead | subsection | 15 | 32 | Finite thickness of the nematic film | In our discussion of a finite film thickness of the nematic phase we shall consider two cases:The anchoring of the nematic director at the surface of the bottom substrate is
perpendicular as it is the case at the upper interface with the air.
This case bears a strong formal resemblance to charged
colloids on water surf... | {
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... | 10.1140/epje/i2008-10360-1 | 0807.1852 | Effective interactions of colloids on nematic films | [
"M. Oettel",
"A. Dominguez",
"M. Tasinkevych",
"S. Dietrich"
] | [
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252878d69cf1679c96ace64a19d7e80e17cc80e8 | subsection | 16 | 32 | Perpendicular anchoring at both interfaces | As discussed above, the presence of the colloid asymptotically generates a
quadrupolar director field
which fulfills the boundary condition at the nematic–air interface.
In order to fulfill the
boundary condition at the substrate–nematic interface, an image quadrupole
of the same strength Q is needed which, however, le... | {
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{
"arxiv_id": "",
"doi": "",
"end": 2224,
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"raw": "T. C. Lubensky, D. Pettey, N. Currier, and H. Stark, Phys. Rev. E 57, 610 (1998).",
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"arxiv_... | 10.1140/epje/i2008-10360-1 | 0807.1852 | Effective interactions of colloids on nematic films | [
"M. Oettel",
"A. Dominguez",
"M. Tasinkevych",
"S. Dietrich"
] | [
"cond-mat.soft"
] | 2,008 | en | Physics | [
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7a4e658a399ac3023548414c26d45a63ccbc757b | subsection | 17 | 32 | Perpendicular anchoring at both interfaces | Using the solution given in Refs. , we have checked
that V_{\rm el}(d) remains repulsive. For d < h the overall magnitude of V_{\rm el}(d)
is somewhat weakened,
whereas for d \gg h a crossover to V_{\rm el}(d) \propto \exp (-d/h) is observedThis result can be obtained more easily by solving the field equations
\Delta n... | {
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"raw": "S. Ramaswamy, R. Nityananda, V. A. Raghunathan, and J. Prost, Mol. Cryst. Liq. Cryst. 288, 175 (1996).",
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... | 10.1140/epje/i2008-10360-1 | 0807.1852 | Effective interactions of colloids on nematic films | [
"M. Oettel",
"A. Dominguez",
"M. Tasinkevych",
"S. Dietrich"
] | [
"cond-mat.soft"
] | 2,008 | en | Physics | [
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278cd7571f76938514bbd61fe348e2dbbe02038f | subsection | 18 | 32 | Parallel anchoring at the bottom substrate | We assume that the substrate induces a preferred in–plane axis for the director
orientation which we take to be the x–axis. With no colloid present at the nematic–air
interface, the equilibrium director field is given by\mathbf {n}_0= \begin{pmatrix} \sin (-q_0 z) \\ 0 \\ \cos (-q_0 z) \end{pmatrix}
\;, \qquad q_0 = \p... | {
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} | 10.1140/epje/i2008-10360-1 | 0807.1852 | Effective interactions of colloids on nematic films | [
"M. Oettel",
"A. Dominguez",
"M. Tasinkevych",
"S. Dietrich"
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5324f17f9156d40ba10bbdff2a878b24db26d574 | subsection | 19 | 32 | Parallel anchoring at the bottom substrate | The nematic free energy of the film up to order O(v^2,w^2) is obtained
by inserting Eq. (REF ) into Eq. (REF ) for the Frank free energy
after dropping the total divergence of the K_{24}–type.
Using the boundary conditions for v and w we obtain
(with the notation introduced in Eq. (REF )):{\cal F}^{\rm film}_{\rm ne} ... | {
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3d05958cd67408d0eef64fa460ea576ccd369ef7 | subsection | 20 | 32 | Parallel anchoring at the bottom substrate | If one
assumes a power–law dependence on h, dimensional analysis for
P_v leads toP_v = O(R^2\;(R/h)^\kappa ) \qquad (\mbox{with}\; \kappa >0)\;.The precise functional form of P_v turns out to be unimportant for
the subsequent calculations.
We note that an asymptotic solution with a nonvanishing
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f70fd2c8efc7595b0364725bb72f580f71c0a3cd | subsection | 21 | 32 | Parallel anchoring at the bottom substrate | Since Q^Y_w \sim q_0^3 Q and Q = O(R^3), Q^Y_w=O([q_0 R]^3) is
a very small number.The Dirichlet boundary conditions for w at the substrate and at the
nematic–air interface
enforce that the contribution to w due to the quadrupole Q^J_w and all corresponding image
quadrupoles is zero. This holds also for the contributio... | {
"cite_spans": []
} | 10.1140/epje/i2008-10360-1 | 0807.1852 | Effective interactions of colloids on nematic films | [
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57680ec17d8330701ea6d59ad93ee0d35033f583 | subsection | 22 | 32 | Parallel anchoring at the bottom substrate | (R/h\sim 10^{-1}, K/(\gamma R)\sim 10^{-4})
we find V_{\rm men}(d) \sim 10^{-11-4\kappa }\, k_B T\;\ln (R/d)
which appears to be undetectably small.
Note that for d<h the direct elastic repulsion remains essentially unchanged
because in this regime the leading term of the elastic interaction is given by the
repulsion b... | {
"cite_spans": []
} | 10.1140/epje/i2008-10360-1 | 0807.1852 | Effective interactions of colloids on nematic films | [
"M. Oettel",
"A. Dominguez",
"M. Tasinkevych",
"S. Dietrich"
] | [
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cbccd8c4324e41bbf10db7e8dfcaf096230a13b9 | subsection | 23 | 32 | Discussion and conclusion | We have investigated the effective potential between two colloidal microspheres
of radius R
floating at asymptotically large distances d on an interface between a nematic film
of thickness h and air (Fig. REF ).
This effective potential is the sum of an elastic interaction caused by
the director distortions around the ... | {
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"raw": "I. I. Smalyukh, S. Chernyshuk, B. I. Lev, A. B. Nych, U. Ognysta, V. G. Nazarenko, and O. D. Lavrentovich, Phys. Rev. Lett. 93, 117801 (2004).",
"source_ref_id": "1ca1fef5ee0161c17301077604b0... | 10.1140/epje/i2008-10360-1 | 0807.1852 | Effective interactions of colloids on nematic films | [
"M. Oettel",
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"M. Tasinkevych",
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584882def70f001ca3d7b887a2f9f2f3daccc3fb | subsection | 24 | 32 | Discussion and conclusion | At a radial distance \rho from the center
of the colloid this implies Q/\rho ^3 \ll 1, which seems to be fulfilled for
the dimensional estimate for the quadrupole moment Q = O(R^3) and for the
distances \rho = 3R \dots 5R under discussion. However, the absolute magnitude of the
director deformations is not fixed by the... | {
"cite_spans": [
{
"arxiv_id": "",
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"end": 627,
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"raw": "H. Stark, Eur. Phys. J. B 10, 311 (1999).",
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"end": 253... | 10.1140/epje/i2008-10360-1 | 0807.1852 | Effective interactions of colloids on nematic films | [
"M. Oettel",
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"S. Dietrich"
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