Quantum decoration transformation for spin models

Braz, Felipe Fortes; Rodrigues, F. Carvalho; Souza, S. M. de; Rojas, Onofre

doi:10.1016/j.aop.2016.07.007

Cited by 4 publications

(4 citation statements)

References 48 publications

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“…Among the studies of such zero-dimensional structures, the works based on exact methods should be especially mentioned [27,[30][31][32][33][34][35][36][37][38][39][40][41][42][43][44]. It is worth emphasizing that rigorous and exact numerical solutions are, so far, available only for a very limited class of models (especially when the quantum version is considered) [45][46][47].…”

Section: Introductionmentioning

confidence: 99%

Ground-state magnetic properties of spin ladder-shaped quantum nanomagnet: Exact diagonalization study

Szałowski

Kowalewska

2018

Journal of Magnetism and Magnetic Materials

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The paper presents a computational study of the ground-state properties of a quantum nanomagnet possessing the shape of a finite two-legged ladder composed of 12 spins S = 1/2. The system is described with isotropic quantum Heisenberg model with nearest-neighbour interleg and intraleg interactions supplemented with diagonal interleg coupling between next nearest neighbours. All the couplings can take arbitrary values. The description of the ground state is based on the exact numerical diagonalization of the Hamiltonian. The ground-state phase diagram is constructed and analysed as a function of the interactions and the external magnetic field. The ground-state energy and spin-spin correlations are extensively discussed. The cases of ferro-and antiferromagnetic couplings are compared and contrasted.

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Section: Introductionmentioning

confidence: 99%

Ground-state magnetic properties of spin ladder-shaped quantum nanomagnet: Exact diagonalization study

Szałowski

Kowalewska

2018

Journal of Magnetism and Magnetic Materials

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“…The Zassenhaus formula [1,2] plays an important role in various fields of physics, such as the Dirac monopole problem [3], quantum spin lattices [4], fluids dynamics [5] and the study of solitary waves [6], statistical mechanics [7], many-body theories or quantum optics. In particle accelerator physics, the Zassenhaus formula was successfully used to compute the relevant maps both in Taylor-series and factorized-product forms [8].…”

Section: Introductionmentioning

confidence: 99%

Closed forms of the Zassenhaus formula

Dupays

2023

J. Phys. A: Math. Theor.

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The Zassenhaus formula finds many applications in theoretical physics or mathematics, from fluid dynamics to differential geometry. The non-commutativity of the elements of the algebra implies that the exponential of a sum of operators cannot be expressed as the product of exponentials of operators. The exponential of the sum can then be decomposed as the product of the exponentials multiplied by a supplementary term which takes generally the form of an infinite product of exponentials. Such a procedure is often referred to as ``disentanglement''. However, for some special commutators, closed forms can be found. In this work, we propose a closed form for the Zassenhaus formula when the commutator of operators $\hX$ and $\hY$ satisfy the relation $[\hX,\hY]=u\hX+v\hY+c\mathbbm{1}$. Such an expression boils down to three equivalent versions, a left-sided, a centered and a right-sided formula:%\begin{eqnarray}e^{\hX+\hY}&=&e^{\hX}e^{\hY}e^{g_{r}(u,v)[\hX,\hY]}=e^{\hX}e^{g_{c}(u,v)[\hX,\hY]}e^{\hY}\nonumber\\&=&e^{g_{\ell}(u,v)[\hX,\hY]}e^{\hX}e^{\hY},\end{eqnarray}%with respective arguments,%\begin{eqnarray}g_{r}(u,v)&=&g_{c}(v,u)e^{u}=g_{\ell}(v,u)\nonumber\\&=&\frac{u\left(e^{u-v}-e^{u}\right)+v\left(e^{u}-1\right)}{vu(u-v)}\end{eqnarray}%for $u\ne v$ and %\begin{eqnarray}g_{r}(u,u)=\frac{u+1-e^u}{u^2}\;\;\;\;\mathrm{with}\;\;\;\; g_r(0,0)=-1/2.\end{eqnarray}With additional special case\begin{eqnarray} g_{r}(0,v)= -\frac{e^{-v}-1+v}{v^{2}}, \quad & g_{r}(u,0)=\frac{e^{u}(1-u)-1}{u^{2}}.\end{eqnarray}

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“…Zassenhaus formula found recently a regain of interest [1,2] due to its numerous applications in various fields such as Soliton physics, Dirac's monopole problem, quantum lattices of spins or even fluids dynamics [3,4,5,6]. It is also studied for mathematical fundamental purpose such as disentangling exponential operators [7,8] or in differential geometry [9].…”

Section: Introductionmentioning

confidence: 99%

Closed forms of Zassenhaus formula

Dupays¹

2021

Preprint

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Zassenhaus formula is used in a wide range of fields in physics or mathematics, from fluid dynamics to differential geometry. The non commutativity of the elements of the algebra implies that the exponential of a sum of operators cannot be splitted in the product of exponential of operators. The exponential of the sum can then be decomposed as the product of the exponentials multiplied by a supplementary term which form is generally an infinite product of exponentials. However, for some special commutators, closed forms can be found. We propose a closed form for Zassenhaus formula when the commutator of the operators X and Y satisfy the relation [X, Y ] = uX + vY + c½. Zassenhaus formula then reduces to the closed form e X+Y = e X e Y e gr (u,v)[X,Y ] = e X e gc(u,v)[X,Y ] e Y = e g l (u,v)[X,Y ] e X e Y ,(1) a left-sided, centered and right-sided formula are found, with respective arguments, gr(u, v) = gc(v, u)e u = g l (v, u) = u e u−v − e u + v (e u − 1) vu(u − v) .(2)

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Quantum decoration transformation for spin models

Cited by 4 publications

References 48 publications

Ground-state magnetic properties of spin ladder-shaped quantum nanomagnet: Exact diagonalization study

Ground-state magnetic properties of spin ladder-shaped quantum nanomagnet: Exact diagonalization study

Closed forms of the Zassenhaus formula

Closed forms of Zassenhaus formula

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