费米子R算子与四面体Zamolodchikov代数的发现
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"费米子Bazhanov-Stroganov-算子的四面体Zamolodchikov代数"
这篇研究论文主要探讨了在自由费米子模型中的一个特殊算子——费米R型算子，它与Bazhanov-Stroganov三参数椭圆参数化以及相关的Yang-Baxter方程和装饰的Yang-Baxter方程有关。作者A.Melikyan在巴西利亚大学物理研究所工作，该研究发表在《Physics Letters B》上，并且是开放访问的，由SCOAP3资助。
费米R型算子是量子场论和统计力学中的一种重要工具，尤其在研究一维费米子系统的可积性方面。在这篇文章中，作者首先介绍了Bazhanov-Stroganov三参数椭圆参数化方法，这是一种用于描述自由费米子模型的复杂方法，它涉及到椭圆函数，可以处理非对称或非平凡的相互作用。
接着，作者讨论了与这些参数化相关的Yang-Baxter方程和装饰的Yang-Baxter方程。Yang-Baxter方程是量子群理论和可积系统的核心，它描述了在多体系统中粒子交换的规则。装饰的Yang-Baxter方程则是在这个基础上添加了额外的结构，允许更复杂的相互作用和更丰富的物理现象。
研究的亮点在于作者找到了费米R型算子的差分形式，这意味着它在特定的谱参数中可以用微分方程来表示。这种表示法对于计算和理解算子的性质非常有用，因为它简化了处理过程。
此外，论文还深入到一个特定的情况，解决了对应的四面体Zamolodchikov代数。四面体Zamolodchikov代数是更高维度的代数结构，它在多体系统和多维量子场论中扮演着关键角色，特别是在构造和分析多边形网络模型时。作者的这一发现为理解和求解这类代数的实例提供了一个新的视角。
这项工作为费米子系统的可积性理论提供了新的洞察，特别是在处理复杂相互作用和非对称情况时。通过引入差分形式的费米R型算子和四面体Zamolodchikov代数的解决方案，它为后续研究开辟了新的途径，可能有助于开发新的计算方法和理论框架，进一步推动量子物理和统计力学领域的研究。
Physics Letters B 801 (2020) 135175
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
The tetrahedral Zamolodchikov algebra for the fermionic
Bazhanov-Stroganov R-operator
A. Melikyan
Instituto de Física, Universidade de Brasília, 70910-900, Brasília, DF, Brazil
a r t i c l e i n f o a b s t r a c t
Article history:
Received
16 December 2019
Accepted
19 December 2019
Available
online 27 December 2019
Editor:
N. Lambert
We find the fermionic R-operator based on Bazhanov-Stroganov three-parameter elliptic parametrization
of the free fermion model, and the corresponding Yang-Baxter and decorated Yang-Baxter equations,
which are of the difference type in one of the spectral parameters. We also find a solution of the
corresponding tetrahedral Zamolodchikov algebra for a specific case.
© 2019 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP
3
.
1. Introduction
The integrable properties of fermionic (1 + 1)-dimensional models have attracted much attention due to their explicit appearance
in string theory (see [1]for a review), and, on the other hand, their relation to the one-dimensional Hubbard model [2], the R-matrix
of which has been proposed to be related to the S-matrix of strings via AdS/CFT correspondence [3]. More specifically, the following
fermionic model arises from the string theory on AdS
5
× S
5
when restricted to the su(1|1) subsector [4]:
L = i
¯
ψ
γ
μ
∂
μ
ψ −m
¯
ψψ +
g
2
4m
αβ
¯
ψ∂
α
ψ
¯
ψ γ
3
∂
β
ψ −∂
α
¯
ψψ ∂
β
¯
ψ
γ
3
ψ
−
g
3
16m
αβ
¯
ψψ
2
∂
α
¯
ψ
γ
3
∂
β
ψ. (1)
It can be shown to be a completely integrable classically, and its various quantum integrability properties has been investigated from
various points of view [5–10]. Despite this, the quantization of this model by direct methods of integrable systems was not successful
as a consequence of the non-ultralocal nature of the algebra of Lax operators, given in [6,7]. Quantization of such non-ultralocal models
remains an open problem, and their essential features can already be captured by considering the free fermionic model. The Lax pair for
the latter can be obtained from that of the full model (1)given in [6], by setting the coupling constants g
2
and g
3
to zero, and has the
form:
L
0
(x;μ) = ξ
(τ )
0
(x;μ)1 +ξ
(τ )
1
(x;μ)σ
3
+
(−)
τ
(x;μ)σ
+
+
(+)
τ
(x;μ)σ
−
, (2)
L
1
(x;μ) = ξ
(σ )
0
(x;μ)1 +ξ
(σ )
1
(x;μ)σ
3
+
(−)
σ
(x;μ)σ
+
+
(+)
σ
(x;μ)σ
−
, (3)
where we have denoted χ
1
=
(ψ
1
−ψ
2
)
/
√
2, χ
2
= i
(ψ
1
+ψ
2
)
/
√
2, χ
3
≡ χ
∗
1
, χ
4
≡ χ
∗
2
; l
0
= 1, l
1
= cosh(2μ), l
2
=− sinh(2μ), l
3
= cosh(μ), l
4
=
sinh(μ); and the functions ξ
(σ ,τ )
j
(x; μ), and t
(±)
σ ,τ
(x; μ) have the following explicit form:
ξ
(σ )
0
=
1
4
−
χ
3
χ
1
+χ
4
χ
2
−χ
1
χ
3
+χ
2
χ
4
,ξ
(σ )
1
=
il
2
2
, (4)
(−)
σ
=
−l
3
χ
2
−il
4
χ
1
,
(+)
σ
=
−l
3
χ
4
+il
4
χ
3
,
(5)
ξ
(τ )
0
=
i
2
[
χ
3
χ
1
+χ
4
χ
2
]
+
1
4
[
−χ
3
˙
χ
1
−χ
1
˙
χ
3
+χ
4
˙
χ
2
+χ
2
˙
χ
4
]
,ξ
(τ )
1
=−
il
1
2
, (6)
(−)
τ
=−i
[
l
3
χ
2
−il
4
χ
1
]
−
[
l
3
˙
χ
2
+il
4
˙
χ
1
]
,
(+)
τ
= i
[
l
3
χ
4
+il
4
χ
3
]
−
[
l
3
˙
χ
4
−il
4
˙
χ
3
]
. (7)
E-mail address: amelik@gmail.com.
https://doi.org/10.1016/j.physletb.2019.135175
0370-2693/
© 2019 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by
SCOAP
3
.
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