The Matrix Product Ansatz for integrable U(1)N models in Lunin-Maldacena backgrounds

Lazo, Matheus J.

doi:10.1590/s0103-97332008000200005

Cited by 4 publications

(31 citation statements)

References 43 publications

(113 reference statements)

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“…Since the several components of the wavefunction should be uniquely related, the above algebra should be associative. This associativity implies that the above S-matrix should satisfy the Yang-Baxter relations [7,32], which is indeed the case [30]. The components of the wavefunction corresponding to the configurations where we have three or four particles in next-neigbouring sites would give in principle new relations involving three or four matrices A (α)…”

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confidence: 90%

“…where ε n is the energy of the eigenfunction (2) (α = 2, 3) is composed of n = n 2 + n 3 spectral parameter dependent matrices [22,24,30]. A second class of solutions is obtained if matrices A (α)…”

Section: The Mpamentioning

confidence: 99%

“…x (α = 2, 3) with different α value are composed of by distinct sets of spectral parameters matrices [22,30]. Here we will consider only the standard solution but our model can be easily extended to the second class problem.…”

Section: The Mpamentioning

confidence: 99%

“…In [30] we choose for this map the trace operation since the model were homogeneous and defined in a periodic chain. However in the present paper due to the inhomogeneities the trace operator do not work although we introduce another auxiliary matrix as in [21].…”

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Integrable inhomogeneous spin chains in generalized Lunin-Maldacena backgrounds

Lazo¹

2008

Braz. J. Phys.

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We obtain through a Matrix Product Ansatz the exact solution of the most general inhomogeneous spin chain with nearest neighbor interaction and with U(1) 2 and U(1) 3 symmetries. These models are related to the one loop mixing matrix of the Leigh-Strassler deformed N = 4 SYM theory, dual to type IIB string theory in the generalized Lunin-Maldacena backgrounds, in the sectors of two and three kinds of fields, respectively. The solutions presented here generalizes the results obtained by the author in a previous work for homogeneous spins chains with U(1) N symmetries in the sectors of N = 2 and N = 3.

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Section: The Mpamentioning

confidence: 90%

Section: The Mpamentioning

confidence: 99%

Section: The Mpamentioning

confidence: 99%

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Integrable inhomogeneous spin chains in generalized Lunin-Maldacena backgrounds

Lazo¹

2008

Braz. J. Phys.

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“…Recently, we introduced a new class of 3-state model that is integrable despite it do not have particle-particle exchange symmetry [22]. In [23] we extend the model [22] and formulate an one-dimensional asymmetric exclusion process with one kind of impurities (ASEPI). This model describes the dynamics of two types of particles (type 1 and 2) on a lattice of L sites, where each lattice site can be occupied by at most one particle.…”

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confidence: 99%

Asymmetric exclusion model with several kinds of impurities

Lazo¹,

Ferreira²

2012

J. Stat. Mech.

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Abstract.We formulate a new integrable asymmetric exclusion process with N −1 = 0, 1, 2, . . . kinds of impurities and with hierarchically ordered dynamics. The model we proposed displays the full spectrum of the simple asymmetric exclusion model plus new levels. The first excited state belongs to these new levels and displays unusual scaling exponents. We conjecture that, while the simple asymmetric exclusion process without impurities belongs to the KPZ universality class with dynamical exponent 3 2 , our model has a scaling exponent 3 2 +N −1. In order to check the conjecture, we solve numerically the Bethe equation with N = 3 and N = 4 for the totally asymmetric diffusion and found the dynamical exponents 7 2 and 9 2 in these cases.

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Asymmetric exclusion model with impurities

Lazo

Ferreira

2010

Phys. Rev. E

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An integrable asymmetric exclusion process with impurities is formulated. The model displays the full spectrum of the stochastic asymmetric XXZ chain plus new levels. We derive the Bethe equations and calculate the spectral gap for the totally asymmetric diffusion at half filling. While the standard asymmetric exclusion process without impurities belongs to the KPZ universality class with an exponent 3 2 , our model has a scaling exponent One-dimensional three-state quantum Hamiltonians and master equations of reaction-diffusion processes have played an important role in describing strongly correlated electrons and nonequilibrium statistical mechanics in the last decades, mainly due to their intrinsic and nontrivial many-body behavior. Remarkably, in one dimension several models in this category are exactly solvable, as the spin-1 Sutherland ͓1͔ and t-J ͓2͔ models, and the asymmetric diffusion of two types of particles ͓3͔. In its formulation in terms of particles with two global conservation laws, these models describe the dynamics of two types of particles on the lattice, where the total number of particles of each type is conserved separately. In order to ensure integrability, all known models in this class satisfy some particle-particle exchange symmetries ͓4,5͔. Recently, we introduced a new class of three-state model in the context of high-energy physics that is integrable despite it does not have particle-particle exchange symmetry ͓5͔. The quantum version is closely related to the strong regime of the t-U Hubbard model and the XXC model ͓6͔. In this work we formulate a stochastic model related to ͓5͔ that describes an asymmetric exclusion process ͑ASEP͒ with impurities. Although our model can be solved by the coordinate Bethe ansatz, we are going to formulate a matrix product ansatz ͑MPA͒ ͓4,7͔ due its simplicity and unifying implementation for arbitrary systems. This MPA introduced in ͓4,7͔ can be seen as a matrix product formulation of the coordinate Bethe ansatz and it is suited to describe all eigenstates of integrable models. We solve this model with periodic boundary condition through the MPA and we analyze the spectral gap for some special cases. Our model displays the full spectrum of the ASEP without impurities ͓8͔ plus new levels. The first excited state belongs to these new levels and has unusual scaling exponents. Although the ASEP without impurities belongs to the KPZ universality class ͓9͔ ͑dy-namic exponent 3 2 ͒, our model displays a scaling exponent 5 2 . The model we propose describes the dynamics of two types of particles ͑types 1 and 2͒ on the lattice, where the total number of particles of each type is conserved. In this model if the neighbor sites are empty, particles of type 1 can jump to the right or to the left with rates ⌫ 0 1 1 0 and ⌫ 1 0 0 1 , respectively. Particles of type 2 "impurities" do not jump to the neighbor sites if they are empty, but can change positions with neighbor particles of type 1 with rates ⌫ 2 1 1 2 and ⌫ 1 2 2 1 if particle 1 is on the left or on the ri...

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The Matrix Product Ansatz for integrable U(1)N models in Lunin-Maldacena backgrounds

Cited by 4 publications

References 43 publications

Integrable inhomogeneous spin chains in generalized Lunin-Maldacena backgrounds

Integrable inhomogeneous spin chains in generalized Lunin-Maldacena backgrounds

Asymmetric exclusion model with several kinds of impurities

Asymmetric exclusion model with impurities

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