The two-dimensional random-bond Ising model, free fermions and the network model
We develop a recently-proposed mapping of the two-dimensional Ising model with random exchange (RBIM), via the transfer matrix, to a network model for a disordered system of non-interacting fermions. The RBIM transforms in this way to a localisation problem belonging to one of a set of non-standard...
| Главные авторы: | , |
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| Формат: | Journal article |
| Опубликовано: |
2001
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| _version_ | 1826289831097925632 |
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| author | Merz, F Chalker, J |
| author_facet | Merz, F Chalker, J |
| author_sort | Merz, F |
| collection | OXFORD |
| description | We develop a recently-proposed mapping of the two-dimensional Ising model with random exchange (RBIM), via the transfer matrix, to a network model for a disordered system of non-interacting fermions. The RBIM transforms in this way to a localisation problem belonging to one of a set of non-standard symmetry classes, known as class D; the transition between paramagnet and ferromagnet is equivalent to a delocalisation transition between an insulator and a quantum Hall conductor. We establish the mapping as an exact and efficient tool for numerical analysis: using it, the computational effort required to study a system of width $M$ is proportional to $M^{3}$, and not exponential in $M$ as with conventional algorithms. We show how the approach may be used to calculate for the RBIM: the free energy; typical correlation lengths in quasi-one dimension for both the spin and the disorder operators; even powers of spin-spin correlation functions and their disorder-averages. We examine in detail the square-lattice, nearest-neighbour $\pm J$ RBIM, in which bonds are independently antiferromagnetic with probability $p$, and ferromagnetic with probability $1-p$. Studying temperatures $T\geq 0.4J$, we obtain precise coordinates in the $p-T$ plane for points on the phase boundary between ferromagnet and paramagnet, and for the multicritical (Nishimori) point. We demonstrate scaling flow towards the pure Ising fixed point at small $p$, and determine critical exponents at the multicritical point. |
| first_indexed | 2024-03-07T02:34:54Z |
| format | Journal article |
| id | oxford-uuid:a8768f4d-a65e-4f9e-ab7f-c10c851b1f79 |
| institution | University of Oxford |
| last_indexed | 2024-03-07T02:34:54Z |
| publishDate | 2001 |
| record_format | dspace |
| spelling | oxford-uuid:a8768f4d-a65e-4f9e-ab7f-c10c851b1f792022-03-27T03:01:37ZThe two-dimensional random-bond Ising model, free fermions and the network modelJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:a8768f4d-a65e-4f9e-ab7f-c10c851b1f79Symplectic Elements at Oxford2001Merz, FChalker, JWe develop a recently-proposed mapping of the two-dimensional Ising model with random exchange (RBIM), via the transfer matrix, to a network model for a disordered system of non-interacting fermions. The RBIM transforms in this way to a localisation problem belonging to one of a set of non-standard symmetry classes, known as class D; the transition between paramagnet and ferromagnet is equivalent to a delocalisation transition between an insulator and a quantum Hall conductor. We establish the mapping as an exact and efficient tool for numerical analysis: using it, the computational effort required to study a system of width $M$ is proportional to $M^{3}$, and not exponential in $M$ as with conventional algorithms. We show how the approach may be used to calculate for the RBIM: the free energy; typical correlation lengths in quasi-one dimension for both the spin and the disorder operators; even powers of spin-spin correlation functions and their disorder-averages. We examine in detail the square-lattice, nearest-neighbour $\pm J$ RBIM, in which bonds are independently antiferromagnetic with probability $p$, and ferromagnetic with probability $1-p$. Studying temperatures $T\geq 0.4J$, we obtain precise coordinates in the $p-T$ plane for points on the phase boundary between ferromagnet and paramagnet, and for the multicritical (Nishimori) point. We demonstrate scaling flow towards the pure Ising fixed point at small $p$, and determine critical exponents at the multicritical point. |
| spellingShingle | Merz, F Chalker, J The two-dimensional random-bond Ising model, free fermions and the network model |
| title | The two-dimensional random-bond Ising model, free fermions and the
network model |
| title_full | The two-dimensional random-bond Ising model, free fermions and the
network model |
| title_fullStr | The two-dimensional random-bond Ising model, free fermions and the
network model |
| title_full_unstemmed | The two-dimensional random-bond Ising model, free fermions and the
network model |
| title_short | The two-dimensional random-bond Ising model, free fermions and the
network model |
| title_sort | two dimensional random bond ising model free fermions and the network model |
| work_keys_str_mv | AT merzf thetwodimensionalrandombondisingmodelfreefermionsandthenetworkmodel AT chalkerj thetwodimensionalrandombondisingmodelfreefermionsandthenetworkmodel AT merzf twodimensionalrandombondisingmodelfreefermionsandthenetworkmodel AT chalkerj twodimensionalrandombondisingmodelfreefermionsandthenetworkmodel |