### Abstract

A complex problem is solved here; we show how to write Dyson's equations for the (Ising) spin-glass that relate the propagator G to the mass operator M. In other words we are able to reduce the inversion of an ultrametric matrix M to the solution of a Dyson's equation in all sectors (for the replicon sector the result had already been derived). It turns out that what renders the problem tractable is using, instead of the components of G (or M), an object called here the 'kernel' from which one can deduce the components themselves after dressing it with ultrametric weights and summing over eigenvalue indices with their appropriate multiplicity. Dyson's equations are then established as stationarity equations of tr 1n M - tr GM, where the kinetic terms are incorporated in M. At each stage we illustrate the calculation by providing explicit answers for the bare system (mean field in M). In particular the introduction of the 'kernel' allows us to construct the bare propagator for a Lagrangean where one retains all quartic invariants. The case of the system in a magnetic field is also treated.

Original language | English |
---|---|

Pages (from-to) | 1287-1308 |

Number of pages | 22 |

Journal | Journal de physique. III |

Volume | 4 |

Issue number | 9 |

Publication status | Published - Sep 1994 |

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### ASJC Scopus subject areas

- Engineering(all)

### Cite this

*Journal de physique. III*,

*4*(9), 1287-1308.

**Dyson's equations for the (Ising) spin-glass.** / De Dominicis, C.; Kondor, I.; Temesvari, T.

Research output: Contribution to journal › Article

*Journal de physique. III*, vol. 4, no. 9, pp. 1287-1308.

}

TY - JOUR

T1 - Dyson's equations for the (Ising) spin-glass

AU - De Dominicis, C.

AU - Kondor, I.

AU - Temesvari, T.

PY - 1994/9

Y1 - 1994/9

N2 - A complex problem is solved here; we show how to write Dyson's equations for the (Ising) spin-glass that relate the propagator G to the mass operator M. In other words we are able to reduce the inversion of an ultrametric matrix M to the solution of a Dyson's equation in all sectors (for the replicon sector the result had already been derived). It turns out that what renders the problem tractable is using, instead of the components of G (or M), an object called here the 'kernel' from which one can deduce the components themselves after dressing it with ultrametric weights and summing over eigenvalue indices with their appropriate multiplicity. Dyson's equations are then established as stationarity equations of tr 1n M - tr GM, where the kinetic terms are incorporated in M. At each stage we illustrate the calculation by providing explicit answers for the bare system (mean field in M). In particular the introduction of the 'kernel' allows us to construct the bare propagator for a Lagrangean where one retains all quartic invariants. The case of the system in a magnetic field is also treated.

AB - A complex problem is solved here; we show how to write Dyson's equations for the (Ising) spin-glass that relate the propagator G to the mass operator M. In other words we are able to reduce the inversion of an ultrametric matrix M to the solution of a Dyson's equation in all sectors (for the replicon sector the result had already been derived). It turns out that what renders the problem tractable is using, instead of the components of G (or M), an object called here the 'kernel' from which one can deduce the components themselves after dressing it with ultrametric weights and summing over eigenvalue indices with their appropriate multiplicity. Dyson's equations are then established as stationarity equations of tr 1n M - tr GM, where the kinetic terms are incorporated in M. At each stage we illustrate the calculation by providing explicit answers for the bare system (mean field in M). In particular the introduction of the 'kernel' allows us to construct the bare propagator for a Lagrangean where one retains all quartic invariants. The case of the system in a magnetic field is also treated.

UR - http://www.scopus.com/inward/record.url?scp=0028497695&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0028497695&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0028497695

VL - 4

SP - 1287

EP - 1308

JO - Journal De Physique, III

JF - Journal De Physique, III

SN - 1155-4320

IS - 9

ER -