arXiv:1510.08774 · Physical review. B./Physical review. B

Composite operators in cubic field theories and link-overlap fluctuations in spin-glass models

Ada Altieri, Giorgio Parisi A, Tommaso Rizzo

Published Jan 27, 2016• 4 citations

Abstract

We present a complete characterization of the fluctuations and correlations of the squared overlap in the Edwards-Anderson spin-glass model in zero field. The analysis reveals that the energy-energy correlation (and thus the specific heat) has a different critical behavior than the fluctuations of the link overlap in spite of the fact that the average energy and average link overlap have the same critical properties. More precisely the link-overlap fluctuations are larger than the specific heat according to a computation at first order in the $6\ensuremath{-}\ensuremath{\epsilon}$ expansion. An unexpected outcome is that the link-overlap fluctuations have a subdominant power-law contribution characterized by an anomalous logarithmic prefactor which is absent in the specific heat. In order to compute the $\ensuremath{\epsilon}$ expansion we consider the problem of the renormalization of quadratic composite operators in a generic multicomponent cubic field theory: the results obtained have a range of applicability beyond spin-glass theory.

DOI: 10.1103/physrevb.93.024422

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arXiv:1510.08774 · Physical review. B./Physical review. B

Composite operators in cubic field theories and link-overlap fluctuations in spin-glass models

Ada Altieri, Giorgio Parisi A, Tommaso Rizzo

Published Jan 27, 2016• 4 citations

arXiv HTML PDF

Abstract

6\ensuremath{-}\ensuremath{\epsilon}

expansion. An unexpected outcome is that the link-overlap fluctuations have a subdominant power-law contribution characterized by an anomalous logarithmic prefactor which is absent in the specific heat. In order to compute the

\ensuremath{\epsilon}

expansion we consider the problem of the renormalization of quadratic composite operators in a generic multicomponent cubic field theory: the results obtained have a range of applicability beyond spin-glass theory.

DOI: 10.1103/physrevb.93.024422

Related papers

Overlap Equivalence in the Edwards-Anderson Model

Pierluigi Contucci, Cristian Giardinà, Claudio Giberti +1 more

We study the relative fluctuations of the link overlap and the square standard overlap in the three-dimensional Gaussian Edwards-Anderson model with zero external field. We first analyze the correlation coefficient and find that the two quantities are uncorrelated above the critical temperature. Below the critical temperature we find that the link overlap has vanishing fluctuations for fixed values of the square standard overlap and large volumes. Our data show that the conditional variance scales to zero in the thermodynamic limit. This implies that, if one of the two random variables tends to a trivial one (i.e., deltalike distributed), then the other does also, and as a consequence, the "trivial-nontrivial" picture should be dismissed. Our results show that the two overlaps are completely equivalent in the description of the low temperature phase of the Edwards-Anderson model.We study the relative fluctuations of the link overlap and the square standard overlap in the three-dimensional Gaussian Edwards-Anderson model with zero external field. We first analyze the correlation coefficient and find that the two quantities are uncorrelated above the critical temperature. Below the critical temperature we find that the link overlap has vanishing fluctuations for fixed values of the square standard overlap and large volumes. Our data show that the conditional variance scales to zero in the thermodynamic limit. This implies that, if one of the two random variables tends to a trivial one (i.e., deltalike distributed), then the other does also, and as a consequence, the "trivial-nontrivial" picture should be dismissed. Our results show that the two overlaps are completely equivalent in the description of the low temperature phase of the Edwards-Anderson model.

arXiv:cond-mat/0510663Physical Review LettersJun 1, 200625 citations

Numerical results for the Edwards-Anderson spin-glass model at low temperature

Julio F. Fernández, Juan J. Alonso

We have simulated Edwards-Anderson (EA) as well as Sherrington-Kirkpatrick systems of ${L}^{3}$ spins. After averaging over large sets of EA system samples of $3\ensuremath{\le}L\ensuremath{\le}10$, we obtain accurate numbers for distributions $p(q)$ of the overlap parameter $q$ at very low-temperature $T$ . We find $p(0)/T\ensuremath{\rightarrow}0.233(4)$ as $T\ensuremath{\rightarrow}0$. This is in contrast with the droplet scenario of spin glasses. We also study the number of mismatched links---between replica pairs---that come with large scale excitations. Contributions from small scale excitations are discarded. We thus obtain for the fractal dimension of outer surfaces of $q\ensuremath{\sim}0$ excitations in the EA model ${d}_{s}\ensuremath{\rightarrow}2.59(3)$ as $T\ensuremath{\rightarrow}0$. This is in contrast with ${d}_{s}\ensuremath{\rightarrow}3$ as $T\ensuremath{\rightarrow}0$ that is predicted by mean-field theory for the macroscopic limit.We have simulated Edwards-Anderson (EA) as well as Sherrington-Kirkpatrick systems of ${L}^{3}$ spins. After averaging over large sets of EA system samples of $3\ensuremath{\le}L\ensuremath{\le}10$, we obtain accurate numbers for distributions $p(q)$ of the overlap parameter $q$ at very low-temperature $T$. We find $p(0)/T\ensuremath{\rightarrow}0.233(4)$ as $T\ensuremath{\rightarrow}0$. This is in contrast with the droplet scenario of spin glasses. We also study the number of mismatched links---between replica pairs---that come with large scale excitations. Contributions from small scale excitations are discarded. We thus obtain for the fractal dimension of outer surfaces of $q\ensuremath{\sim}0$ excitations in the EA model ${d}_{s}\ensuremath{\rightarrow}2.59(3)$ as $T\ensuremath{\rightarrow}0$. This is in contrast with ${d}_{s}\ensuremath{\rightarrow}3$ as $T\ensuremath{\rightarrow}0$ that is predicted by mean-field theory for the macroscopic limit.

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Paola Ranieri

We study the dynamic fluctuations of the soft-spin version of the Edwards-Anderson model in the critical region for $T\rightarrow T_{c}^{+}$ . First we solve the infinite-range limit of the model using the random matrix method. We define the static and dynamic 2-point and 4-point correlation functions at the order $O(1/N)$ and we verify that the static limit obtained from the dynamic expressions is correct. In a second part we use the functional integral formalism to define an effective short-range Lagrangian $L$ for the fields $δQ^{αβ}_{i}(t_{1},t_{2})$ up to the cubic order in the series expansion around the dynamic Mean-Field value $\overline{{Q}^{αβ}}(t_{1},t_{2})$ . We find the more general expression for the time depending non-local fluctuations, the propagators $[\langleδQ^{αβ}_{i}(t_{1},t_{2}) δQ^{αβ}_{j}(t_{3},t_{4})\rangle_ξ]_{J}$ , in the quadratic approximation. Finally we compare the long-range limit of the correlations, derived in this formalism, with the correlations of the infinite-range model studied with the previous approach (random matrices).We study the dynamic fluctuations of the soft-spin version of the Edwards-Anderson model in the critical region for $T\rightarrow T_{c}^{+}$. First we solve the infinite-range limit of the model using the random matrix method. We define the static and dynamic 2-point and 4-point correlation functions at the order $O(1/N)$ and we verify that the static limit obtained from the dynamic expressions is correct. In a second part we use the functional integral formalism to define an effective short-range Lagrangian $L$ for the fields $δQ^{αβ}_{i}(t_{1},t_{2})$ up to the cubic order in the series expansion around the dynamic Mean-Field value $\overline{{Q}^{αβ}}(t_{1},t_{2})$. We find the more general expression for the time depending non-local fluctuations, the propagators $[\langleδQ^{αβ}_{i}(t_{1},t_{2}) δQ^{αβ}_{j}(t_{3},t_{4})\rangle_ξ]_{J}$, in the quadratic approximation. Finally we compare the long-range limit of the correlations, derived in this formalism, with the correlations of the infinite-range model studied with the previous approach (random matrices).

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Scaling Analysis of Domain-Wall Free Energy in the Edwards-Anderson Ising Spin Glass in a Magnetic Field

M. Sasaki, Koji Hukushima, Hajime Yoshino +1 more

The stability of the spin-glass phase against a magnetic field is studied in the three- and four-dimensional Edwards-Anderson Ising spin glasses. Effective couplings J(eff) and effective fields H(eff) associated with length scale L are measured by a numerical domain-wall renormalization-group method. The results obtained by scaling analysis of the data strongly indicate the existence of a crossover length beyond which the spin-glass order is destroyed by field H. The crossover length well obeys a power law of H which diverges as H --> 0 but remains finite for any nonzero H, implying that the spin-glass phase is absent even in an infinitesimal field. These results are well consistent with the droplet theory for short-range spin glasses.The stability of the spin-glass phase against a magnetic field is studied in the three- and four-dimensional Edwards-Anderson Ising spin glasses. Effective couplings J(eff) and effective fields H(eff) associated with length scale L are measured by a numerical domain-wall renormalization-group method. The results obtained by scaling analysis of the data strongly indicate the existence of a crossover length beyond which the spin-glass order is destroyed by field H. The crossover length well obeys a power law of H which diverges as H --> 0 but remains finite for any nonzero H, implying that the spin-glass phase is absent even in an infinitesimal field. These results are well consistent with the droplet theory for short-range spin glasses.

arXiv:cond-mat/0702302Physical Review LettersSep 25, 200724 citations

Low-temperature behavior of the statistics of the overlap distribution in Ising spin-glass models

Matthew Wittmann, Burcu Yücesoy, Helmut G. Katzgraber +2 more

Using Monte Carlo simulations, we study in detail the overlap distribution for individual samples for several spin-glass models including the infinite-range Sherrington-Kirkpatrick model, short-range Edwards-Anderson models in three and four space dimensions, and one-dimensional long-range models with diluted power-law interactions. We study three long-range models with different powers as follows: The first is approximately equivalent to a short-range model in three dimensions, the second to a short-range model in four dimensions, and the third to a short-range model in the mean-field regime. We study an observable proposed earlier by some of us which aims to distinguish the ``replica symmetry breaking'' picture of the spin-glass phase from the ``droplet picture,'' finding that larger system sizes would be needed to unambiguously determine which of these pictures describes the low-temperature state of spin glasses best, except for the Sherrington-Kirkpatrick model, which is unambiguously described by replica symmetry breaking. Finally, we also study the median integrated overlap probability distribution and a typical overlap distribution, finding that these observables are not particularly helpful in distinguishing the replica symmetry breaking and the droplet pictures.Using Monte Carlo simulations, we study in detail the overlap distribution for individual samples for several spin-glass models including the infinite-range Sherrington-Kirkpatrick model, short-range Edwards-Anderson models in three and four space dimensions, and one-dimensional long-range models with diluted power-law interactions. We study three long-range models with different powers as follows: The first is approximately equivalent to a short-range model in three dimensions, the second to a short-range model in four dimensions, and the third to a short-range model in the mean-field regime. We study an observable proposed earlier by some of us which aims to distinguish the ``replica symmetry breaking'' picture of the spin-glass phase from the ``droplet picture,'' finding that larger system sizes would be needed to unambiguously determine which of these pictures describes the low-temperature state of spin glasses best, except for the Sherrington-Kirkpatrick model, which is unambiguously described by replica symmetry breaking. Finally, we also study the median integrated overlap probability distribution and a typical overlap distribution, finding that these observables are not particularly helpful in distinguishing the replica symmetry breaking and the droplet pictures.

arXiv:1408.2482Physical Review BOct 27, 201411 citations

6\ensuremath{-}\ensuremath{\epsilon}

\ensuremath{\epsilon}

We have simulated Edwards-Anderson (EA) as well as Sherrington-Kirkpatrick systems of

{L}^{3}

spins. After averaging over large sets of EA system samples of $3\ensuremath{\le}L\ensuremath{\le}10$, we obtain accurate numbers for distributions

p(q)

of the overlap parameter

q

at very low-temperature

T

. We find $p(0)/T\ensuremath{\rightarrow}0.233(4)$ as $T\ensuremath{\rightarrow}0$. This is in contrast with the droplet scenario of spin glasses. We also study the number of mismatched links---between replica pairs---that come with large scale excitations. Contributions from small scale excitations are discarded. We thus obtain for the fractal dimension of outer surfaces of $q\ensuremath{\sim}0$ excitations in the EA model ${d}_{s}\ensuremath{\rightarrow}2.59(3)$ as $T\ensuremath{\rightarrow}0$. This is in contrast with ${d}_{s}\ensuremath{\rightarrow}3$ as $T\ensuremath{\rightarrow}0$ that is predicted by mean-field theory for the macroscopic limit.

We study the dynamic fluctuations of the soft-spin version of the Edwards-Anderson model in the critical region for

T\rightarrow T_{c}^{+}

. First we solve the infinite-range limit of the model using the random matrix method. We define the static and dynamic 2-point and 4-point correlation functions at the order

O(1/N)

and we verify that the static limit obtained from the dynamic expressions is correct. In a second part we use the functional integral formalism to define an effective short-range Lagrangian

L

for the fields

δQ^{αβ}_{i}(t_{1},t_{2})

up to the cubic order in the series expansion around the dynamic Mean-Field value

\overline{{Q}^{αβ}}(t_{1},t_{2})

. We find the more general expression for the time depending non-local fluctuations, the propagators

[\langleδQ^{αβ}_{i}(t_{1},t_{2}) δQ^{αβ}_{j}(t_{3},t_{4})\rangle_ξ]_{J}

, in the quadratic approximation. Finally we compare the long-range limit of the correlations, derived in this formalism, with the correlations of the infinite-range model studied with the previous approach (random matrices).