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Register a new userMulticritical point on the de Almeida-Thouless line in spin glasses in $d>6$ dimensions
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The de Almeida-Thouless (AT) line in Ising spin glasses is the phase boundary in the temperature $T$ and magnetic field $h$ plane below which replica symmetry is broken. Using perturbative renormalization group (RG) methods, we show that when the dimension $d$ of space is just above $6$ there is a multicritical point (MCP) on the AT line, which separates a low-field regime, in which the critical exponents have mean-field values, from a high-field regime where the RG flows run away to infinite coupling strength; as $d$ approaches $6$ from above, the location of the MCP approaches the zero-field critical point exponentially in $1/(d-6)$. Thus on the AT line perturbation theory for the critical properties breaks down at sufficiently large magnetic field even above $6$ dimensions, as well as for all non-zero fields when $dleq 6$ as was known previously. We calculate the exponents at the MCP to first order in $varepsilon=d-6>0$. The fate of the MCP as $d$ increases from just above 6 to infinity is not known.
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The behavior of two-dimensional Ising spin glasses at the multicritical point on triangular and honeycomb lattices is investigated, with the help of finite-size scaling and conformal-invariance concepts. We use transfer-matrix methods on long strips to calculate domain-wall energies, uniform susceptibilities, and spin-spin correlation functions. Accurate estimates are provided for the location of the multicritical point on both lattices, which lend strong support to a conjecture recently advanced by Takeda, Sasamoto, and Nishimori. Correlation functions are shown to obey rather strict conformal-invariance requirements, once suitable adaptations are made to account for geometric aspects of the transfer-matrix description of triangular and honeycomb lattices. The universality class of critical behavior upon crossing the ferro-paramagnetic phase boundary is probed, with the following estimates for the associated critical indices: $ u=1.49(2)$, $gamma=2.71(4)$, $eta_1= 0.183(3)$, distinctly different from the percolation values.
We use high temperature series expansions to study the $pm J$ Ising spin-glass in a magnetic field in $d$-dimensional hypercubic lattices for $d=5, 6, 7$ and $8$, and in the infinite-range Sherrington-Kirkpatrick (SK) model. The expansions are obtained in the variable $w=tanh^2{J/T}$ for arbitrary values of $u=tanh^2{h/T}$ complete to order $w^{10}$. We find that the scaling dimension $Delta$ associated with the ordering-field $h^2$ equals $2$ in the SK model and for $dge 6$. However, in agreement with the work of Fisher and Sompolinsky, there is a violation of scaling in a finite field, leading to an anomalous $h$-$T$ dependence of the Almeida-Thouless (AT) line in high dimensions, while scaling is restored as $d to 6$. Within the convergence of our series analysis, we present evidence supporting an AT line in $dge 6$. In $d=5$, the exponents $gamma$ and $Delta$ are substantially larger than mean-field values, but we do not see clear evidence for the AT line in $d=5$.
We test for the presence or absence of the de Almeida-Thouless line using one-dimensional power-law diluted Heisenberg spin glass model, in which the rms strength of the interactions decays with distance, r as 1/r^{sigma}. It is argued that varying the power sigma is analogous to varying the space dimension d in a short-range model. For sigma=0.6, which is in the mean field regime regime, we find clear evidence for an AT line. For sigma = 0.85, which is in the non-mean-field regime and corresponds to a space dimension of close to 3, we find no AT line, though we cannot rule one out for very small fields. Finally for sigma=0.75, which is in the non-mean-field regime but closer to the mean-field boundary, the evidence suggests that there is an AT line, though the possibility that even larger sizes are needed to see the asymptotic behavior can not be ruled out.
We test for the existence of a spin-glass phase transition, the de Almeida-Thouless line, in an externally-applied (random) magnetic field by performing Monte Carlo simulations on a power-law diluted one-dimensional Ising spin glass for very large system sizes. We find that an Almeida-Thouless line only occurs in the mean field regime, which corresponds, for a short-range spin glass, to dimension d larger than 6.
The chiral spin-glass Potts system with q=3 states is studied in d=2 and 3 spatial dimensions by renormalization-group theory and the global phase diagrams are calculated in temperature, chirality concentration p, and chirality-breaking concentration c, with determination of phase chaos and phase-boundary chaos. In d=3, the system has ferromagnetic, left-chiral, right-chiral, chiral spin-glass, and disordered phases. The phase boundaries to the ferromagnetic, left- and right-chiral phases show, differently, an unusual, fibrous patchwork (microreentrances) of all four (ferromagnetic, left-chiral, right-chiral, chiral spin-glass) ordered ordered phases, especially in the multicritical region. The chaotic behavior of the interactions, under scale change, are determined in the chiral spin-glass phase and on the boundary between the chiral spin-glass and disordered phases, showing Lyapunov exponents in magnitudes reversed from the usual ferromagnetic-antiferromagnetic spin-glass systems. At low temperatures, the boundaries of the left- and right-chiral phases become thresholded in p and c. In the d=2, the chiral spin-glass system does not have a spin-glass phase, consistently with the lower-critical dimension of ferromagnetic-antiferromagnetic spin glasses. The left- and right-chirally ordered phases show reentrance in chirality concentration p.
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