Maximally random discrete-spin systems with symmetric and asymmetric interactions and maximally degenerate ordering

Bora Atalay and A. Nihat Berker

Phys. Rev. E 97, 052102 – Published 2 May 2018

Abstract

Discrete-spin systems with maximally random nearest-neighbor interactions that can be symmetric or asymmetric, ferromagnetic or antiferromagnetic, including off-diagonal disorder, are studied, for the number of states $q = 3, 4$ in $d$ dimensions. We use renormalization-group theory that is exact for hierarchical lattices and approximate (Migdal-Kadanoff) for hypercubic lattices. For all $d > 1$ and all noninfinite temperatures, the system eventually renormalizes to a random single state, thus signaling $q \times q$ degenerate ordering. Note that this is the maximally degenerate ordering. For high-temperature initial conditions, the system crosses over to this highly degenerate ordering only after spending many renormalization-group iterations near the disordered (infinite-temperature) fixed point. Thus, a temperature range of short-range disorder in the presence of long-range order is identified, as previously seen in underfrustrated Ising spin-glass systems. The entropy is calculated for all temperatures, behaves similarly for ferromagnetic and antiferromagnetic interactions, and shows a derivative maximum at the short-range disordering temperature. With a sharp immediate contrast of infinitesimally higher dimension $1 + ε$ , the system is as expected disordered at all temperatures for $d = 1$ .

Received 20 January 2018

DOI:https://doi.org/10.1103/PhysRevE.97.052102

Physics Subject Headings (PhySH)

Phase transitions

Lattice models in statistical physics Spin glasses

Renormalization group

Statistical Physics & Thermodynamics

Authors & Affiliations

Bora Atalay¹ and A. Nihat Berker^2,3

¹Faculty of Engineering and Natural Sciences, Sabancı University, Tuzla, Istanbul 34956, Turkey
²Faculty of Engineering and Natural Sciences, Kadir Has University, Cibali, Istanbul 34083, Turkey
³Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

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Vol. 97, Iss. 5 — May 2018

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Images

Figure 1
Renormalization-group trajectories for the system with $q = 3$ states in $d = 3$ dimensions, starting at three different temperatures $T = J^{- 1}$ from Eqs. (2) and (3), namely starting with (a) $J = 0.02$ , (b) $J = 0.20$ , (c) $J = 0.50$ . Shown are the second $(J_{2})$ and third $(J_{3})$ largest values and the matrix average of the eight nonleading energies $< J_{2 - 9} >$ of the transfer matrix [Eq. (4)], averaged over the quenched random distribution. The leading energy is $J_{1} = 0$ by subtractive choice. The different starting values can be seen on the left axis of each panel (corresponding to renormalization-group step 0). Starting at any nonzero temperature, the system renormalizes to a state in which the leading energy is totally dominant, all other energies renormalizing to $- \infty$ . The matrix position of the single asymptotically dominant element occurs randomly among the $q \times q$ possibilities including off-diagonal and therefore necessarily asymmetric, but is the same across the quenched random distribution. However, starting at high temperatures, as seen, e.g., in the left and center panels, the system spends many renormalization-group iterations near the infinite-temperature fixed point (where all energies are zero), before crossing over to the ordered fixed point. Since the energies at a specific step of a renormalization-group trajectory directly show the effective couplings across the length scale that is reached at that renormalization-group step, this behavior indicates islands of short-range disorder at the short length scales that correspond to the initial steps of a renormalization-group trajectory. These islands of short-range disorder nested in long-range order have been explicitly calculated and shown in spin-glass systems in Ref. [20]. These islands of short-range disorder occur in the presence of long-range order, since the trajectories eventually flow to the strong-coupling fixed point. As temperature is increased (changing the renormalization-group initial condition), these short-range disordered regions order, giving rise to the smooth specific heat peak but no phase transition singularity, as there is no additional fixed-point structure underlying this short-range ordering.
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Figure 2
Calculated free energy per bond as a function of temperature $T = J^{- 1}$ . The curves are, from top to bottom, for $(q = 4, d = 2), (q = 3, d = 2), (q = 4, d = 3), and (q = 3, d = 3)$ . The expected $T = \infty$ values of $f = F / k N = ln q / (b^{d} - 1)$ are given by the dashed lines and match the calculations.
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Figure 3
Calculated entropy per bond as a function of temperature $T = J^{- 1}$ , for $q = 3, 4$ states in $d = 3, 4$ dimensions. The curves are, from top to bottom, for $(q = 4, d = 2), (q = 3, d = 2), (q = 4, d = 3), and (q = 3, d = 3)$ . The expected $T = \infty$ values of $S / k N = ln q / (b^{d} - 1)$ are given by the dashed lines and match the calculations.
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Figure 4
Calculated specific heat as a function of temperature $T = J^{- 1}$ , for $q = 3, 4$ states in $d = 3, 4$ dimensions. The curves are, from top to bottom, for $(q = 4, d = 2), (q = 3, d = 2), (q = 4, d = 3), (q = 3, d = 3)$ . A specific heat maximum occurs at short-range disordering.
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Figure 5
Calculated specific heat as a function of temperature $T = {| J |}^{- 1}$ for ferromagnetic $(J > 0)$ , full curves, and antiferromagnetic $(J < 0)$ , dashed curves, systems, for $q = 3, 4$ states in $d = 3, 4$ dimensions. The curves are, from top to bottom in each panel, for $d = 2$ and $d = 3$ . The quantitatively same short-range disordering behavior is seen for both ferromagnetic and antiferromagnetic systems.
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