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arXiv:1407.5526 · Physical Review E

Odd q -state clock spin-glass models in three dimensions, asymmetric phase diagrams, and multiple algebraically ordered phases

Efe İlker, A. Nihat Berker

Published Dec 4, 2014• 30 citations
arXivHTMLPDF

Abstract

Distinctive orderings and phase diagram structures are found, from renormalization-group theory, for odd q-state clock spin-glass models in d=3 dimensions. These models exhibit asymmetric phase diagrams, as is also the case for quantum Heisenberg spin-glass models. No finite-temperature spin-glass phase occurs. For all odd q≥5, algebraically ordered antiferromagnetic phases occur. One such phase is dominant and occurs for all q≥5. Other such phases occupy small low-temperature portions of the phase diagrams and occur for 5≤q≤15. All algebraically ordered phases have the same structure, determined by an attractive finite-temperature sink fixed point where a dominant and a subdominant pair states have the only nonzero Boltzmann weights. The phase transition critical exponents quickly saturate to the high q value.

DOI
10.1103/physreve.90.062112

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Distinctive orderings and phase diagram structures are found, from renormalization-group theory, for odd q-state clock spin-glass models in d=3 dimensions. These models exhibit asymmetric phase diagrams, as is also the case for quantum Heisenberg spin-glass models. No finite-temperature spin-glass phase occurs. For all odd q≥5, algebraically ordered antiferromagnetic phases occur. One such phase is dominant and occurs for all q≥5. Other such phases occupy small low-temperature portions of the phase diagrams and occur for 5≤q≤15. All algebraically ordered phases have the same structure, determined by an attractive finite-temperature sink fixed point where a dominant and a subdominant pair states have the only nonzero Boltzmann weights. The phase transition critical exponents quickly saturate to the high q value.
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Antiferromagnetic and asymmetric ppp-state clock models on a square lattice are analyzed with the renormalization group, by studying the effect of ppp-fold symmetry-breaking fields in the planar model. For ppp odd and \ensuremath{\ge}3 (p=3p=3p=3 is the three-state Potts model) these models exhibit two phase transitions with a critical line of continuously varying exponent $\ensuremath{\eta}({\ensuremath{\eta}}_{min}<\ensuremath{\eta}<\frac{1}{4})$ separating them. When the couplings are antiferromagnetic in one direction only, ${\ensuremath{\eta}}_{min}=\frac{2}{{p}^{2}}$, while for the isotropic case ${\ensuremath{\eta}}_{min}=\frac{1}{{p}^{2}}$. In the latter case, the low-temperature phase is characterized by a 2p2p2p-fold degeneracy.
We study zero temperature phase transitions in two classes of random quantum systems---the qqq-state quantum Potts and clock models. For models with purely ferromagnetic interactions in one dimension, we show that for strong randomness there is a second order transition with critical properties that can be determined exactly by use of a renormalization group procedure. Somewhat surprisingly, the critical behavior is completely independent of qqq. For the q>4q>4q>4 clock model, we suggest the existence of a novel multicritical point at intermediate randomness. We also consider the T=0T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0T=0 transition from a paramagnet to a spin glass in an infinite-range model, and find qqq independent exponents.
A spin system is studied with simultaneous permutation-symmetric Potts and spin-rotation-symmetric clock interactions in spatial dimensions d=2 and 3. The global phase diagram is calculated from the renormalization-group solution with the recently improved (spontaneous first-order detecting) Migdal-Kadanoff approximation or, equivalently, with hierarchical lattices with the inclusion of effective vacancies. Five different ordered phases are found: Conventionally ordered ferromagnetic, quadrupolar, antiferromagnetic phases and algebraically ordered antiferromagnetic, antiquadrupolar phases. These five different ordered phases and the disordered phase are mutually bounded by first- and second-order phase transitions, themselves delimited by multicritical points: Inverted bicritical, zero-temperature bicritical, tricritical, second-order bifurcation, and zero-temperature highly degenerate multicritical points. One rich phase diagram topology exhibits all of these phenomena.
By using real-space renormalization group (RG) methods, we show that spin glasses in a field display a new kind of transition in high dimensions. The corresponding critical properties and the spin-glass phase are governed by two nonperturbative zero-temperature fixed points of the RG flow. We compute the critical exponents and discuss the RG flow and its relevance for three-dimensional systems. The new spin-glass phase we discovered has unusual properties, which are intermediate between the ones conjectured by droplet and full replica symmetry-breaking theories. These results provide a new perspective on the long-standing debate about the behavior of spin glasses in a field.
An infinite-range quantum Heisenberg spin glass with uniaxial (D) and cubic (K) anisotropies is studied using the thermo-field-dynamic method. Mean-field theory phase diagrams in the temperature-anisotropy plane (T,D) for quantum spins S=2 and S=5/2 are presented for different values of K. Generally, for integer spin values and large anisotropy D a condensation into a nonmagnetic spin state occurs, accompanied by the destruction of the spin-glass order as indicated by the finite critical value Dc{\mathrm{D}}_{\mathrm{c}}Dc​(T=0). For half-integer S and sufficiently low temperature, the spin-glass phase persists for arbitrary D and K. In both cases the presence of a cubic term induces complicated crossovers from longitudinal to transverse freezing of spins.