We study the anisotropic Heisenberg spin-glass model in a three-dimensional hierarchical lattice (designed to approximate the cubic lattice), within a real-space renormalization-group approach. Two different initial probability distributions for the exchange interaction (
Jij), Gaussian and uniform, are used, with zero mean and width
Jˉ. The
(kT/Jˉ)×Δ0 phase diagram is obtained, where
T is the temperature,
Δ0 is the first moment of the probability distribution for the uniaxial anisotropy, and
k is the Boltzmann constant. For the Ising model (
Δ0=1), there is a spin-glass phase at low temperatures (high
Jˉ) and a paramagnetic phase at high temperatures (low
Jˉ). For the isotropic Heisenberg model (
Δ0=0), our results indicate no spin-glass phase at finite temperatures. The transition temperature between the spin-glass and paramagnetic phase decreases with
Δ0, as expected, but goes to zero at a finite value of the anisotropy parameter, namely
Δ0=Δc∼0.59. Our results indicate that the whole transition line, between the paramagnetic and the spin-glass phases, for
Δc≤Δ0<1, belongs to the same universality class as the transition for the Ising spin glass.
In the present paper we analyze the critical properties of a quantum spherical spin-glass model with short-range random interactions. Since the model allows for rigorous detailed calculations, we can show how the effective partition function calculated with the help of the replica method for the spin-glass fluctuating fields ${Q}_{\ensuremath{\alpha}\ensuremath{\gamma}}(\stackrel{P\vec}{k}{\ensuremath{\omega}}_{1}{\ensuremath{\omega}}_{2})$ separates into a mean-field contribution for ${Q}_{\ensuremath{\alpha}\ensuremath{\alpha}}(0;\ensuremath{\omega};\ensuremath{-}\ensuremath{\omega})$ and a strictly short-range partition function for the fields ${Q}_{\ensuremath{\alpha}\ensuremath{\ne}\ensuremath{\gamma}}(\stackrel{P\vec}{k}{\ensuremath{\omega}}_{1}{\ensuremath{\omega}}_{2})$. Here $\ensuremath{\alpha},\ensuremath{\gamma}=1,\dots{},n$ are replica indices. The mean-field part
WMF coincides with previous results. The short-range part
WSR describes a phase transition in a
Q3-field theory, where the fluctuating fields depend on a space variable $\stackrel{P\vec}{r}$ and times ${\ensuremath{\tau}}_{1}$ and ${\ensuremath{\tau}}_{2}$. This we analyze using the renormalization group with dimensional regularization and minimal subtraction of dimensional poles. By generalizing standard field theory methods to our particular situation, we can identify the critical dimensionality as
dc=5 at very low temperatures due to the dimensionality shift
Dc=dc+1=6. We then perform an ${ϵ}^{\ensuremath{'}}$ expansion to order one loop to calculate the critical exponents by solving the renormalization-group equations.
We propose that real-space properties of the two-impurity Kondo model can be obtained from an effective spin model where two single-impurity Kondo spin chains are joined via an Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between the two impurity spins. We then use a density matrix renormalization group approach, valid in all ranges of parameters, to study its features using two complementary quantum-entanglement measures, the negativity and the von Neumann entropy. This nonperturbative approach enables us to uncover the precise dependence of the spatial extent ξ(K) of the Kondo screening cloud with the Kondo and RKKY couplings. Our results reveal an exponential suppression of the Kondo temperature T(K)~1/ξ(K) with the size of the effective impurity spin in the limit of large ferromagnetic RKKY coupling, a striking display of "Kondo resonance narrowing" in the two-impurity Kondo model. We also show how the antiferromagnetic RKKY interaction produces an effective decoupling of the impurities from the bulk already for intermediate strengths of this interaction, and, furthermore, exhibit how the non-Fermi liquid quantum critical point is signaled in the quantum entanglement between various parts of the system.
Flat conduction bands quench the group velocity and thus challenge conventional, dispersion-driven pictures of the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction, where localized moments are coupled via an effective exchange mediated by conduction electrons. Here we show that RKKY interactions in the flat-band limit are not extinguished by the vanishing group velocity but are instead mediated by the quantum geometry of Bloch states. Starting from a microscopic RKKY derivation, we demonstrate that the Brillouin-zone-averaged quantum metric controls the long-wavelength structure of the static susceptibility, thereby determining the magnetic correlation length and the spin stiffness. As a result, the finite spatial spread of Wannier functions provides an effective long-range coupling channel even when single-particle dispersion is absent. Furthermore, we establish the general principle that the ordering temperature is governed by the quantum metric in finite and low-dimensional samples, effectively circumventing the thermodynamic-limit constraint of the Mermin-Wagner theorem. Specifically, our theoretical investigation reveals that increasing the quantum metric enhances magnetic rigidity and leads to a corresponding rise in the critical temperature within finite-sized systems.