Phase diagram, critical properties and thermodynamic functions of the two-dimensional field-free quantum-spin-1/2 XXZ model has been calculated globally using a numerical renormalisation group theory. The nearest-neighbour spin-spin correlations and entanglement properties, as well as internal energy and specific heat are calculated globally at all temperatures for the whole range of exchange interaction anisotropy, from XY limit to Ising limits, for both antiferromagnetic and ferromagnetic cases. We show that there exists long-range (quasi-long-range) order at low-temperatures, and the low-lying excitations are gapped (gapless) in the Ising-like easy-axis (XY-like easy-plane) regime. Besides, we identify quantum phase transitions at zero-temperature.
We analyze the antiferromagnetic spin-1/2 XXZ model on the kagome lattice at finite external magnetic field with the help of a non-perturbative zero-temperature renormalization group (RG) technique. The exact nature of the ground and excited state properties (e.g. gapped or gapless spectrum etc) of this system are still debated. Approximate methods have typically been adopted towards understanding the low-energy spectrum. Following the work of Kumar et al (2014 Phys. Rev. B 90 174409), we use a Jordan–Wigner transformation to map the spin problem into one of spinless fermions (spinons) in the presence of a statistical gauge field, and with nearest-neighbor interactions. While the work of Kumar et al was confined mostly to the plateau at 1/3-filling (magnetization per site) in the XY regime, we analyze the role of inter-spinon interactions in shaping the phases around this plateau in the entire XXZ model. The RG phase diagram obtained contains three spin liquid phases whose position is determined as a function of the exchange anisotropy and the energy scale for fluctuations arising from spinon scattering. Two of these spins liquids are topologically ordered states of matter with gapped, degenerate states on the torus. The gap for one of these phases corresponds to the one-spinon band gap of the Azbel–Hofstadter spectrum for the XY part of the Hamiltonian, while the other arises from two-spinon interactions. The Heisenberg point of this problem is found to lie within the interaction gapped spin liquid phase, in broad agreement with a recent experimental finding. The third phase is an algebraic spin liquid with a gapless Dirac spectrum for spinon excitations, and possess properties that show departures from the Fermi liquid paradigm. The three phase boundaries correspond to critical theories, and meet at a SU (2)-symmetric multicritical point. This special critical point agrees well with the gap-closing transition point predicted by Kumar et al . We discuss the relevance of our findings to various recent experiments, as well as results obtained from other theoretical analyses.
The anisotropic
XXZ spin-
21 Heisenberg chain is studied using renormalization-group theory. The specific heats and nearest-neighbor spin-spin correlations are calculated throughout the entire temperature and anisotropy ranges in both ferromagnetic and antiferromagnetic regions, obtaining a global description and quantitative results. We obtain, for all anisotropies, the antiferromagnetic spin-liquid spin-wave velocity and the Ising-like ferromagnetic excitation spectrum gap, exhibiting the spin-wave to spinon crossover. A number of characteristics of purely quantum nature are found: The in-plane interaction
sixsjx+siysjy induces an antiferromagnetic correlation in the out-of-plane
siz component, at higher temperatures in the antiferromagnetic
XXZ chain, dominantly at low temperatures in the ferromagnetic
XXZ chain, and, in-between, at all temperatures in the
XY chain. We find that the converse effect also occurs in the antiferromagnetic
XXZ chain: an antiferromagnetic
sizsjz interaction induces a correlation in the
sixy component. As another purely quantum effect, (i) in the antiferromagnet, the value of the specific heat peak is insensitive to anisotropy and the temperature of the specific heat peak decreases from the isotropic (Heisenberg) with introduction of either type (Ising or
XY) of anisotropy; and (ii) in complete contrast, in the ferromagnet, the value and temperature of the specific heat peak increase with either type of anisotropy.
The phase diagram of the anisotropic quantum spin-1/2 Heisenberg antiferromagnet in the presence of competitive nearest
(J1)- and next-nearest $({J}_{2}=\ensuremath{\alpha}{J}_{1})$-neighbor interactions on a simple cubic lattice is studied within the framework of the differential operator technique. The Hamiltonian is solved by using an effective-field theory in a two-spin cluster. We propose a function for the free energy, which is similar to the Landau expansion, in order to find the ground state
(T=0) and the finite-
T phase diagram. At zero temperature, we find a first-order quantum phase transition in the point ${\ensuremath{\alpha}}_{c}(\ensuremath{\Delta})$ between the antiferromagnetic (AF) and superantiferromagnetic (SAF) phases. This is dependent on the anisotropy parameter $\ensuremath{\Delta}$ ($\ensuremath{\Delta}=0$ and 1 correspond to the isotropic Heisenberg and Ising models, respectively). The quantum critical values of ${\ensuremath{\alpha}}_{c}(\ensuremath{\Delta}\ensuremath{\ne}1)={\ensuremath{\alpha}}_{c}^{q}(\ensuremath{\Delta})$ are smaller than those of the classical model ${\ensuremath{\alpha}}_{c}(\ensuremath{\Delta}=1)={\ensuremath{\alpha}}_{c}^{c}=1/4$ (for example, for $\ensuremath{\Delta}=0$, we find ${\ensuremath{\alpha}}_{c}^{q}(0)=0.21$). The former monotonically increases with the anisotropy parameter $\ensuremath{\Delta}$. The phase diagram in $T\text{\ensuremath{-}}\ensuremath{\alpha}$ plane presents a second-order transition at ${T}_{2}(\ensuremath{\alpha})$ between the AF and paramagnetic (P) phases. The phase transition between the SAF and P phases is of first-order ${T}_{1}(\ensuremath{\alpha})$ for all values of $\ensuremath{\Delta}∊[0,1]$. The phase transition lines ${T}_{1}(\ensuremath{\alpha})$ and ${T}_{2}(\ensuremath{\alpha})$ merge in a critical end point (CEP) that is dependent on $\ensuremath{\Delta}$ [i.e., ${T}_{\text{CEP}}(\ensuremath{\Delta})$ and ${\ensuremath{\alpha}}_{\text{CEP}}(\ensuremath{\Delta})$ are the temperature and frustration parameters at the CEP]. The ordered phases (AF and SAF) are separated by first-order transitions and exhibit a nontrivial re-entrant behavior at low temperatures.
We revisit the phase diagram of the dimerized XXZ spin-
21 chain with nearest-neighbor couplings which was studied numerically by Mondal et al. [Phys. Rev. B 106, L201106 (2022)]. The model has isotropic
XY couplings which have a uniform value and
ZZ couplings which have a dimerized form, with strengths
Ja and
Jb on alternate bonds. We find a rich phase diagram in the region of positive
Ja,Jb. We provide a detailed understanding of the different phases and associated quantum phase transitions using a combination of mean-field theory, low-energy effective Hamiltonians, renormalization group calculations employing the technique of bosonization, and numerical calculations using the density-matrix renormalization group (DMRG) method. The phase diagram consists of two Ising paramagnetic phases called
IPM0 and ${\mathrm{IPM}}_{\ensuremath{\pi}}$, and a phase with Ising Neel (IN) order; all these phases are gapped. The phases
IPM0 and ${\mathrm{IPM}}_{\ensuremath{\pi}}$ are separated by a gapless phase transition line $0\ensuremath{\le}{J}_{a}={J}_{b}\ensuremath{\le}1$---the well-known Tomonaga Luttinger liquid captured by a conformal field theory (CFT) with central charge
c=1. In contrast, there are two phase transition lines separating
IPM0 or the ${\mathrm{IPM}}_{\ensuremath{\pi}}$ from IN that are described by
c=21 CFTs and correspond to quantum Ising transitions. In fact, the
c=1 line bifurcates into the two
c=21 lines at the point
Ja=Jb=1; the shape of the bifurcation is found analytically using RG calculations. A symmetry analysis shows that
IPM0 is a topologically trivial phase while ${\mathrm{IPM}}_{\ensuremath{\pi}}$ is a time-reversal symmetry-protected topological phase (SPT) with spin-
21 states at the two ends of an open system. A gap opens in the low-energy spectrum whenever one moves away from one of the phase transition lines; the scaling of the gap with the distance from a transition line is found analytically using the RG method. The numerical results obtained by the DMRG method are in good agreement with the analytical results. Further, we show that useful insights into the nature of the phase diagram of the above Hamiltonian, particularly the SPT as well as the phase transitions, is obtained via Kramers-Wannier duality to a deformed quantum Ashkin-Teller model. Finally, we propose experimental platforms for testing our results.
In this paper, the phase diagrams and the critical behavior of the spin-1/2 anisotropic XXZ ferromagnetic model (the anisotropic parameter Δ\in(-\infty,1]) on two kinds of diamond-type hierarchical (DH) lattices with fractal dimensions d_{f}=2.58 and 3, respectively, are studied via the real-space renormalization group method. It is found that in the isotropic Heisenberg limit (Δ=0), there exist finite temperature phase transitions for the two kinds of DH lattices above. The systems are also investigated in the range of -\infty