We discuss effects of
1/rβ type long-range (LR) interactions in a tight-binding model by utilizing the bosonization technique, renormalization group and conformal field theory (CFT). We obtain the low energy action known for Kibble's model which generates the mass gap in 3 dimension when
β=1, the Coulomb force case. In one dimension, the dispersion relations predict that the system remains gapless even for
β=1 and the existences of Tomonaga-Luttinger liquid (TLL) when
β>1. When
β=1, the LR interactions break TLL in the long wavelength limits, even if the strength is very small. We make the more precise arguments from the stand point of the renormalization group and CFT. Finally we derive the accurate finite size scaling of energies and thermodynamics. Moreover we proceed to numerical calculations, considering the LR umklapp process terms. We conclude that the TLL phase become wider in the strength space of interactions as the power
β approaches to 1.
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.
Long-range interactions can fundamentally reshape the low-energy properties of low-dimensional quantum matter, altering both continuous symmetry breaking and topological phenomena. However, their impact on the quantum criticality separating these regimes remains poorly understood. We determine the ground-state phase diagram and critical properties of the spin-one Heisenberg chain with single-ion anisotropy and staggered antiferromagnetic power-law interactions, using matrix-product state (MPS) calculations complemented by high-order series expansions (pCUT+MC). Such long-range, non-frustrated interactions circumvent the Hohenberg-Mermin-Wagner theorem, thereby stabilizing continuous symmetry breaking (CSB) phases in direct competition with the Haldane phase. We map out the resulting phase diagram and analyze the entanglement entropy scaling behavior in the U(1) and SU(2) CSB phases, finding logarithmic corrections beyond the short-range, universal contributions expected from linearly dispersed Goldstone modes. We further characterize all critical boundaries through finite-size scaling of either the entanglement entropy or the staggered magnetization. In particular, the large-D-to-U(1)-CSB transition exhibits unconventional, continuously varying critical exponents as a function of the long-range decay exponent with a strong dependence on the imposed boundary conditions leading to distinct finite-size scalings for sufficiently long-range potentials. Remarkably, the Haldane-to-U(1)-CSB transition likewise displays unconventional quantum criticality with distinct continuously varying critical exponents. Our work positions this model as a target for near-term atomic platforms with tunable long-range couplings and exhibiting natural single-ion anisotropy, offering a minimal playground for exploring the interplay between long-range interactions, continuous symmetry breaking, and topology.
In this work, we present numerical evidence for the spontaneous breaking of a continuous U ( 1 ) symmetry in a nearest-neighbor interacting spin-1 chain at a quantum critical point separating two XY quasi-long-range ordered phases distinguished by a spontaneously broken Z 2 symmetry. Remarkably, the continuous symmetry breaking emerges precisely at the critical point of the discrete order parameter, suggesting a mechanism beyond currently established scenarios. At criticality, the XY correlations develop true long-range order, accompanied by a finite perpendicular magnetization, a zero-frequency Bragg peak in the transverse dynamical structure factor, and sharp gapless collective excitations. From complementary static and dynamical observables, we quantitatively determine the critical exponents, obtaining a dynamical exponent z = 1.50 ± 0.04 and an anomalous dimension η = 1.04 ± 0.03 . Remarkably, the value of z coincides with the one-dimensional Kardar-Parisi-Zhang exponent despite the system being an equilibrium quantum many-body system. We further show that the noninteracting continuum limit is equivalent to the recently introduced transverse quantum fluid, displaying their off-diagonal long-ranged order in one dimension. Complementing the numerical study, we derive the renormalization-group flow equations of the continuum theory to second order in the ɛ expansion. We identify an interacting fixed point whose critical behavior differs from the Ising universality class already at two-loop order, although the perturbative exponents remain far from the numerical values.
Continuous symmetry breaking (CSB) in low-dimensional systems, forbidden by the Mermin-Wagner theorem for short-range interactions, may take place in the presence of slowly decaying long-range interactions. Nevertheless, there is no stringent bound on how slowly interactions should decay to give rise to CSB in 1D quantum systems at zero temperature. Here, we study a long-range interacting spin chain with U(1) symmetry and power-law interactions V(r)∼1/r^{α}. Using a number of analytical and numerical techniques, we find CSB for α smaller than a critical exponent α_{c}(≤3) that depends on the microscopic parameters of the model. Furthermore, the transition from the gapless XY phase to the gapless CSB phase is mediated by the breaking of conformal and Lorentz symmetries due to long-range interactions, and is described by a universality class akin to, but distinct from, the Berezinskii-Kosterlitz-Thouless transition. Signatures of the CSB phase should be accessible in existing trapped-ion experiments.
The phase diagram of the quantum spin-
21 antiferromagnetic ${J}_{1}^{\phantom{\rule{0.16em}{0ex}}}\text{\ensuremath{-}}{J}_{2}^{\phantom{\rule{0.16em}{0ex}}}$ XXZ chain was obtained by Haldane using bosonization techniques [Haldane, Phys. Rev. B 25, 4925 (1982); Phys. Rev. B 26, 5257 (1982)]. It supports three distinct phases for $0\ensuremath{\le}{J}_{2}^{\phantom{\rule{0.16em}{0ex}}}/{J}_{1}^{\phantom{\rule{0.16em}{0ex}}}<\frac{1}{2}$, i.e., a gapless algebraic spin-liquid phase, a gapped long-range ordered Neel phase, and a gapped long-range ordered dimer phase. Even though the Neel and dimer phases are not related hierarchically by a pattern of symmetry breaking, it was shown that they meet along a line of quantum critical points with a U(1) symmetry and central charge
c=1. Here, we extend the analysis made by Haldane on the quantum spin-
21 antiferromagnetic ${J}_{1}^{\phantom{\rule{0.16em}{0ex}}}\text{\ensuremath{-}}{J}_{2}^{\phantom{\rule{0.16em}{0ex}}}$ XYZ chain using both bosonization and numerical techniques. We show that there are three Neel phases and the dimer phase that are separated from each other by six planes of phase boundaries realizing Gaussian criticality when $0\ensuremath{\le}{J}_{2}^{\phantom{\rule{0.16em}{0ex}}}/{J}_{1}^{\phantom{\rule{0.16em}{0ex}}}<\frac{1}{2}$. We also show that each long-range ordered phase harbors topological point defects (domain walls) that are dual to those across the phase boundary in that a defect in one ordered phase locally binds the other type of order around its core. By using the bosonization approach, we identify the critical theory that describes simultaneous proliferation of these dual point defects, and show that it supports an emergent U(1) symmetry that originates from the discrete symmetries of the XYZ model. To confirm this numerically, we perform exact diagonalization and density-matrix renormalization-group calculations and show that the critical theory is characterized by the central charge
c=1 with critical exponents that are consistent with those obtained from the bosonization approach. Furthermore, we generalize the field-theoretic description of direct continuous phase transition to higher dimensions, especially in
d=3, by using a nonlinear sigma model (NLSM) with a topological term. In particular, we propose the $\ensuremath{\pi}$-flux phase on the cubic lattice with local quartic interactions as a platform for direct continuous phase transition and deconfined criticality. We discuss possible phase diagrams for the $\ensuremath{\pi}$-flux phase on the cubic lattice with these quartic interactions from the renormalization flow of NLSMs.