Asymptotic behavior of the Tomonaga-Luttinger liquid in the vicinity of the phase-separated region is investigated in the one-dimensional $t\ensuremath{-}J$ model to study the universal property of the
c=1 conformal field theory with U(1) symmetry near the $K\ensuremath{\rightarrow}\ensuremath{\infty}$ instability. By analogy to the spinless fermion, we argue that the compressibility behaves as $\ensuremath{\kappa}\ensuremath{\propto}{(J}_{c}\ensuremath{-}{J)}^{\ensuremath{-}1},$ and that the Drude weight is constant and changes to zero discontinuously at the phase boundary. This speculation is confirmed by analyzing the finite-size effect from the result of the exact diagonalization.
We give details of the analysis of critical properties of spin-gap phases in one-dimensional lattice electron models. In Tomonaga-Luttinger (TL) liquid theory, the spin-gap instability occurs when the backward scattering changes from repulsive to attractive. This transition point is shown to be equivalent to that of the level-crossing of the singlet and triplet excitation spectra, using the c=1 conformal field theory and the renormalization group. Based on this notion, the transition point between the TL liquid and the spin-gap phases can be determined with high-accuracy from the numerical data of finite-size clusters. We also discuss the boundary conditions and discrete symmetries to extract these excitation spectra. This technique is applied to the extended Hubbard model, the $t\ensuremath{-}J$ model, and the $t\ensuremath{-}J\ensuremath{-}{J}^{\ensuremath{'}}$ model, and their phase diagrams are obtained. We also discuss the relation between our results and analytical solutions in weak-coupling and low-density limits.
This thesis explores the interaction of symmetries and quantum anomalies on the low-energy dynamics and phase transitions of strongly correlated quantum matter. Beyond the traditional Landau-Ginzburg-Wilson paradigm, we investigate how the embedding of microscopic lattice symmetries into emergent continuum symmetries governs the stability of quantum critical points in (2+1) dimensions. Through the lens of 't Hooft anomalies and anomaly matching, we establish a theoretical framework for understanding the fate of various exotic quantum phase transitions. A central application of this framework is understanding the stability of emergent conformal field theories (CFTs) that describe gapless quantum critical points. The exact nature of these phase transitions is chiefly governed by how the microscopic lattice symmetries impose stringent conditions on the relevant operators allowed in the low-energy effective theory. When physical symmetries suppress relevant perturbations, an interacting conformal fixed point, such as (2+1)-dimensional quantum electrodynamics (QED₃), can stably emerge. Chapters 2 and 3 explore the consequences of this symmetry embedding through the phenomenon of "unnecessary'' quantum critical points, which uniquely describe continuous transitions occurring entirely within a single phase of matter. We demonstrate that the Dirac spin liquid (DSL) in lattice magnets can surprisingly describe a stable continuous phase transition within a single, conventional, symmetry breaking phase. In chapter 4, we apply these tools to a class of quantum phase transitions between featureless bosonic atomic insulators. Despite lacking topological order or protected edge modes, these insulators represent distinct phases because they can be "obstructed,'' meaning their Wannier centers are not pinned to the physical atomic sites. Surprisingly, we find that the transition between these topologically trivial states can host a conformally invariant state described by QED₃, suggesting that even transitions between trivial phases can harbor rich, fractionalized physics. Lastly, chapter 5 analyzes transitions from a composite Fermi liquid (CFL) to a chiral superconductor, employing more traditional field-theoretic techniques and making concrete connections with recent experiments in moire materials.
Quantum criticality in one-dimensional quantum systems is characterized by emergent conformal field theories (CFTs), whose central charge counts independent gapless degrees of freedom. Establishing a microscopic connection between this universal conformal structure and the momentum-space topology of the underlying quasiparticle spectrum remains challenging. Here, we uncover a direct correspondence between Fermi-point topology, conformal criticality, and quantum entanglement in an extended quantum spin chain with competing cluster interactions, exchange anisotropy, and a transverse magnetic field. We show that interaction- and field-driven Lifshitz transitions generate conformal critical phases with effective central charges
ceff=1/2,
1,
3/2,
2, and
3, including a multicritical point where Ising and Luttinger-liquid sectors coexist. Importantly, the central charge is not determined simply by the number of lattice gap closings or Fermi points, but by the number and conformal content of independent low-energy continuum sectors after accounting for lattice symmetries, reciprocal-lattice identifications, and mode equivalences. Thus, Lifshitz transitions may leave the conformal anomaly unchanged or modify the central charge depending on whether spectral reconstruction generates new independent continuum sectors. Real- and momentum-space entanglement spectra provide complementary microscopic signatures, revealing both the conformal content and momentum-space organization of critical modes. Our results establish a microscopic framework linking Fermi-point topology to emergent CFTs and show how interaction-driven spectral reconstruction can generate higher-central-charge criticality and unconventional multicritical behavior.
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.
Universal scaling relations for dissipative Tomonaga-Luttinger (TL) liquids with
SU(N) spin symmetry are obtained for both fermions and bosons, by using asymptotic Bethe-ansatz solutions and conformal field theory (CFT) in one-dimensional non-Hermitian quantum many-body systems with
SU(N) symmetry. We uncover that the spectrum of dissipative TL liquids with
SU(N) spin symmetry is described by the sum of one charge mode characterized by a complex generalization of
c=1 U(1) Gaussian CFT, and $N\ensuremath{-}1$ spin modes characterized by level-
1SU(N) Kac-Moody algebra with the conformal anomaly $c=N\ensuremath{-}1$, and thereby dissipation only affects the charge mode as a result of spin-charge separation in one-dimensional non-Hermitian quantum systems. The derivation is based on a complex generalization of Haldane's ideal-gas description, which is implemented by the
SU(N) Calogero-Sutherland model with inverse-square long-range interactions.