We study the nature of the two-dimensional quantum critical point separating two phases with and without long-range spin-density-wave order, which has been recently observed in cuprate superconductors. We consider the Landau-Ginzburg-Wilson Hamiltonian associated with the spin-density critical modes, perform a mean-field analysis of the phase diagram, and study the corresponding renormalization-group flow in two different perturbative schemes at five and six loops, respectively. The analysis supports the existence of a stable fixed point in the full theory whose basin of attraction includes systems with collinear spin-density-wave order, as observed in experiments. The stable fixed point is characterized by an enlarged $\mathrm{O}(4)\ensuremath{\bigotimes}\mathrm{O}(3)$ symmetry. The continuous transition observed in experiments is expected to belong to this universality class. The corresponding critical exponents are $\ensuremath{\nu}=0.9(2)$ and $\ensuremath{\eta}=0.15(10)$.
We study the nature of the two-dimensional quantum critical point separating two phases with and without long-range spin-density-wave order, which has been recently observed in cuprate superconductors. We consider the Landau-Ginzburg-Wilson Hamiltonian associated with the spin-density critical modes, perform a mean-field analysis of the phase diagram, and study the corresponding renormalization-group flow in two different perturbative schemes at five and six loops, respectively. The analysis supports the existence of a stable fixed point in the full theory whose basin of attraction includes systems with collinear spin-density-wave order, as observed in experiments. The stable fixed point is characterized by an enlarged O(4)\ensuremath⨂O(3) symmetry. The continuous transition observed in experiments is expected to belong to this universality class. The corresponding critical exponents are \ensuremathν=0.9(2) and \ensuremathη=0.15(10).
The Landau-Ginzburg-Wilson (LGW) paradigm is the backbone of the modern understanding of critical phenoma. It rests on the assumption that a continuous phase transition can be described solely in terms of a fluctuating order parameter. In this sandwich thesis, we analyze several quantum phase transitions in two-dimensional systems where this basic assumption is violated. Our analysis is structured in three parts: In the first part, two ordering transitions in metals are studied, where the presence of gapless fermionic modes invalidates the pure order parameter description. Instead, we apply a renormalization group approach which retains both fermionic and bosonic (order paramater) degrees of freedom, using the deviation from an upper critical dimension of d = 5/2 as a control parameter. We first apply this technique to describe the quantum phase transition between a normal metal and an inhomogeneous (FFLO) superconductor. Our analysis rigorously confirms the mean field expectation that this transition is continuous, and shows that interesting non-Fermi liquid physics can arise at the critical point, manifesting itself in unusual scaling of various observables. The second case study is the onset of incommensurate 2kF charge density wave order. While non-Fermi liquid features are less pronounced at this transition, we find a strong dynamical nesting of the Fermi surface, which stabilizes the density wave formation and results in a continuous transition, opposed to early theoretical claims. In the second part, we present a novel case study of “deconfined criticality” for a quantum magnet: Here, symmetries are broken on both sides of the phase transition, which would be first order within a conventional two-order-parameter LGW description. By contrast, the scenario of deconfined criticality predicts a continuous transition, driven by condensation of topological defects which carry quantum numbers of the opposite phase. Its paradigmatic application is the transition from a SU (2)-Neél state to a valence-bond solid (a singlet covering configuration that breaks spatial symmetries) on the square lattice. Here, we propose a novel extension and study a SU (3) antiferromagnet on a triangular lattice, which supports a transition between a magnetic, three-sublattice color-ordered phase and a trimerized SU (3) singlet phase. We provide a critical theory in terms of fractional bosonic degrees of freedom (and with a topological defect interpretation), and study its fixed point properties with functional renormalization group. This yields a critical fixed point in a suitable large-N -limit, implying a continuous transition in accordance with the deconfined criticality framework. In the third and final part, we study the “polaron”-problem of a single impurity coupled to a majority Fermi sea, which is particularly interesting when formation of a bound state (“molecule”) between impurity and majority is allowed for. Starting from the exactly solvable limit of infinite impurity mass, we present a controlled computation of impurity spectra (both single- and two-particle) for heavy impurities based on Feynman diagram techniques, with an eye for experiments on doped semiconductors and ultracold gases. Furthermore, we discuss various aspects of the “molecule-to-polaron” transition that occurs in these systems, which also defies a simple LGW description in the single-impurity limit.
We describe two-dimensional models with a metallic Fermi surface which display quantum phase transitions controlled by strongly interacting critical field theories below their upper critical dimension. The primary examples involve transitions with a topological order parameter associated with dislocations in collinear spin-density-wave (``stripe'') correlations: the suppression of dislocations leads to a fractionalization of spin and charge collective modes, and this transition has been proposed as a candidate for the cuprates near optimal doping. The coupling between the order parameter and long-wavelength volume and shape deformations of the Fermi surface is analyzed by the renormalization group, and a runaway flow to a nonperturbative regime is found in most cases. A phenomenological scaling analysis of simple observable properties of possible second-order quantum critical points is presented, with results quite similar to those near quantum spin-glass transitions and to phenomenological forms proposed by Schr\"oder et al. [Nature (London) 407, 351 (2000)].
We study the competition of spin- and charge-density waves and their quantum multicritical behavior for the semimetal-insulator transitions of low-dimensional Dirac fermions. Employing the effective Gross-Neveu-Yukawa theory with two order parameters as a model for graphene and a growing number of other two-dimensional Dirac materials allows us to describe the physics near the multicritical point at which the semimetallic and the spin- and charge-density-wave phases meet. With the help of a functional renormalization group approach, we are able to reveal a complex structure of fixed points, the stability properties of which decisively depend on the number of Dirac fermions Nf. We give estimates for the critical exponents and observe crucial quantitative corrections as compared to the previous first-order $\ensuremath{\epsilon}$ expansion. For small Nf, the universal behavior near the multicritical point is determined by the chiral Heisenberg universality class supplemented by a decoupled, purely bosonic, Ising sector. At large Nf, a novel fixed point with nontrivial couplings between all sectors becomes stable. At intermediate Nf, including the graphene case (Nf=2), no stable and physically admissible fixed point exists. Graphene's phase diagram in the vicinity of the intersection between the semimetal, antiferromagnetic, and staggered density phases should consequently be governed by a triple point exhibiting first-order transitions.
When a 2D superconductor is subjected to a strong in-plane magnetic field, Zeeman polarization of the Fermi surface can give rise to inhomogeneous FFLO order with a spatially modulated gap. Further increase of the magnetic field eventually drives the system into a normal metal state. Here, we perform a renormalization group analysis of this quantum phase transition, starting from an appropriate low-energy theory recently introduced in Phys. Rev. B 93, 085112 (2016). We compute one-loop flow equations within the controlled dimensional regularization scheme with fixed dimension of Fermi surface, expanding in $\ensuremath{\epsilon}=5/2\ensuremath{-}d$. We find a new stable non-Fermi-liquid fixed point and discuss its critical properties. One of the most interesting aspects of the FFLO non-Fermi-liquid scenario is that the quantum critical point is potentially naked, with the scaling regime observable down to arbitrary low temperatures. In order to study this possibility, we perform a general analysis of competing instabilities, which suggests that only charge density wave order is enhanced in the vicinity of the quantum critical point.
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.
In light of the new experimental and theoretical important developments in high-Tc superconductivity, we revisit the fermionic hot-spot model relevant to the phenomenology of the cuprates. We extend previous results by means of a complete two-loop order renormalization group (RG) framework. Here, we explicitly study the effect of the charge-density-wave (CDW) order parameter with a d-wave form factor with the experimentally observed modulation $(\ifmmode\pm\else\textpm\fi{}{Q}_{0},0)$ and $(0,\ifmmode\pm\else\textpm\fi{}{Q}_{0})$ at the infrared-stable nontrivial fixed point obtained previously for this model. Additionally, we proceed to investigate also the so-called pair-density-wave (PDW) order that was recently proposed in the literature as a possible candidate for the ``hidden'' order to describe the pseudogap phase observed in underdoped cuprates. We confirm that although the above two ordering tendencies are also found to be nearly degenerate both at one-loop and two-loop RG orders and linked by an emergent SU(2) pseudospin symmetry, they turn out to be subleading for weaker couplings in the present model to antiferromagnetism, d-wave bond-density wave (BDW) order with modulation along Brillouin zone diagonals $(\ifmmode\pm\else\textpm\fi{}{Q}_{0},\ifmmode\pm\else\textpm\fi{}{Q}_{0})$, and d-wave singlet superconductivity (SSC). However, as we increase the strength of the initial coupling towards moderate values, we do capture a tendency for the entangled PDW/CDW order to become leading compared to BDW/SSC in the model, which suggests that the former composite order might be indeed a viable concept to describe some cuprate superconductors at high temperatures in the underdoped regime, as has been recently alluded to by many authors in the literature.