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Topology plays a cardinal role in explaining phases and quantum phase transitions beyond the Landau-Ginzburg-Wilson paradigm. In this study, we formulate a set of models of Dirac fermions in 2+1 dimensions with SU($N$)$\times$SU(2)$\times$U(1) symmetry that have the potential to host critical points described by field theories with topological terms. For $N=2$ it shows a rich phase diagram containing semimetallic, quantum spin Hall insulating, Kekul\'e valence bond solid and s-wave superconducting phases and features multiple Landau-Ginzburg-Wilson phase transitions driven by interaction strength. At $N=1$ a deconfined quantum critical point is observed. At $N=2$ one expects the critical theory to correspond to a level 2 Wess-Zumino-Witten theory in 2+1 dimensions. Here the numerical results however show a strong first order transition. Another transition can be governed by a topological $\theta$-term which is rendered irrelevant for even values of $N$ thus leading to Landau-Ginzburg-Wilson behaviour., Comment: 6 pages main text + 7 pages Supplemental Material, 3 figures main text + 9 figures Supplemental Material

While altermagnetic materials are characterized by a vanishing net magnetic moment, their symmetry in principle allows for the existence of an anomalous Hall effect (AHE). Here we introduce a model with altermagnetism in which the emergence of an AHE is driven by interactions. This model is grounded in a modified Kane-Mele framework with antiferromagnetic (AFM) spin-spin correlations. Quantum Monte Carlo simulations show that the system undergoes a finite temperature phase transition governed by a primary AFM order parameter accompanied by a secondary one of Haldane type. The emergence of both orders turns the metallic state of the system, away from half-filling, to an altermagnet with a finite anomalous Hall conductivity. A mean field ansatz corroborates these results, which pave the way into the study of correlation induced altermagnets with finite Berry curvature., Comment: 5 pages, 3 figures

We compute the rotational anisotropy of the free energy of $\alpha$-RuCl$_3$ in an external magnetic field. This quantity, known as the magnetotropic susceptibility, $k$, relates to the second derivative of the free energy with respect to the angle of rotation. We have used approximation-free, auxiliary-field quantum Monte Carlo simulations for a realistic model of $\alpha$-RuCl$_3$ and optimized the path integral to alleviate the negative sign problem. This allows us to reach temperatures down to $30~\rm{K}$ -- an energy scale below the dominant Kitaev coupling. We demonstrate that the magnetotropic susceptibility in this model of $\alpha$-RuCl$_3$ displays unique scaling, $k = Tf(B/T)$, with distinct scaling functions $f$ at high and low temperatures. In comparison, for the XXZ Heisenberg model, the scaling $k = Tf(B/T)$ breaks down at a temperature scale where the uniform spin susceptibility deviates from the Curie law (i.e. at the energy scale of the exchange interactions) and never recovers at low temperatures. Our findings suggest that correlations in $\alpha$-RuCl$_3$ lead to degrees of freedom that respond isotropically to a magnetic field. One possible interpretation for the apparent scale-invariance observed in experiments could be fractionalization of the spin degrees of freedom in the extended Kitaev model., Comment: 6 pages, 4 figures, supplemental material

We carry out large-scale quantum Monte Carlo simulations of a candidate field theory for the onset of superconductivity in magic-angle twisted bilayer graphene. The correlated insulating state at charge neutrality spontaneously breaks U(1) Moir\'e valley symmetry. Owing to the topological nature of the bands, skyrmion defects of the order parameter carry charge $2e$ and condense upon doping. In our calculations we encode the U(1) symmetry by an internal degree of freedom such that it is not broken upon lattice regularization. Furthermore, the skyrmion carries the same charge. The nature of the doping-induced phase transitions depends on the strength of the easy-plane anisotropy that reduces the SU(2) valley symmetry to U(1) $\times \mathbb{Z}_2 $. For large anisotropy, we observe two distinct transitions separated by phase coexistence. While the insulator to superconducting transition is of mean-field character, the U(1) transition is consistent with three-dimensional XY criticality. Hence, the coupling between the gapless charge excitations of the superconducting phase and the XY order parameter is irrelevant. At small anisotropy, we observe a first-order transition characterized by phase separation., Comment: 6 pages, 5 figures, supplemental material

We introduce a model of Dirac fermions in 2+1 dimensions with a semimetallic, a quantum spin-Hall insulating (QSHI), and an s-wave superconducting (SSC) phase. The phase diagram features a multicritical point at which all three phases meet as well as a QSHI-SSC deconfined critical point. The QSHI and SSC orders correspond to mutually anti-commuting mass terms of the Dirac Hamiltonian. Based on this algebraic property, SO(5) symmetric field theories have been put forward to describe both types of critical points. Using quantum Monte Carlo simulations, we directly study the operator that rotates between QSHI and SSC states. The results suggest that it commutes with the low-energy effective Hamiltonian at criticality but has a gap in the ordered phases. This implies an emergent SO(5) symmetry at both the multicritical and the deconfined critical points., Comment: 5 pages, 4 figures, supplemental material

The quantum spin Hall state can be understood in terms of spontaneous O(3) symmetry breaking. Topological skyrmion configurations of the O(3) order parameter vector carry a charge 2e, and as shown previously, when they condense, a superconducting state is generated. We show that this topological route to superconductivity survives easy-plane anisotropy. Upon reducing the O(3) symmetry to O(2)$\times$ Z$_2$, skyrmions give way to merons that carry a unit charge. On the basis of large-scale auxiliary field quantum Monte Carlo simulations, we show that at the particle-hole symmetric point, we can trigger a continuous and direct transition between the quantum spin Hall state and s-wave superconductor by condensing pairs of merons. This statement is valid in both strong and weak anisotropy limits. Our results can be interpreted in terms of an easy-plane deconfined quantum critical point. However, in contrast to the previous studies in quantum spin models, our realization of this quantum critical point conserves $U(1)$ charge, such that skyrmions are conserved.

We provide a negative-sign-free formulation of the auxiliary field quantum Monte Carlo algorithm for generalized Kitaev models with higher symmetries. Our formulation is based on the Abrikosov fermion representation of the spin-1/2 degree of freedom and the phase pinning approach [Phys. Rev. B 104, L081106 (2021)]. Enhancing the number of fermion flavors or orbitals from one to $N$ allows one to generalize the inherent $Z_2$ global symmetry to Z$_2$$\times$SU($N$)$_o$. Using this general approach, we study the Z$_2$$\times$SU(2)$_o$ Kitaev-Heisenberg model reflecting the competition between the isotropic Heisenberg exchange and Kitaev-type bond-directional exchange interactions. We show that the symmetry enhancement provides a path to escape frustration and that the spin liquid phases in the original Z$_2$ symmetric model are not present in this model. Nevertheless, the ground-state phase diagram is extremely rich and has points with higher global and local continuous symmetries as well as de-confined quantum critical points., Comment: 8 pages, 5 figures

Thermodynamic and dynamical properties of a model of Dirac fermions with a deconfined quantum critical point (DQCP) separating an interaction-generated quantum spin-Hall insulator from an s-wave superconductor [Nature Comm.~{\bf 10}, 2658 (2019)] are studied by quantum Monte Carlo simulations. Inside the deconfined quantum critical region bound by the single-particle gap, spinons and spinless charge-2e skyrmions emerge. Since the model conserves total spin and charge, and has a single length scale, these excitations lead to a characteristic linear temperature dependence of the uniform spin and charge susceptibilities. At the DQCP, the order parameter dynamic structure factors show remarkable similarities that support emergent Lorentz symmetry. Above a critical temperature, superconductivity is destroyed by the proliferation of spin-1/2 vortices., Comment: 8 pages and 8 figures

We consider fermions on a honeycomb lattice supplemented by a spin invariant interaction that dynamically generates a quantum spin Hall insulator. This lattice model provides an instance of Gross-Neveu Heisenberg criticality, as realized for example by the Hubbard model on the honeycomb lattice. Using auxiliary field quantum Monte Carlo simulations we show that we can compute with unprecedented precision susceptibilities of the order parameter. In O(N) Gross-Neveu transitions, the anomalous dimension of the bosonic mode grows as a function of N such that in the large-N limit it is of particular importance to consider susceptibilities rather than equal time correlations so as to minimize contributions from the background. For the N=3 case, we obtain $1/\nu=1.11(4)$, $\eta_{\phi}=0.80(9)$, and $\eta_{\psi}=0.29(2)$ for respectively the correlation length exponent, bosonic and fermionic anomalous dimensions., Comment: 11 pages,13 figures

Frustrated spin systems generically suffer from the negative sign problem inherent to Monte Carlo methods. Since the severity of this problem is formulation dependent, optimization strategies can be put forward. We introduce a phase pinning approach in the realm of the auxiliary field quantum Monte Carlo algorithm. If we can find an anti-unitary operator that commutes with the one body Hamiltonian coupled to the auxiliary field, then the phase of the action is pinned to $0$ and $\pi$. For generalized Kitaev models, we can successfully apply this strategy and observe a remarkable improvement of the average sign. We use this method to study thermodynamical and dynamical properties of the Kitaev-Heisenberg model down to temperatures corresponding to half of the exchange coupling constant. Our dynamical data reveals finite temperature properties of ordered and spin-liquid phases inherent to this model., Comment: 5 pages, 5 figures, supplemental material

A unique property of a dynamically generated quantum spin Hall state are Goldstone modes that correspond to the long-wavelength fluctuations of the spin-orbit coupling order parameter whose topological Skyrmion excitations carry charge 2$e$. Within the model considered here, upon varying the chemical potential, we observe two transitions: An s-wave superconducting order parameter develops at a critical chemical potential $\mu_{c1}$, corresponding to the excitation gap of pairs of fermions, and at $\mu_{c2}$ the SO(3) order parameter of the quantum spin Hall state vanishes. Using negative-sign-free, large-scale quantum Monte Carlo simulations, we show that $\mu_{c1}=\mu_{c2}$ within our accuracy -- we can resolve dopings away from half filling down to $\delta = 0.0017$. The length scale associated with the fluctuations of the quantum spin Hall order parameter grows down to our lowest doping, suggesting either a continuous or a weakly first-order transition. Contrary to mean-field expectations, the doping versus chemical potential curve is not linear, indicating a dynamical critical exponent $z > 2$ if the transition is continuous., Comment: 10 pages, 13 Figures

Dirac fermions in $2+1$ dimensions with dynamically generated anticommuting SO(3) antiferromagnetic (AFM) and Z$_2$ Kekul\'e valence-bond solid (KVBS) masses map onto a field theory with a topological $\theta$-term. This term provides a mechanism for continuous phase transitions between different symmetry-broken states: topological defects of one phase carry the charge of the other and proliferate at the transition. The $\theta$-term implies that a domain wall of the Z$_2$ KVBS order parameter harbors a spin-$1/2$ Heisenberg chain, as described by a $1+1$ dimensional SO(3) non-linear sigma model with $\theta$-term at $\theta = \pi$. Using pinning fields to stabilize the domain wall, we show that our auxiliary-field quantum Monte Carlo simulations indeed support the emergence of a spin-$1/2$ chain at the Z$_2$ topological defect. This concept can be generalized to higher dimensions where $2+1$ dimensional SO(4) or SO(5) theories with topological terms are realized at a domain wall., Comment: 5 pages, 3 figures, supplemental material

A key notion in heavy-fermion systems is the entanglement between conduction electrons and localized spin degrees of freedom. To study these systems from this point of view, we compute the mutual information in a ferromagnetic and antiferromagnetic Kondo lattice model in the presence of geometrical frustration. Here the interplay between the Kondo effect, the Ruderman-Kittel-Kasuya-Yosida interaction, and geometrical frustration leads to partial Kondo screened, conventional Kondo insulating, and antiferromagnetic phases. In each of these states the mutual information follows an area law, the coefficient of which shows sharp crossovers (on our finite lattices) across phase transitions. Deep in the respective phases, the area law coefficient can be understood in terms of simple direct product wave functions thereby yielding an accurate measure of the entanglement in each phase. The above-mentioned results stem from approximation-free auxiliary field quantum Monte Carlo simulations., Comment: 8 pages, 8 figures including Supplemental Material; v2: corrected typo; v3: published version

The discovery that spin-orbit coupling can generate a new state of matter in the form of quantum spin-Hall (QSH) insulators has brought topology to the forefront of condensed matter physics. While QSH states from spin-orbit coupling can be fully understood in terms of band theory, fascinating many-body effects are expected if the state instead results from interaction-generated symmetry breaking. In particular, topological defects of the corresponding order parameter provide a route to exotic quantum phase transitions. Here, we introduce a model in which the condensation of skyrmion defects in an interaction-generated QSH insulator produces a superconducting (SC) phase. Because vortex excitations of the latter carry a spin-$1/2$ degree of freedom numbers, the SC order may be understood as emerging from a gapless spin liquid normal state. The QSH-SC transition is an example of a deconfined quantum critical point (DQCP), for which we provide an improved model with only a single length scale that is accessible to large-scale quantum Monte Carlo simulations., Comment: 14 pages, 11 figures, 3 tables

The absence of negative sign problem in quantum Monte Carlo simulations of spin and fermion systems has different origins. World-line based algorithms for spins require positivity of matrix elements whereas auxiliary field approaches for fermions depend on symmetries such as particle-hole. For negative-sign-free spin and fermionic systems, we show that one can formulate a negative-sign-free auxiliary field quantum Monte Carlo algorithm that allows Kondo coupling of fermions with the spins. Using this general approach, we study a half-filled Kondo lattice model on the Honeycomb lattice with geometric frustration. In addition to the conventional Kondo insulator and anti-ferromagnetically ordered phases, we find a partial Kondo screened state where spins are selectively screened so as to alleviate frustration, and the lattice rotation symmetry is broken nematically., Comment: 6 pages, 5 figures, supplemental material

We consider a model of Dirac fermions in $2+1$ dimensions with dynamically generated, anticommuting SO(3) N\'eel and Z$_2$ Kekul\'e mass terms that permits sign-free quantum Monte Carlo simulations. The phase diagram is obtained from finite-size scaling and includes a direct and continuous transition between the N\'eel and Kekul\'e phases. The fermions remain gapped across the transition, and our data support an emergent SO(4) symmetry unifying the two order parameters. While the bare symmetries of our model do not allow for spinon-carrying Z$_3$ vortices in the Kekul\'e mass, the emergent SO(4) invariance permits an interpretation of the transition in terms of deconfined quantum criticality. The phase diagram also features a tricritical point at which N\'eel, Kekul\'e, and semimetallic phases meet. The present, sign-free approach can be generalized to a variety of other mass terms and thereby provides a new framework to study exotic critical phenomena., Comment: 5 pages, 5 figures, to appear in Phys. Rev. Lett

We numerically study optical conductivity $\sigma (\omega )$ near the "antiferromagnetic" phase transition in the square-lattice Hubbard model at half filling. We use a cluster dynamical mean field theory and calculate conductivity including vertex corrections, and to this end, we have reformulated the vertex corrections in the antiferromagnetic phase. We find that the vertex corrections change various important details in temperature- and $\omega$-dependencies of conductivity in the square lattice, and this contrasts sharply the case of the Mott transition in the frustrated triangular lattice. Generally, the vertex corrections enhance variations in the $\omega$-dependence, and sharpen the Drude peak and a high-$\omega$ incoherent peak in the paramagnetic phase. They also enhance the dip in $\sigma (\omega )$ at $\omega$=0 in the antiferromagnetic phase. Therefore, the dc conductivity is enhanced in the paramagnetic phase and suppressed in the antiferromagnetic phase, but this change occurs slightly below the transition temperature. We also find a temperature region above the transition temperature in which the dc conductivity shows an insulating behavior but $\sigma (\omega )$ retains the Drude peak, and this region is stabilized by the vertex corrections. We also investigate which fluctuations are important in the vertex corrections and analyze momentum dependence of the vertex function in detail., Comment: 12 pages, 10 figures

We employ the multi-orbital dynamical mean-field theory to examine the ground state of a three-orbital Hubbard model with a relativistic spin-orbit coupling (SOC) at four electrons per site. We demonstrate that the interplay between the strong electron correlations and the SOC induces a Van Vleck-type nonmagnetic insulator and its magnetic exciton condensation. We also find in the moderate electron correlation regime that the SOC induces another type of a nonmagnetic excitonic insulator, in addition to a relativistic band insulator. The characteristic features among these insulators are manifested in the momentum resolved single-particle excitations, thus accessible by angle-resolved photoemission spectroscopy experiments., Comment: 5 pages, 3 figures

On the basis of the multi-orbital dynamical mean field theory, a three-orbital Hubbard model with a relativistic spin-orbit coupling (SOC) is studied at five electrons per site. The numerical calculations are performed by employing the continuous-time quantum Monte Carlo (CTQMC) method based on the strong coupling expansion. We find that appropriately choosing bases, i.e., the maximally spin-orbit-entangled bases, drastically improve the sign problem in the CTQMC calculations, which enables us to treat exactly the full Hund's coupling and pair hopping terms. This improvement is also essential to reach at low temperatures for a large SOC region where the SOC most significantly affects the electronic structure. We show that a metal-insulator transition is induced by the SOC for fixed Coulomb interactions. The insulating state for smaller Coulomb interactions is antiferromagnetically ordered with the local effective total angular momentum $j=1/2$, in which the $j=1/2$ based band is essentially half-filled while the $j=3/2$ based bands are completely occupied. More interestingly, for larger Coulomb interactions, we find that an excitonic insulating state emerges, where the condensation of an electron-hole pair in the $j=1/2$ and $j=3/2$ based bands occurs. The origin of the excitonic insulator as well as the experimental implication is discussed., Comment: 5 pages, 4 figures, accepted for publication in Phys. Rev. B

We study the dynamics of doublon in the half-filled Hubbard model on the triangular lattice by using the cellular dynamical mean field theory. Investigating the nearest-neighbor dynamical correlations, we demonstrate that a nearest-neighbor doublon-holon pair shows a strong attraction, in particular in the insulating phase. We also calculate the on-site dynamical correlation of doublon and find that the life time of doublon is longer in the metallic phase than in the insulating phase. In the long-time region, the metallic phase has persistent fluctuations in various nearest-neighbor configurations, while the fluctuations are vanishingly small in the insulating phase. Obtained results indicate clear differences of dynamics of doublon between in the metallic and in the insulating phases., Comment: 4 pages, 3 figures

We study electric transport near the Mott metal-insulator transition in a triangular-lattice Hubbard model at half filling. We calculate optical conductivity $\sigma(\omega)$ based on a cellular dynamical mean field theory including vertex corrections inside the cluster. Near the Mott critical end point, a Drude analysis in the metallic region suggests that the change in the Drude weight is important rather than that in the transport scattering rate for the Mott transition. In the insulating region, there emerges an "ingap" peak in $\sigma(\omega)$ at low $\omega$ near the Mott transition, and this smoothly connects to the Drude peak in the metallic region with decreasing Coulomb repulsion. We find that the weight of these peaks exhibits a power-law behavior upon controlling Coulomb repulsion at the critical temperature. The obtained critical exponent suggests that conductivity does not correspond to magnetization or energy density of the Ising universality class in contrast to several previous works., Comment: 14 pages, 17 figures, accepted for publication in Physical Review B

We study electric transport near the Mott metal-insulator transition. Optical conductivity of the half-filled Hubbard model on a triangular lattice is calculated based on a cellular dynamical mean field theory including vertex corrections inside the cluster. By investigating the spectrum at low frequencies, we find that a Drude peak on the metallic side smoothly connects to an "ingap" peak on the insulating side. The optical weight of these peaks exhibits a critical behavior with power-law near the Mott critical end point, $|D-D^*|\propto|U-U^*|^{1/\delta}$. We find that the critical exponent $1/\delta$ differs from the exponents in the thermodynamics., Comment: 4 pages, 4 figures