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The sign problem (SP) is the fundamental limitation to simulations of strongly correlated materials in condensed matter physics, solving quantum chromodynamics at finite baryon density, and computational studies of nuclear matter. As a result, it is part of the reason fields such as ultra-cold atomic physics are so exciting: they can provide quantum emulators of models that could not otherwise be solved, due to the SP. For the same reason, it is also one of the primary motivations behind quantum computation. It is often argued that the SP is not intrinsic to the physics of particular Hamiltonians, since the details of how it onsets, and its eventual occurrence, can be altered by the choice of algorithm or many-particle basis. Despite that, we show that the SP in determinant quantum Monte Carlo (DQMC) is quantitatively linked to quantum critical behavior. We demonstrate this via simulations of a number of fundamental models of condensed matter physics, including the spinful and spinless Hubbard Hamiltonians on a honeycomb lattice and the ionic Hubbard Hamiltonian, all of whose critical properties are relatively well understood. We then propose a reinterpretation of the low average sign for the Hubbard model on the square lattice when away from half-filling, an important open problem in condensed matter physics, in terms of the onset of pseudogap behavior and exotic superconductivity. Our study charts a path for exploiting the average sign in QMC simulations to understand quantum critical behavior, rather than solely as an obstacle that prevents quantum simulations of many-body Hamiltonians at low temperature.
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According to Landau criterion, a phase transition should be first order when cubic terms of order parameters are allowed in its effective Ginzburg-Landau free energy. Recently, it was shown by renormalization group (RG) analysis that continuous transition can happen at putatively first-order $Z_3$ transitions in 2D Dirac semimetals and such non-Landau phase transitions were dubbed fermion-induced quantum critical points (FIQCP) [Li et al., Nature Communications 8, 314 (2017)]. The RG analysis, controlled by the 1/$N$ expansion with $N$ the number of flavors of four-component Dirac fermions, shows that FIQCP occurs for $Ngeq N_c$. Previous QMC simulations of a microscopic model of SU($N$) fermions on the honeycomb lattice showed that FIQCP occurs at the transition between Dirac semimetals and Kekule-VBS for $Ngeq 2$. However, precise value of the lower bound $N_c$ has not been established. Especially, the case of $N=1$ has not been explored by studying microscopic models so far. Here, by introducing a generalized SU($N$) fermion model with $N=1$ (namely spinless fermions on the honeycomb lattice), we perform large-scale sign-problem-free Majorana quantum Monte Carlo simulations and find convincing evidence of FIQCP for $N=1$. Consequently, our results suggest that FIQCP can occur in 2D Dirac semimetals for all positive integers $Ngeq 1$.
We study high frequency response functions, notably the optical conductivity, in the vicinity of quantum critical points (QCPs) by allowing for both detuning from the critical coupling and finite temperature. We consider general dimensions and dynamical exponents. This leads to a unified understanding of sum rules. In systems with emergent Lorentz invariance, powerful methods from conformal field theory allow us to fix the high frequency response in terms of universal coefficients. We test our predictions analytically in the large-N O(N) model and using the gauge-gravity duality, and numerically via Quantum Monte Carlo simulations on a lattice model hosting the interacting superfluid-insulator QCP. In superfluid phases, interacting Goldstone bosons qualitatively change the high frequency optical conductivity, and the corresponding sum rule.
For a system near a quantum critical point (QCP), above its lower critical dimension $d_L$, there is in general a critical line of second order phase transitions that separates the broken symmetry phase at finite temperatures from the disordered phase. The phase transitions along this line are governed by thermal critical exponents that are different from those associated with the quantum critical point. We point out that, if the effective dimension of the QCP, $d_{eff}=d+z$ ($d$ is the Euclidean dimension of the system and $z$ the dynamic quantum critical exponent) is above its upper critical dimension $d_C$, there is an intermingle of classical (thermal) and quantum critical fluctuations near the QCP. This is due to the breakdown of the generalized scaling relation $psi= u z$ between the shift exponent $psi$ of the critical line and the crossover exponent $ u z$, for $d+z>d_C$ by a textit{dangerous irrelevant interaction}. This phenomenon has clear experimental consequences, like the suppression of the amplitude of classical critical fluctuations near the line of finite temperature phase transitions as the critical temperature is reduced approaching the QCP.
We consider the finite-temperature phase diagram of the $S = 1/2$ frustrated Heisenberg bilayer. Although this two-dimensional system may show magnetic order only at zero temperature, we demonstrate the presence of a line of finite-temperature critical points related to the line of first-order transitions between the dimer-singlet and -triplet regimes. We show by high-precision quantum Monte Carlo simulations, which are sign-free in the fully frustrated limit, that this critical point is in the Ising universality class. At zero temperature, the continuous transition between the ordered bilayer and the dimer-singlet state terminates on the first-order line, giving a quantum critical end point, and we use tensor-network calculations to follow the first-order discontinuities in its vicinity.
We address the quantum-critical behavior of a two-dimensional itinerant ferromagnetic systems described by a spin-fermion model in which fermions interact with close to critical bosonic modes. We consider Heisenberg ferromagnets, Ising ferromagnets, and the Ising nematic transition. Mean-field theory close to the quantum critical point predicts a superconducting gap with spin-triplet symmetry for the ferromagnetic systems and a singlet gap for the nematic scenario. Studying fluctuations in this ordered phase using a nonlinear sigma model, we find that these fluctuations are not suppressed by any small parameter. As a result, we find that a superconducting quasi-long-range order is still possible in the Ising-like models but long-range order is destroyed in Heisenberg ferromagnets.
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