No abstract

It is shown that the criticism presented in the Comment by Galanakis et al [1] on the paper by Efetov et al [2] is irrelevant to the bosonization approach.In spite of the far going conclusion that our bosonization scheme cannot help to overcome the negative sign problem, the Comment does not really address the suitability of the method for Monte Carlo (MC) calculations. We wrote in the paper that our mapping of the fermionic model onto a bosonic one was exact having in mind the continuous with respect to (imaginary) time limit. Any quantum MC scheme implies generically a discrete time with extremely strong variation of the HubbardStratonovich (HS) field φ on different slices. In this case, our bosonization scheme is not necessarily exact for a given HS field and we make an approximation. If everything remained exact also in this case, we would not be able to overcome the sign problem because the partition function Z [φ] for a given φ would be the same for both bosonic and fermionic representations and could be negative.In our opinion, the comment of Galanakis et al. is based on a misunderstanding of the definition of Z b . Indeed Z b should not be understood as coming from Eqn (9) but from Eqns(12-13). [We apologize for this confusion, which is the result of writing the paper several times in order to improve the presentation]. Having failed to understand this point, Galanakis et al. re-derive the same steps leading to Z f and jump on the far reaching conclusion that our technique does not solve the MC sign problem. However, the bosonization method starts later with Eqs. (12, 13), which means that Galanakis et al make their conclusions not about the bosonized model but still about the original fermionic one.They fail as well to understand that for finite time slices, our method is not exact but requires an approximation. The crucial approximation is made when writing Eq. (13). According to Eq. (11), the function A r,r ′ (τ ) is expressed in terms of the Green functions at slightly different times τ and τ + δ,where δ → +0. Therefore, deriving the equation for the function A r,r ′ and making no approximations one would have the fields φ r and φ r ′ at slightly different times τ and τ + δ. Nevertheless, we put δ = 0 in Eq. (13). In the continuous limit, (implying subsequent averaging over φ r (τ )), this approximation becomes exact. Therefore our field theory based on the introduction of superfields is exact.However, working with finite slices of time and putting δ = 0 changes the function Z b [φ] for a given configuration of φ. Taking Eq. (13) with the fields φ r (τ ) and φ r ′ (τ ) at coinciding times τ results in a symmetry of the solution A r,r ′ (τ ) under the replacement r ⇆ r ′ . We argue in the paragraph after Eq. (23) that any possible singularity in the integral over u in the exponent should be absent due to this symmetry. This means that the imaginary part in the exponent does not arise and the function Z b [φ] must be positive.In order to make everything well defined one needs a regularization, otherwis...

We study the timely issue of charge order checkerboard patterns observed in a variety of cuprate superconductors. We suggest a minimal model in which strong quantum fluctuations in the vicinity of a single antiferromagnetic quantum critical point generate the complexity seen in the phase diagram of cuprates superconductors and, in particular, the evidenced charge order. The Fermi surface is found to fractionalize into hotspots and antinodal regions, where physically different gaps are formed. In the phase diagram, this is reflected by three transition temperatures for the formation of pseudogap, charge density wave, and superconductivity (or quadrupole density wave if a sufficiently strong magnetic field is applied). The charge density wave is characterized by modulations along the bonds of the CuO lattice with wave vectors connecting points of the Fermi surface in the antinodal regions. These features, previously observed experimentally, are so far unique to the quantum critical point in two spatial dimensions and shed a new light on the interplay between strongly fluctuating critical modes and conduction electrons in high-temperature superconductors.