Effects of disorder on lattice GinzburgLandau model of
wave superconductors and superfluids
Abstract
We study the effects of quenched disorder on the twodimensional wave superconductors (SC’s) at zero temperature by MonteCarlo simulations. The model is defined on the threedimesional (3D) lattice and the SC pair field is put on each spatial link as motivated in the resonatingvalencebond theory of the high SC’s. For the nonrandom case, the model exhibits a secondorder phase transition to a SC state as density of charge carriers is increased. It belongs to the universality class different from that of the 3D XY model. Quenched disorders (impurities) are introduced both in the hopping amplitude and the plaquette term of pair fields. Then the secondorder transition disappears at a critical concentration of quenched disorder, . Implication of the results to cold atomic systems in optical lattices is also discussed.
pacs:
74.81.g, 11.15.Ha, 74.25.DwEffects of disorder on phase structures and phase transitions have been studied for various systems. In particular, the high materials are microscopically highly nonuniform and it is suggested that there exists a spinglass like phase near the phase transition point of superconductivity (SC) at low temperatures ()sg (). Furthermore, existence of a Bose glass was recently suggested in the Mottinsulating phase of cold atomic systems in random potentialsBoseglass (). In the present paper, being motivated in part by these observations, we shall study effects of quenched disorders on SC’s by using a lattice GinzburgLandau (GL) model of unconventional wave SC’s. The model in its pure case was introduced as the GL theory of the resonatingvalencebond (RVB) field for the tJ modelnm (), and the case of including interaction with the electromagnetic (EM) field has been investigated recentlysmi (). We expect that this model also describes superfluid phase of fermionic atoms in cold atomic systems in optical lattices. There, the effects of disorders can be investigated well under control.
We are interested in quantum SC phase transition of the twodimensional (2D) model at . The model in pathintegral representation is described in terms of the following RVBtype Cooper pair field :
(1) 
where is the site of the 3D cubic lattice of size with periodic boundary condition and denotes the spatial direction and also the unit vector in th direction. is the electron annihilation operator at site and spin . In the wave SC, the Cooper pair amplitudes in and have the opposite signatures as .
Below we shall neglect the effects of the EM field and focus on the effects of quenched disorders. The action of the clean system is then given as follows:
(2) 
where we consider the London limit and set a U(1) variable, . The overall factor plays a role of and controls quantum fluctuations. When we fix , can be taken as an increasing function of the carrier concentration . (In the tJ model is the hole density.) The terms controls fluctuation of flux of ’s around each spatial plaquette. The and terms represent the spatial hopping of , whereas the term describes the hopping in the imaginarytime (th) direction. The partition function is given by . We consider the parameter region to expect , although because of the change of variables . The action is related to the action of the U(1) Higgs gauge theory that is obtained from Eq.(2) by the replacement , where is the U(1) Higgs field. is invariant under timeindependent local gauge transformation where is an independent function. Actually, is viewed as the gaugefixed version of in the unitary gauge .
Queched disorder is introduced in the system (2) by replacing the coefficinets as spatialplaquette dependent ones. First, we consider the 2D spatial plane with fixed , say . Among plaquettes in the plane we choose plaquettes randomly as ones at which impurities reside. We call it a sample. We consider that the configuration of “wrong” plaquettes is independent because the location of impurities are fixed along the imaginary time. Then we reverse the values of for interaction terms contained in these plaquettesdisorder (). Thus the new plaquettedependent coefficients are given by
(3) 
for each spatial plaquette. Please note . Then the partition function and the free energy per site of one sample are given by
(4) 
where is obtained from Eq.(2) by replacing by . To obtain an ensemble average of an observable in the disordered system, we first prepare samples and calculate the quantummechanical average for the th sample (. Then we average it over samples,
(5) 
For the MC simulations, we used the standard Metropolis algorithmma (). The typical statistics used was sweeps per block and the average and MC errors were estimated over 10 blocks for each sample. Then we take quenched averages over samples. We estimated standard deviation of physical quantities like “specific heat” over samples (we call it sample error) as a function of , which becomes stable for .
Let us first consider the nonrandom case, . We studied the phase structure by calculating the “internal energy” per site and the “specific heat” per site . Typical behavior of is shown in Fig.1, which indicates a secondorder phase transition at . This transition has been predicted in Ref.nm () and also observed in the presence of the EM gauge interactionssmi (). In order to verify that it is a transition to a SC phase, we measured the correlation function of , the order parameter field of SC,
(6) 
We present in Fig.2. It is obvious that there exists SC longrange order (LRO) for , whereas no LRO for lro (). Therefore the observed transition is nothing but the SC phase transition.
The critical exponents for were estimated by the finitesize scaling analysis for . We obtained and the critical coupling . When , the system (2) reduces to a set of decoupled 2D XY spin models [ plays the role of an XY spin], each of which has the KosterlitzThouless transition. The and terms couple these 2D XY spins. From the above values of critical exponents, we judge that the present phase transition does not belong to the universality class of the 3D XY model having exponent ().
Let us next turn to the random case, . At first, we consider the plane where , and search for the location of the peak of . In Fig.3, we present the peak location of for given disorder concentration For , the peak line expresses the secondorder transition as we saw in Figs.1 and 2. We see that the region of the normal (non SC) state increases as increases, although we need to check whether the location of this peak expresses genuine SC transition for . Below we examine it
by focusing on the specific case with the varying parameter and calculate , etc. Density of quenched disorder that we studied is and .
We first present the result of in Fig.4. The signal of the secondoder phase transition at is getting weaker as is increased, and also the location of the peak moves to larger .
Next, in Fig.5, we present the systemsize dependence (SSD) of for and . We also calculated the derivative of , , to identify the order of the phase transition. Results are shown in Fig.6. From these results, we judge that there exists a secondorder transition for whereas it changes to a crossover for . The case of seems to have no SSD in but have certain SSD for . Therefore, the transition in the case might be of third order.
To study if the SC state persists even for , we measured the SC correlation averaged over samples with randomly generated . In Fig.7 we present the results for and . It supports that for there exists the SC LRO for whereas for no LRO for any values of . This observation leads to the conclusion that the SC phase disappears for , i.e., there is a multicritical point near in the phase diagram.
It is interesing to ask what kind of phase is realized in the region and , where is the crossover coupling determined by the peak location of the specific heat. Probably, in the case of the doped AF magnets with strong inhomogeneity, this phase may simply correspond to dirty “normal” metallic state. We think that the present model given by Eq.(2) also describes the superfluid wave RVB state of fermionic atoms in 2D optical lattice, which was recently proposed by sereval authorsdwave (). Random disorders can be introduced into the systems by a laser speckle or by an incommensurate bichromatic potentail. In that experimental setup, there is an interesting possibility that a Bose glass phase, which is an analogue of the spin glass phase, is realized there. For ultracold strongly interacting Rb bosons, a Bose glass phase is suggested by experimentsBoseglass (). In the experiments of cold atom systems in optical lattices, the Bose glass phase has no longrange coherence but excitations are gapless. In Fig.8, an expected phase diagram of the present lattice model of dirty wave SC and superfluidity (SF) is shown. We think that existence of the Bose glass phase is examined theoretically by the standard analytical and numerical methods applied for spin glasssg (); z2 ().
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 (12) It is interesting that the study of 3D Ising model with random couplings by M. Hasenbusch, F. P. Toldin, A. Pelissetto, E. Vicari, arXiv:0810.0685, also shows the similar phase diagram as Fig.8 with a glass phase. L. Dang, M. Boninsegni, and L. Polletar, arXiv:0812.3417, also discuss the disorderinduced glassy SF state.