Quantum Criticality and Incipient Phase Separation in the Thermodynamic Properties of the Hubbard Model
D. Galanakis, E. Khatami, K. Mikelsons, A. Macridin, J. Moreno, D. A. Browne, M. Jarrell
aa r X i v : . [ c ond - m a t . s t r- e l ] S e p Quantum Criticality and Incipient PhaseSeparation in the ThermodynamicProperties of the Hubbard Model
By D. Galanakis , E. Khatami , K. Mikelsons , A. Macridin , J.Moreno , D. A. Browne and M. Jarrell Department of Physics and Astronomy, Louisiana State University, BatonRouge, Louisiana, 70803, USA Department of Physics, Georgetown University, Washington, District ofColumbia, 20057, USA Fermilab, P. O. Box 500, Batavia, Illinois, 60510, USA
Transport measurements on the cuprates suggest the presence of a quantum criti-cal point hiding underneath the superconducting dome near optimal hole doping.We provide numerical evidence in support of this scenario via a dynamical clus-ter quantum Monte Carlo study of the extended two-dimensional Hubbard model.Single particle quantities, such as the spectral function, the quasiparticle weightand the entropy, display a crossover between two distinct ground states: a Fermiliquid at low filling and a non-Fermi liquid with a pseudogap at high filling. Bothstates are found to cross over to a marginal Fermi-liquid state at higher tempera-tures. For finite next-nearest-neighbor hopping t ′ we find a classical critical point attemperature T c . This classical critical point is found to be associated with a phaseseparation transition between a compressible Mott gas and an incompressible Mottliquid corresponding to the Fermi liquid and the pseudogap state, respectively.Since the critical temperature T c extrapolates to zero as t ′ vanishes, we concludethat a quantum critical point connects the Fermi-liquid to the pseudogap region,and that the marginal-Fermi-liquid behavior in its vicinity is the analogous of thesupercritical region in the liquid-gas transition. Keywords: Quantum criticality, DCA, Cluster methods
1. Introduction (a) Relevance of quantum criticality in the cuprates
The unusually high superconducting transition temperature of the hole dopedcuprates (Bednorz & Müller, 1986) remains an unsolved puzzle, despite more thantwo decades of intense theoretical and experimental research. Pairing, which hasa d − wave symmetry and short coherence length, but too high of a T c to be ac-counted by BCS (Bardeen et al. , 1957), is not the only unconventional property ofthese materials. Their phase diagram, shown in Fig. 1, is a landscape of exotic statesof matter. Undoped cuprates are Mott insulators with antiferromagnetic long-rangeorder (Néel, 1949). Antiferromagnetism collapses upon small doping and it is re-placed by a pseudogap state characterized by a suppression of spectral weight along Article submitted to Royal Society
TEX Paper
Galanakis, Khatami, Mikelsons, Macridin, Moreno, Browne, Jarrell the antinodal direction. Further doping turns the system into a conventional Fermi-liquid metal. Between the Fermi-liquid and the pseudogap region lies a strange metalphase with linear- T resistivity. The superconducting dome emerges in the cross-overbetween the pseudogap and the Fermi-liquid regions at lower temperatures.Strong electronic correlations are the cause of the rich phase diagram of cupratesuperconductors (Phillips, 2010). The same strong correlations render traditionaltheoretical approaches, such as perturbation theory and Fermi-liquid theory, in-applicable. Some recent conceptual progress has been achieved by associating theoptimal T c with a quantum critical point (QCP), lying underneath the supercon-ducting dome and connecting the pseudogap and the Fermi-liquid regions (Broun,2008; Sachdev, 2010). Unlike a classical critical point, a QCP affects the behaviorof the system in a wide range of temperatures and might explain the emergence ofa linear- T resistivity up to room temperature.Experimental evidence for a QCP comes from transport (van der Marel et al. ,2003; Daou et al. , 2009; Balakirev et al. , 2009) and thermodynamic measurements(Bernhard et al. , 2001). Angle-resolved photo-emission spectroscopy (ARPES) (Shen et al. ,2005; Platé et al. , 2005) and quantum oscillation measurements (Doiron-Leyraud et al. ,2007) show that in the pseudogap region, the Fermi surface consists of small pocketswhich have different topology than the large Fermi surface present in the Fermi liq-uid. It is reasonable to assume that those two states are orthogonal to one anotherand are connected through a transition. Additional evidence in support of quantumcriticality comes from measurements of the Kerr signal in YBCO by Jing Xia etal. (Xia et al. , 2008). They find that at the pseudogap crossover temperature, T ∗ anon-zero Kerr signal develops sharply and persists even inside the superconductingdome. This is consistent with earlier neutron scattering measurements by Fauqué et al. (Fauqué et al. , 2006), which show the development of magnetic order in thepseudogap phase.In this manuscript we review numerical evidence of quantum criticality in theHubbard model, the de-facto model for the cuprates, that appeared in earlier pub-lications. In those cited works, the Hubbard model is solved using the dynamicalcluster approximation (DCA) in conjunction with several quantum Monte Carlo(QMC) cluster solvers. In all calculations relevant for the phase diagram we neglectthe superconducting transition. The interplay between the QCP and superconduc-tivity will be discussed in a future publication. (Yang et al. , 2010) In this reviewwe focus on the thermodynamic quantities, such as the entropy and the chemicalpotential, and also on single-particle quantities, such as the spectral weight andthe quasiparticle weight. The thermodynamic properties give unbiased evidence ofquantum criticality, whereas single-particle properties may be used to gain moredetailed insight on the ground state. Both set of quantities rely on the evaluationof the self-energy which can be calculated using quantum cluster methods.At a critical interaction-dependent filling, we find that the entropy exhibitsa maximum, the quasiparticle weight displays a crossover from Fermi liquid topseudogap behavior, and the spectral function shows a wide saddle point regioncrossing the chemical potential. This is consistent with the presence of a QCP, sincethe lack of an energy scale results in an enhanced entropy at low temperatures.We also find that by tuning an appropriate control parameter, the next-nearest-neighbor hoping, t ′ , the QCP becomes a classical critical point associated with aphase separation transition. We present our findings in two sections. In section 2, Article submitted to Royal Society uantum Criticality and Incipient Phase Separation... Pseudogap Metal T e m p e r a t u r e Strange Metaldoping T * Superconductivity A n t i f e rr o m a g n e t QCP T X Figure 1. The phase diagram of thecuprates. As a function of temperature anddoping, the cuprates display antiferromag-netic order at low doping, a non-Fermi liq-uid pseudogap region at intermediate dop-ing and a metallic region at higher doping.Around optimal doping, superconductivitydevelops, and above the superconductingdome, a strange metal with non-Fermi liq-uid properties appears. T ∗ separates thepseudogap from the marginal Fermi-liq-uid phase. T X is the crossover tempera-ture between the Fermi and the marginalFermi-liquid regions. A quantum criticalpoint hides underneath the superconduct-ing dome near optimal hole doping. we discuss the single-particle spectra and the thermodynamics properties of the t ′ = 0 Hubbard model. In section 3, we discuss the phase separation in the t ′ > Hubbard model. (b) Hubbard Model
Short after the discovery of high- T c superconductors, Anderson (Anderson,1987) suggested that the Hubbard model captures the basic properties of the hightemperature superconductors and Zhang and Rice (Zhang & Rice, 1988) demon-strated that only a single band is needed. The single-band Hubbard model is rep-resented by the Hamiltonian: H = − t X h i,j i ,σ h c † iσ c jσ + H.c. i + U X i n i ↓ n i ↑ , (1.1)where c † iσ ( c iσ ) is the creation (annihilation) operator of an electron at site i and spin σ , n iσ is the corresponding number operator, t is the hopping parameter betweennearest-neighbor sites, and U the on-site Coulomb repulsion. Despite its apparentsimplicity, the Hubbard model is notoriously difficult to solve. No analytical so-lutions exist except in one dimension (Lieb & Wu, 1968; Frahm & Korepin, 1990;Kawakami & Yang, 1990). However, tremendous theoretical and computational ef-forts have resulted in approximation schemes that provide access to the physics ofthis model in higher dimensions. In this manuscript we also discuss results for thegeneralized Hubbard model which includes hopping between next-nearest neighborwith amplitude t ′ : H = − t X h i,j i ,σ h c † iσ c jσ + H.c. i − t ′ X hh i,l ii h c † iσ c lσ + H.c. i + U X i n i ↓ n i ↑ . (1.2)Important progress in our understanding of strongly correlated models has beenachieved by the development of finite size methods, including exact diagonalizationand QMC. The latter works well in the simulation of bosonic systems where creationand annihilation operators commute. However, due to the minus sign problem asso-ciated with the anticommutation relations of fermionic operators, QMC is limitedto small lattice sizes and consequently give questionable predictions for correlatedelectronic systems in the thermodynamic limit. Article submitted to Royal Society
Galanakis, Khatami, Mikelsons, Macridin, Moreno, Browne, Jarrell
Another successful approach is the dynamical mean-field approximation (DMFA)which treats the local dynamical correlations explicitly and non-local (inter-site)correlations in a mean-field approximation (Georges et al. , 1996; Metzner & Vollhardt,1989; Müller-Hartmann, 1989 a , b ). This technique becomes exact in the limit of in-finite dimensions (Georges & Kotliar, 1992; Jarrell, 1992). However, when appliedto finite dimensions, the DMFA fails to describe the renormalization effects dueto momentum-dependent modes and the transitions to phases with non-local orderparameters. Thus, DMFA misses physical phenomena that are abundant in stronglycorrelated systems, such as the development of spin or charge density wave phases,localization in the presence of disorder, spin-liquid physics, unconventional super-conductivity, etc.The limitations of the DMFA are addressed by cluster mean-field theories. Thosefall into two categories (Maier et al. , 2005): the cluster dynamical mean field the-ory (CDMFT) (Kotliar et al. , 2001), which is formulated in real space, and theDCA (Hettler et al. , 1998) which is formulated in momentum space. In both casesthe system is viewed as a cluster embedded in an effective medium. The formaldifference between DCA and CDMFT is that in real space, the DCA cluster sat-isfies periodic boundary conditions whereas the CDMFT cluster is open. The twomethods should give the same results for large enough clusters. In this work wepresent DCA (Hettler et al. , 1998, 2000) results.DCA treats short-ranged correlations explicitly, while longer ranged ones areapproximated by the mean field. By increasing the cluster size, the length-scaleof the explicitly treated correlations can be gradually increased while the calcula-tion remains in the thermodynamic limit. In momentum space, the DCA can easilybe conceptualized as the approximation in which the self-energy calculated by thecoarse grained green function. Quantum Monte Carlo based solvers such as Hirsch-Fye (HFQMC) (Hirsch & Fye, 1986), continuous-time (CTQMC) (Rubtsov et al. ,2005) and determinantal quantum Monte Carlo (DQMC) (Blankenbecler et al. ,1981) are used to solve the cluster problem. QMC methods are often formulated inimaginary time and an analytic continuation to real time is necessary to evaluatephysical quantities. Fortunately, powerful techniques such as the maximum entropymethod (MEM) (Gubernatis et al. , 1991; Jarrell & Gubernatis, 1996) are able tosuccessfully select the most likely solution.Even though quantum cluster schemes have provided a tremendous breakthroughin our understanding of the Hubbard model, they are also subject to limitations.Quantum Monte Carlo solvers suffer from the sign problem, which scales exponen-tially with inverse temperature, interaction strength and cluster size. This limits theapplication of the method to relatively small cluster sizes, higher temperatures andintermediate interactions. The limitation in the cluster size is particularly problem-atic close to a phase transition where the correlation length diverges. The coarse-graining also limits the momentum resolution, which for typical cluster sizes is toosmall to capture detail features of the spectra, such as van Hove singularities. Fora Fermi liquid, this is not a limitation since the physics is dominated by the lowfrequencies in which the self-energy is momentum independent. However, intrinsi-cally anisotropic states, such as the pseudogap, or possibly the quantum criticalregion, can be captured only approximately. Finally, MEM uses Bayesian statisticsto find the most likely spectra for the QMC data, subject to sum rules, such as Article submitted to Royal Society uantum Criticality and Incipient Phase Separation...
2. From Fermi Liquid to Pseudogap
A great advantage of the DCA is its ability to evaluate the self-energy as a func-tion of momentum k and Matsubara frequency iω n , Σ( k , iω n ) . From the self-energyvarious single-particle quantities, such as the spectral function, A ( k , ω ) , the quasi-particle weight, Z k , and the energy can be derived. All those quantities provideinsight about the ground state of the system. In this section we will show howthe transition from the Fermi-liquid to the pseudogap state is reflected in suchsingle-particle quantities. (a) Spectral Function The single-particle spectral function shows a clear evolution from a Fermi-liquidto a pseudogap state as the filling increases towards half filling. Fig. 2 displays adensity plot of the spectral function, A ( k , ω ) = − π ℑ G ( k , ω ) , which is extracted byanalytically continuing the imaginary time Green function. At low filling, n < . ,the spectral function exhibits a typical Fermi-liquid form. A notable characteristicis the presence of a wide saddle point region, reminiscent of a van Hove singular-ity, (Radtke & Norman, 1994) along the antinodal direction. Around the criticalfilling of n = 0 . this saddle point feature crosses the chemical potential. Thiscrossing results in a sharp peak in the density of states (Vidhyadhiraja et al. , 2009),which displays low-energy particle-hole symmetry (Chakraborty et al. , 2008). Weare currently exploring the influence of the van Hove singularity on the supercon-ducting transition. (Pathak et al. ) At higher filling, n > . , the spectral weightcollapses along the antinodal direction and a pseudogap opens. The Fermi surfaceobtained by extremizing |∇ n k | shows a similar evolution (see lower panels in Fig. 2).The Fermi-liquid region consists of a large hole pocket, which extends and touchesthe edges of the Brillouin zone (0 , ± π ) , ( ± π, at n = 0 . . In the pseudogap regionthe Fermi surface consists of four Fermi arcs centered around the nodal points, sim-ilar to the ones obtained from ARPES. These results clearly demonstrate that theDCA can capture qualitatively the evolution of the ground state from a Fermi-liquidto a pseudogap phase. (b) Quasiparticle Weight Whereas the spectral function gives a qualitative understanding of the groundstate, it relies on the analytic continuation of numerical data. Since extractingquantitative information from analytically continued data is difficult, a more robustway is to rely on imaginary time quantities, such as the quasiparticle weight Z ( k ) .Since the quasiparticle weight is finite across a Fermi surface, but vanishes if thespectrum is incoherent, it will allows to clearly distinguish between a Fermi liquidand a pseudogap state. The quasiparticle weight can be directly obtained from theMatsubara frequency self-energy as Z ( k ) = (cid:18) − ℑ Σ ( k , iω ) ω (cid:19) − , where ω = πT Article submitted to Royal Society
Galanakis, Khatami, Mikelsons, Macridin, Moreno, Browne, Jarrell ω G M X G−6−4−20246 G M X G G M X Gn=0.75 n=0.85 n=0.95GXM
Figure 2. Upper panels: Density plots of the spectral function A ( k , ω ) for the Fermiliquid (left), marginal Fermi liquid (middle) and pseudogap region (right) for filling n = 0 . , . and . , respectively. (The dashed feature seen in the regions of steep-est dispersion, especially for n = 0 . , is a plotting artifact). The momentum is along thepath G (0 , → M ( π, π ) → X ( π, → G (0 , . A wide saddle point region between X andG sits above the chemical potential in the Fermi-liquid region and crosses it around thecritical filling ( n = 0 . ). In the pseudogap region this features sits below the chemicalpotential leaving a gap along the antinodal direction behind it. Note that the fact that thedispersion looks discontinuous along G (0 , → M ( π, π ) in the left and middle panels isan artifact of our interpolation algorithm. Lower panels: Fermi surface as extracted from |∇ n k | in the Fermi liquid (left), marginal Fermi liquid (middle) and pseudogap (right) re-gion showing the development of the pseudogap in the antinodal direction. The Coulombrepulsion is U = 6 t , the temperature T = 0 . t , and the cluster size N c = 16 . The energyunit is t . is the lowest fermionic Matsubara frequency. At the limit T → and for a well-behaved self-energy, Z ( k ) converges to the quasiparticle weight, Z ( k ) . Fig. 3 (a)displays Z AN = Z ( ω = πT, k k (0 , → (0 , π )) , the Matsubara quasiparticleweight along the antinodal momentum direction for U = 6 t and a cluster of size N c = 16 (Vidhyadhiraja et al. , 2009). The momentum k at the Fermi surface isdetermined by maximizing |∇ n ( k ) | . Z AN exhibits two distinguishable behaviors:for n > n c = 0 . the quasiparticle weight vanishes, whereas it approaches a finitevalue for n < n c . The n > n c region corresponds to the pseudogap state in which Article submitted to Royal Society uantum Criticality and Incipient Phase Separation... T Z AN n=1.00n=0.95n=0.90n=0.85n=0.80n=0.75n=0.70X fitMFL fitFL fit T Z N / Z AN T x T * a) Filling T T * T X MFLNFL FL b) Figure 3. a) The antinodal quasiparticle fraction Z AN as a function of temperature fordifferent values of filling, U = 6 t and cluster size N c = 16 (the unit of energy is t ). Theonset of the pseudogap region is determined by the vanishing of the antinodal spectralweight at zero temperature. The dashed and solid lines represent fits of the low tem-perature ( T < . ) data to marginal Fermi liquid (red solid curves), Fermi liquid (blacksolid curves) and crossover forms (dashed black curves), respectively. The arrows showthe corresponding crossover temperatures T X and T ∗ . The value of T ∗ presented here isobtained from the spin susceptibility as explained in (Vidhyadhiraja et al. , 2009), but isconsistent with the one extracted from the from the fitting forms. The ratio Z N /Z AN of the quasiparticle weight in the nodal ( ( π, π ) ) and antinodal ( (0 , π ) ) directions (inset)diverges as the pseudogap develops in accordance with Fig. 2. b) The crossover tempera-tures T X and T ∗ as a function of filling as extracted from the temperature dependence of Z AN (Vidhyadhiraja et al. , 2009) for the same parameters. the spectral weight collapses along the antinodal direction, while the n < n c regionbehaves as a Fermi liquid.The temperature dependence of Z AN (Fig. 3 (a)) not only provides informationabout the ground state but also allows the extraction of relevant energy scales.By comparing the numerical results with analytical expressions derived from par-ticular phenomenological forms of the self-energy, we obtain T X and T ∗ . At lowfilling, n < n c , the high T dependence of Z AN is best fit by a marginal Fermi-liquidform, whereas for low T the data is best fit by a Fermi liquid. The crossover oc-curs at a temperature T X , which is extracted by fitting with a crossover function,and is accompanied by a change in the sign of the curvature of Z AN . At higherfilling ( n > . ), the high temperature Z AN can also be fit by a marginal Fermiliquid, whereas at low temperatures, it cannot. The crossover temperature T ∗ canbe extracted as the lowest temperature where the marginal Fermi liquid fit lieswithin the statistical error. However a more accurate value can be obtained fromthe bulk spin susceptibility which exhibits a peak at T ∗ and the two values arefound to be consistent (Vidhyadhiraja et al. , 2009). The crossover temperatures T X and T ∗ are shown in Fig. 3 (b). Both of them converge to zero as the fillingapproaches n c = 0 . , which is the same value for which the peak in the density ofstates (Vidhyadhiraja et al. , 2009) crosses the chemical potential. (c) Thermodynamics A different perspective at the transition from a Fermi liquid to the pseudogap
Article submitted to Royal Society
Galanakis, Khatami, Mikelsons, Macridin, Moreno, Browne, Jarrell n S / T a) b) T/t µ /t T/t=4.002.001.000.500.3330.2000.129T/t=0.077 T * T FL n=0.950.900.870.850.830.800.75n=0.70 Figure 4. a) The filling dependence of the entropy divided by temperature,
S/T , for varioustemperatures at U = 6 t and N c = 16 . With decreasing temperature a peak develops aroundthe critical filling of n c = 0 . . b) The temperature dependence of the chemical potential µ for different fillings. At the critical filling, n c , µ becomes temperature independent atlow temperatures. state comes from the evaluation of the entropy. We obtain the entropy by integratingthe energy using the formula: S ( β, n ) = S (0 , n ) + βE ( β, n ) − ˆ β E ( β ′ , n ) dβ ′ , (2.1)where β is inverse temperature and S (0 , n ) is the infinite temperature entropy.Equation 2.1 is appropriate for QMC calculations, because the integration reducesthe statistical error. The challenge is to have good enough statistics to control theerror of the surface term, βE ( β, n ) . In Mikelsons et al. (Mikelsons et al. , 2009) largestatistics was possible simply by using large computational resources. The entropydivided by the temperature, shown in Fig. 4 (a), exhibits a maximum at exactly thesame critical filling that was identified before from the spectral function and thequasiparticle weight. In Fig. 4 (b), we show the chemical potential, µ , as a functionof temperature. We note that at the critical filling dµ/dT = 0 , since the entropyand the chemical potential are related by the Maxwell relation: (cid:18) ∂S∂n (cid:19) T,U = − (cid:18) ∂µ∂T (cid:19) U,n . (2.2)Also the temperature dependence of the chemical potential can be used as apractical criterion to identify the location of the critical filling, because evaluat-ing the chemical potential is much less computationally intensive than evaluatingthe entropy. Using this criterion we investigate the important question of the de-pendence of n c on the Coulomb repulsion U . As it is shown in Fig. 5, we find Article submitted to Royal Society uantum Criticality and Incipient Phase Separation... µ / t U=7tU=8tU=9t0.850.90 U=6t0.850.900.850.90n=0.850.90a)
96 7 80.90.850.860.870.880.89 U/t C r i t i c a l F illi n g b) Figure 5. a) The chemical potential as a function of temperature for fillings of n = 0 . and . and for a variety of interaction strengths U for N c = 12 . b) The critical filling,defined by the filling in which ∂µ/∂T = 0 versus U . The critical filling decreases with U monotonically and is projected to reach the atomic limit value of n c = 2 / at U c = 30 t . that increasing U reduces the critical filling and thus enlarges the pseudogap re-gion in the phase diagram. Our results follow the trend proposed in earlier argu-ments (Chakraborty et al. , 2008) according to which the critical filling decreases inorder to reach the atomic limit value of n c = 2 / .In this section we have shown that several single-particle quantities are con-sistent with the presence of a QCP. The qualitative form of the single-particlespectrum shown in Fig. 2 is fundamentally different in the Fermi-liquid and thepseudogap regions, which points to orthogonal ground states. The temperaturedependence of the quasiparticle weight reveals the presence of two crossover tem-peratures T ∗ and T X , which converge to zero at n c as shown in Fig. 3 (b). If thecrossover temperatures T X and T ∗ constitute energy scales that suppress degreesof freedom, their vanishing at n c means that there are no relevant energy scales toquench the entropy and therefore it collapses at a slower rate, which is consistentwith the peak of the entropy observed at n c . The natural next step to investigatequantum criticality is to access the QCP. However the fermion sign problem severelylimits the applicability of quantum Monte Carlo techniques close to a QCP. It ispossible however, as we will discuss in the next section, that by tuning an appro-priate control parameter, the critical point may be lifted to finite temperature andthus studied with QMC.
3. Phase Separation and Quantum Criticality
Experiments suggest that cuprate superconductors are susceptible to charge in-homogeneities, such as stripes or checkerboard modulations (Hinkov et al. , 2004).These inhomogeneous charge patterns have stimulated intense theoretical and ex-perimental research. Here we will consider the possibility that those charge insta-bilities are evidence that the cuprates are close to a phase separation transition,and this proximity may be related to the nature of the QCP.Our findings suggest that the Hubbard model displays a phase diagram similarto the one for the gas-liquid transition with Mott liquid (ML) and Mott Gas (MG)regions. Fig. 6(a) shows a possible phase diagram for the Hubbard model as a
Article submitted to Royal Society Galanakis, Khatami, Mikelsons, Macridin, Moreno, Browne, Jarrell (T C ,μ C ) μn Tn c CP T C (T C ,n C )MLMG Figure 6. a) The schematic phase diagram in the presence of charge separation. This phasediagram describes the transition between two states labeled Mott Liquid (ML) and MottGas (MG) as a function of temperature, T , chemical potential, µ , and filling, n . The redsurface represents the coexistence region, which terminates in a critical point (CP). Aswe go around the critical point the state changes smoothly from ML to MG. Along thefirst order transition line and for a fixed T and µ , the filling has two values. b) Fillingas a function of chemical potential for several temperatures in the vicinity of the chargeseparation critical point. The number next to each curve represents the temperature. Thecoexisting phases are an incompressible Mott liquid at n ≈ and a compressible Mottgas at n ≈ . . The critical temperature is T c = 0 . t . The blue dashed line representsthe surface of metastability which is not accessible within the DCA. The green dotted linerepresents the isothermal of the metastable state inside the phase coexistence region (grayzone). At the critical point the isothermals for T > T c cross. The inset shows the scalingcurve ( n − n c )( T − T c ) − β vs ( µ − µ c )( T − T c ) − βδ in arbitrary units for µ c = 3 t , n c = 0 . , T c = 0 . t . The scaling exponents, β = 0 . ± . and βδ ∼ , are roughly consistent withthe Ising universality class. function of T , | µ | , and n . The red-colored surface is a schematic of the region wherethe Mott liquid and Mott gas states, characterized by different densities, coexistfor T < T c . The critical point is located at temperature T c , filling n c , and chemicalpotential µ c . One can go from one state to the other either smoothly, by avoidingthe phase separation region, or through a first-order transition by crossing it. Righton the phase separation region, the density has two values for a given value of µ and T .Macridin et al. (Macridin et al. , 2006) provided compelling evidence of phaseseparation in the case of the generalized Hubbard model (Eq. (1.2)) with positivenext-near-neighbor hopping t ′ = 0 . t and U = 8 t . Using the DCA in a N c = 8 cluster with HFQMC as the cluster solver, they showed that below a critical tem-perature T c ∼ . t a first order transition occurs, which is identified by a hysteresisin the n versus µ curve for T < T c . As shown in Fig. 6(b) with more precise dataobtained using DQMC as the cluster solver, the hysteresis is between two statesof different filling, the Mott liquid at half filling and the Mott gas at a filling ofabout 0.93 for T = 0 . t . The Mott liquid is incompressible and insulating. Itscompressibility, which is the slope of the filling vs µ curve in the high filling sideof the hysteresis curve, is small and decreases with temperature. Also the densityof states of the ML phase, shown in Fig. 7(a), exhibits a gap as expected for an Article submitted to Royal Society uantum Criticality and Incipient Phase Separation... -3 -2 -1 0 1 2 3 ω N ( ω ) T=0.057tT=0.077t -4 -2 0 2 4 ω a) ML b) MG Figure 7. The density of states of the a) Mott liquid and b) Mott gas states at T = 0 . t (dotted line) and T = 0 . (solid line). The Mott liquid is an incompressible insulatorwith a pseudogap while the Mott gas is weakly compressible with a Fermi liquid peak inthe DOS. insulator. On the other hand, the Mott gas is compressible and metallic; the densityof states is finite at the chemical potential ( µ = ω = 0 ), as displayed in Fig. 7(b).The analogy to the well-known phase diagram of a liquid-gas mixture, such aswater and steam, is useful to understand this phase transition. At low temperatures,there is a region in the pressure-volume phase diagram in which water and steamcoexist for a range of pressures. As the temperature is increased, the region of coex-istence contracts and finally terminates at a critical point where the compressibilitydiverges. In the pressure-temperature phase diagram, this region of coexistence be-comes a line of first order transitions which terminates at a second order point wherethe water and gas become indistinguishable and the compressibility diverges. Sincethe line terminates, it is possible for the system to evolve adiabatically from steamto water without crossing a phase transition line; therefore, the steam and watermust have the same symmetry.In the Mott liquid and Mott gas system the chemical potential µ replaces thepressure and the density n replaces the volume of the water-gas mixture. Becausethe order parameter separating the ML from the MG, the density n , does nothave a continuous symmetry, order may occur at finite temperatures, and the ML-MG transition will most likely be in the Ising or lattice gas universality class.Within this context, one may then understand the hysteresis of Fig. 6(b). The solidlines are isotherms which show how the system evolves with increasing density.At the temperature T = T c , the compressibility diverges at the critical filling.As the temperature is lowered further, there is a region where the ML and MGcoexist. Inside this region the isothermals contain unphysical regions of negativecompressibility (dashed green line in Fig. 6(b)) along with metastable regions ofpositive compressibility. The metastable branch of the isothermal in the vicinityof the ML is a "supercooled" ML, whereas the one in the vicinity of the MG is a"superheated" MG. The translational invariance of DCA along with the stabilizingeffect of the mean-field host enable access to those metastable states. However Article submitted to Royal Society Galanakis, Khatami, Mikelsons, Macridin, Moreno, Browne, Jarrell
Tμ (a)1st 2ndML (n≈1)MG (n<1) T C Tμ (b)QCP ML (n>n c )MG (n
4. Conclusions
The presence of a QCP at finite filling in the cuprate phase diagram is a topic ofactive theoretical and experimental research. Quantum cluster methods are ableto shed some light in this phase diagram. By studying single-particle quantitiesfor t ′ = 0 , such as the spectral function and the entropy, it can be shown that aFermi-liquid region at low filling and the pseudogap region at higher filling havedifferent spectral signatures, and are connected through an intermediate "marginalFermi-liquid" region of maximal entropy. Due to limitations of quantum MonteCarlo, the ground state and quantum criticality are not accessible. We also neglectthe superconducting phase transition. The connection with quantum criticality isestablished by switching on t ′ . For positive t ′ a classical critical point emerges atfinite temperature T c , which increases with t ′ . We note that t ′ is not the only controlparameter that may be able to tune the critical point to finite temperatures, butother parameters, such as phonon coupling, may have the same effect. The phasediagram around the critical point is similar to that of the gas-liquid transition, wherethe incompressible Mott liquid and the compressible Mott gas are the coexistingphases. The strange metal region in this context may be viewed as the supercriticalregion lying in the vicinity of the critical point. Within the scenario we presented,the pseudogap region is not characterized by an order parameter, rather it musthave the same symmetry as the Fermi-liquid and the marginal Fermi-liquid, sincethese regions are connected by an adiabatic path in the T − µ phase diagram. Article submitted to Royal Society uantum Criticality and Incipient Phase Separation...
5. Acknowledgements
We would like to thank R. Gass, S. Kivelson, D. J. Scalapino, A. M. Tremblay,C. Varma, M. Vojt, S. R. White, J. Zaanen and for useful discussion that helpedduring the development of the presented work. This research was supported byNSF DMR-0706379, DOE CMSN DE-FG02-08ER46540, and by the DOE SciDACgrant DE-FC02-06ER25792. This research used resources of the National Centerfor Computational Sciences at Oak Ridge National Laboratory, which is supportedby the Office of Science of the U.S. Department of Energy under Contract No.DE-AC05-00OR22725.
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