Quantum phase transitions and ground-state correlations in BCS-like models

# Quantum phase transitions and ground-state correlations in BCS-like models

Mariusz Adamski, Janusz Jȩdrzejewski and Taras Krokhmalskii
Institute of Theoretical Physics, University of Wrocław,
pl. Maksa Borna 9, 50–204 Wrocław, Poland
Institute for Condensed Matter Physics,
1 Svientsitski Street, 79011 Lviv, Ukraine
###### Abstract

We study ground-state correlation functions in one- and two-dimensional lattice models of interacting spinful fermions – BCS-like models, which exhibit continuous quantum phase transitions. The considered models originate from a two-dimensional model of d-wave superconductivity proposed by Sachdev. Due to the exact diagonalizability of the considered models in any dimensionality, exact phase diagrams, with several kinds of quantum-critical points, are constructed and closed-form analytic expressions for two-point correlation functions are obtained. In one- and two-dimensional cases we provide analytic expressions for the asymptotic behavior of those correlation functions at large distances and in neighborhoods of quantum-critical points. The novelty of our results is that in two-dimensions explicit expressions for direction-dependent correlation lengths in terms of model parameters and the values of direction-dependent universal critical indices , that characterize the divergence of correlation lengths on approaching critical points, are determined. Moreover, specific scaling properties of correlation functions with respect to parameters of underlying Hamiltonians are revealed. Besides enriching the knowledge of properties of lattice fermion systems exhibiting continuous quantum phase transitions, especially in two dimensions, our results open new possibilities of testing unconventional methods of studying quantum phase transitions, as the promising fidelity approach or the entanglement approach, beyond one-dimension and beyond the realm of paradigmatic XY and Ising chains in transverse magnetic fields.

## 1 Introduction

In recent years, quantum phase transitions and quantum critical phenomena constitute a subject of great interest and vigorous studies in condensed matter physics. Both, experimental and theoretical developments point out to the crucial role that quantum phase transitions play in physics of frequently studied high- superconductors, rare-earth magnetic systems, heavy-fermion systems or two-dimensional electrons liquids exhibiting fractional quantum Hall effect [1], [2]. Quantum critical phenomena have been also observed in exotic systems as magnetic quasicrystals [3] and in artificial systems of ultracold atoms in optical lattices [4]. The so called classical, thermal phase transitions originate from thermal fluctuations, a competition of internal energy and entropy, and are mathematically manifested as singularities in temperature and other thermodynamic parameters of various thermodynamic functions, and such characteristics of correlation functions as the correlation length, at nonzero temperatures. In contrast, quantum phase transitions originate from purely quantum fluctuations and are mathematically manifested as singularities in system parameters of the ground-state energy density, which is also the zero-temperature limit of the internal energy density. Naturally, singularities of thermodynamic functions appear only in the thermodynamic limit. The importance of quantum phase transitions for physics and the related wide interest in such transitions stems from the fact that, while a quantum phase transition is exhibited by ground states, hence often termed a zero-temperature phenomenon, its existence in a system exerts a great impact on the behavior of that system also at nonzero temperatures, in some case at unexpectedly high temperatures [5], [6].

Theoretically, quantum phase transitions can be studied in quite complex quantum systems by qualitative and approximate methods, or in relatively simple but exactly solvable models by means of analytic methods and high-accuracy numerical calculations [1]. Naturally, for the purpose of testing and illustrating general or new ideas the second route is most suitable. Traditionally, this route involves studying the eigenvalue problem of a Hamiltonian, the ground state and excitation gaps, determining quantum critical points and symmetries, constructing local-order parameters, calculating two-point correlation functions and their asymptotic behavior at large distances and in vicinities of quantum critical points, with correlation lengths and the universal critical indices that characterize the divergence of correlation lengths on approaching critical points. Carrying out such a programme is a hard task, which has been accomplished only in a few one-dimensional models. Among those models, there are quantum spin chains as the isotropic and anisotropic XY models in an external transverse magnetic field, including their extremely anisotropic version - the Ising model [1]. Only in one dimension those models are equivalent to lattice gases of spinless fermions, which can exactly be diagonalized, and exact results concerning the phase diagram, quantum critical points, correlation functions and dynamics have been obtained (concerning XY model see [7], [8], [9], concerning the Ising model see [10], and for both models [11]). Needless to say that parallel results for a higher-dimensional model are desirable; this is the first motivation of our investigations presented in this paper.

In the last decade, fresh ideas coming from quantum-information science entered the field of quantum phase transitions. One of them is the hypothesis, the so called fidelity approach. It claims that it is possible to locate critical points [12], [13], [14] and to determine the correlation lengths and universal critical indices by studying (typically numerically) scaling properties, with respect to the size of the system and the parameters of the underlying Hamiltonian, of the so called quantum fidelity of two ground states in a vicinity of a critical point [15], [16]. To extract the index , the scaling laws of quantum fidelity, derived by renormalization group arguments [18], [19], are needed. The task of determining scaling properties of the quantum fidelity should be much easier than the task of calculating large-distance behavior of two-point correlation functions. The most comprehensive results concerning the verification of the fidelity approach have been obtained for one-dimensional quantum spin systems in a perpendicular magnetic field [15] (the case of Ising model), [16] (the case of XY model). These results are very promising: except a vicinity of a multicritical point, the fidelity approach works fine. In order to verify the effectiveness of this approach in dimensions higher than one, we need an at least two-dimensional exactly solvable model, whose quantum critical points, correlation lengths and critical indices in their vicinities are known; this is the second motivation of our investigations reported in this paper.

To go beyond the one-dimensional case, we consider lattice fermion models which originate from the two-dimensional model of d-wave superconductivity proposed by Sachdev [17](see also [1]), which are spinful BCS-like models. General, mathematical considerations of some classes of such models, but without specifying hopping intensities or coupling constants, which therefore do not reach such subtleties as quantum critical points or critical behavior of correlation functions, can be found in [20], [21]. For translation-invariant hopping intensities and coupling constants the considered models are exactly diagonalizable in any dimension. Consequently, it is possible to derive analytical formulae for correlation functions of finite systems and in the thermodynamic limit. To limit further the great variety of possible models, we restrict the hopping intensities to nearest neighbors and dimensionality . For , we choose the underlying lattice as a square one and require the hopping intensities to be invariant under rotations by . Similarly, we require that the interactions of our systems do not extend beyond nearest neighbors and for they are either invariant under rotations by (the symmetric case) or change sign after such a rotation (the antisymmetric case). In this way, we end up with a unique model in one dimension and with only two models in two dimensions.

The general plan of the paper is as follows. In section 2 we define the models considered in the paper and give closed-form formulae for two basic two-point correlation functions. In sections and subsections that follow we limit our considerations to one of those correlation functions – an offdiagonal matrix element of the ground-state one-body reduced density operator. For a fixed lattice direction, it depends on the distance and the parameters of the underlying Hamiltonian. We focus on the behavior of that function in three regions of its variables: as the distance grows indefinitely, as parameters of the Hamiltonian approach a quantum critical point, and as besides the distance being sufficiently large the parameters approach a quantum critical point. In particular, we determine the correlation lengths and the critical indices , providing analytical and numerical results, which are in excellent agreement.

For completeness, in section 3 we present results obtained for the unique one-dimensional model. Some of the results of section 3 have already been used in [22] to verify the effectiveness of the fidelity approach in the one-dimensional version of our models, similarly to [15] and [16]. Then, section 4 is devoted to the two-dimensional models, the symmetric and the antisymmetric ones. As compared to most frequently considered one-dimensional models – spin chains, the novel feature of two-dimensional systems is that the correlation functions depend on lattice directions. We present their asymptotic behavior separately for the diagonal direction and for offdiagonal directions. As a result, section 4 splits into four independent subsections, each one referring to a different case specified by the model and lattice direction. To facilitate finding a particular result, the contents of each subsection and the order of presenting the results parallel that of section 3. For more detailed comments concerning the contents of subsections see the last two paragraphs of next section. In section 5 we give a resume of numerous scaling laws derived for the two-point correlation function and its correlation length in all the considered models. Some general observations concerning the correlation length are included in this section. Finally, in section 6, we formulate again our motivations, summarize our results, pointing out those that we consider most important.

## 2 The models, their ground states and ground-state correlation functions

We consider a -dimensional spinful fermion model, given by the Hamiltonian,

 (1)

where , stand for creation and annihilation operators, respectively, of a spin fermion, with spin projection on a chosen axis , in a state localized at site of a -dimensional hypercubic lattice. The edge of the lattice in the direction given by the unit vector , , whose -th component is , consists of equidistant sites, labeled by , for . In all the considerations that refer to finite systems, special boundary conditions, specified below, are chosen. The sums over in (1) amount to the sum over pairs of nearest neighbors, with each pair counted once. The real and positive parameter is the nearest-neighbor hopping intensity, – the chemical potential, – the coupling constant of the gauge-symmetry breaking interaction, and , , stand for direction-dependent, in general complex, dimensionless constants. Naturally, we can express the parameters and in units of , while the lengths of the underlying lattice in units of the lattice constant, preserving the original notation. We emphasize that in distinction to [17], [1], where Hamiltonian (1) was derived, are constants independent of and . We note that Hamiltonian (1) is not gauge invariant unless . It is also not hole-particle invariant unless and , , are real. The latter condition can be assumed to hold without any loss of generality, since Hamiltonian (1) with any complex is unitarily equivalent to that with replaced by .

Imposing, independently in each direction , , periodic or antiperiodic boundary conditions, Hamiltonian (1) can be simplified by passing from the site-localized to the plane-wave basis labeled by suitable wave vectors (quasimomenta) ,

 H=∑k,σεkc†k,σck,σ−J∑k,icoski(Δic†k,↑c†−k,↓+h.c.), (2)

where stands for the dispersion relation of the hopping term,

 εk=∑icoski−μ, (3)

with in the case of periodic boundary condition with an odd , and in the case of antiperiodic boundary condition with an even . Formally, Hamiltonian (2) differs from the well-known BCS Hamiltonian of s-wave superconductivity by the presence of factors in the gauge-symmetry breaking term. Such Hamiltonians can readily be diagonalized by means of the Bogoliubov transformation. The dispersion relation of quasi-particles reads

 Ek+∑k(εk−Ek), (4)

where is the ground-state energy, and , given by

 Ek= ⎷ε2k+∣∣ ∣∣J∑iΔicoski∣∣ ∣∣2, (5)

are the single quasi-particle energies. For our choice of boundary conditions, as long as our system is finite the excitation energies remain strictly positive: for all values of , and this is assumed to hold in the sequel.

The Hamiltonian (1) preserves parity; therefore without any loss of generality we can restrict the state-space to the subspace of even number of particles (electrons). In this subspace, the state of an unspecified (but even) number of electrons, defined by

 |0⟩qp=∏k(uk+vkc†k,↑c†−k,↓)|0⟩, (6)

where is the electron vacuum, with real and positive,

 uk=√12(1+εkEk), (7)

and, in general, complex ,

 |vk|=√12(1−εkEk),argvk=arg(J∑iΔicoski), (8)

is the eigenstate of (2) to the lowest eigenenergy, . As long as for all values of , the unique ground state is the vacuum of elementary excitations (quasi-particles). However, on passing to the thermodynamic limit, when the system’s linear sizes in all directions tend to infinity, the minimum of over (the excitation gap in the spectrum of quasi-particles) may approach zero at special values of the chemical potential and the coupling constant , and then the ground state becomes degenerate. Those special points in the -plane are the quantum-critical points, where the system undergoes continuous quantum phase transitions.

All the correlation functions of considered systems can be expressed in terms of two basic two-point correlation functions. Since we are interested only in ground-state correlation functions, taking into account the lattice-translation invariance of our system these two basic two-point correlation functions can be chosen as follows:

 qp⟨0|a†0,σar,σ|0⟩qpandqp⟨0|a0,σar,−σ|0⟩qp, (9)

with some . The first correlation function, , is gauge and spin-flip invariant; for it represents offdiagonal matrix elements of the ground-state one-body reduced density operator, and amounts to

 qp⟨0|a†0,σar,σ|0⟩qp=−12Ld∑kεkEkexpikr, (10)

which, upon using the invariance of and with respect to reflections of in coordinate axes, in the thermodynamic limit becomes

 limL→∞qp⟨0|a†0,σar,σ|0⟩qp≡G(r)=−12πd∫0≤kj≤πdkεkEkd∏j=1coskjrj. (11)

Choosing the spin projection , the second correlation function, measuring the degree of gauge-symmetry breaking, amounts to

 qp⟨0|a0,↑ar,↓|0⟩qp=−12Ld∑kJ∑iΔicoskiEkexpikr, (12)

which, by the above arguments, in the thermodynamic limit becomes

 limL→∞qp⟨0|a0,↑ar,↓|0⟩qp≡h(r)=−12πd∫0≤kj≤πdkJ∑iΔicoskiEkd∏j=1coskjrj. (13)

Both the above defined two-point correlation functions are used to define the order parameters in the ground-state phase diagrams presented in the sections that follow. However, analytic results will be given only for the gauge-invariant correlation function , defined in (11). For a fixed lattice direction and the parameters , depends on three parameters: - the distance between the two points of the correlation function, the chemical potential and the coupling constant . The ground-state phase diagrams are presented in the -plane, in particular the quantum-critical points of the considered models are uniquely defined by pairs .

In the sequel, we will discuss three kinds of asymptotic behavior of correlation functions, as their variables enter a region specified by the conditions imposed on the variables. First, the large-distance asymptotic behavior, that is, for a fixed point , distance tends to infinity. Second, the critical-asymptotic behavior, that is, for an arbitrary fixed distance , -points approach a quantum-critical point along some path in the -plane. Two kinds of paths will be considered: -paths that are parallel to the -axis and -paths that are parallel to the -axis. Third, the doubly-asymptotic behavior, that is, for a fixed but sufficiently large , -points approach a quantum-critical point along a -path or a -path.

Anticipating the results presented in the sections that follow, in the one-dimensional model and the two-dimensional symmetric model the critical points are located in a symmetric interval at the -axis and at the -axis in the parameter space of -points. In all the analytic asymptotic formulae to be derived, a vicinity of the multicritical point is excluded. In particular, for the -coordinates of -paths have to be away from zero; analogous condition applies to -paths.

In gapped phases, a decay with increasing of a two-point correlation function is dominated by an exponential factor, , which defines the correlation length . If additionally -points approach a quantum-critical point, i.e. the distance between them tends to zero (that is in a doubly-asymptotic region), then diverges as , which in turn defines a universal critical index associated with a particular quantum-critical point. Below we demonstrate, providing explicit formulae, that in two-dimensional systems the decay of two-point correlation functions, hence and , depends on lattice directions.

In what follows we shall focus on the gauge-invariant correlation function , defined in (11). For a reader’s convenience, in section 3 (-system) and in four subsections of section 4 (four cases of -system), we report the obtained results in the same order. First, we present the large-distance asymptotic behavior of , with explicit expressions for . Then, we give the doubly-asymptotic behavior, with the values of shown in the phase diagram, for each kind of quantum-critical points. After that, we derive specific scaling laws, with respect to and , satisfied by in doubly-asymptotic regions. Finally, we provide numerical arguments that in critical regions, that is for -points sufficiently close to a quantum-critical point, these scaling laws hold for any distances (not only for sufficiently large ones).

## 3 The one-dimensional case

As mentioned in Introduction, this section is included for reasons of completeness. All the expressions of previous section can be adapted to the one-dimensional case by setting for , , and . Moreover, in all the formulae and figures of this section we make the identification .

We can distinguish four ground-state phases labeled by two order parameters, and . The order parameter is given by

 O1=G(0)−12. (14)

When is real, the system is hole-particle invariant at , and then is a deviation of the number density with given spin projection from its value at the hole-particle symmetry line ; it is an odd function of . Since depends only on the latter property of holds as well if is complex. The order parameter , measuring the degree of gauge symmetry breaking, is defined as

 O2=−Δ∗h(1). (15)

The quantum critical points of the one-dimensional system are located at the –axis and in the closed interval of the –axis. There are two critical end points and a multicritical point . The ground-state phase diagram of the one-dimensional system is shown in Fig. 1.

We start with presenting a summary of analytic results concerning the large-distance asymptotic behavior of . It is convenient to introduce an auxiliary function :

 ~G(r)=−12π∫π0dkcos(rk)√(cosk−μ)2+J2cos2k, (16)

in terms of which

 G(r)=12[~G(r+1)+~G(r−1)]−μ~G(r). (17)

In the stripe of the –plane, but excluding the and lines of quantum critical points, the large-distance asymptotic behavior of , derived by heuristic arguments, reads

 ~G(r)≈−√1+J22π|μJ|c1/4exp(−r/ξ)√rcos(θr−ϕ), (18)

where

 1ξ=ln(a++√a2+−1),θ=arccos(a−),ϕ=14arctan2μ|μJ|(1+J2)2−μ2(1−J2), (19)

with

 a±=12(1+J2)[√(μ+1+J2)2+μ2J2±√(μ−1−J2)2+μ2J2], (20)

and

 c=[(1+J2)2−μ2(1−J2)]2+4μ4J2. (21)

We note that the correlation length, given by formulae (19), (20) and (21), is in excellent agreement with the correlation length determined numerically, which is demonstrated in Fig. 2.

As might be expected, the doubly asymptotic formulae (i.e. holding in a doubly asymptotic region of sufficiently large distances and -points sufficiently close to a quantum critical point) are considerably simpler than the large-distance asymptotic formula (18). Specifically, in doubly asymptotic regions, where -points approach along a -path a point belonging to any one of the two half lines of quantum critical points, and , formula (18) gives

 |~G(r)|≈√12π|μJ|exp(−r/ξ)√r∣∣ ∣∣cosr(π2−1|J|ξ)∣∣ ∣∣,with1ξ≈|μJ|1+J2, (22)

then if -points approach along the -path one of the two end critical points, and ,

 |~G(r)|≈(1π2|2J|3)1/4exp(−r/ξ)√r∣∣∣cos(rξ−π8)∣∣∣,with1ξ≈√|J|, (23)

and finally if -points approach along a -path any point of the two line segments of quantum critical points, and ,

 |~G(r)|≈√12π|μJ|(1−μ2)1/2exp(−r/ξ)√r× ∣∣ ∣∣cos[(π2−arcsin|μ|+|μ|(2−μ2)J22(1−μ2)3/2)r−12|J|ξ2]∣∣ ∣∣,with1ξ≈|μJ|√1−μ2. (24)

From the expressions for the correlation lengths in neighborhoods of quantum critical points, given in (22), (23) and (24), one readily obtains the values of indices displayed in the phase diagram, Fig. 1. We note also, that irrespectively of the value of (but separated from ), tends asymptotically to as , see Fig. 2.

Apparently, in each one of the above doubly asymptotic formulae there are three factors: a distance-independent positive coefficient , a damping factor , determining decay of correlations with distance (a product of an exponential and power factors), and an oscillating factor , so that . We note that in doubly asymptotic regions such factorizations hold as well for the whole correlation function . An inspection of doubly asymptotic formulae (22), (23) and (24) for the auxiliary function reveals interesting scaling properties, with respect to distance and one of the parameters, or , of the correlation function or of some of its factors. Specifically, in the corresponding doubly asymptotic regions, formula (22) implies

 |G(r)|≈|μ|gJ(|μ|r), (25)

then from (23) we derive

 |G(r)|≈√|J|g|μ|=1(√|J|r), (26)

and finally, formula (24) gives

 CD≈|J|gμ(|J|r), (27)

where , and stand for some functions. From scaling formulae (25), (26) and (27) one can infer the values of critical indices in the corresponding critical regions: , and , respectively.

Numerical calculations of reveal a remarkable fact. Namely, the scaling laws (25), (26) and (27), derived only in doubly asymptotic regions, hold in the whole range of distances; only a proximity of -point to a quantum critical one is required. To demonstrate this, consider for definiteness the scaling of with respect to and . Let us write scaling formula (25) as . For two values of the chemical potential, and , – a positive number, we readily find the relation

 λ−1|Gλμ(r)|≈|Gμ(λr)|. (28)

In Figs. 3 and 4, we demonstrate that relation (28) holds surprisingly well even for relatively small distances. In an analogous way we verified the scaling formula (26), see Fig. 5, and (27), see Fig. 6.

## 4 The two-dimensional case

All the expressions of section 2 can be adapted to the two-dimensional case by setting for . But in distinction to the one-dimensional case, due to the freedom in choosing the relation between the parameters and , formula (1) represents a great variety of models. In this paper we limit our considerations to only two cases. Namely, the symmetric case, with the interaction term invariant under rotations by , where , and the antisymmetric case, with the interaction term that changes sign under a rotation by , where . We note that in both cases the correlation functions of our systems are invariant not only with respect to lattice translations but also with respect to rotations by .

As compared to the one-dimensional case, a novel feature of two-dimensional models is that the two-point correlation functions depend not only on the distance but also on the direction of . Expressing by its Cartesian coordinates, , we can parameterize directions by the ratio . Then, for a given critical point, we can expect -dependent doubly-asymptotic behaviors of correlations. Unfortunately, the analytic asymptotic formulae for , which we have been able to obtain, apply only to points such that or, by symmetry, , that is for offdiagonal directions which form a sufficiently small angle with the axial directions. Therefore, the asymptotic formulae in the diagonal direction are derived separately. These formulae define -dependent correlation lengths in offdiagonal directions satisfying the conditions specified above and the correlations length in the diagonal direction, where the superscript plus refers to the symmetric model and minus – to the antisymmetric one. Interestingly, our analytical and numerical results show that, for each critical point, there are only two kinds of universal critical indices : for all offdiagonal directions and for the diagonal direction.

Similarly to the one-dimensional case, in the formulae and figures of this section we make the identification ,

### 4.1 The symmetric model

We can distinguish four ground-state phases labeled by two order parameters, and . The parameter is defined as in the one-dimensional case, formula (14), while the new definition of is

 O2=−Δ∗h(1,0). (29)

The comments of previous section related to and remain valid.

The quantum-critical points of the symmetric two-dimensional system are located at the –axis and in the closed interval of the –axis. There are two critical end points and a multicritical point . The ground-state phase diagram of the symmetric two-dimensional system is shown in Fig. 7.

(i) the diagonal direction:

The correlation function is given by the general formula (11),

 G(r′,r′)=−12π2∫0≤k1,k2≤πdk1dk2cos(r′k1)cos(r′k2)ε(k1,k2)E(k1,k2), (30)

where

 ε(k1,k2)=cosk1+cosk2−μ,E(k1,k2)=√(cosk1+cosk2−μ)2+J2(cosk1+cosk2)2. (31)

Naturally, determining its large-distance asymptotic behavior is a harder task, then in the one-dimensional case, considered in section 3. In the stripe of the –plane, but excluding the and lines, the large-distance asymptotic behavior of is given by

 G(r′,r′)≈−sgn(μ)12π(J21+J2)1/4exp(−r′/ξ(+))r′cos(θr′+ϕ), (32)

where

 1ξ(+)=2ln(a++√a2+−1),θ=2arccos(a−),ϕ=π4−12arctan|J|, (33)
 a±≡12(1+J2)[√(|μ|/2+1+J2)2+μ2J2/4±√(|μ|/2−1−J2)2+μ2J2/4]. (34)

Note that, since the distance between the points and is , is not the correlation length; the correlation length in the diagonal direction . The formulae (33) and (34) for the correlation length , are in excellent agreement with correlation lengths determined numerically, which is demonstrated in Fig. 8.

The large-distance asymptotic formula (32) assumes a lot simpler form in doubly asymptotic regions; we give only simplified expressions for and . Specifically, in doubly asymptotic regions, where -points approach along a -path a point belonging to any one of the two half lines of quantum critical points, and , formulae (32), (33) and (34) imply

 1ξ(+)≈|μJ|1+J2,θ≈π−|μ|1+J2, (35)

then if -points approach along the -path one of the two end critical points, and ,

 1ξ(+)≈2|J|1/2,θ≈2|J|1/2, (36)

and finally if -points approach along a -path any point of the two line segments of quantum critical points, and ,

 1ξ(+)≈2|μJ|√4−μ2,θ≈2arccos|μ|2. (37)

From the expressions for the correlation lengths in neighborhoods of quantum critical points, given in (35), (36) and (37), one readily obtains the values of indices displayed in the phase diagram, Fig. 7. We note also that as , tends to , irrespectively of separated from zero, clarify Fig. 8.

As in the one-dimensional case, discussed in previous section, the above doubly asymptotic formulae imply interesting scaling relations with respect to distance and one of the parameters, or , of the correlation function , or of some of its factors. Specifically, in the corresponding doubly asymptotic regions, formulae (32) and (35) imply that

 |G(r′,r′)|≈|μ|dJ(|μ|r′), (38)

from (32) and (36) we obtain

 |G(r′,r′)|≈|J|d|μ|=2(√|J|r′). (39)

Finally, from (32) and (37) we derive

 CD≈|J|3/2dμ(|J|r′), (40)

with , defined as in section 3, and where , and stand for some functions. We note that the scaling formulae (38), (39) and (40) immediately imply that in the corresponding critical neighborhoods the critical indices assume the values , and , respectively. Remarkably, similarly to the one-dimensional case numerical calculations show that the scaling relations (38), (39) and (40), derived only for sufficiently large distances, hold in the whole range of distances. Specifically, scaling property (38) is demonstrated in Figs. 9, 10, then scaling property (39) in Fig. 11, and finally scaling property (40) in Figs. 12, 13.

(ii) offdiagonal directions:

This case is a bit more involved than the diagonal case; the presentation follows that of previous subsection. The correlation function , given by (11), assumes the form

 G(r1,r2)=−12π2∫0≤k1,k2≤πdk1dk2cos(r1k1)cos(r2k2)ε(k1,k2)E(k1,k2), (41)

where and are defined by (31). In the stripe of the –plane, but excluding the and lines, the large-distance asymptotic behavior of reads

 G(r1,r2)≈−Cr2π(μ2J21+J2)1/4exp(−r1/ξ1−r22/(ξ2r1))r1cos(θ1r1+θ2r22/r1+ϕ), (42)

provided the points are located between the ray , with being a sufficiently large rational (in fact it is enough that , see Fig. 17), and the -axis. The function of , , is defined as follows

 Cr={1,%if$μ>0$,(−1)(r1+r2+1),if μ<0. (43)

The numerous constants (independent of coordinates and ) in (42) are given as follows:

 1ξ1=2ln(a++√a2+−1),θ1=2arccosa−,ϕ=−π4+arctan1|J|, (44)
 a±=12(√(u+1)2+v2±√(u−1)2+v2), (45)
 |u+iv|=√|μ|2√1+J2,arg(u+iv)=12arctan|J|, (46)
 ∣∣∣θ2−i1ξ2∣∣∣=√|μ|2(1+J2)⎡⎣(1−|μ|2(1+J2))2+(μJ2(1+J2))2⎤⎦1/4, (47)
 arg(θ2−i1ξ2)=12arctan|J|−12arctan|μJ|2(1+J2)−|μ|. (48)

In particular, if the points become remote from the origin along a ray , then in terms of the distance (), the asymptotic formula (42) can be rewritten as

 G(r1,r2)≈−Cr2π(μ2J21+J2)1/4(1+n2n2)1/2exp(−r/ξ(+)offdiag)rcos(rθoffdiag+ϕ), (49)

defining – the correlation length in an offdiagonal direction specified by ,

 1ξ(+)offdiag=(n21+n2)1/2(1ξ1+1n21ξ2), (50)

and ,

 θoffdiag=(n21+n2)1/2(θ1+1n2θ2). (51)

The above formula for the correlation length is in excellent agreement with the correlation length determined numerically, which is demonstrated in Fig. 14, where plots of , in an axial direction, against , for a number of -values, are displayed.

To reveal the dependence on direction , in Figs. 15 and 16 we display similar plots, but now, for a fixed we show plots for a number of directions .

Naturally, the large-distance asymptotic formula (42) can be simplified in doubly asymptotic regions; we provide only simplified expressions for , , and . Specifically, in doubly asymptotic regions, where -points approach along a -path a point belonging to any one of the two half lines of quantum critical points, and , formulae (44)-(48) imply that

 1ξ1≈√2|μ|√1+J2sin(12arctan|J|)=−2ξ2,θ1≈π−2θ2, (52)

with

 θ2≈√|μ|2√1+J2cos(12arctan|J|), (53)

then if -points approach along the -path one of the two end critical points, and ,

 1ξ1≈√2|J|=2ξ2,θ1≈√2|J|=2θ2, (54)

and finally if -points approach along a -path any point of the two line segments of quantum critical points,