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Phase structure of a U(1) lattice gauge theory with dual gauge fields
Nuclear Physics B 764 [FS] (2007) 168–182
Phase structure of a U(1) lattice gauge theory with dual gauge fields Tomoyoshi Ono a , Yuki Moribe a , Shunsuke Takashima a , Ikuo Ichinose a,∗ , Tetsuo Matsui b , Kazuhiko Sakakibara c a Department of Applied Physics, Nagoya Institute of Technology, Nagoya 466-8555, Japan b Department of Physics, Kinki University, Higashi-Osaka 577-8502, Japan c Department of Physics, Nara National College of Technology, Yamatokohriyama 639-1080, Japan
Received 3 August 2006; accepted 6 December 2006 Available online 3 January 2007
Abstract We introduce a U(1) lattice gauge theory with dual gauge fields and study its phase structure. This system is partly motivated by unconventional superconductors like extended s-wave and d-wave superconductors in the strongly-correlated electron systems and also studies of the t–J model in the slave-particle representation. In this theory, the “Cooper-pair” (or RVB spinon-pair) field is put on links of a cubic lattice due to strong on-site repulsion between original electrons in contrast to the ordinary s-wave pair field on sites. This pair field behaves as a gauge field dual to the U(1) gauge field coupled with the hopping of electrons or quasi-particles of the t–J model, holons and spinons. By Monte Carlo simulations we study this lattice gauge model and find a first-order phase transition from the normal state to the Higgs (superconducting) phase. Each gauge field works as a Higgs field for the other gauge field. This mechanism requires no scalar fields in contrast to the ordinary Higgs mechanism. An explicit microscopic model is introduced, the lowenergy effective theory of which is viewed as a special case of the present model. © 2006 Elsevier B.V. All rights reserved.
1. Introduction The Ginzburg–Landau (GL) theory has proved itself a powerful tool to describe the phase transitions and low-energy excitations of conventional s-wave superconductors. In field-theory * Corresponding author.
E-mail address: [email protected] (I. Ichinose). 0550-3213/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2006.12.004
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terminology, the GL theory takes a form of Abelian Higgs model (AHM), and its phase structure has been studied by field-theoretical techniques and Monte Carlo (MC) simulations of lattice gauge theory. These studies are partly motivated by the work of Halperin, Lubensky, and Ma [1] which predicts a first-order phase transition. At present, it is established that the phase structure of three-dimensional (3D) AHM on the lattice strongly depends on a parameter controlling fluctuations of the amplitudes |ϕ(x)| of the Higgs (Cooper-pair) field [2]. At the London limit in which |ϕ(x)| is fixed, there is only the confinement phase in the lattice model. As the fluctuations of |ϕ(x)| are increased, a second-order phase transition to the Higgs phase appears, and for further fluctuations, the transition becomes of first-order. In recent years, it becomes clear that phase transitions in certain strongly-correlated electron systems have an order parameter which has an “unconventinal” symmetry like d-wave as a result of, e.g., the strong on-site Coulomb repulsion. For example, some strongly-correlated electron systems exhibit d-wave superconductivity at low temperatures (T ). The first d-wave superconductor CeCu2 Si2 was discovered in 1979 [3]. In 1986, the cuprate high-Tc superconductors were discovered [4], and later, it was found that they are d-wave superconductors. In order to describe the d-wave superconducting phase transition by a GL theory defined on a lattice, the Cooperpair field must be put on links because it changes its sign under a π/2 rotation. It is interesting and also important to set up and study the GL theory of the unconventional superconductivity (UCSC) like d-wave SC [5]. In the framework of weak-coupling theory, such studies already have appeared [6]. However, the strong-coupling region remains to be studied [7]. Another example of the appearance of the order with the d-wave symmetry is the spin-gap state in the high-Tc cuprates above the SC state. In the t–J model in the slave-boson representation, the spin-gap state is described by the condensation of the resonating-valence-bond (RVB) amplitude of the spinon-pair link field. Low-T effective theory similar to the above GL theory for the UCSC is obtained by integrating over the holon and spinon fields. In this paper, motivated by the above examples of the strongly-correlated electron systems, we shall introduce a GL theory with order-parameter field defined on links and study its phase structure by means of MC simulations.We shall see that the order parameter, a bilocal field, is regarded as a gauge field, and the knowledge and method of gauge theory are useful to study this GL lattice gauge theory. We find that this new type of gauge theory has a very interesting phase structure. The present paper is organized as follows. In Section 2, we shall introduce the GL gauge theory and explain its relation to the UCSC and the t–J model. In Section 3, we show the phase diagram obtained by a mean-field theory. Sections 4 and 5 are main part of this paper, and show the results of the numerical study for two typical parameter regions. The model exhibits various phases which include the SC or spin-gap phase. In Section 6, quantum phase transition at T = 0 is studied. Section 7 is devoted for conclusion. In Appendix A, we give a microscopic model of strongly-correlated electrons, the low energy effective theory of which is viewed as a special case of the present GL theory. 2. Lattice gauge model with dual gauge fields Let us first introduce the GL gauge theory on a three-dimensional (3D) lattice and then we shall explain its relation to the UCSC and the t–J model. We put a “Cooper-pair field” Vxj on the link (x, x + j ) of the lattice, where x is the site index and j (= 1, 2, 3) is the direction index (it also denotes the unit vector of the j th direction). Vxj is related to fermionic quasi-particles
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Fig. 1. Action AGL of the GL theory (2.4).
like the electrons in the UCSC or the spinons in the t–J model as Vxj ∝ ψx↑ ψx+j,↓ − ψx↓ ψx+j,↑ ,
(2.1)
where ψxσ is the fermion operator at x with spin σ =↑, ↓. In the rest of paper, we focus on the v ). London limit of the Vxj and put |Vxj | = 1, i.e., Vxj = exp(iθxj There is another link field Uxj , a compact U(1) gauge field, which appears in the hopping of ψxσ as † ψx+j,σ Uxj ψxσ ,
(2.2)
u iθxj
where Uxj = e . We require the invariance of the system under a local gauge transformation ψxσ → eiϕx ψxσ . Under this transformation, Vxj and Uxj transform as Vxj → eiϕx+j Vxj eiϕx ,
Uxj → eiϕx+j Uxj e−iϕx .
(2.3)
It is easily seen that under the replacement ϕx → −ϕx (x ∈ odd sites) Vxj transforms just like the original Uxj and vice versa. As the action of the GL theory AGL must respect the local gauge invariance under Eq. (2.3), it is straightforward to “derive” AGL in the local expansion as 1 cu U 4 + cv V 4 + cm (U V U V + V U V U ) AGL = 2 pl + (d1 U U V V + 3 permutations) + c.c., (2.4) where cu , etc. are effective parameters and some of them are increasing functions of 1/T . Each term in AGL is depicted in Fig. 1. For example, the U 4 term stands for the usual plaquette term † † Ux+i,j Ux+j,i Uxj of lattice gauge theory. The partition function Z is given by Uxi Z = [dU ][dV ] exp(AGL ), (2.5) u /2π , etc. We consider a 3D cubic lattice of the size L3 with the periodic where [dU ] ≡ xj dθxj boundary condition. (The results given below are for L = 16 and 24.) We also note that the system has a symmetry Z(cu , cv , cm , dm ) = Z(cv , cu , cm , dm ) for the case d1 = · · · = d4 ≡ dm , corresponding to the interchange Uxj ↔ Vxj . Let us explain how AGL is related to the UCSC and the t–J model. For the UCSC, ψxσ is nothing but the electron operator Cxσ itself, and then “condensation of Vxj ” means the SC phase transition. We consider the compact U(1) gauge field Uxj , which corresponds to the electromagnetic field, instead of the ordinary noncompact one, because the studies on the compact
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gauge models reveal also the phase structure of the noncompact models. (See later discussion.) At present it is believed that in the underdoped region of the high-Tc materials the SC phase transition is a sort of the Bose–Einstein condensation (BEC) of the Cooper-pair field Vxj , i.e., Vxj has a finite amplitude and its phase fluctuation induces the SC phase transition. Therefore taking the London limit is a good approximation. A spatial lattice introduced in order to set up the GL theory and study its strong-coupling region should be regarded as a “coarse grained” lattice although some of the UCSC are granular systems. In Appendix A, an explicit microscopic model is introduced, and its low-energy effective theory is derived. It can be viewed as a special case of the present GL gauge model on a lattice. Let us turn to the t–J model of the high-Tc superconductors. The Hamiltonian of the t–J model on a two-dimensional square lattice is given by HtJ = −t
2 1 † ˜ ˜ Cx+i,σ Cxσ + H.c. + J Sx · Sx+i − nx nx+i , 4 x
2 x
i=1 σ =↑,↓
(2.6)
i=1
† † Cx,−σ )Cxσ , nx ≡ σ Cxσ Cxσ , and where i = 1, 2 is the spatial direction, C˜ xσ ≡ (1 − Cx,−σ τ are the Pauli spin matrices). The physical states are restricted to those satisfyS x = Cx† τ Cx /2 ( ing the local constraint, † Cxσ Cxσ |phys 1. phys| (2.7) σ
In the slave-boson representation, the electron operator is expressed as Cxσ = bx† fxσ , where bx is a bosonic holon operator and fxσ is a fermionic spinon operator. The above constraint becomes equivalent to the equality
† † (2.8) fxσ fxσ + bx bx − 1 |phys = 0. σ
In this representation, there appear quartic terms b† bf † f and f † f † ff in HtJ . We adopt the path-integral method of the imaginary-time formalism, and decouple these terms by introducing auxiliary complex fields as [8]
† χxi J (2.9) fx+i,σ fxσ + tbx+i bx† , Dxi fx+i,↑ fx↓ − fx+i,↓ fx↑ , σ
where χxi is the holon and spinon hopping amplitude and Dxi is the spin-gap amplitude. It is then straightforward to see that phase degrees of freedom of χxi and Dxi correspond to the gauge fields Uxi and Vxi as χxi = |χxi |Uxi ,
Dxi = |Dxi |Vxi ,
(2.10)
under a “gauge transformation” (fxσ , bx ) → eiαx (fxσ , bx ) [see Eq. (2.3)]. There is also the timecomponent Uxj =3 , which works as the Lagrange multiplier for the local constraint (2.8). The gauge dynamics of the composite gauge fields Uxμ and Vxi strongly influences lowenergy excitations in the t–J model. In particular, in the deconfinement phase of Uxμ , holons and spinons acquire independent hopping amplitudes (∝ Uxμ ), so the phenomenon of chargespin separation (CSS) takes place. In the deconfinement phase of Vxi , the RVB amplitude Vxi develops, inducing a spin gap.
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The effective gauge theory for the CSS and the spin-gap state is obtained by integrating holon and spinon variables in the path-integral representation (we do not think the possibility of the Bose condensation of holons). From the gauge invariance its action should take the form of AGL of Eq. (2.4) with the third direction as the imaginary time. By making the hopping expansion at finite T [8], the parameters in the AGL are estimated as cu ∼ (J 2 + t 2 δ)T −3 , cv ∼ T −4 , cm , d1 , . . . , d4 ∼ J 2 T −4 where δ is the hole concentration [9].1 After obtaining the phase structure of the present gauge model, we shall comment on its implication to the physics of the t–J model more precisely. 3. Mean-field theory Before going into details of the numerical simulations, we first study the model by the meanfield theory (MFT) for dm = 0. We choose the action A0 of MFT as the following single-link form: 1 (λU Uxj + λV Vxj + c.c.). A0 = (3.1) 2 xj
The partition function Z0 and the free energy F0 of MFT is given by Z0 = [dU ][dV ] exp(A0 ) = I0 (λU )3N I0 (λV )3N ≡ exp(−F0 ), F0 = −3N log I0 (λU )I0 (λV ) ,
(3.2)
where N is the number of the sites and I0 (a) is the modified Bessel function (n = 0) defined by 2π In (a) =
dθ exp a cos(θ ) + inθ . 2π
(3.3)
0
The MFT is based on the following variational principle (Jensen inequality) [10]: F ≡ − log Z, F Fv ≡ F0 + A0 − A0 , −1 [dU ][dV ]O exp(A0 ). O0 ≡ Z0
(3.4)
Then we minimize Fv (λU , λV ) by optimizing the variational parameters λU and λV . The minimum of Fv is regarded as the best approximation of F . Fv is calculated explicitly as Fv /(3N ) = − log I0 (λU ) − log I0 (λV ) − cu m4U − cv m4V − 2cm m2U m2V + λU mU + λV mV , I1 (λU ) I1 (λV ) , mV ≡ Vxj 0 = , mU ≡ Uxj 0 = I0 (λU ) I0 (λV ) 2π dθ cos θ exp(a cos θ ). I1 (a) = (3.5) 2π 0
1 At finite T , the gauge fields U and V must satisfy the periodic boundary condition with respect to the imaginary xj xj time, but the coefficients cu , etc. in AGL depend on T more strongly than the fields and they give the leading contribution to the phase transition.
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Fig. 2. Phase structure for dm = 0 in the mean-filed theory. It is symmetric about the axis cu = cv . The symbols C, U , V , UV indicate the confinement phase, U phase, V phase, and UV phase, respectively.
The solutions of λU , λV are classified into the following four cases: (1) confinement phase: (2) U -phase: (3) V -phase: (4) UV-phase:
λU λU λU λU
= 0, λV = 0, λV = 0, λV = 0, λV
= 0 → Uxj 0 = 0, Vxj 0 = 0, = 0 → Uxj 0 = 0, Vxj 0 = 0, = 0 → Uxj 0 = 0, Vxj 0 = 0, = 0 → Uxj 0 = 0, Vxj 0 = 0.
(3.6)
In Fig. 2, we show the phase diagram in the cu –cv plane for the fixed values of cm = 0, 0.3, 0.6, and 1.0 (dm = 0). We note that all the phase transitions studied here are of first order, i.e., the internal energy E and (at least one of) λU and λV has a jump across the boundary. In Fig. 3, we plot a typical first-order behavior of λU and λV across the transition point. 4. Extended s-wave Higgs phase (dm = 0 and cm > 0) Let us turn to the numerical studies. For vanishing cm and di (i = 1, . . . , 4), the two gauge fields Uxj and Vxj decouple with each other and the system reduces to two 3D pure U(1) gauge systems. As no phase transition takes place in the 3D U(1) pure gauge theory [11], the present system for cm = di = 0 (i = 1, . . . , 4) has only a single (confinement) phase. Then we assign various nonvanishing values to cm and/or di , and determine the phase structure in the cu –cv plane for fixed cm and di . Hereafter we often consider the symmetric case di = dm (i = 1, . . . , 4). Below we present the results for two typical cases: (i) dm = 0 and cm > 0 and (ii) cm = 0 and dm = di < 0.
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Fig. 3. (a) Free energy Fv /(3N ) vs. λU = λV for cu = cv , cm = 0.5, dm = 0 near the transition point cu = 1.323. (b) U vs. cu = cv for cm = 0.5, dm = 0.
Let us first consider the case (i), i.e., the extended s-wave Higgs phase with the “spherical symmetry” (i.e., the symmetry under a π/2-rotation). To study the phase structure of the model, we measure the internal energy E and the specific heat C (fluctuation of E) by the MC simulations, E ≡ −AGL /L3 ,
2 C ≡ AGL − AGL /L3 .
(4.1)
We considered the cases dm = 0, cm = 0.2, 0.4, 0.6, 0.8, 1.0, and 1.2. In Fig. 4(a) and (b) we show E and C for dm = 0, cm = 0.6 and cv /cu = 0.1. At cv ∼ 1, E shows a hysteresis, which implies a first-order phase transition. The cm term works as a “Higgs coupling” of the “Higgs” field Vxj (Uxj ) to the gauge field Uxj (Vxj ) to stabilize their fluctuations and induce such a transition [8]. Thus the transition is expected from the confinement phase where Uxj and Vxj fluctuate violently to the Higgs (superconducting) phase where their fluctuations are small. The data for cm 0.6 show signals of first-order phase transitions, while the data of cm 0.4 show no signals of phase transitions. In Fig. 4(c) we show the phase diagram in the cu –cv plane for cm = 0.6. Similar phase diagram is obtained for cm = 0.8–1.2. To confirm the above interpretation of each phase, we measured instanton densities. We consider two kinds of instantons, i.e., U -instantons and V -instantons, and denote their average densities per cube as ρU and ρV , respectively. For the U -instantons we employ the definition given in Ref. [12], which measures magnetic fluxes emanating from each smallest cube. Similar gauge-invariant definition is possible for the V -instantons.2 In Fig. 4(d) we plot ρU and ρV . ρV shows a discontinuity at the first-order transition point cv ∼ 1.0 just like E, and decreases very rapidly as cv increases. On the other hand, ρU decreases very rapidly and almost vanishes for cv > 0.2. This result indicates that the small cusp in C of Fig. 4(b) at cv ∼ 0.17 reflects a crossover from the dilute to dense instanton “phases” of the gauge field Uxj [11–13]. Then we conclude that the system changes from the confinement phase to the Higgs phase as cv (cu ) increases. 2 To define the V -instantons we consider the replacement ϕ → −ϕ for the odd sites in Eq. (2.3). Then V transforms x x xj just like Uxj before the replacement, so it is straightforward to define the V -instantons.
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Fig. 4. Results for dm = 0, cm = 0.6. (a) Internal energy E for cv /cu = 0.1 shows a hysteresis around cv ∼ 1.0, a signal of first-order transition. (b) Specific heat C for cv /cu = 0.1 shows a small cusp at cv ∼ 0.17, which is interpreted as a crossover (see the text). (c) Phase structure determined by E and C for L = 16. The first-order transition line separates the confinement phase and the Higgs phase. The cross symbols denote crossover. (d) Instanton densities ρU and ρV for cv /cu = 0.1. ρV exhibits a discontinuity at cv ∼ 1.0, while ρU decreases rapidly at cv ∼ 0.17.
In order to support the above conclusions, we also calculated expectation values of the U and V -Wilson loops, WU (Γ ) = (4.2) U , WV (Γ ) = V , Γ
Γ
where Γ is a closed loop on the lattice, and the products of Uxj and Vxj in Eq. (4.2) are formed to be gauge-invariant. From the above results of instanton densities, we expect that WU (V ) (Γ ) obey the area law in the instanton-plasma phase for small cu(v) and the perimeter law in the instanton-dipole phase for large cu(v) . In calculating WU (V ) (Γ ), we consider various shapes of Γ . For example, we take Γ ’s having a fixed area and various perimeters, and vice versa. In Fig. 5 we show the results for cm = 0.6. In the case cu = cv = 1.4, WU (Γ ) fits the area law, while the case cu = cv = 2.0 fits the perimeter law. These results support the previous conclusion obtained from instanton densities. Because all the coefficients in AGL are positive in the present case, we expect that the observed “ordered” state is an extended s-wave Higgs phase (superconductor). We calculated U U V V in order to verify this expectation for cm 0.6 and dm = 0 cases and found that U U V V has vanishing value in the normal state whereas it has a finite value in the Higgs phase showing hysteresis loop. Furthermore its value takes a negative as well as positive value depending on samples. Turning on a small but finite positive dm term, U U V V becomes positive. This result indicates that positive dm term is necessary to produce a genuine extended s-wave Higgs phase (superconductor). In this section, we have reported the results of the MC simulations for the compact U(1) gauge theory with dual gauge fields in which Uxj as well as Vxj is the compact gauge field. In this
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Fig. 5. Wilson loops for dm = 0, cm = 0.6 with cu = cv , which means WU (Γ ) = WV (Γ ). (a) The case cu = cv = 1.4 fits the area law. (b) The case cu = cv = 2.0 fits the perimeter law rather than the area law. Fig. 4(c) indicates that the transition between the two laws occurs at cu = cv ∼ 1.6.
system, the phase transition is interpreted as the one from the confining to Higgs phases. Behavior of the Wilson loops and the instanton densities support this conclusion. From the numerical calculations, we can readily expect that there exists a Coulomb–Higgs phase transition in the system of the noncompact U(1) gauge field Uxj for sufficiently large cm . In fact as it is seen from Fig. 4(d), the U -instanton density is almost vanishing for cu > 2.5 (cv > 0.25) and therefore Uxj behaves substantially like the noncompact gauge field with small fluctuations in that parameter region. Sufficiently large cm (dm ) terms with the noncompact Uxj suppress the phase fluctuations of Vxj because the following terms effectively appear, † cm (4.3) Uxi Ux+j,i Vxj Vx+i,j + ···, xj
and a phase transition to the Higgs phase occurs as cv increases. 5. d-wave Higgs phase (cm = 0 and dm < 0) We expect and verified that the dm -term with dm < 0 enhances d-wave condensation of Vxj ; † Vxj < 0. For large cu , the fluctuations of Uxj are suppressed as Uxj ∼ 1 Δij ≡ Uxi Ux+i,j Vx+j,i [up to the gauge transformation Eq. (2.3)], hence the d-wave configuration Vxi† Vx+i,j < 0 (i = j ) is preferred, although there are no configurations with all negative Δij in three dimensions. We considered the cases dm = −0.4, −0.6, −0.8, −1.0, and measured E, C, ρU , and ρV as before. No signals of phase transitions are found for dm = −0.4 and −0.6, whereas signals of phase transitions to the d-wave Higgs phase are obtained for dm < −0.6. In Fig. 6(a) and (b) we present E and ρ for cu /cv = 1.0 and dm = −0.8. E and ρ show three first-order phase transitions along cv = cu . In Fig. 6(c) we present the phase structure determined by the measurement of E. There are four phases (I)–(IV). The phase (I) is the confinement phase. (II) is the “staggered state” which is generated by the frustration of strong negative d-term. It breaks the translational symmetry by the unit lattice spacing as supported by the Wilson loop of Fig. 6(d) which has two branches, one for even areas and one for odd areas. (III) is the disorder state connecting (II) and (IV). (IV) is the Higgs phase corresponding to d-wave superconductor. These interpretations are consistent with the behavior of ρ in Fig. 6(b). The measurement of Δij shows that the cubic symmetry Δ23 = Δ12 = Δ13 is maintained in (IV), whereas it is reduced to the square symmetry, e.g., Δ23 < Δ12 = Δ13 in (II).
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Fig. 6. Results for cm = 0, dm = −0.8. (a) E and (b) ρU = ρV for cu /cv = 1.0. They show three hysteresis loops. (c) Phase structure in the cu –cv plane. (d) Wilson loop of M × M square for cu = cv = 0.9 shows two smooth branches for even M and odd M, i.e., staggered perimeter law.
We also studied an anisotropic system in which the intraplane dm term is larger than the interplane dm term because many of the real materials have a layered structure. In Fig. 7 we show E and the instanton density for cu /cv = 1.0, cm = 0, intraplane dm = −1.0 and interplane dm = −0.8. There is a first-order phase transition near cu = cv = 1.7, whereas the other transitions existing in the isotropic case in Fig. 6 disappeared. Thus the phase (I) and (II) disappear whereas the two phases (III) and (IV) survive. This phase transition in the anisotropic case should correspond to the SC transition observed in the high Tc cuprates, the heavy-fermion materials, etc. As explained in Section 2, the couplings cu etc are increasing functions of 1/T in the effective gauge theory of the t–J model. The result of the present section showing the existence of the Higgs phase for sufficiently large couplings indicates that the spin gap Δ develops at low T as observed in the experiments. More precise estimation of the couplings cu etc as a function of t , J , δ and T is required to identify the spin-gap region in the phase diagram of the t–J model [14]. 6. Quantum phase transition We have considered the gauge model AGL (2.4) defined on the cubic lattice and studied its phase structure. From the viewpoint of the UCSC, the phase transitions which we found in the previous discussion correspond to thermal phase transitions, i.e., the coefficients in AGL are increasing functions of 1/T and the SC (Higgs) phase appears as T is lowered. Recently, in the studies of the strongly-correlated electron systems like the high-Tc cuprates and the heavyfermion materials, significance of quantum phase transition (QPT) has been recognized [15]. In particular, the QPT in ordinary s-wave charged SC was studied by using a XY model coupled with a U(1) gauge field [16] and very recently this model was applied for the QPT in the high-Tc
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Fig. 7. (a) Internal energy for an anisotropic system. Parameters are cu /cv = 1, cm = 0, intraplane dm = −1.0 and interplane dm = −0.8. There is a first-order phase transition near cu = cv = 1.7. (b) Instanton density also shows hysteresis loop near cu = cv = 1.7.
q
Fig. 8. E and instanton density of the quantum system AGL with (cu , cv , cm , dm ) = g(1, 1, 1, 0). They exhibit first-order phase transition.
cuprates in order to explain the anomalous behavior of the superfluid density near the quantum critical point (QCP) at T = 0 [17]. Quantum theory of the present GL theory can be constructed straightforwardly. In the continuum imaginary-time formalism, Uxj and Vxj depend on the imaginary time τ . Then we discretize the imaginary-time axis and define the quantum GL theory on the 4D hypercubic lattice. The acq tion AGL of the quantum system has a similar form of AGL in Fig. 1, but it is defined on the 4D lattice. It involves the time component Uxτ but does not contain the terms including Vxτ as the Cooper-pair field lives on links of the spatial lattice. q We also studied this quantum system AGL by the MC simulations. In the practical experiments, external conditions and properties of samples like external pressure, doping parameter, q etc., change the effective parameters contained in AGL . Here we consider typical two cases in q which the coefficients in AGL are scaled as (cu , cv , cm , dm ) = g(1, 1, 1, 0) and g(1, 1, 1, −1) where g is a positive parameter. In Fig. 8, we show E and the instanton density for the former case. The result shows a first-order phase transition from the normal phase to the SC phase. We found that the other case also has a similar phase structure. From this numerical result, we can study the SC QPT more precisely by analytical methods in which a complex link field ψxj (τ ) is introduced as a SC “order parameter” field by the Hubbard–Stratanovich transformation [16]. The results will be published in a future publication.
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7. Conclusion In this paper, we have introduced the GL gauge theory with the “order-parameter field” put on links. This system is partly motivated by the observed phase transitions with the d-wave order parameter in the condensed matter physics which include the d-wave SC and the spin-gap state. We showed that the present GL theory can be regarded a new type of lattice gauge model which contains two kinds of gauge fields. By means of MC simulations, we clarified its phase structure. For the case of cm > 0, dm = 0, there are two phases for sufficiently large cm , which correspond to the normal and Higgs phases. They are separated by a first-order phase transition line. On the other hand, for the case of cm = 0, dm −0.8 there are four phases in the isotropic case. Only the two of them survive in the anisotropic case which corresponds to the layered structure. We also studied the QPT at T = 0 in the present model and found that the first-order phase transition also occurs as the parameters of the Hamiltonian are changed. It is quite interesting and also important to apply the obtained results for the UCSC and the spin-gap state in the t–J model and compare the results with experiments. This is under study and the result will be reported in a future publication. Acknowledgements We are very grateful to M.N. Chernodub for several stimulating discussions. Appendix A. A model of UCSC and the dual gauge theory In this appendix we consider a model of strongly-correlated electron system on a 3D lattice as a typical example of the UCSC. Let us consider the following Hamiltonian written in terms of the electron operator Cxσ at the site x and spin σ (= 1, 2): HJ K = HJ + HK ,
† 1 HJ = J Rxj Rxj , Sx+j Sx − nx+j nx = −J 4 x,j
x,j
1 † † Cxσ σ σ σ Cxσ , nx = Cxσ Cxσ , S x = 2 σ σ,σ
1 Rxj = √ σ σ Cx+j,σ Cxσ , 2 σ,σ
† 1 Qxj Qxj , HK = K S x+j S x + nx+j nx = −K 4 x,j
Qxj
1 † =√ Cx+j,σ Cxσ . 2 σ
xj
(A.1)
The first term HJ favors formations of the RVB at nearest-neighbor (NN) pair (we consider the AF spin coupling J > 0). Rxj is the annihilation operator of RVB on the link (x, x + j ) (j = 1, 2, 3). The second term HK represents the AF coupling and the repulsion between the NN pair of electrons (K > 0). We note that a pair of spin-singlet two electrons at the same site has the energy zero while the singlet pair at the NN sites has the energy −J − K/2. Thus two electrons prefer to sitting on the NN sites instead of on the same site.
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In path-integral representation of the partition function Z = Tr exp(−βH ) (β = (kT )−1 ), Z is rewritten for small J and K by introducing a pair of U(1) fields Uxj , Vxj as Z= L=−
β
[dU ][dV ][dC] exp
dτ L , 0
C¯ xσ (∂τ − μ)Cxσ − H ,
xσ
√ √ ( J Vxj R¯ xj + KUxj Qxj − H.c.), H =i
(A.2)
x,j
where μ is the chemical potential controlling the electron density as nx = n. One may prove this expression directly by expanding the exponent in powers of J and K and integrate over Uxj and Vxj . Eq. (A.2) is rewritten in terms of the SU(2) field Wxj ,
χxj , Dxj Wxj = , −D¯ xj , χ¯ xj W † Wxj = 1, (A.3) det Wxj = 1, χ¯ xj χxj + D¯ xj Dxj = 1, xj
and the fermion doublet,
Cx1 ψx = , † Cx2 as
dτ Lψ,W , Z = [dW ][dψ] exp √ (ψ¯ x+j Wxj ψx − H.c.) + L, −ψ¯ x (∂τ − μσ3 )ψx − i J + K Lψ,W = x
L = −γ
2 |χxj | − ρU ,
ρU ≡
xj
(A.4)
j
K , J +K
(A.5)
with the limit γ → ∞. [dW ] is the SU(2) Haar measure. In fact in this limit, one has J |Dxj | → ρV ≡ , J +K Dxj → ρV Vxj = ρV exp(iθV xj ), χxj → ρU Uxj = ρU exp(iθU xj ).
(A.6)
Thus Eq. (A.2) is recovered. Except for the chemical potential term and L, the system has the SU(2) gauge invariance under ψx → Ωx ψx ,
Wxj → Ωx+j Wxj Ωx† ,
Ωx ∈ SU(2).
(A.7)
The case of L = 0, i.e., γ = 0 is discussed in Ref. [18]. The Hamiltonian in this case is H ∝ − xj S x+j S x , which corresponds to V = J in Eq. (A.1).
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By integrating over ψxσ we obtain an effective gauge theory for general γ . The partition function Z is given by Z = [dW ] exp A(W ) , A(W ) = Tr Log(∂τ − μσ3 + W ). (A.8) By making a local(hopping) expansion of A(W ) in powers of W we obtain the action AGL of the Ginzburg–Landau theory, † A(W ) = c0 + c2 (A.9) Tr Wxi Wx+i,j Wx+j,i Wxj + H.c. + O W 6 , x,j
where we consider high-T region and the T -dependence is included in the coefficients c0 , c2 , . . . . In the limit γ → ∞, one obtains a U(1) lattice gauge theory with a pair of dual compact U(1) gauge variables Uxj , Vxj . In fact, by calculating the c2 term of Eq. (A.9) explicitly, it becomes equivalent to the AGL model with the coefficients, K2 , (J + K)2 J2 cv = c2 ρV4 = c2 , (J + K)2 JK cm = c2 ρU2 ρV2 = c2 , (J + K)2 JK d1 = −c2 ρU2 ρV2 = −c2 , (J + K)2 d2 = −d1 , d3 = d 1 , d4 = −d1 . cu = c2 ρU4 = c2
(A.10)
We note the following two points concerning to the signatures of these coefficients: (i) The four dmα ’s have alternative signatures in contrast with the choice of Section 5 where all the di ’s are set negative (di = dm < 0). (ii) The hopping term of electrons in a conventional model like the Hubbard model is written as † Ht = −t (A.11) Cx+j,σ Cx,σ + H.c. x,j,σ
This corresponds to the case that the average of Uxj is Uxj = i. This motivates us to introduce Uxj,new via Uxj,old = iUxj,new . In terms of Uxj,new , the coefficients of A(W ) read cu = c2 ρU4 ,
cv = c2 ρV4 ,
cm = −c2 ρU2 ρV2 ,
di = c2 ρU2 ρV2 .
(A.12)
Thus all the di ’s become positive (for positive c2 ), but cm becomes negative. These considerations show that the concrete model considered in this appendix fix the parameters of the action AGL of the GL theory (2.4) introduced in the text. The MC study of the GL theory in the text surveys more general parameter regions. References [1] B.I. Halperin, T.C. Lubensky, S. Ma, Phys. Rev. Lett. 32 (1972) 292; See also, S. Coleman, E. Weinberg, Phys. Rev. D 7 (1973) 1888.
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[2] M.N. Chernodub, E.-M. Ilgenfritz, A. Schiller, Phys. Lett. B 547 (2002) 269; S. Wenzel, E. Bittner, W. Janke, A.M.J. Schakel, A. Schiller, Phys. Rev. Lett. 95 (2005) 051601, and references cited therein. [3] F. Steglich, J. Aarts, C.D. Brell, W. Lieke, D. Meschede, W. Franz, H. Schafer, Phys. Rev. Lett. 43 (1979) 1892. [4] J.G. Bednorz, K.A. Mueller, Z. Phys. B 64 (1986) 189. [5] V.P. Mineev, K.V. Samokhin, Introduction to Unconventional Superconductivity, Gordon and Breach Science Publishers, 1999. [6] M. Sigrist, K. Ueda, Rev. Mod. Phys. 63 (1993) 308. [7] Recently, M.N. Chernodub, E.-M. Ilgenfritz, A. Schiller, Phys. Rev. B 73 (2006) 100506, studied a related gauge theory, where a doubly-charged scalar field on sites is introduced instead of the V -field in the present paper. [8] I. Ichinose, T. Matsui, Phys. Rev. B 51 (1995) 11860; I. Ichinose, T. Matsui, M. Onoda, Phys. Rev. B 64 (2001) 104516. [9] In the hopping expansion [8], dm > 0 for small δ’s and dm < 0 for δ 0.15. In the perturbative analysis of M. Onoda, I. Ichinose, T. Matsui, J. Phys. Soc. Jpn. 69 (2000) 3497, dm < 0 for δ 0. [10] R.P. Feynman, Statistical Mechanics, A Set of Lectures, W.A. Benjamin, 1972, Chapter 8. [11] A.M. Polyakov, Nucl. Phys. B 120 (1977) 429. [12] T.A. DeGrand, D. Toussaint, Phys. Rev. D 22 (1980) 2478; R.J. Wensley, J.D. Stack, Phys. Rev. Lett. 63 (1989) 1764. [13] S. Takashima, I. Ichinose, T. Matsui, Phys. Rev. B 72 (2005) 075112. [14] I. Ichinose, T. Matsui, K. Sakakibara, in preparation. [15] S. Sachdev, Quantum Phase Transitions, Cambridge Univ. Press, Cambridge, 1999. [16] M.P.A. Fisher, G. Grinstein, Phys. Rev. Lett. 60 (1988) 208. [17] M. Franz, A.P. Iyengar, Phys. Rev. Lett. 96 (2006) 047007. [18] E. Dagotto, E. Fradkin, A. Moreo, Phys. Rev. B 38 (1988) 2926.
Phase structure of a U(1) lattice gauge theory with dual gauge fields Tomoyoshi Ono a , Yuki Moribe a , Shunsuke Takashima a , Ikuo Ichinose a,∗ , Tetsuo Matsui b , Kazuhiko Sakakibara c a Department of Applied Physics, Nagoya Institute of Technology, Nagoya 466-8555, Japan b Department of Physics, Kinki University, Higashi-Osaka 577-8502, Japan c Department of Physics, Nara National College of Technology, Yamatokohriyama 639-1080, Japan
Received 3 August 2006; accepted 6 December 2006 Available online 3 January 2007
Abstract We introduce a U(1) lattice gauge theory with dual gauge fields and study its phase structure. This system is partly motivated by unconventional superconductors like extended s-wave and d-wave superconductors in the strongly-correlated electron systems and also studies of the t–J model in the slave-particle representation. In this theory, the “Cooper-pair” (or RVB spinon-pair) field is put on links of a cubic lattice due to strong on-site repulsion between original electrons in contrast to the ordinary s-wave pair field on sites. This pair field behaves as a gauge field dual to the U(1) gauge field coupled with the hopping of electrons or quasi-particles of the t–J model, holons and spinons. By Monte Carlo simulations we study this lattice gauge model and find a first-order phase transition from the normal state to the Higgs (superconducting) phase. Each gauge field works as a Higgs field for the other gauge field. This mechanism requires no scalar fields in contrast to the ordinary Higgs mechanism. An explicit microscopic model is introduced, the lowenergy effective theory of which is viewed as a special case of the present model. © 2006 Elsevier B.V. All rights reserved.
1. Introduction The Ginzburg–Landau (GL) theory has proved itself a powerful tool to describe the phase transitions and low-energy excitations of conventional s-wave superconductors. In field-theory * Corresponding author.
E-mail address: [email protected] (I. Ichinose). 0550-3213/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2006.12.004
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terminology, the GL theory takes a form of Abelian Higgs model (AHM), and its phase structure has been studied by field-theoretical techniques and Monte Carlo (MC) simulations of lattice gauge theory. These studies are partly motivated by the work of Halperin, Lubensky, and Ma [1] which predicts a first-order phase transition. At present, it is established that the phase structure of three-dimensional (3D) AHM on the lattice strongly depends on a parameter controlling fluctuations of the amplitudes |ϕ(x)| of the Higgs (Cooper-pair) field [2]. At the London limit in which |ϕ(x)| is fixed, there is only the confinement phase in the lattice model. As the fluctuations of |ϕ(x)| are increased, a second-order phase transition to the Higgs phase appears, and for further fluctuations, the transition becomes of first-order. In recent years, it becomes clear that phase transitions in certain strongly-correlated electron systems have an order parameter which has an “unconventinal” symmetry like d-wave as a result of, e.g., the strong on-site Coulomb repulsion. For example, some strongly-correlated electron systems exhibit d-wave superconductivity at low temperatures (T ). The first d-wave superconductor CeCu2 Si2 was discovered in 1979 [3]. In 1986, the cuprate high-Tc superconductors were discovered [4], and later, it was found that they are d-wave superconductors. In order to describe the d-wave superconducting phase transition by a GL theory defined on a lattice, the Cooperpair field must be put on links because it changes its sign under a π/2 rotation. It is interesting and also important to set up and study the GL theory of the unconventional superconductivity (UCSC) like d-wave SC [5]. In the framework of weak-coupling theory, such studies already have appeared [6]. However, the strong-coupling region remains to be studied [7]. Another example of the appearance of the order with the d-wave symmetry is the spin-gap state in the high-Tc cuprates above the SC state. In the t–J model in the slave-boson representation, the spin-gap state is described by the condensation of the resonating-valence-bond (RVB) amplitude of the spinon-pair link field. Low-T effective theory similar to the above GL theory for the UCSC is obtained by integrating over the holon and spinon fields. In this paper, motivated by the above examples of the strongly-correlated electron systems, we shall introduce a GL theory with order-parameter field defined on links and study its phase structure by means of MC simulations.We shall see that the order parameter, a bilocal field, is regarded as a gauge field, and the knowledge and method of gauge theory are useful to study this GL lattice gauge theory. We find that this new type of gauge theory has a very interesting phase structure. The present paper is organized as follows. In Section 2, we shall introduce the GL gauge theory and explain its relation to the UCSC and the t–J model. In Section 3, we show the phase diagram obtained by a mean-field theory. Sections 4 and 5 are main part of this paper, and show the results of the numerical study for two typical parameter regions. The model exhibits various phases which include the SC or spin-gap phase. In Section 6, quantum phase transition at T = 0 is studied. Section 7 is devoted for conclusion. In Appendix A, we give a microscopic model of strongly-correlated electrons, the low energy effective theory of which is viewed as a special case of the present GL theory. 2. Lattice gauge model with dual gauge fields Let us first introduce the GL gauge theory on a three-dimensional (3D) lattice and then we shall explain its relation to the UCSC and the t–J model. We put a “Cooper-pair field” Vxj on the link (x, x + j ) of the lattice, where x is the site index and j (= 1, 2, 3) is the direction index (it also denotes the unit vector of the j th direction). Vxj is related to fermionic quasi-particles
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Fig. 1. Action AGL of the GL theory (2.4).
like the electrons in the UCSC or the spinons in the t–J model as Vxj ∝ ψx↑ ψx+j,↓ − ψx↓ ψx+j,↑ ,
(2.1)
where ψxσ is the fermion operator at x with spin σ =↑, ↓. In the rest of paper, we focus on the v ). London limit of the Vxj and put |Vxj | = 1, i.e., Vxj = exp(iθxj There is another link field Uxj , a compact U(1) gauge field, which appears in the hopping of ψxσ as † ψx+j,σ Uxj ψxσ ,
(2.2)
u iθxj
where Uxj = e . We require the invariance of the system under a local gauge transformation ψxσ → eiϕx ψxσ . Under this transformation, Vxj and Uxj transform as Vxj → eiϕx+j Vxj eiϕx ,
Uxj → eiϕx+j Uxj e−iϕx .
(2.3)
It is easily seen that under the replacement ϕx → −ϕx (x ∈ odd sites) Vxj transforms just like the original Uxj and vice versa. As the action of the GL theory AGL must respect the local gauge invariance under Eq. (2.3), it is straightforward to “derive” AGL in the local expansion as 1 cu U 4 + cv V 4 + cm (U V U V + V U V U ) AGL = 2 pl + (d1 U U V V + 3 permutations) + c.c., (2.4) where cu , etc. are effective parameters and some of them are increasing functions of 1/T . Each term in AGL is depicted in Fig. 1. For example, the U 4 term stands for the usual plaquette term † † Ux+i,j Ux+j,i Uxj of lattice gauge theory. The partition function Z is given by Uxi Z = [dU ][dV ] exp(AGL ), (2.5) u /2π , etc. We consider a 3D cubic lattice of the size L3 with the periodic where [dU ] ≡ xj dθxj boundary condition. (The results given below are for L = 16 and 24.) We also note that the system has a symmetry Z(cu , cv , cm , dm ) = Z(cv , cu , cm , dm ) for the case d1 = · · · = d4 ≡ dm , corresponding to the interchange Uxj ↔ Vxj . Let us explain how AGL is related to the UCSC and the t–J model. For the UCSC, ψxσ is nothing but the electron operator Cxσ itself, and then “condensation of Vxj ” means the SC phase transition. We consider the compact U(1) gauge field Uxj , which corresponds to the electromagnetic field, instead of the ordinary noncompact one, because the studies on the compact
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gauge models reveal also the phase structure of the noncompact models. (See later discussion.) At present it is believed that in the underdoped region of the high-Tc materials the SC phase transition is a sort of the Bose–Einstein condensation (BEC) of the Cooper-pair field Vxj , i.e., Vxj has a finite amplitude and its phase fluctuation induces the SC phase transition. Therefore taking the London limit is a good approximation. A spatial lattice introduced in order to set up the GL theory and study its strong-coupling region should be regarded as a “coarse grained” lattice although some of the UCSC are granular systems. In Appendix A, an explicit microscopic model is introduced, and its low-energy effective theory is derived. It can be viewed as a special case of the present GL gauge model on a lattice. Let us turn to the t–J model of the high-Tc superconductors. The Hamiltonian of the t–J model on a two-dimensional square lattice is given by HtJ = −t
2 1 † ˜ ˜ Cx+i,σ Cxσ + H.c. + J Sx · Sx+i − nx nx+i , 4 x
2 x
i=1 σ =↑,↓
(2.6)
i=1
† † Cx,−σ )Cxσ , nx ≡ σ Cxσ Cxσ , and where i = 1, 2 is the spatial direction, C˜ xσ ≡ (1 − Cx,−σ τ are the Pauli spin matrices). The physical states are restricted to those satisfyS x = Cx† τ Cx /2 ( ing the local constraint, † Cxσ Cxσ |phys 1. phys| (2.7) σ
In the slave-boson representation, the electron operator is expressed as Cxσ = bx† fxσ , where bx is a bosonic holon operator and fxσ is a fermionic spinon operator. The above constraint becomes equivalent to the equality
† † (2.8) fxσ fxσ + bx bx − 1 |phys = 0. σ
In this representation, there appear quartic terms b† bf † f and f † f † ff in HtJ . We adopt the path-integral method of the imaginary-time formalism, and decouple these terms by introducing auxiliary complex fields as [8]
† χxi J (2.9) fx+i,σ fxσ + tbx+i bx† , Dxi fx+i,↑ fx↓ − fx+i,↓ fx↑ , σ
where χxi is the holon and spinon hopping amplitude and Dxi is the spin-gap amplitude. It is then straightforward to see that phase degrees of freedom of χxi and Dxi correspond to the gauge fields Uxi and Vxi as χxi = |χxi |Uxi ,
Dxi = |Dxi |Vxi ,
(2.10)
under a “gauge transformation” (fxσ , bx ) → eiαx (fxσ , bx ) [see Eq. (2.3)]. There is also the timecomponent Uxj =3 , which works as the Lagrange multiplier for the local constraint (2.8). The gauge dynamics of the composite gauge fields Uxμ and Vxi strongly influences lowenergy excitations in the t–J model. In particular, in the deconfinement phase of Uxμ , holons and spinons acquire independent hopping amplitudes (∝ Uxμ ), so the phenomenon of chargespin separation (CSS) takes place. In the deconfinement phase of Vxi , the RVB amplitude Vxi develops, inducing a spin gap.
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The effective gauge theory for the CSS and the spin-gap state is obtained by integrating holon and spinon variables in the path-integral representation (we do not think the possibility of the Bose condensation of holons). From the gauge invariance its action should take the form of AGL of Eq. (2.4) with the third direction as the imaginary time. By making the hopping expansion at finite T [8], the parameters in the AGL are estimated as cu ∼ (J 2 + t 2 δ)T −3 , cv ∼ T −4 , cm , d1 , . . . , d4 ∼ J 2 T −4 where δ is the hole concentration [9].1 After obtaining the phase structure of the present gauge model, we shall comment on its implication to the physics of the t–J model more precisely. 3. Mean-field theory Before going into details of the numerical simulations, we first study the model by the meanfield theory (MFT) for dm = 0. We choose the action A0 of MFT as the following single-link form: 1 (λU Uxj + λV Vxj + c.c.). A0 = (3.1) 2 xj
The partition function Z0 and the free energy F0 of MFT is given by Z0 = [dU ][dV ] exp(A0 ) = I0 (λU )3N I0 (λV )3N ≡ exp(−F0 ), F0 = −3N log I0 (λU )I0 (λV ) ,
(3.2)
where N is the number of the sites and I0 (a) is the modified Bessel function (n = 0) defined by 2π In (a) =
dθ exp a cos(θ ) + inθ . 2π
(3.3)
0
The MFT is based on the following variational principle (Jensen inequality) [10]: F ≡ − log Z, F Fv ≡ F0 + A0 − A0 , −1 [dU ][dV ]O exp(A0 ). O0 ≡ Z0
(3.4)
Then we minimize Fv (λU , λV ) by optimizing the variational parameters λU and λV . The minimum of Fv is regarded as the best approximation of F . Fv is calculated explicitly as Fv /(3N ) = − log I0 (λU ) − log I0 (λV ) − cu m4U − cv m4V − 2cm m2U m2V + λU mU + λV mV , I1 (λU ) I1 (λV ) , mV ≡ Vxj 0 = , mU ≡ Uxj 0 = I0 (λU ) I0 (λV ) 2π dθ cos θ exp(a cos θ ). I1 (a) = (3.5) 2π 0
1 At finite T , the gauge fields U and V must satisfy the periodic boundary condition with respect to the imaginary xj xj time, but the coefficients cu , etc. in AGL depend on T more strongly than the fields and they give the leading contribution to the phase transition.
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Fig. 2. Phase structure for dm = 0 in the mean-filed theory. It is symmetric about the axis cu = cv . The symbols C, U , V , UV indicate the confinement phase, U phase, V phase, and UV phase, respectively.
The solutions of λU , λV are classified into the following four cases: (1) confinement phase: (2) U -phase: (3) V -phase: (4) UV-phase:
λU λU λU λU
= 0, λV = 0, λV = 0, λV = 0, λV
= 0 → Uxj 0 = 0, Vxj 0 = 0, = 0 → Uxj 0 = 0, Vxj 0 = 0, = 0 → Uxj 0 = 0, Vxj 0 = 0, = 0 → Uxj 0 = 0, Vxj 0 = 0.
(3.6)
In Fig. 2, we show the phase diagram in the cu –cv plane for the fixed values of cm = 0, 0.3, 0.6, and 1.0 (dm = 0). We note that all the phase transitions studied here are of first order, i.e., the internal energy E and (at least one of) λU and λV has a jump across the boundary. In Fig. 3, we plot a typical first-order behavior of λU and λV across the transition point. 4. Extended s-wave Higgs phase (dm = 0 and cm > 0) Let us turn to the numerical studies. For vanishing cm and di (i = 1, . . . , 4), the two gauge fields Uxj and Vxj decouple with each other and the system reduces to two 3D pure U(1) gauge systems. As no phase transition takes place in the 3D U(1) pure gauge theory [11], the present system for cm = di = 0 (i = 1, . . . , 4) has only a single (confinement) phase. Then we assign various nonvanishing values to cm and/or di , and determine the phase structure in the cu –cv plane for fixed cm and di . Hereafter we often consider the symmetric case di = dm (i = 1, . . . , 4). Below we present the results for two typical cases: (i) dm = 0 and cm > 0 and (ii) cm = 0 and dm = di < 0.
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Fig. 3. (a) Free energy Fv /(3N ) vs. λU = λV for cu = cv , cm = 0.5, dm = 0 near the transition point cu = 1.323. (b) U vs. cu = cv for cm = 0.5, dm = 0.
Let us first consider the case (i), i.e., the extended s-wave Higgs phase with the “spherical symmetry” (i.e., the symmetry under a π/2-rotation). To study the phase structure of the model, we measure the internal energy E and the specific heat C (fluctuation of E) by the MC simulations, E ≡ −AGL /L3 ,
2 C ≡ AGL − AGL /L3 .
(4.1)
We considered the cases dm = 0, cm = 0.2, 0.4, 0.6, 0.8, 1.0, and 1.2. In Fig. 4(a) and (b) we show E and C for dm = 0, cm = 0.6 and cv /cu = 0.1. At cv ∼ 1, E shows a hysteresis, which implies a first-order phase transition. The cm term works as a “Higgs coupling” of the “Higgs” field Vxj (Uxj ) to the gauge field Uxj (Vxj ) to stabilize their fluctuations and induce such a transition [8]. Thus the transition is expected from the confinement phase where Uxj and Vxj fluctuate violently to the Higgs (superconducting) phase where their fluctuations are small. The data for cm 0.6 show signals of first-order phase transitions, while the data of cm 0.4 show no signals of phase transitions. In Fig. 4(c) we show the phase diagram in the cu –cv plane for cm = 0.6. Similar phase diagram is obtained for cm = 0.8–1.2. To confirm the above interpretation of each phase, we measured instanton densities. We consider two kinds of instantons, i.e., U -instantons and V -instantons, and denote their average densities per cube as ρU and ρV , respectively. For the U -instantons we employ the definition given in Ref. [12], which measures magnetic fluxes emanating from each smallest cube. Similar gauge-invariant definition is possible for the V -instantons.2 In Fig. 4(d) we plot ρU and ρV . ρV shows a discontinuity at the first-order transition point cv ∼ 1.0 just like E, and decreases very rapidly as cv increases. On the other hand, ρU decreases very rapidly and almost vanishes for cv > 0.2. This result indicates that the small cusp in C of Fig. 4(b) at cv ∼ 0.17 reflects a crossover from the dilute to dense instanton “phases” of the gauge field Uxj [11–13]. Then we conclude that the system changes from the confinement phase to the Higgs phase as cv (cu ) increases. 2 To define the V -instantons we consider the replacement ϕ → −ϕ for the odd sites in Eq. (2.3). Then V transforms x x xj just like Uxj before the replacement, so it is straightforward to define the V -instantons.
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Fig. 4. Results for dm = 0, cm = 0.6. (a) Internal energy E for cv /cu = 0.1 shows a hysteresis around cv ∼ 1.0, a signal of first-order transition. (b) Specific heat C for cv /cu = 0.1 shows a small cusp at cv ∼ 0.17, which is interpreted as a crossover (see the text). (c) Phase structure determined by E and C for L = 16. The first-order transition line separates the confinement phase and the Higgs phase. The cross symbols denote crossover. (d) Instanton densities ρU and ρV for cv /cu = 0.1. ρV exhibits a discontinuity at cv ∼ 1.0, while ρU decreases rapidly at cv ∼ 0.17.
In order to support the above conclusions, we also calculated expectation values of the U and V -Wilson loops, WU (Γ ) = (4.2) U , WV (Γ ) = V , Γ
Γ
where Γ is a closed loop on the lattice, and the products of Uxj and Vxj in Eq. (4.2) are formed to be gauge-invariant. From the above results of instanton densities, we expect that WU (V ) (Γ ) obey the area law in the instanton-plasma phase for small cu(v) and the perimeter law in the instanton-dipole phase for large cu(v) . In calculating WU (V ) (Γ ), we consider various shapes of Γ . For example, we take Γ ’s having a fixed area and various perimeters, and vice versa. In Fig. 5 we show the results for cm = 0.6. In the case cu = cv = 1.4, WU (Γ ) fits the area law, while the case cu = cv = 2.0 fits the perimeter law. These results support the previous conclusion obtained from instanton densities. Because all the coefficients in AGL are positive in the present case, we expect that the observed “ordered” state is an extended s-wave Higgs phase (superconductor). We calculated U U V V in order to verify this expectation for cm 0.6 and dm = 0 cases and found that U U V V has vanishing value in the normal state whereas it has a finite value in the Higgs phase showing hysteresis loop. Furthermore its value takes a negative as well as positive value depending on samples. Turning on a small but finite positive dm term, U U V V becomes positive. This result indicates that positive dm term is necessary to produce a genuine extended s-wave Higgs phase (superconductor). In this section, we have reported the results of the MC simulations for the compact U(1) gauge theory with dual gauge fields in which Uxj as well as Vxj is the compact gauge field. In this
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Fig. 5. Wilson loops for dm = 0, cm = 0.6 with cu = cv , which means WU (Γ ) = WV (Γ ). (a) The case cu = cv = 1.4 fits the area law. (b) The case cu = cv = 2.0 fits the perimeter law rather than the area law. Fig. 4(c) indicates that the transition between the two laws occurs at cu = cv ∼ 1.6.
system, the phase transition is interpreted as the one from the confining to Higgs phases. Behavior of the Wilson loops and the instanton densities support this conclusion. From the numerical calculations, we can readily expect that there exists a Coulomb–Higgs phase transition in the system of the noncompact U(1) gauge field Uxj for sufficiently large cm . In fact as it is seen from Fig. 4(d), the U -instanton density is almost vanishing for cu > 2.5 (cv > 0.25) and therefore Uxj behaves substantially like the noncompact gauge field with small fluctuations in that parameter region. Sufficiently large cm (dm ) terms with the noncompact Uxj suppress the phase fluctuations of Vxj because the following terms effectively appear, † cm (4.3) Uxi Ux+j,i Vxj Vx+i,j + ···, xj
and a phase transition to the Higgs phase occurs as cv increases. 5. d-wave Higgs phase (cm = 0 and dm < 0) We expect and verified that the dm -term with dm < 0 enhances d-wave condensation of Vxj ; † Vxj < 0. For large cu , the fluctuations of Uxj are suppressed as Uxj ∼ 1 Δij ≡ Uxi Ux+i,j Vx+j,i [up to the gauge transformation Eq. (2.3)], hence the d-wave configuration Vxi† Vx+i,j < 0 (i = j ) is preferred, although there are no configurations with all negative Δij in three dimensions. We considered the cases dm = −0.4, −0.6, −0.8, −1.0, and measured E, C, ρU , and ρV as before. No signals of phase transitions are found for dm = −0.4 and −0.6, whereas signals of phase transitions to the d-wave Higgs phase are obtained for dm < −0.6. In Fig. 6(a) and (b) we present E and ρ for cu /cv = 1.0 and dm = −0.8. E and ρ show three first-order phase transitions along cv = cu . In Fig. 6(c) we present the phase structure determined by the measurement of E. There are four phases (I)–(IV). The phase (I) is the confinement phase. (II) is the “staggered state” which is generated by the frustration of strong negative d-term. It breaks the translational symmetry by the unit lattice spacing as supported by the Wilson loop of Fig. 6(d) which has two branches, one for even areas and one for odd areas. (III) is the disorder state connecting (II) and (IV). (IV) is the Higgs phase corresponding to d-wave superconductor. These interpretations are consistent with the behavior of ρ in Fig. 6(b). The measurement of Δij shows that the cubic symmetry Δ23 = Δ12 = Δ13 is maintained in (IV), whereas it is reduced to the square symmetry, e.g., Δ23 < Δ12 = Δ13 in (II).
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Fig. 6. Results for cm = 0, dm = −0.8. (a) E and (b) ρU = ρV for cu /cv = 1.0. They show three hysteresis loops. (c) Phase structure in the cu –cv plane. (d) Wilson loop of M × M square for cu = cv = 0.9 shows two smooth branches for even M and odd M, i.e., staggered perimeter law.
We also studied an anisotropic system in which the intraplane dm term is larger than the interplane dm term because many of the real materials have a layered structure. In Fig. 7 we show E and the instanton density for cu /cv = 1.0, cm = 0, intraplane dm = −1.0 and interplane dm = −0.8. There is a first-order phase transition near cu = cv = 1.7, whereas the other transitions existing in the isotropic case in Fig. 6 disappeared. Thus the phase (I) and (II) disappear whereas the two phases (III) and (IV) survive. This phase transition in the anisotropic case should correspond to the SC transition observed in the high Tc cuprates, the heavy-fermion materials, etc. As explained in Section 2, the couplings cu etc are increasing functions of 1/T in the effective gauge theory of the t–J model. The result of the present section showing the existence of the Higgs phase for sufficiently large couplings indicates that the spin gap Δ develops at low T as observed in the experiments. More precise estimation of the couplings cu etc as a function of t , J , δ and T is required to identify the spin-gap region in the phase diagram of the t–J model [14]. 6. Quantum phase transition We have considered the gauge model AGL (2.4) defined on the cubic lattice and studied its phase structure. From the viewpoint of the UCSC, the phase transitions which we found in the previous discussion correspond to thermal phase transitions, i.e., the coefficients in AGL are increasing functions of 1/T and the SC (Higgs) phase appears as T is lowered. Recently, in the studies of the strongly-correlated electron systems like the high-Tc cuprates and the heavyfermion materials, significance of quantum phase transition (QPT) has been recognized [15]. In particular, the QPT in ordinary s-wave charged SC was studied by using a XY model coupled with a U(1) gauge field [16] and very recently this model was applied for the QPT in the high-Tc
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Fig. 7. (a) Internal energy for an anisotropic system. Parameters are cu /cv = 1, cm = 0, intraplane dm = −1.0 and interplane dm = −0.8. There is a first-order phase transition near cu = cv = 1.7. (b) Instanton density also shows hysteresis loop near cu = cv = 1.7.
q
Fig. 8. E and instanton density of the quantum system AGL with (cu , cv , cm , dm ) = g(1, 1, 1, 0). They exhibit first-order phase transition.
cuprates in order to explain the anomalous behavior of the superfluid density near the quantum critical point (QCP) at T = 0 [17]. Quantum theory of the present GL theory can be constructed straightforwardly. In the continuum imaginary-time formalism, Uxj and Vxj depend on the imaginary time τ . Then we discretize the imaginary-time axis and define the quantum GL theory on the 4D hypercubic lattice. The acq tion AGL of the quantum system has a similar form of AGL in Fig. 1, but it is defined on the 4D lattice. It involves the time component Uxτ but does not contain the terms including Vxτ as the Cooper-pair field lives on links of the spatial lattice. q We also studied this quantum system AGL by the MC simulations. In the practical experiments, external conditions and properties of samples like external pressure, doping parameter, q etc., change the effective parameters contained in AGL . Here we consider typical two cases in q which the coefficients in AGL are scaled as (cu , cv , cm , dm ) = g(1, 1, 1, 0) and g(1, 1, 1, −1) where g is a positive parameter. In Fig. 8, we show E and the instanton density for the former case. The result shows a first-order phase transition from the normal phase to the SC phase. We found that the other case also has a similar phase structure. From this numerical result, we can study the SC QPT more precisely by analytical methods in which a complex link field ψxj (τ ) is introduced as a SC “order parameter” field by the Hubbard–Stratanovich transformation [16]. The results will be published in a future publication.
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7. Conclusion In this paper, we have introduced the GL gauge theory with the “order-parameter field” put on links. This system is partly motivated by the observed phase transitions with the d-wave order parameter in the condensed matter physics which include the d-wave SC and the spin-gap state. We showed that the present GL theory can be regarded a new type of lattice gauge model which contains two kinds of gauge fields. By means of MC simulations, we clarified its phase structure. For the case of cm > 0, dm = 0, there are two phases for sufficiently large cm , which correspond to the normal and Higgs phases. They are separated by a first-order phase transition line. On the other hand, for the case of cm = 0, dm −0.8 there are four phases in the isotropic case. Only the two of them survive in the anisotropic case which corresponds to the layered structure. We also studied the QPT at T = 0 in the present model and found that the first-order phase transition also occurs as the parameters of the Hamiltonian are changed. It is quite interesting and also important to apply the obtained results for the UCSC and the spin-gap state in the t–J model and compare the results with experiments. This is under study and the result will be reported in a future publication. Acknowledgements We are very grateful to M.N. Chernodub for several stimulating discussions. Appendix A. A model of UCSC and the dual gauge theory In this appendix we consider a model of strongly-correlated electron system on a 3D lattice as a typical example of the UCSC. Let us consider the following Hamiltonian written in terms of the electron operator Cxσ at the site x and spin σ (= 1, 2): HJ K = HJ + HK ,
† 1 HJ = J Rxj Rxj , Sx+j Sx − nx+j nx = −J 4 x,j
x,j
1 † † Cxσ σ σ σ Cxσ , nx = Cxσ Cxσ , S x = 2 σ σ,σ
1 Rxj = √ σ σ Cx+j,σ Cxσ , 2 σ,σ
† 1 Qxj Qxj , HK = K S x+j S x + nx+j nx = −K 4 x,j
Qxj
1 † =√ Cx+j,σ Cxσ . 2 σ
xj
(A.1)
The first term HJ favors formations of the RVB at nearest-neighbor (NN) pair (we consider the AF spin coupling J > 0). Rxj is the annihilation operator of RVB on the link (x, x + j ) (j = 1, 2, 3). The second term HK represents the AF coupling and the repulsion between the NN pair of electrons (K > 0). We note that a pair of spin-singlet two electrons at the same site has the energy zero while the singlet pair at the NN sites has the energy −J − K/2. Thus two electrons prefer to sitting on the NN sites instead of on the same site.
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In path-integral representation of the partition function Z = Tr exp(−βH ) (β = (kT )−1 ), Z is rewritten for small J and K by introducing a pair of U(1) fields Uxj , Vxj as Z= L=−
β
[dU ][dV ][dC] exp
dτ L , 0
C¯ xσ (∂τ − μ)Cxσ − H ,
xσ
√ √ ( J Vxj R¯ xj + KUxj Qxj − H.c.), H =i
(A.2)
x,j
where μ is the chemical potential controlling the electron density as nx = n. One may prove this expression directly by expanding the exponent in powers of J and K and integrate over Uxj and Vxj . Eq. (A.2) is rewritten in terms of the SU(2) field Wxj ,
χxj , Dxj Wxj = , −D¯ xj , χ¯ xj W † Wxj = 1, (A.3) det Wxj = 1, χ¯ xj χxj + D¯ xj Dxj = 1, xj
and the fermion doublet,
Cx1 ψx = , † Cx2 as
dτ Lψ,W , Z = [dW ][dψ] exp √ (ψ¯ x+j Wxj ψx − H.c.) + L, −ψ¯ x (∂τ − μσ3 )ψx − i J + K Lψ,W = x
L = −γ
2 |χxj | − ρU ,
ρU ≡
xj
(A.4)
j
K , J +K
(A.5)
with the limit γ → ∞. [dW ] is the SU(2) Haar measure. In fact in this limit, one has J |Dxj | → ρV ≡ , J +K Dxj → ρV Vxj = ρV exp(iθV xj ), χxj → ρU Uxj = ρU exp(iθU xj ).
(A.6)
Thus Eq. (A.2) is recovered. Except for the chemical potential term and L, the system has the SU(2) gauge invariance under ψx → Ωx ψx ,
Wxj → Ωx+j Wxj Ωx† ,
Ωx ∈ SU(2).
(A.7)
The case of L = 0, i.e., γ = 0 is discussed in Ref. [18]. The Hamiltonian in this case is H ∝ − xj S x+j S x , which corresponds to V = J in Eq. (A.1).
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By integrating over ψxσ we obtain an effective gauge theory for general γ . The partition function Z is given by Z = [dW ] exp A(W ) , A(W ) = Tr Log(∂τ − μσ3 + W ). (A.8) By making a local(hopping) expansion of A(W ) in powers of W we obtain the action AGL of the Ginzburg–Landau theory, † A(W ) = c0 + c2 (A.9) Tr Wxi Wx+i,j Wx+j,i Wxj + H.c. + O W 6 , x,j
where we consider high-T region and the T -dependence is included in the coefficients c0 , c2 , . . . . In the limit γ → ∞, one obtains a U(1) lattice gauge theory with a pair of dual compact U(1) gauge variables Uxj , Vxj . In fact, by calculating the c2 term of Eq. (A.9) explicitly, it becomes equivalent to the AGL model with the coefficients, K2 , (J + K)2 J2 cv = c2 ρV4 = c2 , (J + K)2 JK cm = c2 ρU2 ρV2 = c2 , (J + K)2 JK d1 = −c2 ρU2 ρV2 = −c2 , (J + K)2 d2 = −d1 , d3 = d 1 , d4 = −d1 . cu = c2 ρU4 = c2
(A.10)
We note the following two points concerning to the signatures of these coefficients: (i) The four dmα ’s have alternative signatures in contrast with the choice of Section 5 where all the di ’s are set negative (di = dm < 0). (ii) The hopping term of electrons in a conventional model like the Hubbard model is written as † Ht = −t (A.11) Cx+j,σ Cx,σ + H.c. x,j,σ
This corresponds to the case that the average of Uxj is Uxj = i. This motivates us to introduce Uxj,new via Uxj,old = iUxj,new . In terms of Uxj,new , the coefficients of A(W ) read cu = c2 ρU4 ,
cv = c2 ρV4 ,
cm = −c2 ρU2 ρV2 ,
di = c2 ρU2 ρV2 .
(A.12)
Thus all the di ’s become positive (for positive c2 ), but cm becomes negative. These considerations show that the concrete model considered in this appendix fix the parameters of the action AGL of the GL theory (2.4) introduced in the text. The MC study of the GL theory in the text surveys more general parameter regions. References [1] B.I. Halperin, T.C. Lubensky, S. Ma, Phys. Rev. Lett. 32 (1972) 292; See also, S. Coleman, E. Weinberg, Phys. Rev. D 7 (1973) 1888.
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