Lattice Gauge Theory for Condensed Matter Physics: Ferromagnetic Superconductivity as its Example
aa r X i v : . [ c ond - m a t . s t r- e l ] S e p October 29, 2018 5:51 WSPC/INSTRUCTION FILE mplb20140730
Modern Physics Letters Bc (cid:13)
World Scientific Publishing Company
LATTICE GAUGE THEORY FOR CONDENSED MATTERPHYSICS: FERROMAGNETIC SUPERCONDUCTIVITY AS ITSEXAMPLE
IKUO ICHINOSE
Department of Applied Physics, Nagoya Institute of Technology,Nagoya, 466-8555, Japan
TETSUO MATSUI
Department of Physics, Kinki University,Higashi-Osaka, 577-8502, Japan
Received (Day Month Year)Revised (Day Month Year)Recent theoretical studies of various strongly-correlated systems in condensed matterphysics reveal that the lattice gauge theory(LGT) developed in high-energy physics isquite a useful tool to understand physics of these systems. Knowledges of LGT are tobecome a necessary item even for condensed matter physicists. In the first part of thispaper, we present a concise review of LGT for the reader who wants to understand itsbasics for the first time. For illustration, we choose the abelian Higgs model, a typical andquite useful LGT, which is the lattice verison of the Ginzburg-Landau model interactingwith a U(1) gauge field (vector potential). In the second part, we present an account ofthe recent progress in the study of ferromagnetic superconductivity (SC) as an exampleof application of LGT to topics in condensed matter physics, . As the ferromagnetism(FM) and SC are competing orders with each other, large fluctuations are expected totake place and therefore nonperturbative methods are required for theoretical investiga-tion. After we introduce a LGT describing the FMSC, we study its phase diagram andtopological excitations (vortices of Cooper pairs) by Monte-Carlo simulations.
Keywords : Lattice gauge theory; Strongly-correlated systems; Ginzburg-Landau theory;Ferromagnetic superconductivity; Monte-Carlo simulation; Path-integral
1. Introduction
A strongly-correlated system is a set of either fermions or bosons which interactstrongly each other. Such a system may exhibit collective behaviors that cannot beexpected from the behaviors of each single particle, and has been the subject of greatinterest in various fields of condensed matter physics. For example, the t − J model,which is a typical model of electrons interacting via strong Coulomb repulsion, istaken as a standard model of high-temperature superconductors1. Another exampleis the Bose Hubbard model3 and the related bosonic t − J model4, which are tobe used to study cold bosonic atoms put on an optical lattice5 and tuned to have ctober 29, 2018 5:51 WSPC/INSTRUCTION FILE mplb20140730 I.Ichinose and T.Matsui large interaction parameters. In the last two decades, theoretical approach usinggauge theory and the associated concepts has proved itself a convincing way tounderstand essentials of their physics 6 , V ( r )between a pair of static quark and anti-quark separated by the distance r increaseslinearly in r and costs infinite energy to separate them ( r → ∞ ). These three phasesare classified by the magnitude of fluctuations of gauge field (See Table I below).The gauge field put on the lattice can be treated as periodic variables (one calls thiscase a compact gauge field; see Sect.2.1), and the gauge-field configurations reflect-ing this periodicity may involve large spatial variations beyond the conventionalperturbative ones. Monopoles are a typical and important example of such config-urations (monopoles in the continuum gauge theory cost infinite energy and so notpossible). In the confinement phase, such monopoles condense10, and the resultingstrong fluctuations of gauge field are argued to generate squeezed one-dimensionalline of electric flux (variables conjugate the the gauge field itself). Then, in theconfinement phase, such an electric flux gives rise to the linearly-rising confiningpotential. This phenomenon of formation of electric fluxes is sometimes called the“dual” Meissner effect in contrast to the Meissner effect in ordinary superconduc-tors where magnetic field is squeezed into magnetic fluxes due to Cooper-pair con-densation. One way to recognize the relevance of monopoles among other possible(topological) excitations is to perform the duality transformation11, which rewritesthe system as an ensemble of monopoles or monopole loops.hase fluctuation of gauge field potential energy V ( r )Higgs small ∝ exp( − mr ) r Coulomb medium ∝ r Confinement large ∝ r Table 1. Three phases of gauge dynamics: V ( r ) is the potential energy stored between apair of point charges with the opposite signs and separated by the distance r . ctober 29, 2018 5:51 WSPC/INSTRUCTION FILE mplb20140730 Lattice gauge theory for condensed matter physics Also, in some restricted fields of condensed matter physics, LGT has been ap-plied successfully. One example is the t − J model of high-T c superconductivity(SC)6. Here the strong correlations are implemented by excluding the double occu-pancy state of electrons, the state of two electrons with the opposite spin directionsoccupying the same site, from the physical states. This may be achieved by view-ing an electron as a composite of fictitious particles, a so-called holon carrying thecharge degree of freedom of the electron and a spinon carrying the spin degree offreedom12. By the method of auxiliary field in path-integral, one may introduceU(1) gauge field which binds holon and spinon. Then the possible phases may beclassified according to the dynamics of this U(1) gauge field. In the confinementphase, the relevant quasiparticles are the original electrons in which holons andspinons are confined. This corresponds to the state at the overdoped high temper-ature region. In the deconfinement Coulomb phase, quasiparticles are holons andspinons in which charge and spin degrees of freedom behave independently. Thisphenomenon is called the charge-spin separation13 and are to describe the anoma-lous behavior of the normal state such as the liner-T resistivity. Further correspon-dence between the gauge dynamics and the observed phases are also satisfactoryas discussed in Refs.1 , ,
13. In particular, possibility of experimental observation ofspinons is discussed in Ref.2.Another application of LGT to relate the dynamics of auxiliary gauge field andthe observed phases is the (fractional) Hall effect14. Among a couple of parallelformulations15, one may view an electron as a composite of a so called bosonicfluxon carrying fictitious (Chern-Simons) gauge flux and a so called fermionic char-gon carrying the electric charge7. In the deconfinement phase, these fluxons andchargons are separated, i.e., particle-flux separation takes place. If fluxons Bosecondense at sufficiently low temperatures, the resulting system is fermionic char-gons in a reduced uniform magnetic field. These chargons are nothing but the Jain’scomposite fermions, and the integer Hall effect of them explain the fractional Halleffect of the original electron system.In this paper, we shall see yet another example of LGT approach to condensedmatter physics. Namely, we consider a lattice model of ferromagnetic SC, which ex-hibits the ferromagnetism (FM) and/or SC, two typical and important phenomenain condensed matter physics. We note here that the introduction of a spatial latticein the original LGT in high-energy theory is traced back, as mentioned, to the ne-cessity to define a nonperturbative field theory. The lattice spacing there should betaken as a running cutoff(scale) parameter in the renormalization-group theory18and the continuum limit should be taken carefully. However, in condensed matterphysics, introduction of a lattice has often a practical support independent of theabove meaning, i.e., the system itself has a lattice structure. One may naturallyregard the lattice of the model as the real lattice of material in question. In thispaper, we shall not touch the continuum limit in the sense of renormalization group.The layout of the paper is as follows: In Sect. 2, we review LGT. The reader whois not familiar with this subject may acquire necessary knowledge to understandctober 29, 2018 5:51 WSPC/INSTRUCTION FILE mplb20140730 I.Ichinose and T.Matsui r r + j r + i ij r r + i ^(cid:30)r ^Eri ^(cid:18)ri (r; i) (r;j) ^(cid:30)r+i Fig. 1. i − j plane of the 3d lattice and ˆ φ r defined on the site r and ˆ θ ri and ˆ E ri defined on thelink ( ri ) = ( r, r + i ). the successive sections. In Sect. 3, we explain the lattice model of ferromagneticsuperconductivity (FMSC). It is based on the known Ginzburg-Landau theory de-fined in the continuum space. In Sect. 4, we present numerical results of MonteCarlo simulations for the lattice model in Sect. 3. In Sect. 5, we give conclusion.
2. Introduction to lattice gauge thoery
Formulating field-theoretical models by using space(-time) lattice is quite usefulfor studying their properties nonperturbatively. In particular, because of the latticeformulation itself, the high-temperature expansion (strong-coupling expansion) canbe performed analytically even for the case in which fluctuations of the fields arevery strong. Furthermore, numerical studies like the Monte-Carlo simulation canbe applicable for wide range of the lattice field theories. In this section, we reviewLGT consisting of a U(1) gauge field and a charged bosonic matter field.
Lattice gauge theory in Hamiltonian formalism
In this subsection, we explain the Hamiltonian formalism of LGT16. The reasonsare two fold: (i) it gives rise to an intuitive picture of the relevant states in termsof electric field and magnetic field, and (ii) it plays an important role in recentdevelopment in quantum simulations of the dynamical gauge field by using ultra-cold atomic systems put on an optical lattice.Let us start with an example of Hamiltonian system defined on a three-dimensional ( d = 3) spatial cubic lattice with lattice spacing a . Its sites are la-beled by r , and its links (bonds) are labeled as ( r, i ) where i is the direction index i = 1 , ,
3. We sometimes use i also for the unit vector in the i th-direction, such as( r, i ) = ( r, r + i ) (See Fig.1).One may put matter fields on each site r , which may be bosons/fermions com-posing materials in question, e.g., electrons, atoms in an optical lattice, Coope-pairfield of SC, spins, or even more complicated ones. For definiteness, we put belowa one-component canonical nonrelativistic boson field17, whose annihilation andctober 29, 2018 5:51 WSPC/INSTRUCTION FILE mplb20140730 Lattice gauge theory for condensed matter physics creation operators ˆ φ r , ˆ φ † r satisfy [ ˆ φ r , ˆ φ † r ′ ] = δ rr ′ . (2.1)Gauge fields are put on each link ( r, i ) in LGT (See Fig.1). They may be associatedwith the gauge group U(1), SU(N), etc, and meditate interactions between mat-ters. They are viewed as connections in differential geometry that relate two localframes of the internal space of matter field at neighboring lattice sites r and r + i .For definiteness, we consider below a U(1) gauge field operator ˆ θ ri (= ˆ θ † ri ). This isviewed as the lattice version of well-known vector potential field ˆ A i ( r ) (we use thesame letter r for the coordinate of the continuum space) describing the electromag-netic(EM) interaction, but may have different origins, e.g., mediating interactionsbetween holons and spinons, as explained in Section 1. The momentum operatorˆ E ri conjugate to ˆ θ ri describes the electric field and satisfy[ˆ θ ri , ˆ E r ′ j ] = iδ rr ′ δ ij , [ˆ θ ri , ˆ θ r ′ j ] = [ ˆ E ri , ˆ E r ′ j ] = 0 . (2.2)To understand the LGT Hamiltonian, it is helpful to start with a model Hamil-tonian ˆ H c defined in the continuum space. Let us consider the following Ginzburg-Landau(GL)-type model: H c = Z d r h X i | D i ˆ φ ( r ) | + m ˆ ρ ( r ) + λ ˆ ρ ( r ) + 12 X i ˆ E i ( r ) + 14 X i,j ˆ F ij ( r ) i ,D i = ∂ i + ig ˆ A i ( r ) , ˆ ρ ( r ) = ˆ φ † ( r ) ˆ φ ( r ) , ˆ F ij ( r ) = ∂ i ˆ A j ( r ) − ∂ j ˆ A i ( r ) , (2.3)where D i is the covariant derivative expressing the minimal coupling of φ -particlewith the EM field, g is the coupling constant (charge of φ -particle20), m, λ areGL parameters whose meaning is explained later, and ˆ F ij ( r ) is the magnetic field B i ( r ) = ǫ ijk F jk ( r ). One may check that the each term of H c is invariant under thelocal gauge transformation,ˆ φ ( r ) → e i Λ( r ) ˆ φ ( r ) , ˆ A i ( r ) → ˆ A i ( r ) − g ∂ i Λ( r ) , ˆ E i ( r ) → ˆ E i ( r ) , (2.4)for an arbitrary function Λ( r ). By using the canonical commutation relations[ ˆ φ ( r ) , ˆ φ † ( r ′ )] = δ (3) ( r − r ′ ) and [ ˆ A i ( r ) , ˆ E j ( r ′ )] = iδ ij δ (3) ( r − r ′ ) , one may checkthat its generator ˆ Q ( r ) is given byˆ Q ( r ) = X i ∂ i ˆ E i ( r ) − g ˆ ρ ( r ) , (2.5)and ˆ Q ( r ) commutes with ˆ H c , [ ˆ Q ( r ) , ˆ H c ] = 0. So one may define the physical states | phys i so that they are eigenstates of ˆ Q ( r ) (superselection rule). In the usual case,i.e., when there are no external electric field, one imposes the local constraint,ˆ Q ( r ) | phys i = 0 , (2.6)which leads to the lattice version of the Gauss’ law ˆ Q r = 0 [See Eq.(2.17].Now we consider the Hamiltonian on the cubic lattice. In LGT, the operatorˆ θ ri defined on link ( r, i ) is assumed to approach as ˆ θ ri → ag ˆ A i ( r ) as the latticectober 29, 2018 5:51 WSPC/INSTRUCTION FILE mplb20140730 I.Ichinose and T.Matsui spacing a becomes small. Furthermore, as we shall see, the natural U(1) gauge fieldoperator in LGT is not ˆ θ ri but its exponential,ˆ U ri ≡ exp( i ˆ θ ri ) , ˆ U † ri ˆ U ri = ˆ1 , (2.7)and so ˆ θ ri is viewed as an angle operator (we discuss this point in detail later). Then,by using Eq.(2.2) we have ˆ E ri → a E i ( r ) as a → E ri is dimensionless and δ (3) ( r ) ≃ δ r a − ) and [ ˆ E ri , ˆ U r ′ j ] = δ ij δ rr ′ ˆ U ri . (2.8)(Note that Eq.(2.2) implies the replacement ˆ E ri → − i∂/∂θ ri .) To construct a setof eigenstates satisfying the completeness conditions, let us start with a pair ofoperators ˆ U and ˆ E sitting on certain link (we suppress their link index). Then wehave ˆ U | θ i = e iθ | θ i , θ ∈ [0 , π } , ˆ E | n i = n | n i , n ∈ Z , h θ | n i = e inθ p π , ˆ1 U = Z π dθ | θ ih θ | , ˆ1 E = ∞ X n = −∞ | n ih n | . (2.9)So ˆ θ and ˆ E correspond to the position and momentum operators in the ordinaryquantum mechanics respectively, but the significant difference is that the theory istaken to be compact , i.e., the physical state | phys i is imposed to be periodic in θ , h θ | phys i = h θ + 2 π | phys i , (2.10)which implies that the eigenvalues are an angle θ ∈ [0 , π ) defined by mod 2 π , andintegers n , instead of two real numbers. Because one may write | n i = ˆ1 U | n i = Z dθ p π e inθ | θ i = e in ˆ θ · Z dθ p π | θ i = e in ˆ θ | n = 0 i = ˆ U n | n = 0 i , (2.11)ˆ U ( † ) is just the creation (annihilation) operator of the electric field of unit electricflux (the signature of n distinguishes the direction of the flux), which is called a“string bit”. These relations remind us the analogy with the angular momentumoperators ~L as ˆ E ↔ ˆ L z ( n ↔ L z / ~ ) , ˆ U ↔ ˆ L x + i ˆ L y , θ ↔ ϕ (azimuthal angle)with the limit of large angular momentum ℓ → ∞ . We note that there holds theuncertainty principle, ∆ θ · ∆ n ≥ . (2.12)The Hamiltonian ˆ H of LGT should (i) reduce to ˆ H c of Eq.(2.3) in the naivecontinuum limit a →
0, and (ii) respect the U(1) local gauge symmetry as in thecontinuum. A simple example of ˆ H satisfying these two conditions is given byˆ H = t X r,i ( ˆ φ † r + i − ˆ U ri ˆ φ † r )( ˆ φ r + i − ˆ U † ri ˆ φ r ) + X r V ( ˆ φ r ) + ˆ H g , ˆ H g = g a X r,i ( ˆ E ri ) + 12 g a X r X i F F as U U U U + c . c . → exp[ iga ( A + A − A − A )] + c . c . = exp[ iga ( ∂ i A j ( r ) − ∂ j A i ( r ))] + c . c . → ga F ij ( r )) → − g a F ij ( r ) . (2.15)For gauge invariance, one may define the gauge transformation on the lattice asˆ U ri → V ∗ r + i ˆ U ri V r , V r = e i Λ r ∈ U ( ) , ˆ φ r → V r ˆ φ r , (2.16)which reduces to Eq.(2.4) by scaling Λ r → Λ( r ) as a → 0, and check that eachterm of ˆ H is invariant under Eq.(2.16). We define the physical states in the samemanner as in the continuum case, Q r ≡ X i ∇ i ˆ E ri − ˆ φ † r ˆ φ r , [ ˆ H, ˆ Q r ] = 0 , ˆ Q r | phys i = 0 , (2.17)where the forward difference operator ∇ i is defined as ∇ i f r ≡ f r + i − f r . Both ∇ i ˆ E ri and ˆ φ † r ˆ φ r have integer eigenvalues. We note that ˆ H of Eq.(2.13) has the periodicityˆ θ ri → ˆ θ ri + 2 π , which may be regarded as a gauge symmetry under a special gaugetransformation Λ r = 2 πr i . Physical properties of LGT Let us discuss the expected properties of the model (2.13). First, we focus on thecase of pure gauge theory ˆ H g without matter field. In te continuum theory, thecorresponding system,ˆ H g → Z d r X i (cid:16) ˆ E i ( r ) ˆ E i ( r ) + ˆ B i ( r ) ˆ B i ( r ) (cid:17) , (2.18)is well known to describe the ensemble of free photons. In contrast, ˆ H g on thelattice is an interacting theory. One can confirm it from cos( ga F ij ( r )) in Eq.(2.15).It contains, in addition to the leading F ij ( r ) term, F ij ( r ) and higher interactionterms describing scattering of gauge bosons. These terms are traced back to thecompactness of the model (the periodicity (2.10)).For large g ≫ 1, which is called the strong coupling region, one may take theelectric term as the unperturbed Hamiltonian and the magnetic term as a perturba-tion. By recalling Eq.(2.17), the unperturbed eigenstate | eigen i is the eigenstate ofˆ E ri with their eigenvalues satisfying the divergenceless condition. A simple examplectober 29, 2018 5:51 WSPC/INSTRUCTION FILE mplb20140730 I.Ichinose and T.Matsui is an electric flux of strength n along a closed loop C on the 3D lattice. The generaleigenstate may be composed of such loops. So one may write | eigen i = Y k Y ℓ k ∈ C k ( ˆ U ℓ k ) n k | i , (2.19)where C k denotes the k -th flux loop, ℓ k denotes links composing C k , and n k is thestrength of the k -th flux. If there are two external sources of charge g at r and − g at r , the lowest-energy state is an electrix flux state forming a straight line ˜ C (on the lattice) connecting these two charges and written as Y ℓ ∈ ˜ C ˆ U ℓ | i , (2.20)(See Fig.2a). This state has an energy g / (2 a ) × number of links in C , thus pro-portional to the distance r . If these two charges are quark and antiquark, isolationof each quark from the other implies r → ∞ and costs infinite amount of energy,being impossible to be realized. This is Wilson’s explanation of quark confinement8.On the other hand, for the weak-coupling region, g ≪ 1, the magnetic term isleading. Up to the gauge transformation, the energy eigenstates in the limit g → θ ri in contrast to the strong-coupling limit. By the un-certainty principle (2.12), the eigenstate there should be a superposition of electricfluxes of various strength and locations as in the ordinary classical Coulombic elec-tromagnetism shown in Table.1. So the physical state corresponding to the stateconsidered in Eq.(2.20) is replaced by X C Γ( C ) Y ℓ ∈ C ˆ U ℓ | i , (2.21)where the coefficient Γ( C ) is the weight for the path C connecting the sites 1 and2. For the lowest-energy state, Γ( C ) is to be determined by the minimum-energycondition, and the energy stored in the two charges here should be proportional to1 /r as in the Coulomb potential energy (See Fig.2b).The above qualitative consideration may indicate that there exists a phaseboundary separating the strong and weak-coupling regions. Dynamics of the pure a b c Fig. 2. Illustration of electric flux lines between two oppositely charged particles in the threephases. (a) confinement phase; (b) Coulomb phase; (c) Higgs phase. Dashed lines in (c) implythat the strength of fluxes are not conserved as Eq.(2.17) with h ˆ φ † r ˆ φ r i = const = 0 shows. ctober 29, 2018 5:51 WSPC/INSTRUCTION FILE mplb20140730 Lattice gauge theory for condensed matter physics gauge system H g in (2.13) is quite nontrivial and careful investigation such as nu-merical study is required to obtain the phase diagram. At present, it is known thatthe 3D pure U(1) gauge system H g has two phases; The phase at strong couplingregion is called the confinement phase, and the phase at weak coupling region iscalled Coulomb phase. The phase transition is said to be weak first order26. Thesetwo phases are characterized by the behavior of the potential energy V ( r ) for a pairof static charges of opposite signs and separated by a distance r ; ∝ r or ∝ /r . Alsothey are distinguished by the fluctuation of gauge field ˆ θ ri , ∆ θ , as ∆ θ ≫ θ ≪ H ofEq.(2.13). By referring to Eq.(2.3), one may write V ( ˆ φ r ) as V ( ˆ φ r ) = λ ( ˆ φ † r ˆ φ r − v ) . (2.22)Let us recall that the GL theory with the gauge interaction being turned off bysetting g = 0 and for sufficiently large and positive λ , the system exhibits a second-order phase transition by varying v . It is signaled by the order parameter φ = h ˆ φ r i ,(or more strictly defined as | φ | ≡ lim | r − r ′ |→∞ h ˆ φ † r ˆ φ r ′ i ), and separates the ordered(condensed) phase h φ i 6 = 0 and the disordered phase h φ i = 0. In the mean-fieldtheory, the transition temperature T c is determined as the vanishing point of v as v ∝ ( T c − T ). However, the fluctuations of the phase degree of freedom ϕ r ofˆ φ r = | ˆ φ r | exp( iϕ r ), which are controlled by the hopping t -term of Eq.(2.13), playan essentially important role for realization of a condensation. In fact, these twophases may be characterized by the magnitude of fluctuation ∆ ϕ of ϕ r as ∆ ϕ ≪ ϕ ≫ T c , more detailed study such as MC simulation is required as we discusslater.For the full fledged gauge-matter system ˆ H of Eq.(2.13), one may naively con-sider four combinations such as (∆ θ ≪ ≫ 1) and (∆ ϕ ≪ ≫ 1) for apossible phase. However, one may easily recognize that the combination ∆ ϕ ≪ θ ≫ ϕ r appears in the hopping term with thecombination ∇ i ˆ ϕ r − ˆ θ ri and the condition ∆ θ ≫ ∇ i ˆ ϕ r ≃ 0. In fact, the numerical study and the mean field theory predict the threephases listed in Table 1; (1) confinement phase ∆ θ ≫ , ∆ ϕ ≫ 1, (2) Coulombphase ∆ θ ≪ , ∆ ϕ ≫ 1, (3) Higgs phase ∆ θ ≪ , ∆ ϕ ≪ t [ t is thehopping amplitude in Eq.(2.13)] and large g regardless of the values of λ and v . In a sense, this phase corresponds to the high-temperature ( T ) region andctober 29, 2018 5:51 WSPC/INSTRUCTION FILE mplb20140730 I.Ichinose and T.Matsui there exist no long-range orders. The parameter region of the confinement phaseis enlarged by decreasing a value of v . For pure U(1) gauge theory withoutmatter field ˆ φ r , a potential V ( r ) for a pair of oppositely charged static sourcesseparated with distance r behaves as V ( r ) ∝ r as explained.(2) The Coulomb phase appears for small g and small t . Fluctuations of the gaugefield is suppressed by the plaquette terms in Eq.(2.13) and the compactness ofˆ θ ri plays no role. The potential of a pair of two static charges is given by theCoulomb law, which is V ( r ) ∝ /r for 3D space.(3) In the Higgs phase, a coherent condensation of the boson field ˆ φ r takes place.Then the hopping term in Eq.(2.13) gives the following term of U xµ , t ˆ φ † r + i ˆ U † ri ˆ φ r + H . c . → tv ( ˆ U † ri + U ri ) + · · · ≃ − tva g ˆ θ ri + · · · . (2.23)As a result, the gauge field acquires a mass m θ = p tv and a force mediated bythe gauge field ˆ θ ri becomes short range like h ˆ θ r + Ri,j ˆ θ tj i = e − m θ R (See Fig.2c).The Higgs phase is realized for large value of t and a finite v (in addition tosmall g ). Superconducting (SC) phase is a kind of the Higgs phase in which theboson field ˆ φ r describes Cooper pairs, and the gauge field becomes short-rangedby the Meissner effect.So far, we studied the compact LGT. In some literatures, so called noncom-pact LGT is considered22. The noncompact U(1) LGT is given by the followingHamiltonian for the gauge part,ˆ H ′ g = g a X r,i ( ˆ E ri ) + 12 g a X r X i 1, the compact puregauge theory (2.13) reduces to above ˆ H ′ g . However, for large fluctuations, these twomodels behave quite differently and they generally have different phase diagrams.In the noncompact gauge theories with the gauge part (2.24), the Coulomb andHiggs phases are realized but the confinement phase is impossible because of thesuppression of large fluctuations. In other words, the large fluctuations of ˆ θ ri inthe confinement phase are achieved by large-field configurations called topologicalexcitations, the typical examples of which are instantons and monopoles. They areallowed only for a system with periodicity such as compact LGT. To describe a SCphase transition with the ordinary EM interactions, one should use the noncompactU(1) gauge theory. The gauge field ˆ θ ri there is nothing but the vector potential forthe electro-magnetism, and the single-component boson field ˆ φ r corresponds to the s -wave Cooper-pair field as mentioned. Therefore, as we have explained above, theHiggs phase is nothing but the SC phase. In order to describe a multi-componentSC state such as the p -wave SC, introduction of a multi-component boson field isnecessary. This system will be discussed rather in detail in the subsequent sections.ctober 29, 2018 5:51 WSPC/INSTRUCTION FILE mplb20140730 Lattice gauge theory for condensed matter physics Partition function in the path-integral representation on theEuclidean lattice Let us consider the partition function Z for ˆ H of Eq.(2.13) at the temperature T , Z = Tr ˆ P exp( − β ˆ H ) , β ≡ k B T , ˆ P ≡ Y r δ ˆ Q r , , (2.25)where Tr is over the space of all the values of ˆ Q r , and ˆ P is the projection operatorto the physical states. As usual, we start by factorizing the Boltzmann factor into N factors but with care of ˆ P asˆ P exp( − β ˆ H ) = [ ˆ P exp( − ∆ β ˆ H )] N , ∆ β ≡ βN , (2.26)where we used ˆ P = ˆ P , [ ˆ P , ˆ H ] = 0, and then insert the complete set between everysuccessive factors.For ˆ φ r , we use the following coherent states | φ i :ˆ φ | φ i = φ | φ i , | φ i = exp( − ¯ φ ˆ φ + ˆ φ † φ ) | i , ˆ φ | i = 0 , h φ ′ | φ i = exp( − ¯ φ ′ φ ′ − ¯ φφ φ ′ φ ′ ) , |{ φ }i = Y r | φ r i , Z d φ = Y r Z d φ r π , ˆ1 φ = Z d φ |{ φ }ih{ φ }| , (2.27)where φ is a complex number and ¯ φ is its complex conjugate (The bar denotes thecomplex-conjugate quantity).The eigenstates and the completeness of gauge field for the entire lattice arewritten by using Eqs.(2.9) asˆ1 θ = Z dθ |{ θ }ih{ θ }| , ˆ1 n = X n |{ n }ih{ n }| , |{ θ }i = Y r,i | θ ri i , |{ n }i = Y r,i | n ri i , Z dθ = Y r,i Z dθ ri , X n = Y r,i X n ri ∈ Z . (2.28)(We abbreviate the symbol ⊗ of tensor product in |{ θ }i and |{ n }i .)For example, the matrix element of ˆ P is calculated as δ ˆ Q r , = Z π dθ r π exp( iθ r ˆ Q r ) , h{ φ ′ , n ′ }| ˆ P |{ φ, n }i = Y r Z dθ r π Y i δ n ′ ri n ri · Y r exp( − 12 ¯ φ ′ r φ ′ r − 12 ¯ φ r φ r + ¯ φ ′ r e − iθ r φ r ) exp( iθ r X i ∇ i E ri ) . (2.29)The Lagrange multiplier θ r will be interpreted as the imaginary-time componentof gauge field. Then we obtain the following expression of Z for sufficiently largectober 29, 2018 5:51 WSPC/INSTRUCTION FILE mplb20140730 I.Ichinose and T.Matsui N : Z = Z [ dU ] Z [ dφ ] X { n } exp( A H ) , [ dU ] = Y x,µ dU xµ , [ dφ ] = Y x dφ x , X { n } = Y x,i X n xi ,A H = ∆ β X x h i ∆ β X i ( E xi ∇ θ xi + θ x ∇ i E xi ) − β ¯ φ x +0 ( φ x +0 − e − iθ x φ x ) − V ( φ x ) − t X i | φ x + i − ¯ U xi φ x | − g a X i E xi + 12 g a X i Phase structure of various gauge-field models has been investigated by both analyticand numerical methods. In particular, for the gauge-Higgs model, which describesthe SC phase transitions, interesting phase diagrams have been obtained. To thisend, the numerical study by the Monte-Carlo (MC) simulations plays a very im-portant role as they provide us reliable results including all nonperturbative effects.Here we present some known schematic phase diagrams restricting the system tothe relativistic Higgs coupling and the frozen radial degrees of freedom of φ x field,ctober 29, 2018 5:51 WSPC/INSTRUCTION FILE mplb20140730 Lattice gauge theory for condensed matter physics x + (cid:23) c1 c2x x + (cid:22) (cid:30)x (cid:22)(cid:30)x+(cid:22) (cid:22)Ux(cid:22) ( )+ + ( + )x x + (cid:22) (cid:22)Ux(cid:22) Ux(cid:23) Fig. 3. Illustration of the action A of the compact LGT (2.34). Open (filled) circle denotes φ x =exp( iϕ x )( ¯ φ x ) and straight line with an arrow in the negative (positive) µ − direction denotes U xµ =exp( iθ xµ )( ¯ U xµ ) i.e., φ x = exp( iϕ x ), but with spatial dimensions d = 2 and 3 (dimension of thecorresponding Euclidean lattice D = d + 1 is 3 and 4). Explicitly, we consider thefollowing partition function; Z = Z [ dϕ ][ dθ ] G exp( A ) , A = c X x,µ cos( ϕ x + µ + θ xµ − ϕ x ) + A ( ′ ) g ,A g = c X x,µ<ν cos θ xµν , A ′ g = − c X x,µ<ν θ xµν , θ xµν ≡ θ xν + θ x + ν,µ − θ x + µ,ν − θ xµ , (2.34)where µ = 0 , · · · , d, and c and c are the parameters. A g is for the compact LGTand A ′ g is for the noncompact LGT. Each term of the action A is depicted in Fig.3.This model is sometimes called the Abelian Higgs model11. Note that the Higgscoupling (the c term) along the imaginary time µ = 0 direction has the same formas the spatial directions, i.e., every direction has couplings both in the positive andnegative directions. This is in contrast to the nonrelativistic model (2.31) in whichthe coupling for µ = 0 is only in the positive direction. In short, the coupling in thenegative µ = 0 direction describes the existence of antiparticles which are allowedin the relativistic theory but not in nonrelativistic theory27.Here we comment on the integration measure [ dθ ] G and gauge fixing. Thepath integral (2.34) involves the same contributions from a set of configurationsof { ϕ, θ } (so called gauge copies) that are connected with each other by gaugetransformations. So one may reduce this redundancy by choosing only one repre-sentative from each set of gauge copies without changing physical contents. Thisprocedure is called gauge fixing and achieved by inserting the gauge fixing func-tion G ( { ϕ, θ } ), which is not gauge invariant under Eq.(2.33), to the measure,[ dϕ ][ dθ ] → [ dϕ ][ dθ ] G ≡ G ( { ϕ, θ } )[ dϕ ][ dθ ]. The difference between the gauge fixedcase [ dθ ] G and the nonfixed case [ dθ ] appears as a multiplicative factor in Z , Z nonfixed = C gv Z fixed , C gv = Y x,µ Z d Λ xµ , (2.35)where the constant C gv is called gauge volume. In the compact case, the region of θ xµ and Λ xµ are both compact [0 , π ). Therefore C gv is finite and the gauge fixingis irrelevant (either choice of fixing gauge or not will do). In the noncompact case,the region of θ xµ and Λ xµ is ( −∞ , ∞ ), so C gv diverges. Thus, one must fix thectober 29, 2018 5:51 WSPC/INSTRUCTION FILE mplb20140730 I.Ichinose and T.Matsui gauge in formal argument in the noncompact case, [ dϕ ][ dθ ] G . We said “in formalargument” here because the choice G ( { ϕ, θ } ) = 1 works even in the noncompactcase as long as one calculate the average of gauge-invariant quantities, e.g., in theMC simulations. This is because the average does not suffer from the overall factorin Z . We shall present in Sect.4.1. another reason supporting this point, which isspecial in MC simulations. c1 c2 CoulombHiggs (a) Noncompact 3D c1 c2 CoulombHiggs (b) Noncompact 4D c1 c2 confinement (c) Compact 3D c1 c2 CoulombHiggsconfinement (d) Compact 4D Fig. 4. Schematic phase diagrams in the c − c plane of the 3D and 4D LGT with compact andnoncompact action. All the Higgs-Coulomb transitions are of second order. The Higgs-confinementone in (d) is first order and the confinemnet-Coulomb one in (d) is weak first order. The partitionfunction along the line c = 0 is evaluated exactly by single link integral. It is an analytic functionof c and so no transitions exist along this line. In Fig.4 we present the schematic phase diagrams in the c − c plane. Thenoncompact 3D and 4D models have the Coulomb phase and the Higgs phase. Thecompact 3D model has only the confinement phase. The compact 4D model has allthe three phases. These points are understood as follows; (i) the Higgs phase mayexist at large c above the Coulomb phase in 3D and 4D, and above the confinementphase in 4D, (ii) the Coulomb phase may exist at sufficiently large c in 4D case,(iii) the confinement phase exists in the compact case, but not in the noncompactphase.We note that, as c decreases, all the existing confinement-Higgs transitioncurves terminate at points with c > 0. This can be understood by calculating Z for c = 0 exactly. Because the integration over θ xµ there is factorized as Z ( c , c = 0) = Y x,µ Z dθ ′ xµ e c cos θ ′ xµ = I ( c ) N ℓ , θ ′ xµ ≡ ϕ x + µ + θ xµ − ϕ x , (2.36)ctober 29, 2018 5:51 WSPC/INSTRUCTION FILE mplb20140730 Lattice gauge theory for condensed matter physics where I ( c ) is the modified Bessel function and N ℓ ( ≡ P x,µ 1) is the total numberof links. Z ( c , c = 0) is then an analytic function of c and no phase transition canexists along the c axis. This fact is known as complementarity and implies thatthe confinement phase and the Higgs phase are connected with each other withoutcrossing a phase boundary.Another limiting case c → ∞ may imply ∆ θ → 0. Therefore the resultingsystem becomes Z ( c , c → ∞ ) = Y x,µ Z dϕ ′ x exp[ c X x,µ cos( ϕ ′ x + µ − ϕ ′ x )] , (2.37)by setting as θ xµ = ∇ µ Λ x (pure gauge configuration) with ϕ ′ x ≡ ϕ x + Λ x . Thissystem is just the 3D and 4D XY spin models with the XY spin (cos ϕ x , sin ϕ x ),which are known to exhibit a second-order transition. This corresponds to theHiggs-Coulomb transition in Fig.4.Let us discuss a general correlation function h O ( { θ } , { ϕ } ) i (the expectationvalue of the function O ( { θ } , { ϕ } )) of the model (2.34) defined as h O ( { θ } , { ϕ } ) i = 1 Z Z [ dϕ ][ dθ ] O ( { θ } , { ϕ } ) exp( A ) . (2.38)First, h O ( { θ } , { ϕ } ) i is known to satisfy the Elitzur theorem24, which says h O ( { θ } , { ϕ } ) i = 0 if O ( { θ } , { ϕ } ) is not gauge invariant . (2.39)Its proof is rather simple and given by making a change of variables in a formof gauge transformation and noting the measure [ dθ ][ dϕ ] is invariant under it. Afamous example of gauge-invariant correlation function is the Wilson loop definedon a closed loop C along successive links as W ( C ) = h Y ℓ ∈ C U ℓ i . (2.40)It is used by Wilson as a nonlocal “order parameter” to distinguish a confinementphase and a deconfinement phase for pure gauge system (i.e., involving no dynamicalcharged particles). If W ( C ) satisfies the “area law” as C becomes large, W ( C ) ∼ exp( − aS ( C )) where a > S ( C ) is the minimum area bound by C , the systemis in the confinement phase. On the other hand, if it satisfies the perimeter law, W ( C ) ∼ exp( − a ′ L ( C )) where L ( C ) is the length of C , it is in the deconfinementphase. This interpretation comes from the fact that for the rectangular C of thehorizontal size R and the vertical size T [The coordinate ( x , x ) of four corners inthe 0-1 plane, e.g., are (0,0), (0,R), (T,0), (T,R)], V ( R ) defined through W ( C ) ∼ exp( − V ( R ) T ) at sufficiently large T expresses the lowest potential energy storedby a pair of external charges separated by the distance R . The area law implies theconfining potential V ( R ) = aR .Second, one may estimate Z and the correlation functions of gauge-invariantobjects by fixing the gauge because they are gauge invariant. Fixing implies tofix one degree of freedom per site x corresponding to Λ x whatever one wants.ctober 29, 2018 5:51 WSPC/INSTRUCTION FILE mplb20140730 I.Ichinose and T.Matsui For example, the choice ϕ x = 0 is called the unitary gauge, and another choice θ x = 0 is called the temporal gauge. In the mean field theory for the compactLGT, one choose gauge variant quantities h exp( iθ xµ ) i and h exp( iϕ x ) i as a set oforder parameters and predicts the phase structure involving the above mentionedthree phases. They should be considered as a result after gauge fixing. If one wantsto make it compatible with the Elitzur theorem, one simply need to superpose overthe gauge copies of these order parameters as Drouffe suggested25. For h exp( iθ xµ ) i ,it implies Y x Z π d Λ x π · h exp( iθ ′ xµ ) i = 0 , θ ′ xµ = θ xµ − ∇ µ Λ x , (2.41)without changing the gauge-invariant results such as phase diagram, etc.. 3. Ferromagnetic Superconductivity and its Lattice Gauge Model In this section and Sect.4, we shall consider a theoretical model for the FMSC,which can be regarded as a kind of LGT. In some materials such as UGe , it hasbeen observed that SC can coexist with a FM long-range order29. In UGe andURhGe, a SC appears only within the FM state in the pressure-temperature ( P - T )phase diagram, whereas, in UCoGe, the SC exists both in the FM and paramagneticstates. Soon after their discovery, phenomenological models of FMSC materials wereproposed30. In those studies, the FMSC state is characterized as a spin-triplet p -wave state of electron pairs as suggested experimentally31. In this section, weintroduce the GL theory of FMSC materials defined on the 3d spatial lattice (4DEuclidean lattice). As we see, the FM order naturally induces nontrivial vectorpotential (gauge field) and it competes with another order, the SC order, thereforea nonperturbative study is required in order to clarify the phase diagram etc. Ginzburg Landau theory of FMSC in the continuum space Before considering a lattice model of FMSC itself, let us start with the GL theoryof FMSC in the 3d continuum space30. It was introduced phenomenologically forthe case with strong spin-orbit coupling, although the origin of SC, etc. had notbeen clarified. Its free energy density f GL contains a pair of basic 3d-vector fields, ~ψ ( r ) and ~m ( r ), which are to be regarded as c-numbers appearing in the path-integral expression for the partition function of the underlying quantum theory asin Eq.(2.31) [We discuss the partition function in detail later; see Eq.(3.16)].The three-component complex field ~ψ = ( ψ , ψ , ψ ) t is the SC order parameter,describing the degrees of freedom of electrons participating in SC. More explicitly, ~ψ ( r ) describes Cooper-pairs of electrons with total spin s = 1 (triplet) and therelative angular momentum ℓ = 1 ( p -wave) (Note that only odd ℓ ’s are allowe for s = 1 by Pauli principle). The complex Cooper-pair field Φ q ( r ) for a s = 1 and ℓ = 1 state generally has 3 × q = 1 , · · · , ~S ( r ) and the angular momentum ~L ( r ) made of Φ q ( r )ctober 29, 2018 5:51 WSPC/INSTRUCTION FILE mplb20140730 Lattice gauge theory for condensed matter physics parallel each other, and reduces their degrees of freedom down to 3. Such a Cooperpair is directly expressed in terms of the amplitudes defined as∆ σσ ′ ( r ) ≡ Z dr Γ( r ) h ˆ C σ ( r + r C σ ′ ( r − r i . (3.1)ˆ C σ ( r ) is the annihilation operator of electron at r and spin σ = ↑ , ↓ . Γ( r ) with r being the relative coordinate of electrons is the weight reflecting the attractiveinteraction and the p -wave nature. Because we assign ~ψ ( r ) as a 3d vector, it isconvenient to use the so called ~d -vector, which transforms also as a vector in the3d space; ~ψ ( r ) ∝ ~d ( r ) = d x ( r ) d y ( r ) d z ( r ) ≡ − (∆ ↑↑ ( r ) − ∆ ↓↓ ( r )) − i (∆ ↑↑ ( r ) + ∆ ↓↓ ( r ))∆ ↑↓ ( r )(= ∆ ↓↑ ( r )) . (3.2)The real vector field ~m ( r ) is the FM order parameter, describing the degrees offreedom of electrons participating in the normal state. More explicitly, it describestheir magnetization, ~m ( r ) = h ˆ C ′ σ ( r ) (cid:16) − iδ σσ ′ ~r × ~ ∇ + 12 ~σ σσ ′ (cid:17) ˆ C ′ σ ′ ( r ) i , (3.3)where ˆ C ′ σ ( r ) denotes the annihilation operator of electrons not participating in theSC state. Because there holds div ~m ( r ) = 0 as in the usual magnetic field, ~m ( r ) isexpressed by using the vector potential ~A ( r ) (gauge field) as ~m ( r ) = rot ~A ( r ) . (3.4)Therefore, the fundamental fields may be ~ψ ( r ) and ~A ( r ),The GL free energy density f GL is then given by ? f GL = f ψ + f m + f Z ,f ψ = K X i ( D i ~ψ ) ∗ · ( D i ~ψ ) + α ( T − T ) ~ψ ∗ · ~ψ + λ ( ~ψ ∗ · ~ψ ) ,f m = K ′ X i ( ∂ i ~m ) + α ′ ( T − T ) ~m + λ ′ ( ~m ) ,f Z = − J ~m · ~S, D i = ∂ i − ieA i , ~S = − i ~ψ ∗ × ~ψ. (3.5) K ( ′ ) , α ( ′ ) , λ ( ′ ) are GL parameters; real positive parameters characterizing each ma-terial. ~S ( r ) is a real vector field describing the spin of Cooper pairs. For example, S = ( | ∆ ↑↑ | −| ∆ ↓↓ | ). f Z with J > ~S and ~m , which enhances the coexistence of the FM and SC orders as one easily expects.Although the existence of f Z term is supported phenomenologically, its microscopicorigin should be clarified by detailed study of the interactions of electrons (andnuclei) in each material. f GL is gauge invariant under the following U(1) gauge transformation, ~ψ ( r ) → ~ψ ′ ( r ) = e iλ ( r ) ~ψ ( r ) , ~A ( r ) → ~A ′ ( r ) = ~A ( r ) + 12 e ~ ∇ λ ( r ) . (3.6)ctober 29, 2018 5:51 WSPC/INSTRUCTION FILE mplb20140730 I.Ichinose and T.Matsui So ~m ( r ) and ~S ( r ) are gauge invariant. This gauge invariance reflects the originalU(1) gauge invariance of the EM interaction of electrons. For example, the K termwith the covariant derivative D i describes the interaction of SC Cooper pairs ofcharge − e with the vector potential ~A made of the normal electrons. Its micro-scopic origin should be traced back to the repulsive Coulombic interactions betweenelectrons.Before introducing the lattice GL theory, let us list up some main characteristicssuggested by the continuum GL theory (3.5). • In the mean-field approximation with ignoring the Zeeman coupling f Z , T and T are critical temperatures of the FM and SC phase transitions, respec-tively. • Existence of a finite magnetization, h ~m i 6 = 0, means a nontrivial configuration of A i ( r ). This induces nontrivial spatial dependence of the Cooper pair ~ψ throughthe kinetic term K ( D i ~ψ ) ∗ · ( D i ~ψ ) in f GL , because this term favors D i ~ψ ≃ 0. Atypical example of such configurations is a vortex configuration characterizedby a nonvanishing vorticity ~v ( r ) of ~ψ . The 3d vector ~v ( r ) of a complex field φ ( r ) = | φ ( r ) | exp( iϕ ( r )) is defined generally as the winding number of its phase ϕ ( r ). Explicitly, the component of ~v ( r ) along the direction of a normal vector ~n is defined by the circle integral over ϕ ( r ), ~n · ~v ( r ′ ) = 12 π I C ~n ( r ′ ) dϕ ( r ) ( ∈ Z ) , (3.7)where C ~n ( r ′ ) is a circle lying in the plane parpendicular to ~n with its centerat r ′ . ~n · ~v ( r ) takes integers due to the single-valuedness of φ ( r ). In the latersections, we shall consider two vorticities ~v ± ( r ) corresponding to ψ ( r ) ± iψ ( r )respectively. Then the simple mean-field like estimation of the SC critical tem-perature has to be reexamined by more reliable analysis. • The Zeeman coupling f Z obviously enhances the coexisting phase of the FMand SC orders, since it favors a set of antiparallel ~S and ~m with large | ~ψ | and | ~m | . • In the FM phase, ~A produces a nonvanishing magnetic field ~m ( r ) inside thematerials. Because the K term in f ψ of Eq.(3.5) disfavors the coherent (spatiallyuniform) condensation of ~ψ ( ∂ i ~ψ = 0), due to rot A = 0 (favoring nontrivial onessuch as vortices as explained above), the SC in a magnetic field is disfavored.In this sense, the system (3.5) is a kind of a frustrated system (the K termdisfavors coexistence although the Zeeman term does).Then a detailed study by using numerical methods is needed to obtain thecorrect phase diagram. To this end, we shall introduce a lattice version of the GLtheory (3.5) that is suitable for the investigation by using MC simulations.ctober 29, 2018 5:51 WSPC/INSTRUCTION FILE mplb20140730 Lattice gauge theory for condensed matter physics GL theory on the lattice In this subsection we explain the lattice GL theory introduced in Ref.32. In con-structing the theory, we start with the following two simplifications:(1) We consider the “London” limit of the SC, i.e., we assume ~ψ ∗ ( r ) · ~ψ ( r ) =const . ignoring the fluctuations of the radial degrees of freedom | ~ψ ( r ) | . Thisassumption is legitimate as the phase degrees of freedom ~ψ ( r ) themselves playan essentially important role in the SC transition (Recall the account of phasecoherence for a Bose-Einstein condensate) 28.(2) We assume that the third component ψ ( r ) ∝ ∆ ↑↓ ( r ) is negligibly small com-pared to the remaining ones ψ , ( r ), ψ ± ≡ p ψ ± iψ ) , ψ ↑↑ ≡ ψ − ∝ ∆ ↑↑ , ψ ↓↓ ≡ − ψ + ∝ ∆ ↓↓ . (3.8)This simplification is consistent with the fact that the real materials exhibitFM orders of Ising-type with the i = 3-direction as the easy axis. In fact, ~S = − i ~ψ ∗ × ~ψ is calculated with ψ ↑↓ ≡ ψ as ~S = ( − p ψ ↑↑ + ψ ↓↓ ) ψ ∗↑↓ , p ψ ↑↑ − ψ ↓↓ ) ψ ∗↑↓ , | ψ ↑↑ | − | ψ ↓↓ | ) t . (3.9)So the Zeeman coupling f Z requires large ψ , compared to ψ to favor ~m ∝ (0 , , m ).In summary, we parametrize the Cooper-pair field in the London limit, | ψ ( r ) | =[ α/ (2 λ )]( T − T ), at low T ( < T ) with the third easy axis as ψ ( r ) ψ ( r ) ψ ( r ) = r α λ ( T − T ) × z ( r ) z ( r )0 , | z ( r ) | + | z ( r ) | = 1 . (3.10)The “normalized” two-component complex field z a ( r ) ( a = 1 , 2) satisfying the con-straint (3.10) is called CP variable (CP stands for complex projective group). Interms of the SC order-parameter field z a ( r ), the first K term of (3.5) is rewrittenas Kα (2 λ ) − ( T − T ) P i P a D i z a · D i z a .Now let us introduce the lattice GL theory of FMSC defined on the 3d cubiclattice. Its free energy density per site f r is given by32 f r = − c X i =1 2 X a =1 (cid:0) ¯ z r + i,a ¯ U ri z ra + c . c . (cid:1) − c ~m r − c ~m r · ~S r + c ( ~m r ) − c X i ~m r + i · ~m r , X a =1 ¯ z ra z ra = 1 , U ri ≡ exp( iθ ri ) . (3.11)The five coefficients c i ( i = 1 ∼ 5) in (3.11) are real nonnegative parameters that areto distinguish various materials in various environments. z ra is the CP variable puton the site r and plays the role of SC order-parameter field. U ri is the exponentiatedctober 29, 2018 5:51 WSPC/INSTRUCTION FILE mplb20140730 I.Ichinose and T.Matsui vector potential, θ ri , put on the link ( r, r + i ). ~m r = ( m r , m r , m r ) t is the magneticfield made out of θ ri as m ri ≡ X j,k =1 ǫ ijk ∇ j θ rk , ∇ j θ rk ≡ θ r + j,k − θ rk . (3.12) ~m r serves as the FM order-parameter field. Following Eq.(3.5), the spin ~S r of theCooper pair is defined as ~S r = (0 , , S r ) , S r ≡ − i (¯ z r z r − ¯ z r z r ) ( ∝ | ψ ↑↑ | − | ψ ↓↓ | ) , (3.13)where we absorbed the normalization of ~S ( r ) into the coefficient c . ~m r , ~S r and f r are invariant under the following gauge transformation, z ra → z ′ ra = e iλ r z ra , U ri → U ′ ri = e − iλ r + i U ri e iλ r . (3.14)Here we comment on the way of putting a 3d continuum vector field on the3d lattice. It sounds natural that a vector field ~B ( r ) should be put on the link( r, r + i ) as B ri . However, it is too naive. In fact, the gauge field ~A ( r ) has thenature of connection as explained and the connection between two points separatedby finite distance such as nearest-neighbor pair of sites cannot be implemented by θ ri itself but by its exponentiated form such as U ri . This is the reason why onehas ¯ z r + i,a ¯ U ri z ra term in Eq.(3.11). On the other hands, concerning to the SC orderfield ~ψ ( r ), its suitable lattice version depends on the coherence length ξ . If ξ is ofthe same order of lattice spacing a , its vector nature should be respected and thelink field ψ ri is adequate. If ξ ≫ a , the detailed lattice structure is irrelevant and asimpler version ψ ra ( a = 1 , , 3) on the site (or its Ising counterpart z ra ( a = 1 , ξ ≫ a as some materials show.Let us see the meaning of each term in f r . The c -term describes a hopping ofCooper pairs from site r to r + i (and from r + i to r ). As the Cooper pair hasthe electric charge − e , it minimally couples with the vector potential θ ri via U ri as explained above. It is important to observe the relation between the hoppingparameter c and T . From the expression given in the paragraph below Eq. (3.5),we have c ∼ Kαλ − ( T − T ) a. (3.15)At sufficiently large βc = c /T (we set k B = 1) that corresponds to low T ’s, the c -term stabilizes the combination of phases of ¯ z r + i,a ¯ U ri z ra , and then, if U ri is sta-bilized already by other mechanism, a coherent condensation of the phase degrees offreedom of z r is realized inducing the superconductivity . The c and c -terms are thequartic GL potential of ~m r , favoring a finite amount of local magnetization h ~m r i 6 = 0(note that we take c > c -term represents the NN coupling of the magnetization and corresponds tothe K ′ term of Eq.(3.5). It enhances uniform configurations of ~m r , i.e., a FM long-range order signaled by a finite magnetization m , m ≡ lim | r − r ′ |→∞ h ~m r · ~m r ′ i 6 = 0.ctober 29, 2018 5:51 WSPC/INSTRUCTION FILE mplb20140730 Lattice gauge theory for condensed matter physics The c -term is the Zeeman coupling, which favors collinear configurations of ~m r and ~S r , i.e., enhances the coexistence of FM and SC orders.The partition function Z at T for the energy density f r of Eq.(3.11) is given bythe integral over a set of two fundamental fields z ra and θ ri as Z = Z [ dz ][ dθ ] exp( − βF ) , β = T − , F = X r f r , [ dz ] = Y r d z r d z r δ ( | z r | + | z r | − , [ dθ ] = G ( { θ } ) Y r,i dθ ri , θ ri ∈ ( −∞ , ∞ ) . (3.16)where G ( { θ } ) is a gauge fixing function[see the paragraph containing Eq.(2.35)].We stress here that the free energy F and the integration variables in Eq.(3.16)have no dependence on the imaginary time x in contrast with the expression (2.34).The ordinary GL theory explained in the literature also shares this properties.This implies that we ignore the x -dependent modes of would-be ϕ x and θ xµ as A ( { ϕ x } , { θ xµ } ) → βF ( { ϕ r } , { θ ri } ). This is an approximation applicable for small β ’s, i.e., for high T ’s. This procedure allows us to make use of MC method that isbased on the probabilistic process. In fact, the form (3.16) has a real function F and the Boltzmann factor exp( − βF ) /Z is able to be interpreted as a probability.This is in contrast with the original action A corresponding to F ; A is certainly acomplex function as Eq.(2.34) giving rise to a complex probability. We come backto this point later.The coefficients c i ( i = 1 , · · · , 5) in f r may have nontrivial T -dependence asEqs.(3.5) and (3.15) suggest. However, in the present study we consider the re-sponse of the system by varying the “temperature” T ≡ /β defined by β , anoverall prefactor in Eq.(3.16), while keeping c i fixed. This method corresponds towell-known studies such as the FM transition by means of the O (3) nonlinear- σ model33 and the lattice gauge-Higgs models discussed in Sec.2.1, and is sufficientto determine the critical temperature [See, e.g., Eq.(4.7)].In the following section, we shall show the results obtained by MC numericalevaluation of Eq.(3.16). Physical quantities O ( z, θ ) like the internal energy, specificheat, correlation functions, etc. are also calculated by the MC simulations as hO ( z, θ ) i = 1 Z Z [ dz ][ dθ ] O ( z, θ ) exp( − βF ) . (3.17)From the obtained results, we shall clarify the phase diagram of the system andphysical properties of each phase. 4. Results of Monte Carlo simulation In the present section, we introduce and discuss the results of numericalcalculations32 , 34 for the lattice GL model (3.11), which include the phase diagramctober 29, 2018 5:51 WSPC/INSTRUCTION FILE mplb20140730 I.Ichinose and T.Matsui and Meissner effect in the SC phase. Figs.5 ∼ ∼ 13 are from Ref.34. Phase diagram and Meissner effect We first seek for a suitable boundary condition (BC) of θ ri as the magnetizationis expressed by θ ri as Eq.(3.12). A familiar condition is the periodic boundarycondition such as θ r + Li,j = θ rj for the linear system size L . However, this conditionnecessarily gives rise to a vanishing net mean magnetization due to the latticeStorkes theorem. So we consider the 3d lattice of the size (2 + L + 2) × L and takea “free BC” on z r in the i = 1 , z r + i,a − ¯ U ri z ra = 0 for r = (0 , r , r ) , i = 1 , ( L + 2 , r , r ) , i = 1 , ( r , , r ) , i = 2 , ( r , L + 2 , r ) , i = 2 , (4.1)whereas we impose the free boundary condition on θ ri . It is easily shown that theBC (4.1) implies that the suppercurrent j SC ri , j SC ri ∝ Im ( X a ¯ z r + i,a ¯ U ri z ra ) , (4.2)satisfies j SC r = 0 on the boundary surfaces in the 2(1) − z ra , but we expect, as usual, thatthe qualitative bulk properties of the results are not affected by the BC seriously.[The above BC (4.1) means that the magnetization is just like the Ising type and h ~m x i = (0 , , m ), which describes the real materials properly.]In the simulation made in Refs.32 , 34 we use the standard Metropolisalgorithm35 for the lattice size up to L = 30. The typical number of sweeps for ameasurement is (30000 ∼ × 10 and the acceptance ratio is 40% ∼ U , the specific heat C of the central region R of the size L , the magnetization m i , which are defined as follows, U = 1 L h F L i , C = 1 L h ( F L − h F L i ) i , F L ≡ X r ∈ R f r , m i ≡ L h| X r ∈ R m ri |i , (4.3)and the normalized correlation functions, G m ( r − r ) = h ~m r · ~m r ih ~m r · ~m r i , G S ( r − r ) = h S r S r , ih S r , S r , i , (4.4)ctober 29, 2018 5:51 WSPC/INSTRUCTION FILE mplb20140730 Lattice gauge theory for condensed matter physics where r is chosen on the boundary of R such as (3 , L/ , z ). Singular behaviorsof U and/or C indicate the existence of phase transitions, and the magnetization m i and the correlation functions in Eqs.(4.4) clarify the physical meaning of theobserved phase transitions. As the present system is a frustrated system as weexplained above, G m and G S may exhibit some peculiar behavior in the FMSCstate (the state in which FM and SC orders coexist).To verify that the model (3.16) actually exhibits a FM phase transition as T is lowered, we put c = c = 0 and ( c , c , c ) = (0 . , . , . C increasing β in the Boltzmann factor of Eq.(3.16). In Fig.5 we show C and m i ,which obviously indicate that a second-order phase transition to the FM state takesplace at β FM = 1 /T FM ≃ . 0. We observed that other cases with various values of c , , exhibit similar FM phase transitions.Let us see the SC phase transition and how it coexists with the FM. We recallthat the case of all c i = 0 except for c was studied in the previous paper Ref.19.There it was found that the phase transition from the confinement phase to theHiggs phase takes place at c ≃ . 85. This result suggests that the SC state existsat sufficiently large c also in the present system with c i ( =1) = 0.To study this possibility, we use ( c , c , c ) = (0 . , . , . 0) as in the FM sectorabove and ( c , c ) = (0 . , . C vs β , which exhibits a largeand sharp peak at β ≃ . β ≃ . 5. To understand thephysical meaning of the second broad peak, it is useful to measure “partial specificheat” C i for each term F i in the free energy (3.11) defined by C i = 1 L h ( F i − h F i i ) i , F i = X r ∈ R f ir , (4.5)where f ir is the c i -term in f r of Eq.(3.11).Figs.6b,c show that the partial specific heat C , , have a sharp peak at β ≃ . C there should indicate the FM phase transition. The analysisof Meissner effect (see the discussion below using Fig.8) supports this point. On theother hand, C of the c -term has a relatively large and broad peak at β ≃ . C also shows a broader peak there. Then we judge that the SC phase transition =β (a) m m m β m µ (b) Fig. 5. (a) Specific heat for ( c , c , c ) = (0 . , . , . 0) and c = c = 0. At β ≃ . C exhibits asharp peak indicating a second-order FM phase transition. (b) Each component of magnetization m µ vs β . For T < T FM , m develops, whereas m and m are zero within the errors as expected. ctober 29, 2018 5:51 WSPC/INSTRUCTION FILE mplb20140730 I.Ichinose and T.Matsui C β C C (b) β C C C (c) Fig. 6. (a) Specific heat vs β for ( c , c ) = (0 . , . 2) and ( c , c , c ) = (0 . , . , . 0) ( L = 20).There are a large peak at β ≃ . β ≃ . 5. (b,c) Partial specific heat C i ofEq.(4.5) vs β . The small and broad peak at β ≃ . C is related to fluctuations of c -term. takes place at β SC ≃ . 5. To support these conclusions, we show G m ( r ) and G S ( r )in Fig.7. At β = 2 . G m ( r ) exhibits a finite amount of the FM order, whereas G S ( r ) decreases very rapidly to vanish. This means that, as T is decreased, theFM transition takes place first and then the SC transition does. Therefore, for β ≥ β SC ≃ . 5, the FM and SC orders coexist. In this way, the partial specific heatmay be used to judge the nature of each transition found by the peak of full specificheat C (We note C = P i C i in general due to interference).As we explained in the previous section, the bare transition temperature T inEq.(3.5) and the genuine transition temperature T SC are different. Then it is inter-esting to clarify the relation between them. From Eq.(3.16), any physical quantityis a function of βc i . In the numerical simulations, we fix the values of c i and vary β as explained. Then the result β SC ≃ . βc | T = T SC = 4 . × . . (4.6) r G m (r) r G S (r) Fig. 7. Correlation functions G m ( r ) and G S ( r ) at various T ’s for L = 20. c i ’s are the same as inFig.6. ctober 29, 2018 5:51 WSPC/INSTRUCTION FILE mplb20140730 Lattice gauge theory for condensed matter physics M G of the external magnetic field propagating in x - y plane vs β for thesame c i as in Fig.6 and c ′ = 3 . L = 16). At the SC phase transition point β SC ≃ . C , M G starts to increase from small values. By using Eq.(3.15), this gives the following relation;1 T SC Kα ( T − T SC ) aλ = 0 . → T SC = (cid:16) . λKα a (cid:17) − T . (4.7)Eq.(4.7) shows that the transition temperature is lowered by the fluctuations of thephase degrees of freedom of Cooper pairs. We expect that a relevant contributionto lowering the SC transition temperature comes from vortices that are generatedspontaneously in the FMSC as we shall show in Sec.4.2.After having confirmed that the genuine critical temperature can be calculatedby the critical value of β with fixed c i , we use the word temperature in the rest of thepaper just as the one defined by T ≡ /β while c i are T -independent parameters.One of the most important phenomena of the SC is the appearance of a finitemass of the electromagnetic field, i.e., the Meissner effect. In theoretical study ofthe SC that includes the effect of fluctuations of the Cooper-pair wave function asin the present one, a finite mass of the photon is the genuine order parameter ofthe SC. To study it, we follow the following steps19; (i) introduce a vector potential θ ex ri for an external magnetic field, (ii) couple it to Cooper pairs by replacing U ri → U ri exp( iθ ex ri ) in the c term of f x and add its magnetic term f ex r = + c ′ ( ~m ex r ) ( c ′ > 0) to f r with ~m ex r defined in the same way as (3.12) by using θ ex ri , (iii) let θ ex ri fluctuatetogether with z xa and θ ri and measure an effective mass M G of θ ex ri via the decayof correlation functions of ~m ex r . The result of ~m ex r propagating in the 1-2 plane isshown in Fig.8. It is obvious that the mass M G starts to develop at the SC phasetransition point, and we conclude that Meissner effect takes place in the SC state. Vortices in FMSC state In this section, we study the FMSC state observed in the previous section in detail.In particular, we are interested in whether there exist vortices and their densityif any. There are two kinds of vortices as the present SC state contains two gapsdescribed by ψ + r ∝ z + r and ψ − r ∝ z − r , where z ± r are defined as z ± r ≡ p z r ± iz r ) ≡ q ρ ± x exp( iγ ± r ) . (4.8)ctober 29, 2018 5:51 WSPC/INSTRUCTION FILE mplb20140730 I.Ichinose and T.Matsui Corresponding to the above Cooper pair fields, one may define the following twokinds of gauge-invariant vortex densities V + r and V − r in the 1-2 plane; V ± r ≡ π [mod( γ ± r +1 − γ ± r − θ r ) + mod( γ ± r +1+2 − γ ± r +1 − θ r +1 , ) − mod( γ ± r +1+2 − γ ± r +2 − θ r +2 , ) − mod( γ ± r +2 − γ ± r − θ r )] , (4.9)where mod( x ) ≡ mod( x, π ). In short, V ± r describes vortices of electron pairs withthe amplitude z ± r .To study the behaviors of vortices step by step, we first consider the case ofconstant magnetization (magnetic field), i.e., we set θ ri by hand so that m r = m r = 0 , m r = f = constant , (4.10)freezing their fluctuations. As the above numerical studies show that a typicalmagnitude of the magnetization h m r i = 0 . · · · , we put f = π . In this case, theSC phase transition takes place at β = 4 . 8. Snapshots of vortices are shown in Fig.9.It is obvious that at lower- T , i.e., β = 7 . 0, densities of vortices are low compared tothe higher- T ( β = 3 . 0) case but they are still nonvanishing. This result obviouslycomes from the existence of the finite magnetization, i.e., the internal magneticfield. A B C D Fig. 9. Snapshots of vortex densities V ± r at ( c , c , c , c , c ) = (0 . , . , . , . , . 0) for a fixedmagnetization ~m r = (0 , , π/ 4) ( L = 16) from Ref.32. Black dots; V ± r = 0.875, Dark gray dots;-1.125, Light gray dots; -0.125. (A) V + r at β = 3 . 0, (B) V − r at β = 3 . 0, (C) V + r at β = 7 . 0, (D) V − r at β = 7 . 0. The average magnitude h| V r ± |i is (A) 0.387, (B) 0.380, (C) 0.331 and (D) 0.335. Thepoints V ± r = − . 125 = − m / (2 π ) reflect ~m itself, and corresponds to the state without genuinevortices. ctober 29, 2018 5:51 WSPC/INSTRUCTION FILE mplb20140730 Lattice gauge theory for condensed matter physics β=2.0 + density+ vortex - density- vortexM Fig. 10. Snapshot at β = 2 . dynamically fluctuating ~m r . The values of c i are( c , c , c , c , c ) = (0 . , . , . , . , . It is interesting to study behavior of vortices and the magnetization in theFMSC state rather in detail. We now allow for fluctuations of θ ri . In this state, themagnetization m r fluctuates around its mean value. As we showed in the previoussection, the FM and SC phase transitions take place at β FM ≃ . β SC ≃ . β -dependence of various quantities such asthe magnetic field h m r i , the vortex densities V ± r , and the Cooper-pair densities h ρ ± r i .In Figs. 10 ∼ 13 we show these quantities as snapshots at β = 2 . , . , . , . c , c , c , c , c ) = (0 . , . , . , . , . β = 2 . m r fluctuates rather strongly, and also the density ofCooper pairs ρ + r and ρ − r are almost equal. Densities of vortices V ± r are also fairlylarge. At β = 3 . m r tends to have a positive value. This correspondsto the fact that the system has the FM long-range order. At β = 4 . ρ + r β=3.0 + vortex - vortexM - density+ density Fig. 11. Snapshot at β = 3 . dynamically fluctuating ~m r . The values of c i are the same asin Fig.10. ctober 29, 2018 5:51 WSPC/INSTRUCTION FILE mplb20140730 I.Ichinose and T.Matsui β=4.0 + density - density+ vortex - vortexM Fig. 12. Snapshot at β = 4 . dynamically fluctuating ~m r . The values of c i are the same asin Fig.10. is getting larger compared to ρ − r . Similarly, V − r is sightly larger than V + r . Theseresults reflect the Zeeman coupling. At β = 5 . V − r is larger than V + r . As in the case of the constant magnetic field,the densities of vortices are nonvanishing even in the SC phase. 5. Conclusion In this paper, we present a brief review of LGT with Higgs matter field, and thenapply it to the GL lattice model of FMSC. The detailed MC simulations of themodel reveal the global structure such as the phase diagram and the physical char-acteristics of each phase in an explicit manner. The concepts and knowledges ofLGT considerably help us to build an appropriate lattice model as well as to in-terpret the results of MC simulations. We want to add this example to the rather β=5.2 M + vortex - vortex- density+ density Fig. 13. Snapshot at β = 5 . dynamically fluctuating ~m r . The values of c i are the same asin Fig.10. ctober 29, 2018 5:51 WSPC/INSTRUCTION FILE mplb20140730 Lattice gauge theory for condensed matter physics long list of studies that have utilized the following common three-step method: (i)Establishing a lattice gauge model of the phenomenon of one’s interest, (ii) Per-forming MC simulations of the model, (iii) Intepreting the MC results accordingto LGT. Here we list up a couple of tips to remove obstacles in carrying out thisprogram. • MC simulations become more effective with less variables. To avoid unneces-sary inflation of degrees of freedom, one should identify what are the relevantvariables. Neglecting the radial degrees of freedom of the Higgs field φ r is anexample28. • MC simulations require positive measure to make use of a probabilistic(Markoff) process. Fermionic systems in their original forms suffer from thewell known negative-sign problem, which generates negative probabilities. Byrestricting the region of parameters of the model, it could be possible to avoidthis problem. For example, the critical region around a continuous phase tran-sition, the order-parameters are small and the GL expansion may assure us apositive provability6. • Another example to avoid nonpositive provability is the case of bosonic systemat finite temperatures. The path integral of bosonic variables usually containthe imaginary kinetic term like i R dτ ¯ φ ( dφ/dτ ) in the action, where τ ∈ (0 , β =1 / ( k B T )) is the imaginary time. By focusing in the high-temperature region,the τ -dependent modes of φ ( τ ) may be ignored and so is this kinetic term. Thisis the case of FMSC studied in Sect. 3 and 4, and some of the cases studied inRef.6 , 7. This procedure has an extra merit that the dimension of the effectivelattice system is reduce to d -dimensions instead of D = d + 1 dimensions dueto the absence of the x -direction.Nowadays, it is well known that a quite wide variety of systems of cold atomsput on an optical lattice may serve as a quantum simulator of certain definitequantum model with almost arbitrary values of interaction parameters. Such asimulator is to clarify the dynamical (time-dependent) properties of the quantummodel. Therefore, the importance of theoretical analysis of such models becomeincreased as a guide for experiments. For example, information on the global phasestructure of a quantum model is quite useful to perform experiments in an effectivemanner, although the former is confined to the static (time-independent) propertiesin thermal equilibrium. We expect that the above path (i-iii) with the successivetips may be useful for this purpose, in particular to simulate the model includingcompact or noncompact gauge fields. We hope as many readers as possible makeuse of this practical but powerful tool to strengthen and expand their researches. Acknowledgments ctober 29, 2018 5:51 WSPC/INSTRUCTION FILE mplb20140730 I.Ichinose and T.Matsui We thank Mr. T. Noguchi, our coauthor of Ref.34, for his help and discus-sions. We also thank Dr. K. Kasamatsu for reading the manuscript and presentingcomments and suggestions. This work was supported by JSPS KAKENHI Grant-numbers 23540301, 26400246, 26400412. References 1. See, e.g. a review; P. A. Lee, N. Nagaosa, and X-G Wen, Rev. Mod. Phys. , 17(2006).2. P. A. Lee, Science 321, (2008).3. M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D. S. Fisher, Phys. Rev. B 40 ,546 (1989); D. Jaksch, C. Bruder, J. I. Cirac, C. W. Cardiner, and P. Zoller, Phys.Rev. Lett. , 3108 (1998).4. M. W. 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