aa r X i v : . [ m a t h - ph ] D ec Two No-Go Theorems on Superconductivity
Yasuhiro Tada , and Tohru Koma U (1) phase of electrons for almost all gaugefields. These observations suggest that the nature of superconductivity is the emergenceof massive photons rather than the symmetry breaking of the U (1) phase of electrons. KEY WORDS:
Superconductivity, Persistent current, U (1) symmetry breaking, Bloch’stheorem, Elitzur’s theorem, Meissner effect Institute for Solid State Physics, The University of Tokyo, Kashiwa 277-8581, JAPAN, e-mail: [email protected] Max Planck Institute for the Physics of Complex Systems, N¨othnitzer Str. 38, 01187 Dresden,GERMANY Department of Physics, Gakushuin University, Mejiro, Toshima-ku, Tokyo 171-8588,JAPAN, e-mail: [email protected]
Introduction
Superconductivity still has some mysteries to solve although it has been one of the centralissues in condensed matter physics. Bloch’s theorem [1, 2, 3, 4, 5, 6, 7] states absence ofbulk persistent current in ground states and equilibrium states for general fermion systems,whereas experiments of superconductors [8] show persistent current with toroidal geom-etry. Elitzur’s theorem [9, 10, 11] states that local gauge symmetries including quantummechanical U (1) symmetry cannot be broken spontaneously, whereas superconductivityhas been often explained by using the idea that the U (1) symmetry is spontaneously bro-ken with a nonvanishing expectation value of an order parameter such as the Cooper pairamplitude. Although these two theorems have a long history, it is still unclear how thetwo theorems are reconciled with the conventional theory of superconductivity.If the superconducting state which carries the persistent current in the experiments isneither a ground state nor a thermal equilibrium state, then the state must be an excitedstate or a nonequilibrium state which has a very long lifetime. Such a state is often said tobe metastable [2]. What mechanism makes the lifetime of the metastable state carrying thepersistent current very long? One of the important stuffs is multiply-connected geometrysuch as toroid. In fact, we can expect that, when magnetic fields are wound up the surfaceof the large toroid in a stable state which is realized by Meissner effect, changing thetopology of the global structure needs a large energy which is determined by the systemsize.In the present paper, we consider possibility of a ground/equilibrium state which carriesa persistent current, and show that Bloch’s theorem cannot exclude such a state if thepersistent current is localized near the surface of the sample with boundary magnetic fieldswhich are stabilized by Meissner effect. Therefore, we can expect that a surface persistentcurrent is realized for a ground/equilibrium state in the sense of stability against localperturbations. Such a current carrying state has a long lifetime even in a finite sizesystem, because a transition from the corresponding state to other states costs larger andlarger energy as the system size increases.In order to explain Meissner effect in the above argument, one has to deal with theinteractions between electrons and electro-magnetic fields. As mentioned above, however,Elitzur’s theorem [9] implies that the local U (1) gauge symmetry cannot be broken sponta-neously. Actually, we can prove, by relying on the argument of [9], that the fluctuations ofthe gauge fields yield absence of the U (1) symmetry breaking for superconductors at finitetemperatures. Surprisingly, even for the systems with a fixed configuration of gauge fields,we can prove absence of symmetry breaking of the global U (1) phase of electrons for almostall configurations of gauge fields in the sense of macroscopic spontaneous magnetization.This is slightly different from the former consequence of Elitzur’s theorem.These observations in the present paper strongly suggest that the nature of supercon-ductivity is the emergence of massive photons rather than the symmetry breaking of the U (1) phase of electrons. In fact, it is known that, in Higgs models coupled with gaugefields [11, 12, 13, 14], a transition from massless photons to massive photons is possible forvarying the coupling constant, irrespective of whether or not the U (1) symmetry breakingoccurs.The present paper is organized as follows: In Sec. 2, we give the precise definition of2he models which we consider, and examine Bloch’s theorem in a mathematically rigorousmanner. As a result, we show that Bloch’s theorem cannot exclude the possibility ofsurface persistent current. In Sec. 3, we apply Elitzur’s theorem to two situations, thepresent models with annealed and quenched gauge fields. In both cases, we prove absenceof the U (1) symmetry breaking. In this section, we show that Bloch’s theorem does not exclude possibility of a surfacecurrent in a ground/equilibrium state, by examining the proof in a mathematically rigorousmanner.
Consider a d -dimensional connected graph G := (Λ , B ), where Λ is a set of lattice sitesand B is a set of bonds, i.e., pairs of lattice sites. The graph-theoretic distance, dist( x, y ),between two lattice sites x, y is defined as the minimum number of bonds in B that oneneeds to connect x and y .We consider a tight-binding model on the lattice Λ. The Hamiltonian H Λ is given by H Λ := X x,y ∈ Λ X α,β t α,βx,y c † x,α c y,β + X I ≥ X x ,σ X x ,σ · · · X x I ,σ I W x ,σ ; x ,σ ; ... ; x I ,σ I n x ,σ n x ,σ · · · n x I ,σ I , (2.1)where c † x,σ , c x,σ are, respectively, the creation and annihilation fermion operators at thesite x ∈ Λ with the internal degree of freedom, σ , such as spin or orbital; the hoppingamplitudes t α,βx,y are complex numbers which satisfy the Hermitian conditions, t β,αy,x = (cid:0) t α,βx,y (cid:1) ∗ , and the coupling constants W x ,σ ; x ,σ ; ... ; x I ,σ I of the interactions are real numbers. Asusual, we have written n x,σ = c † x,σ c x,σ for the number operators of the fermion with σ at the site x . We assume that both of the hopping amplitudes and the interactions areof finite range in the sense of the graph-theoretic distance, and assume that all of thestrengths are uniformly bounded as (cid:12)(cid:12) t β,αy,x (cid:12)(cid:12) ≤ t and | W x ,σ ; x ,σ ; ... ; x I ,σ I | ≤ W with some positive constants, t and W . Consider a cylindrical region Ω which is written a disjoint union of ( d − D j with a common radius R asΩ = D ∪ D ∪ D ∪ · · · ∪ D L ⊂ Λ3ith a large positive integer L . Namely, the region Ω consists of L -layers of the segments D j of the ( d − D i and D j satisfies dist( D i , D j ) = | i − j | for i, j = 1 , , . . . , L .In order to define the current operator on the region Ω, we consider a larger region ˜Ωwhich is written a disjoint union of large discs ˜ D j as˜Ω = ˜ D − ˜ L ∪ ˜ D − ˜ L +1 ∪ · · · ∪ ˜ D − ∪ ˜ D ∪ ˜ D ∪ ˜ D ∪ · · · ∪ ˜ D ˜ L − ∪ ˜ D ˜ L ⊂ Λ , where ˜ L is a large positive integer satisfying ˜ L > L . We choose the discs ˜ D j so that thefollowing conditions satisfy: ˜ D j ⊃ D j for j = 0 , , , . . . , L, dist( ˜ D i , ˜ D j ) = | i − j | for i, j = − ˜ L, − ˜ L + 1 , . . . , − , , , , . . . , ˜ L − , ˜ L, and dist(Ω , ∂ ˜Ω) ≥ r , where the boundary ∂ ˜Ω of the region ˜Ω is given by ∂ ˜Ω := { x ∈ ˜Ω | dist( x, Λ \ ˜Ω) = 1 } and r is the maximum hopping range, i.e., r := max { dist( x, y ) | t α,βx,y = 0 } . Clearly, one has Ω ⊂ ˜Ω.Let ϕ ∈ ℓ (Λ , C M ) be a wavepacket of single fermion, where C M is the space of theinternal degree of freedom with a finite dimension M . We assume that the support of thewavepacket ϕ is contained in the region ˜Ω. We define the position operator X as( Xϕ )( x ) = jϕ ( x ) for x ∈ ˜ D j . We introduce the step functions, ϑ j ( x ) := (cid:26) , x ∈ ˜Ω [ j, ˜ L ] ;0 , otherwise,with the kink ˜ D j , where we have written˜Ω [ j, ˜ L ] := ˜ D j ∪ ˜ D j +1 ∪ · · · ∪ ˜ D ˜ L . Then, the position operator X is written as X = ˜ L X j = − ˜ L +1 ϑ j − ˜ L
4n terms of the step functions. We write H (1)˜Ω for the single-fermion Hamiltonian on theregion ˜Ω. The current operator is given by [15, 16] J (1) := i [ H (1)˜Ω , X ] . From the above expression of the position operator X , we have J (1) := i [ H (1)˜Ω , X ] = ˜ L X j = − ˜ L +1 i [ H (1)˜Ω , ϑ j ] . for the above wavepacket ϕ . Since the summand in the right-hand side is nothing but thecurrent density operator across the disc ˜ D j , we write J (1) j := i [ H (1)˜Ω , ϑ j ]for the operator. In order to obtain the corresponding current density operator for manyfermions, we introduce the step function asΘ j := X σ X x ∈ ˜Ω [ j, ˜ L ] n x,σ by using the number operators n x,σ of fermions. In consequence, the current densityoperator across the disc ˜ D j for many fermions is written J j := i [ H ˜Ω , Θ j ] (2.2)for the wavefunction Φ which is supported by the region ˜Ω, where the Hamiltonian H ˜Ω isthe restriction of the Hamiltonian H Λ of (2.1) to the region ˜Ω. By using the expression(2.1) of the Hamiltonian H Λ , we obtain the explicit form of the operator J j as J j = X α,β X x ∈ ˜Ω [ − ˜ L,j − X y ∈ ˜Ω [ j, ˜ L ] i ( t α,βx,y c † x,α c y,β − t β,αy,x c † y,β c x,α ) , where ˜Ω [ − ˜ L,j − := ˜Ω \ ˜Ω [ j, ˜ L ] . Clearly, the above summand is a local current density. Relying on this expression, wedefine the current operator J Ω per volume on the region Ω as J Ω := 1 | Ω | L X j =1 X α,β X x ∈ Ω [0 ,j − X y ∈ ˜Ω [ j, ˜ L ] i ( t α,βx,y c † x,α c y,β − t β,αy,x c † y,β c x,α ) , (2.3)where | · · · | stands for the number of the elements in the set, and we have writtenΩ [0 ,j ] := D ∪ D ∪ · · · ∪ D j for j = 0 , , . . . , L − . .3 Bloch’s theorem In order to realize persistent current in the ground state of the present model, we considera toroidal geometry for the graph as an example. We impose a boundary condition whichmimics the plus boundary condition for ferromagnetic Ising models leading to the sym-metry breaking phases with plus spontaneous magnetization. To be specific, we apply amagnetic field tangential to the surface ∂ Λ of the toroidal lattice Λ so that surface currentperpendicular to the magnetic field can appear along the boundary of the toroid.Clearly, the total number operator P σ P x ∈ Λ n x,σ of the fermions commutes with theHamiltonian H Λ . We consider the N -fermion ground state Φ ( N )Λ , of the Hamiltonian H Λ with the above boundary condition. The ground-state expectation is given by ω ( N )Λ , ( · · · ) := D Φ ( N )Λ , , ( · · · )Φ ( N )Λ , E with the norm k Φ ( N )Λ , k = 1. We take the infinite-volume limit | Λ i | → ∞ by using thesequence of the lattice Λ i and of the total number N i of the fermions so that the den-sity Λ i /N i of the fermions converges to a positive constant ρ as i → + ∞ , and that theexpectation value ω ( N )Λ , ( a ) converges to the expectation value ω ( a ) with respect to theinfinite-volume ground state ω for all the local observables a which we consider.The statement of Bloch’s theorem is as follows: Theorem 2.1
The expectation value of the current operator J Ω per volume is vanishingfor the infinite-volume ground state ω , i.e., lim L ր∞ lim R ր∞ ω ( J Ω ) = 0 , (2.4) where Ω is the cylindrical region with length L and radius R .Remark: (i) For the two or higher dimensional systems, the order of the double limit in(2.4) is not interchangeable. In fact, for a fixed R , the upper bound of the expectationvalue in the limit L ր ∞ becomes infinity and meaningless in the proof below. If wewant to measure the strength of surface current which is localized near the surface of thesample, we must take R to be finite so that the support Ω of the current operator J Ω islocalized near the surface. Then, Bloch’s theorem tells us absolutely nothing about thesurface current.(ii) Clearly, in one dimensional systems, the surface current itself is meaningless, andthe net current along the direction of the chain always vanishes, in contrast to higherdimensional systems. This is a consequence of lack of a mechanism which stabilizes thecurrent against local perturbations. In other words, any current is destroyed by localperturbations in one dimension.(iii) The boundary magnetic fields are expected to be realized by Meissner effect, andonce it is generated, the surface current remains stable in a ground states, i.e., a persistentcurrent.(iv) The extension of the statement of Bloch’s theorem to the systems at finite tempera-tures is straightforward by relying on “passivity” which is a stability property of thermal6quilibrium states. More precisely, an infinite-volume state ω is said to be passive if ω ( U ∗ [ H, U ]) ≥ U , where H is the Hamiltonian. As is well known, all ofthermal equilibrium states are passive [17]. This fact was pointed out to us by Hal Tasaki. Proof:
We introduce two local unitary transformations, U ( ± )Ω := Y σ Y x ∈ Ω exp [ iθ ± ( x ) n x,σ ] , (2.5)where θ ± ( x ) := ± πℓ ( x ) L (2.6)with ℓ ( x ) := (cid:26) j, x ∈ D j for j = 1 , , . . . , L ;0 , otherwise. (2.7)Using these local transformations, we set ω ( N )Λ , ± ( · · · ) := D Φ ( N )Λ , , ( U ( ± )Ω ) ∗ ( · · · ) U ( ± )Ω Φ ( N )Λ , E . Since these two states are a local perturbation for the ground state, we have ω ( N )Λ , ± ( H Λ ) − ω ( N )Λ , ( H Λ ) = ω ( N )Λ , ([( U ( ± )Ω ) ∗ H Λ U ( ± )Ω − H Λ ]) ≥ . (2.8)Note that exp[ − iθ ± ( x ) n x,α ] c † x,α exp[ iθ ± ( x ) n x,α ] = exp[ − iθ ± ( x )] c † x,α and exp[ − iθ ± ( y ) n y,β ] c y,β exp[ iθ ± ( y ) n y,β ] = exp[ iθ ± ( y )] c y,β By using these relations and the expression (2.1) of the Hamiltonian, one has( U ( ± )Ω ) ∗ H Λ U ( ± )Ω − H Λ = X α,β X x,y t α,βx,y { exp[ − i ( θ ± ( x ) − θ ± ( y ))] − } c † x,α c y,β . The right-hand side can be decomposed into three parts as( U ( ± )Ω ) ∗ H Λ U ( ± )Ω − H Λ = ∆ H Ω + ∆ H (1) ∂ Ω + ∆ H (2) ∂ Ω , (2.9)where ∆ H Ω = X α,β X x,y ∈ Ω t α,βx,y { exp[ − i ( θ ± ( x ) − θ ± ( y ))] − } c † x,α c y,β , ∆ H (1) ∂ Ω = X α,β X x ∈ Ω X y / ∈ Ω t α,βx,y { exp[ − iθ ± ( x )] − } c † x,α c y,β , and ∆ H (2) ∂ Ω = X α,β X x/ ∈ Ω X y ∈ Ω t α,βx,y { exp[ iθ ± ( y )] − } c † x,α c y,β , H (1) ∂ Ω . For this purpose, we decompose ˜Ω intothree parts as ˜Ω = ˜Ω ( − ) ∪ ˜Ω (0) ∪ ˜Ω (+) with ˜Ω ( − ) := ˜ D − ˜ L ∪ · · · ∪ ˜ D − , ˜Ω (0) := ˜ D ∪ ˜ D ∪ · · · ∪ ˜ D L , and ˜Ω (+) := ˜ D L +1 ∪ · · · ∪ ˜ D ˜ L . Using this decomposition, the operator ∆ H (1) ∂ Ω further decompose into three parts as∆ H (1) ∂ Ω = ∆ H (1 , − ) ∂ Ω + ∆ H (1 , ∂ Ω + ∆ H (1 , +) ∂ Ω , where ∆ H (1 , − ) ∂ Ω := X α,β X x ∈ Ω X y ∈ ˜Ω ( − ) t α,βx,y { exp[ − iθ ± ( x )] − } c † x,α c y,β , ∆ H (1 , ∂ Ω := X α,β X x ∈ Ω X y ∈ ˜Ω (0) \ Ω t α,βx,y { exp[ − iθ ± ( x )] − } c † x,α c y,β , and ∆ H (1 , +) ∂ Ω := X α,β X x ∈ Ω X y ∈ ˜Ω (+) t α,βx,y { exp[ − iθ ± ( x )] − } c † x,α c y,β . We assume that all of the discs D j satisfy | D j | = O ( R d − ), where R is the radius of thediscs. Then, the volume | Ω | of the region Ω satisfies | Ω | = O ( R d − × L ). We also assumethat the area of the side of the cylindrical region Ω satisfies | ∂ Ω \ ( D ∪ D L ) | = O ( R d − × L ).Since the hopping range is of finite, we have (cid:13)(cid:13)(cid:13) ∆ H (1 , ± ) ∂ Ω (cid:13)(cid:13)(cid:13) ≤ Const . × R d − L from the definition (2.6) of θ ± ( x ). In the same way, one has (cid:13)(cid:13)(cid:13) ∆ H (1 , ∂ Ω (cid:13)(cid:13)(cid:13) ≤ Const . × R d − L. Combining these, we obtain L | Ω | (cid:13)(cid:13)(cid:13) ∆ H (1) ∂ Ω (cid:13)(cid:13)(cid:13) ≤ Const . × L + Const . × LR . (2.10)In the same way, we also have L | Ω | (cid:13)(cid:13)(cid:13) ∆ H (2) ∂ Ω (cid:13)(cid:13)(cid:13) ≤ Const . × L + Const . × LR . (2.11)8ext, consider ∆ H Ω . From (2.6) and (2.7), one has∆ H Ω = ± πL X α,β X x,y ∈ Ω: ℓ ( x ) <ℓ ( y ) i [ ℓ ( y ) − ℓ ( x )] (cid:16) t α,βx,y c † x,α c y,β − t β,αy,x c † y,β c x,α (cid:17) + O ( | Ω | × L − ) . Write k = ℓ ( y ) − ℓ ( x ) for the factor in the summand in the right-hand side. In theexpression (2.3) of the current J Ω on the region Ω, the nonvanishing contributions for x, y satisfying k = ℓ ( y ) − ℓ ( x ) appear k times in the sums. From these observations, we have12 π L | Ω | ∆ H Ω ∓ J Ω = O ( L − ) + O ( L/R ) . Combining this, (2.8), (2.9), (2.10) and (2.11), we obtain ± lim L ր∞ lim R ր∞ ω ( J Ω ) ≥ , where ω is the infinite-volume ground state, and the double limit is taken so that L/R → U (1) symmetry breaking In this section, we extend Elitzur’s theorem to fermionic models. Consider the Hamiltonian H Λ of (2.1) on a d -dimensional finite lattice Λ ⊂ Z d . For simplicity, we assume that theHamiltonian H Λ contains only the nearest neighbor hopping, and that the fermions haveonly spin-1/2 as the internal degree of freedom, i.e., we consider usual electrons. In order to take into account the fluctuation of an electromagnetic field, we introduce a U (1) gauge field A as follows: For each nearest neighbor pair h x, y i of sites x, y ∈ Λ, thegauge field A x,y takes the value A x,y ∈ R mod 2 π , and satisfies the conditions, A y,x = − A x,y mod 2 π. The hopping amplitudes of the Hamiltonian H Λ of (2.1) is replaced with˜ t α,βx,y ( A ) := t α,βx,y e iA x,y for each nearest neighbor pair h x, y i of sites. Then, the Hamiltonian of electrons coupledto the gauge field A is given by˜ H Λ ( A ) := X x,y ∈ Λ X α,β ˜ t α,βx,y ( A ) c † x,α c y,β + X I ≥ X x ,σ X x ,σ · · · X x I ,σ I W x ,σ ; x ,σ ; ... ; x I ,σ I n x ,σ n x ,σ · · · n x I ,σ I . (3.1)9s a local order parameter to detect the U (1) symmetry breaking, we consider the Cooperpair c u, ↑ c v, ↓ of electrons for fixed two sites u, v , where we assume that the site v is in aneighborhood N u of the site u . The total energy containing the energy of the gauge fieldsand the terms of the symmetry breaking fields is H ( N )Λ ( A ) := ˜ H Λ ( A ) − µN Λ − κ X p cos B p − hH ext , Λ − ˜ h ˜ H ext , Λ ( A ) , (3.2)where N Λ is the total number operator of electrons with the chemical potential µ , i.e., N Λ := X σ X x ∈ Λ n x,σ , and B p is the magnetic flux through the plaquette p (unit square cell); the Hamiltonian H ext , Λ of the symmetry breaking field is given by the sum of the local order parameters as H ext , Λ := X x ∈ Λ X y ∈N x (cid:16) c x, ↑ c y, ↓ + c † y, ↓ c † x, ↑ (cid:17) , and the last term in the energy (3.2) is also the Hamiltonian of the symmetry breakingfield, ˜ H ext , Λ ( A ) := X b ∈B cos A b , (3.3)for the gauge field A . Here, B is the set of the bonds (the nearest neighbor pairs of sites),and we have written A b = A x,x + e i with the unit vector e i in the i -th direction. The threeparameters, κ, h and ˜ h , are taken to be positive.The expectation value of the Cooper pair at an inverse temperature β is given by h c u, ↑ c v, ↓ i Λ ( β, h, ˜ h, κ ) := 1 Z Λ ( β ) Z π − π Y b ∈B dA b Tr c u, ↑ c v, ↓ exp[ − β H ( N )Λ ( A )] , (3.4)where Z Λ ( β ) is the partition function. In the following, we take the limit κ ↑ ∞ so that allthe magnetic flux B p through the plaquette p are vanishing. We stress that our result stillholds for a finite κ . But we impose this strong condition because a nonvanishing magneticflux generally is believed to suppress superconducting states. In the infinite limit κ ↑ ∞ ,the gauge fixing degree of freedom still remains. Therefore, in order to lift the gaugefixing degree of the freedom, we have introduced the Hamiltonian ˜ H ext , Λ ( A ) of (3.3) whichprefers the gauge fixing A b = 0 for all the bonds b .Let us consider the spontaneous magnetization for the Cooper pair in the infinite-volume limit and the zero magnetic field limit. Namely, we define h c u, ↑ c v, ↓ i ( β ) := lim ˜ h ↓ lim h ↓ lim Λ ր Z d lim κ ր∞ h c u, ↑ c v, ↓ i Λ ( β, h, ˜ h, κ ) . (3.5)More precisely, the left-hand side denotes all of the accumulation points in the limit.We have: 10 heorem 3.1 The spontaneous magnetization for the Cooper pair is vanishing in theinfinite-volume limit and in the zero magnetic field, i.e., h c u, ↑ c v, ↓ i ( β ) = 0 , (3.6) for any inverse temperatures β .Remark: (i) Instead of the Cooper pair amplitude, the statement of Theorem 3.1 holds forany local observable which transforms nontrivially under the local gauge transformations.(ii) Consider the situation without the symmetry breaking fields. But, instead of thesymmetry breaking fields, we introduce gauge fixing terms into the Hamiltonian as inKennedy and King [20] and Borgs and Nill [11, 13]. For the Higgs models, they provedthat the two-point correlations for the Higgs fields do not exhibit long-range order indimensions d ≤ U (1) Higgs model has a phase transition indimensions d ≥
3. However, when there appears the remaining gauge degree of freedomwhich is called the Gribov ambiguity, the two-point correlations vanish at noncoincidingpoints even in the Landau gauge as Borgs and Nill [11, 13] pointed out.In the same situation for the present lattice fermion systems, we can prove that the two-point Cooper pair correlations do not show long-range order in spatial dimensions d ≤ Proof:
We introduce the unitary transformation with twisting angle θ ∈ R mod 2 π at thesite u for the Cooper pair c u, ↑ c v, ↓ as U u, ↑ ( θ ) := exp[ iθn u, ↑ ] . Since one has ( U u, ↑ ( θ )) ∗ c u, ↑ U u, ↑ ( θ ) = e iθ c u, ↑ , the hopping amplitudes in the Hamiltonian ˜ H Λ ( A ) of (3.1) for electrons change under thetransformation as ˜ t α,βx,y ( A ) → ˜ t α,βx,y ( A ′ ) , where A ′ x,y = A x,y − θ for x = u and y = u + e i ,A ′ y,x = A y,x + θ for x = u and y = u − e i , and A ′ x,y = A x,y , otherwise. Therefore, the energy operator of (3.2) is transformed as(( U u, ↑ ( θ )) ∗ H ( N )Λ U u, ↑ ( θ ))( A ) = ˜ H Λ ( A ′ ) − µN Λ − κ X p cos B p − h ( U u, ↑ ( θ )) ∗ H ext , Λ U u, ↑ ( θ ) − ˜ h ˜ H ext , Λ ( A ) . H ext := ( U u, ↑ ( θ )) ∗ H ext , Λ U u, ↑ ( θ ) − H ext , Λ and ∆ ˜ H ext := ˜ H ext , Λ ( A ) − ˜ H ext , Λ ( A ′ ) . Clearly, both of these are local. Then, we have(( U u, ↑ ( θ )) ∗ H ( N )Λ U u, ↑ ( θ ))( A ) = H ( N )Λ ( A ′ ) − h ∆ H ext − ˜ h ∆ ˜ H ext . Using this unitary transformation and changing the variables in the integral for the ex-pectation value of (3.4), we have h c u, ↑ c v, ↓ i Λ ( β, h, ˜ h, κ ) = e iθ Z Λ ( β ) Z π − π Y b ∈B dA ′ b Tr c u, ↑ c v, ↓ exp[ − β H ( N )Λ ( A ′ ) − β ∆ H ext ] , (3.7)where we have written ∆ H ext := h ∆ H ext + ˜ h ∆ ˜ H ext for short.Let O be an observable. Note that∆ N ( O ) := Tr O exp[ − β H ( N )Λ ( A ′ ) − β ∆ H ext ] − Tr O exp[ − β H ( N )Λ ( A ′ )]= Z dλ ddλ Tr O exp[ − β H ( N )Λ ( A ′ ) − λβ ∆ H ext ]= ( − β ) Z dλ ( O, ∆ H ext )Tr exp[ − β H ( N )Λ ( A ′ ) − λβ ∆ H ext ] , (3.8)where ( O , O ) is the Duhamel two-point function [18, 19] which is given by( O , O ) := 1 Z Z dz Tr O e − zβ H O e − (1 − z ) β H for the observables O , O . In order to evaluate the right-hand side of (3.8), let us considerlog Tr exp[ − β H ( N )Λ ( A ′ ) − λβ ∆ H ext ] − log Tr exp[ − β H ( N )Λ ( A ′ )]= Z dλ ′ ddλ ′ log Tr exp[ − β H ( N )Λ ( A ′ ) − λ ′ λβ ∆ H ext ]= ( − λβ ) Z dλ ′ Tr ∆ H ext exp[ − β H ( N )Λ ( A ′ ) − λ ′ λβ ∆ H ext ]Tr exp[ − β H ( N )Λ ( A ′ ) − λ ′ λβ ∆ H ext ] . (3.9)Therefore, one has (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) log Tr exp[ − β H ( N )Λ ( A ′ ) − λβ ∆ H ext ]Tr exp[ − β H ( N )Λ ( A ′ )] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ | λ | β k ∆ H ext k . This implies e −| λ | β k ∆ H ext k ≤ Tr exp[ − β H ( N )Λ ( A ′ ) − λβ ∆ H ext ]Tr exp[ − β H ( N )Λ ( A ′ )] ≤ e | λ | β k ∆ H ext k . (3.10)12ote that the Duhamel two-point function satisfies [18, 19] | ( O , O ) | ≤ k O k k O k . Using this and the inequality (3.10), ∆ N ( O ) of (3.8) can be estimated as | ∆ N ( O ) | ≤ β k O k k ∆ H ext k e β ( ah +˜ a ˜ h ) Tr exp[ − β H ( N )Λ ( A ′ )] ≤ β ( ah + ˜ a ˜ h ) k O k e β ( ah +˜ a ˜ h ) Tr exp[ − β H ( N )Λ ( A ′ )] , (3.11)where a and ˜ a are some positive constant.By using ∆ N ( c u, ↑ c v, ↓ ) of (3.8) for the right-hand side of (3.7), we obtain h c u, ↑ c v, ↓ i Λ ( β, h, ˜ h, κ ) = e iθ h c u, ↑ c v, ↓ i Λ ( β, h, ˜ h, κ ) + e iθ Z Λ ( β ) Z π − π Y b ∈B dA ′ b ∆ N ( c u, ↑ c v, ↓ ) . (3.12)From the inequality (3.11), the second term in the right-hand side is estimated as (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z Λ ( β ) Z π − π Y b ∈B dA ′ b ∆ N ( c u, ↑ c v, ↓ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Z Λ ( β ) Z π − π Y b ∈B dA ′ b β ( ah + ˜ a ˜ h ) e β ( ah +˜ a ˜ h ) k c u, ↑ c v, ↓ k Tr exp[ − β H ( N )Λ ( A ′ )] ≤ β ( ah + ˜ a ˜ h ) e β ( ah +˜ a ˜ h ) k c u, ↑ c v, ↓ k . (3.13)Then, from (3.5), (3.12) and (3.13), we obtain h c u, ↑ c v, ↓ i ( β ) = e iθ h c u, ↑ c v, ↓ i ( β ) . This implies the desired result (3.6).
Next, we extend Elitzur’s theorem to the present models whose configurations of gaugefields are quenched. For a given gauge fields A , let us consider the Hamiltonian,ˆ H ( N )Λ ( A ) = ˜ H Λ ( A ) − µN Λ − hH ext , Λ (3.14)with the Hamiltonian of the symmetry breaking field, H ext , Λ = X x ∈ Λ X y ∈N x (cid:16) c x, ↑ c y, ↓ + c † y, ↓ c † x, ↑ (cid:17) . Then, the expectation value is given by h· · ·i Λ ( β, h, A ) = 1 Z Λ ( β, A ) Tr( · · · ) exp[ − β ˆ H ( N )Λ ( A )] , (3.15)13here Z Λ ( β, A ) is the partition function. As an example, we consider the expectation valueof the Cooper pair h c u, ↑ c v, ↓ i Λ ( β, h, A ). Here, we have not introduced the Hamiltonian ofthe symmetry breaking field (3.3) because the term plays no role in the expectation value.The average with respect to the gauge fields A is given by E Λ ,κ [ · · · ] := Z π − π Y b ∈B dA b ρ Λ ,κ ( A ) ( · · · ) (3.16)with the probability density, ρ Λ ,κ ( A ) := exp[ κ P p cos B p ] R π − π Q b ∈B dA b exp[ κ P p cos B p ] . In the limit κ ր ∞ , the zero magnetic field is realized. In the following, we take the limitas in the preceding section.We write M Λ := 1 | Λ | X x ∈ Λ X y ∈N x (cid:16) c x, ↑ c y, ↓ + c † y, ↓ c † x, ↑ (cid:17) , (3.17)and M ( β, h, A ) := lim Λ ր Z d hM Λ i Λ ( β, h, A ) . We define the spontaneous magnetization M ( β, A ) and the long-range order σ ( β, A ) as M ( β, A ) := lim h ↓ M ( β, h, A )and σ ( β, A ) := lim Λ ր Z d p h ( M Λ ) i Λ ( β, , A ) . Here, we consider all of the accumulation points again.We obtain:
Theorem 3.2
Both of the spontaneous magnetization M ( β, A ) and the long-range order σ ( β, A ) for the Cooper pair are vanishing for almost all gauge fields A .Remark: (i) Theorem 3.2 does not exclude the possibility that the U (1) symmetry break-ing occurs for some particular gauge fixing. As mentioned in Remark (ii) of Theorem 3.1,Kennedy and King [20, 21] showed that there appears U (1) symmetry breaking in a non-compact U (1) Higgs model in Landau gauge. However, there are some differences betweennoncompact and compact gauge theories. In fact, except for this special case which wastreated by Kennedy and King, several cases in a certain gauge fixing show absence of the U (1) symmetry breaking in Higgs models [11, 12, 13].(ii) Remark (i) of Theorem 3.1 holds for the present case, too. Proof:
Using the gauge transformation in the same way as in the preceding subsection,we have E Λ ,κ [ h c u, ↑ c v, ↓ i Λ ( β, h, · · · )] = e iθ Z π − π Y b ∈B dA ′ b ρ Λ ,κ ( A ′ ) Z ′ Λ ( β, A ′ ) Tr c u, ↑ c v, ↓ exp[ − β ( ˆ H ( N )Λ ( A ′ )+∆ H ext )] , (3.18)14here the deformed partition function is given by Z ′ Λ ( β, A ′ ) = Tr exp[ − β ( ˆ H ( N )Λ ( A ′ ) + ∆ H ext )] . and ∆ H ext = h ( U u, ↑ ( θ )) ∗ H ext , Λ U u, ↑ ( θ ) − hH ext , Λ . This expectation value can be written as E Λ ,κ [ h c u, ↑ c v, ↓ i Λ ( β, h, · · · )]= e iθ Z π − π Y b ∈B dA ′ b ρ Λ ,κ ( A ′ ) Z ′ Λ ( β, A ′ ) n Tr c u, ↑ c v, ↓ exp[ − β ( ˆ H ( N )Λ ( A ′ ) + ∆ H ext )] − Tr c u, ↑ c v, ↓ exp[ − β ˆ H ( N )Λ ( A ′ )] o + e iθ Z π − π Y b ∈B dA ′ b ρ Λ ,κ ( A ′ ) Z Λ ( β, A ′ ) (cid:20) Z Λ ( β, A ′ ) Z ′ Λ ( β, A ′ ) − (cid:21) Tr c u, ↑ c v, ↓ exp[ − β ˆ H ( N )Λ ( A ′ )]+ e iθ E Λ ,κ [ h c u, ↑ c v, ↓ i Λ ( β, h, · · · )] . (3.19)The first and the second terms in the right-hand side can be evaluated in the same wayas in (3.11) and (3.10), respectively. In consequence, we obtain | E Λ ,κ [ h c u, ↑ c v, ↓ i Λ ( β, h, · · · )] | ≤ Const . βh | − e iθ | e βah (3.20)with some positive constant a . This implieslim h ↓ lim Λ ր Z d lim κ ↑∞ E Λ ,κ [ h c u, ↑ c v, ↓ i Λ ( β, h, · · · )] = 0for any pair of two sites u, v . Similarly, for M Λ of (3.17), we have | E Λ ,κ [ hM Λ i Λ ( β, h, · · · )] |≤ | Λ | X x ∈ Λ X y ∈N x n(cid:12)(cid:12)(cid:12) E Λ ,κ [ h c x, ↑ c y, ↓ i Λ ( β, h, · · · )] (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) E Λ ,κ [ h c † y, ↓ c † x, ↑ i Λ ( β, h, · · · )] (cid:12)(cid:12)(cid:12)o ≤ Const . βh | − e iθ | e βah by using inequalities similar to the above inequality (3.20). Immediately,lim h ↓ lim Λ ր Z d lim κ ↑∞ E Λ ,κ [ hM Λ i Λ ( β, h, · · · )] = 0 . (3.21)Note that the global U (1) transformation contains the reversal of the orientation of theexternal symmetry breaking field. Under this transformation, the free energy is invariant.Combining this fact with the concavity of the free energy, one can show the positivity ofthe magnetization, i.e., hM Λ i Λ ( β, h, A ) ≥ h ≥
0. Let ε be a small positive number, and consider the probability that the eventssatisfying hM Λ i Λ ( β, h, A ) ≥ ε occur. Then, Markov’s inequality yields E Λ ,κ [ hM Λ i Λ ( β, h, · · · )] ≥ ε Prob [ hM Λ i Λ ( β, h, · · · ) ≥ ε ] . From the above result (3.21), we obtain0 = lim h ↓ lim Λ ր Z d lim κ ↑∞ E Λ ,κ [ hM Λ i Λ ( β, h, · · · )] ≥ ε lim h ↓ lim Λ ր Z d lim κ ↑∞ Prob [ hM Λ i Λ ( β, h, · · · ) ≥ ε ]= ε lim h ↓ Prob [ M ( β, h, · · · ) ≥ ε ] , (3.22)where Prob [ M ( β, h, · · · ) ≥ ε ] is the probability that the magnetization M ( β, h, A ) in theinfinite-volume limit satisfies M ( β, h, A ) ≥ ε . This implies that the probability that thespontaneous magnetization M ( β, A ) is greater than or equal to ε is vanishing. Sincethe small positive number ε is arbitrary, we obtain that the spontaneous magnetization M ( β, A ) is vanishing with probability one.In order to show that the long-range order σ ( β, A ) for the Cooper pair is vanishing, werecall the relation [22] between the spontaneous magnetization and the long-range orderas M ( β, A ) ≥ σ ( β, A ) ≥ . (3.23)In their derivation, they relied on the existence of the thermodynamic limit of the freeenergy per volume. However, for an inhomogeneous system such as the present system witha fixed configuration of random gauge fields, the thermodynamic limit of the free energyper volume may not be unique. Therefore, we consider all of the accumulation points inthe limit, by relying on the boundedness of the free energy per volume. Combining theinequality (3.23) with the vanishing of the spontaneous magnetization M ( β, A ), we obtainthe desired result, i.e., the vanishing of the long-range order with probability one. Acknowledgements:
We would like to thank Peter Fulde, Hosho Katsura, MasaakiShimozawa, Hal Tasaki and Masafumi Udagawa for helpful discussions. YT was partlysupported by JSPS/MEXT Grant-in-Aid for Scientific Research (Grant No. 26800177) andby Grant-in-Aid for Program for Advancing Strategic International Networks to Acceleratethe Circulation of Talented Researchers (Grant No. R2604) “TopoNet.”
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