Infinite-range correlations in 1D systems with continuous symmetry
IInfinite-range correlations in 1D systems with continuous symmetry
Tobias Rindlisbacher ∗ University of Helsinki, Department of Physics, P.O. Box 64, FI-00014 University of Helsinki, Finland O( N ) -symmetric lattice scalar fields are considered, coupled to a chemical potential and source terms. Atthe example of N = 2 , it is shown that such systems can even in (0+1) dimensions produce infinite-rangecorrelations and a non-zero vacuum expectation value whenever the chemical potential assumes certain discretevalues. Different mechanisms for how the latter phenomena are produced are discussed, depending on whethersource terms are set to zero or non-zero values. In the conclusion, the relation of these findings to the Mermin-Wagner theorem is addressed. Keywords: O( N ) scalar field; chemical potential; finite density; long-range order; Mermin-Wagner theorem. I. INTRODUCTION
Mermin and Wagner showed in 1966 [1], usingBogoliubov’s inequality [2], that in one and twodimensions the ferromagnetic (antiferromagnetic)isotropic Heisenberg model with finite-range inter-action cannot undergo spontaneous magnetization(sub-lattice magnetization) at any non-zero temper-ature. Isotropic here refers to the internal space inwhich the Heisenberg spins ¯ s x = ( s x , s x , s x ) takevalue, and means that if we write the interactionbetween two spins ¯ s x , ¯ s y , located on sites x and y , as (cid:80) i =1 α i ( x − y ) s ix s iy , then α ( r ) = α ( r ) = α ( r ) for all r . If, on the other hand, the couplings satisfy α ( r ) = α ( r ) (cid:54) = α ( r ) , so that the global O(3) -symmetry of the isotropic case is reduced to
O(2) ,the authors mention that the same line of reasoningwould then only rule out spontaneous magnetization(sub-lattice magnetization) in the s - s -plane (whereit would break the reduced O(2) -symmetry). Theauthors also stressed, that the impossibility of sponta-neous magnetisation does not exclude the existence ofother types of phase transitions in these models. Forthe theorem to apply, it is important that long-rangeinteractions are sufficiently suppresssed [3, 4].In 1967 Wegner [5] confirmed this for the ferro-magnetic Heisenberg model with reduced
O(2) -symmetry ( ∼ XY-model), by showing, based on alow-temperature expansion, that the spin-spin cor-relation function in one and two dimensions alwaysconverges to zero at sufficiently large distances. TheMermin-Wagner theorem has in the following yearsbeen generalized to many other (non-relativistic)classical and quantum systems, cf. [6–12], to mentiononly a few. In 1973 Coleman [13] then proved theanalogue of the Mermin-Wager theorem [1] and thework of Wegner [5] and Berezinsky [10, 11] for arelativistic scalar field theory; showing that in two (i.e. (1 + 1) ) dimensions, the vacuum expectation valueof a scalar field with a continuous global symmetry isalways zero, so that no Goldstone bosons can occur. ∗ tobias.rindlisbacher@helsinki.fi In this article, we consider non-linear O( N ) lattice models in the presence of a chemical potentialthat couples to the conserved charge of a U(1) sub-symmetry of O( N ) . On an Euclidean lattice,one can in these models for a discrete but infiniteset of values of the chemical potential even in theone-dimensional case (i.e. in (0+1) dimensions)observe infinite range correlations and the formationof a non-zero vacuum expectation value. From afield theoretic point of view, this (0+1)-dimensionalcase is, of course, not particularly interesting, and itdoes also not conflict with Coleman’s "no Goldstonebosons" theorem, as the latter addresses only (1 + 1) dimensional systems. However, the Euclidean latticeformulations of our O( N ) spin models can also beinterpreted in a solid state physics context, with thethe Euclidean action S playing the role of β H , i.e. ofthe product of the inverse temperature β = 1 / ( k B T ) and the Hamiltonian H . From this point of view, theformation of long-range order in a one-dimensionalsystem seems to conflict with the Mermin-Wagnertheorem. As the one-dimensional lattice is in this casespatial, the chemical potential parameter µ couples toa spatial current and could be interpreted as a negativeresistance .The paper is organized as follows: in the followingsection, Sec. II, the lattice model as well as itsformulation in terms of dual flux-variables will beintroduced in detail. In Sec. III we then show ana-lytically that on a one-dimensional lattice, the modelcan for a discrete but infinite set of values of thechemical potential develop infinite range correlationsand a non-zero vacuum expectation value. Sec. IVsummarizes the findings and discusses their relationto the Mermin-Wagner theorem [1]. II. THE MODEL
The model of interest to us in the present workis the non-linear O( N ) spin model on a 1D (one-dimensional) Euclidean lattice, coupled to a chemicalpotential and source terms. In order to get an idea of a r X i v : . [ h e p - l a t ] D ec how the lattice model is related to the correspondingcontinuum theory, we will first write down the actionin d = ( d s + 1) -dimensional continuous Minkowskispace ( d s being the number of spatial dimensions).We then go through the steps of performing the Wick-rotation to Euclidean time, and then putting the model on a d = ( d s + 1) -dimensional Euclidean lattice. Inthis way, we can keep track of how the lattice parame-ters are related to their continuum counterparts and thelattice spacing a .So, in ( d s + 1) -dimensional Minkowski space, the ac-tion for our non-linear O( N ) -model reads: S M [ φ ] = (cid:90) d d x (cid:110) f π (cid:0)(cid:0) ∂ ν + 2 i µ c τ δ ν (cid:1) φ c ( x ) (cid:1) (cid:62) g νρ (cid:0)(cid:0) ∂ ρ + 2 i µ c τ δ ρ (cid:1) φ c ( x ) (cid:1) − j ( x ) φ c ( x ) (cid:111) , (II.1)where g νρ = diag(1 , − , . . . , − is the Minkowskispace metric, φ c ∈ S N − ⊂ R N , and f π is a (cou-pling) constant of mass dimension ( d − / , nec-essary to render the action (in natural units) dimen-sion less while φ ∈ S N − is itself dimensionless.The field j = ( j , . . . , j N ) ∈ R N is a N -vector ofsource terms. The quantity τ that comes with thechemical potential µ c is an O( N ) -generator ( τ ij ) ab = − i( δ ai δ j,b − δ aj δ i,b ) , so that µ c couples to the conservedcharge, corresponding to the U(1) -symmetry of rota-tions in the φ c - φ c -plane . The subscript c (for con-tinuum) in φ c and µ c is just there to avoid confus-ing these continuum quantities with their lattice ana-logues that will be introduced shortly. If the chemi-cal potential is non-zero, the global O( N ) -symmetryis explicitly broken to O( N − × U(1) . By setting j ( x ) = J , with J ∈ R N to a position-independent,non-zero value, the symmetry can be reduced further. Setting the components J or J to non-zero valuescan be problematic in combination with a non-zerochemical potential, as these source components ex-plicitly break the U(1) symmetry in the φ c - φ c -planeand thereby render the charge to which the chemicalpotential couples non-conserved. However, as longas ˜ J = (cid:112) ( J ) + ( J ) is small, the U(1) -chargecan still be considered approximately conserved. Non-zero values for the components J i with i ≥ are onthe other hand never a problem and will just break the O( N − part of the global symmetry further downto O( N − .We now perform a Wick rotation, so that i S M [ φ ] →− S E [ φ ] . With our sign convention for the Minkowskimetric, this is achieved by setting x = − i x d (imply-ing d x = − i d x d and ∂ = i ∂ d ), and the Euclideanaction is then obtained as: S E [ φ ] = (cid:90) d d x (cid:110) f π (cid:0) ( ∂ ν + 2 µ τ δ ν,d ) φ ( x ) (cid:1) (cid:62) g νρ (cid:0) ( ∂ ρ + 2 µ τ δ ρ,d ) φ ( x ) (cid:1) + j ( x ) φ ( x ) (cid:111) , (II.2)where g νρ and φ ( x ) are now the Euclidean met-ric and field, respectively, with the latter being ob-tained from the Minkowski space field by substituting φ ( − i x d , x , . . . , x d − ) → φ ( x , . . . , x d ) .Finally, we put the theory on a lattice with finite latticespacing a , substituting (cid:82) d d x → a d (cid:80) x and (cid:0) ∂ ν + 2 µ c τ δ dν (cid:1) φ c ( x ) → a (cid:0) φ x + (cid:98) ν − e − µ τ δ ν,d φ x (cid:1) , (II.3)where on the right-hand side of (II.3), factors of a ,resp. /a have been absorbed into µ and x by re-defining a µ c → µ and x/a → x , in order to ren-der the quantities dimensionless. The coordinate x on the right-hand side of (II.3) takes then values in Z d , and the relation between φ and φ c is given by φ x = φ c ( a x ) and φ x + (cid:98) ν = φ c ( a ( x + e ν )) with e ν being the unit vector in ν -direction. After setting also β = f π a ( d − and s = J a d , the Euclidean latticeaction then takes the standard form: S [ φ ] = − (cid:88) x (cid:26) β d (cid:88) ν =1 (cid:0) φ x e µ τ δ ν,d φ x + (cid:98) ν + φ x e − µ τ δ ν,d φ x − (cid:98) ν (cid:1) + ( s · φ x ) (cid:27) . (II.4) The in front of the chemical potential in (II.1) is convention. Note, that for d = ( d s + 1) = 1 (i.e. in the caseof spatial dimensionality d s = 0 ), the parameter f π has mass dimension -1/2 and its lattice counterpart isgiven by β = f π /a . If f π is non-zero, we thereforehave, that ( a → implies that ( β → ∞ ) , i.e. theparameter β has to diverge in the continuum limit. A. Dual formulation
The action (II.4) is in general complex for non-zerovalues of the chemical potential µ , which gives rise toa so-called sign-problem when trying to evaluate the partition function, Z = (cid:90) D (cid:2) φ (cid:3) e − S [ φ ] , (II.5)numerically by means of Monte Carlo simulations: acomplex action S implies that the Boltzmann weight e − S is complex as well and therefore lacks the proba-bilistic interpretation, required to do importance sam-pling. This problem can be overcome by changing rep-resentation and expressing (II.5) in terms of new, dis-crete variables. Such a change of representation is car-ried out in detail in appendix A 1, following the stepsdescribed in [15–18]. After dualization, the partitionfunction (II.5) reads: Z = (cid:88) { k,l,χ,p,q,n } (cid:89) x (cid:26)(cid:18) d (cid:89) ν =1 β | k x,ν | +2 l x,ν + (cid:80) Ni =3 χ ix,ν ( | k x,ν | + l x,ν )! l x,ν ! (cid:81) Ni =3 χ ix,ν ! (cid:19) · ( s + ) ( | p x | + p x )+ q x ( s − ) ( | p x |− p x )+ q x e µ k x,d ( | p x | + q x )! q x ! (cid:18) N (cid:89) i =3 ( s i ) n ix n ix ! (cid:19) δ (cid:0) p x − (cid:88) ν (cid:0) k x,ν − k x − (cid:98) ν,ν (cid:1)(cid:1) · W (cid:0) A x + | p x | + 2 q x , B x + n x , . . . , B Nx + n Nx (cid:1)(cid:27) , (II.6)with A x = (cid:88) ν (cid:0)(cid:12)(cid:12) k x,ν (cid:12)(cid:12) + (cid:12)(cid:12) k x − (cid:98) ν,ν (cid:12)(cid:12) + 2 (cid:0) l x,ν + l x − (cid:98) ν,ν (cid:1)(cid:1) , B ix = (cid:88) ν (cid:0) χ ix,ν + χ ix − (cid:98) ν,ν (cid:1) (II.7)and W (cid:0) A, B , . . . , B N (cid:1) = Γ (cid:0) A (cid:1) N (cid:81) i =3 1+( − Bi Γ (cid:0) B i (cid:1) A / Γ (cid:0) N + A + (cid:80) Ni =3 B i (cid:1) , (II.8)and where, for all sites x and all links ( x, ν ) : k x,ν , p x ∈ Z , (II.9a) l x,ν , q x , χ x,ν , n x , . . . , χ Nx,ν , n Nx ∈ N . (II.9b)The delta function at the end of the second line of (II.6)is just a Kronecker delta, δ ( x ) = δ ,x , and the new pa-rameters s ± = √ ( s ∓ i s ) are the sources for the U(1) -charged φ ± = √ ( φ ± i φ ) .While in (II.5) the partition function was given interms of an integral over a continuum of configu-rations { φ x } x with complex weights, in the flux-representation (II.6) it is given in terms of an in-finite sum over configurations of discrete variables { k, l, χ i , p, q, n i } , which all have real and non- negative weights .In the present work we will primarily be interested inthe (0 + 1) -dimensional case, which could also in thestandard formulation be solved using transfer-matrixmethods, regardless of whether the action is real orcomplex. The reformulation of the partition func-tion in terms of flux-variables is nevertheless useful asit simplifies the necessary computations considerably.Even more important is, however, that as discussed in For a configuration to have a real and non-negative weight, the weight factor W s = (cid:81) x ( s + ) ( | p x | + p x )+ q x ( s − ) ( | p x |− p x )+ q x has to be realand non-negative. That this is the case can be seen, by writing s ± = ˜ s e ∓ i φs √ , so that W s = ˜ s (cid:80) x ( | p x | +2 q x ) e i φ s (cid:80) x p x ,and noting that due to the Kronecker delta on the second line of(II.6), we have (cid:80) x p x = (cid:80) x,ν ( k x,ν − k x − (cid:98) ν,ν ) = 0 the next section, the flux-variables have a direct phys-ical meaning, which can be useful in identifying pro-cesses underlying different thermodynamic properties. B. Physical meaning of dual variables
The variables k , l and χ i live on the links of the lat-tice and are called flux-variables . They can in generalbe interpreted as counting the number of particle worldlines passing along the links. So, for example χ ix,ν ,with i ∈ { , . . . , N } , counts the number of φ i -worldlines traversing the link that connects the sites x and x + (cid:98) ν . Because the φ i -particles (with i ∈ { , . . . , N } )are neutral, they are their own anti-particles, so thattheir world lines don’t need an orientation. For the φ ± particles, which carry a U(1) -charge, the situation isdifferent: it matters whether a φ + moves from x to x + (cid:98) ν or from x + (cid:98) ν to x , as the two cases lead todifferent charge displacements. On the other hand, itdoesn’t make a difference whether a φ + moves from x to x + (cid:98) ν , or a φ − moves form x + (cid:98) ν to x , as the charge-displacement is in both cases the same ( CT symme-try is preserved). The k and l variables are thereforepicked in such a way, that k x,ν counts the net-positivecharge that moves from site x to site x + (cid:98) ν , whereas l x,ν counts the number of neutral pairs of φ ± worldlines that pass along the link between the two sites.The remaining variables p , q and n i , with i ∈{ , . . . , N } , live on the lattice sites and are called monomer numbers . In analogy to the flux-variables,the monomer numbers are organized so that p x countsthe net-positive charge on site x and q x the numberof neutral pairs of φ ± -monomers, while n ix counts thenumber of φ i -monomers on site x .Due to the delta-function constraints in (II.6), the k -and p -variables are on each site x subject to a conser-vation law: (cid:88) ν ( k x,ν − k x − (cid:98) ν,ν ) ! = p x , (II.10)which is a lattice manifestation of Gauss’ law ∂ ν j ν ( x ) = ρ s ( x ) , with k x,ν being related to thecurrent density, j ν ( x ) = i ( φ + ( x )( ∂ ν φ − ( x )) − φ − ( x )( ∂ ν φ + ( x ))) , and p x (which can be non-zero only if | s ± | > ) being related to the U(1) -breakingfield ρ s ( x ) = i ( s + φ + − s − φ − ) . While showing, thatfor | s ± | > , charge conservation is locally violated,equation (II.10) also implies that for periodic lattices,we have: (cid:88) x p x = (cid:88) x,ν ( k x,ν − k x − (cid:98) ν,ν ) = 0 , (II.11)meaning that the overall charge carried by φ ± -monomers must be zero ( ∼ integral-form of Gauss’law in a space without boundary). Note, however,that in this quantum field theory context, the numberof φ ± -monomers has nothing to do with the electriccharge density in the system! The latter, which in thecontinuum would be given by j (or j d in Euclideanspace-time), is represented by the flux variables k x,d that live on the time-like links.The l - and q -variables are completely unconstrainedas they represent the numbers of neutral pairs of φ ± -world line segments or monomers, which can at anypoint be created and annihilate. This is not the casefor the χ i - and n i -variables, which are on each site x subject to the evenness-constraint , (cid:0) n ix + (cid:88) ν ( χ ix,ν − χ ix − (cid:98) ν,ν ) (cid:1) mod 2 ! = 0 , (II.12)encoded in the site weight (II.8). This constraintreflects the fact that also neutral particles can becreated and annihilated only in pairs, so that a worldline, that enters a site, has either to continue and leavethe site, or to annihilate with a monomer. III. RESULTS
For simplicity we discuss in the following the caseof N = 2 , which corresponds to an XY-model that iscoupled to source terms and a chemical potential forthe U(1) charge. The qualitative findings generalizeto
N > , as well as to the case of linear spin modelswith arbitrary on-site potentials.With N = 2 , the site weight (II.8) simplifies to W ( A ) = 2 − A/ . The summations over the individ-ual l - and q -variables in the partition function (II.6)then decouple, and the partition function simplifies to: Z = (cid:88) { k,p } (cid:89) x (cid:26)(cid:18) d (cid:89) ν =1 I k x,ν ( β ) (cid:19) I p x (˜ s ) e i φ s p x e µ k x,d δ (cid:0) p x − (cid:88) ν (cid:0) k x,ν − k x − (cid:98) ν,ν (cid:1)(cid:1)(cid:27) , (III.1)with I n ( α ) being the modified Bessel function of thefirst kind and ˜ s and φ s are magnitude and phase ofthe sources, so that s ± = ˜ s e ∓ i φs √ .In (0+1) dimensions, i.e. when the lattice extendsonly in the Euclidean time direction, the partitionfunction (III.1) simplifies further. Setting for the mo-ment ˜ s = 0 (which implies p x = 0 ), it reduces to asingle sum: Z = ∞ (cid:88) k = −∞ (cid:0) I k ( β ) e µ k (cid:1) N t , (III.2)with N t being the temporal extent (number of sites)of the periodic lattice. The summation variable k rep-resents the collective value of all the k x,d -variables inthe system and each value of k therefore represents adistinct homogeneous (in terms of flux variables) con-figuration. As discussed in Sec. II B, the values of the k x,d -variables can be identified with quanta of the lo-cal charge density n = j d in a dual configuration andthe value of k in (III.2) is therefore not just a label,but represents the charge density carried by the cor-responding homogeneous configuration. This can beverified by taking the derivative of the logarithm ofthe partition function with respect to µ : in arbitrarydimensions, this yields the charge density: (cid:104) n (cid:105) = 12 V ∂ log( Z ) ∂µ = 1 V (cid:88) x (cid:104) k x,d (cid:105) dual ( d =(0+1)) = (cid:104) k (cid:105) dual , (III.3)and we see that in (0+1) dimensions, using the parti-tion function (III.2), the result is given by the expecta-tion value of k .The partition function (III.2) is therefore a sum overdistinct homogeneous (in terms of dual variables) con-figurations, which are labelled by the integer-valuedcharge (density) k that they carry. A. Transitions between distinct ground states
For each fixed value of β , the modified Besselfunction of the first kind, I n ( β ) , decays super-exponentially as function of increasing | n | (otherwise,the sum in (III.2) would diverge), and it is thereforeevident, that for µ = 0 , the term in (III.2) with k = 0 The modified Bessel function of the first kind can be written as I n ( α ) = (cid:80) ∞ m =0 ( α ) n +2 m ( n + m )! m ! , which is precisely the form of thesums over l and q in (II.6) when N = 2 , with the replacements( m → | k | , α → β ) and ( m → | p | , α → ˜ s ). Note also thatthe factorials in the denominator of the expression for the Besselfunction are understood in terms of gamma-functions, i.e. n ! =Γ(1 + n ) , so that I n ( α ) = I | n | ( α ) and we therefore don’t haveto write the modulus in the subscript of I n ( α ) . μ μ μ μ 〈 n 〉 N t = N t = N t = N t = N t = - - μ μ μ μ N t ( 〈 n 〉 - 〈 n 〉 ) N t = N t = N t = N t = N t = Figure 1. The figure shows the charge density (top) and cor-responding susceptibility (bottom) as functions of the chem-ical potential µ for the O(2) -model in (0+1) dimensions for ˜ s = 0 (c.f. eq. (III.2)), β = 0 . and lattice sizes N t = µ = µ , , µ , , . . . , the system undergoes transitions between vacuacarrying different integer-valued charge densities. The tran-sitions become apparently stronger with increasing systemsize, as one would expect from (III.3). dominates if N t is sufficiently large. With increas-ing µ , however, the factor e µ k will at some point beable to compensate for the decrease in I k ( β ) when k changes from zero to one, and if µ exceeds this point,the term in (III.3) with k = 1 will become the dom-inant one. As all the configurations with different k are homogeneous configurations (in terms of our fluxvariables), each of them qualifies in the ( N t → ∞ ) -limit as vacuum state if its weight dominates the sumin (III.3). The transitions between states with differ-ent index k can therefore be understood as transitionsbetween different ground states which carry differentinteger-valued charge density. These transitions occurwhenever the value of µ is such that two successiveterms in (III.3) become degenerate, i.e. when we havefor some k ∈ Z , that µ is such that: I k ( β ) e µ k = I k +1 ( β ) e µ ( k +1) . (III.4)Upon solving for µ , this yields the critical value µ k,k +1 = 12 log (cid:18) I k ( β ) I k +1 ( β ) (cid:19) , (III.5)for the transition between the states carrying chargedensities k and k + 1 , respectively. That the systemdescribed by (III.2) indeed undergoes such transitionsbetween successive discrete states when the chemi-cal potential is increased, is illustrated in in the upperpanel of Fig. 1 for systems of different size N t andwith β = 0 . . The quantity in the lower panel is thecorresponding charge-susceptibility, χ n = V (cid:0) (cid:104) n (cid:105) − (cid:104) n (cid:105) (cid:1) = 14 V ∂ log( Z )( ∂µ ) = V (cid:0) (cid:104) k x,d (cid:105) dual − (cid:104) k x,d (cid:105) dual (cid:1) ( d =1) = N t (cid:0) (cid:104) k (cid:105) dual − (cid:104) k (cid:105) dual (cid:1) , (III.6)which, with increasing N t , becomes more and morepeaked at the transition points µ k,k +1 , k ∈ Z . - - N t ( μ - μ ) 〈 n 〉 N t = N t = N t = N t = N t = - - N t ( μ - μ ) 〈 n 〉 - 〈 n 〉 N t = N t = N t = N t = N t = Figure 2. The figure shows for the
O(2) -model in (0+1)dimensions with ˜ s = 0 (c.f. eq. (III.2)), β = 0 . and latticesizes N t = 16 , , , , , scaling collapses for thecharge density (top) and the charge susceptibility (bottom)around the first critical point µ = µ , . The collapses areobtained according to (III.7) and (III.8). In the neighbourhood of each critical point µ k,k +1 ,one can rescale (cid:104) n (cid:105) ( µ, N t ) and χ n ( µ, N t ) , so that thecurves for different N t coincide. This is achieved bysetting for each k : (cid:104) ˜ n (cid:105) k (∆˜ µ ) = (cid:104) n (cid:105) (cid:0) µ k,k +1 + N − t ∆˜ µ, N t (cid:1) (III.7)and ˜ χ n,k (∆˜ µ ) = N − t χ n (cid:0) µ k,k +1 + N − t ∆˜ µ, N t (cid:1) , (III.8) - - N t ( μ - μ ) 〈 n 〉 N t = N t = N t = N t = N t = - - N t ( μ - μ ) 〈 n 〉 - 〈 n 〉 N t = N t = N t = N t = N t = Figure 3. Same as Fig. 2, but here the scaling collapses areperformed at the critical point µ = µ , instead of µ = µ , . where for each k , ∆˜ µ is, in terms of the original,unscaled µ , given by ∆˜ µ = N t ( µ − µ k,k +1 ) .Examples for such scaling collapses are shown inFigs. 2-3 for β = 0 . and ˜ s = 0 , i.e. for the samesystems that were used in Fig. 1. As the rescaledcharge-susceptibility (III.8) is independent of N t , theform of the rescaling implies that the original χ n converges towards a sum of Dirac delta-functionswhen the thermodynamic limit ( N t → ∞ ) is taken atfixed β (corresponding to taking the zero-temperaturelimit at finite lattice spacing). The charge density (cid:104) n (cid:105) becomes therefore discontinuous in this limit and thetransition is of first order.The situation changes if we take the thermodynamiclimit not by sending the temperature to zero whilekeeping the lattice spacing fixed, but instead sendingthe lattice spacing to zero while keeping the temper-ature fixed. As β is given by β = f π /a , and f π hasa physical meaning (i.e. it should assume a constantvalue if one enters the scaling window), we can dothis by keeping κ := β/N t = T f π constant whilesending N t to infinity. This is illustrated in Figs. 4-5,which show for different values of the temperature (i.e.different values of κ = β/N t ), how the charge den-sity (III.3) and the corresponding susceptibility (III.6)(multiplied by β − to cancel the dependency on thelattice spacing) as functions of β µ behave when N t is changed: at low temperatures (cf. Fig. 4) one canagain observe the formation of plateaus at integer val-ues of the charge density and for ( N t → ∞ ) the curvesconverge towards a fixed shape that represents the con-tinuum limit. In contrast to the situation we had inFig. 1, the charge density as function µ now remainssmooth when κ instead of β is held fixed while N t issent to infinity.Although decreasing the value of κ (lowering the tem-perature) makes the crossovers between the sectors ofdifferent integer-valued charge density become moreabrupt, the crossovers are turned into first order transi-tions only if in the end also the zero-temperature limitis taken, i.e. when κ is sent to zero. For increasing κ ,on the other hand (cf. Fig. 5), the step-like behavior ofthe charge density as function of µ slowly disappearsand turns into a linear behavior: (cid:104) n (cid:105) ≈ µ β . β μ 〈 n 〉 β / N t = N t = N t = N t = N t = N t = β μ β - N t ( 〈 n 〉 - 〈 n 〉 ) β / N t = N t = N t = N t = N t = N t = Figure 4. The figure shows for the
O(2) model in (0+1) di-mensions the charge density (top) and charge susceptibility(bottom) as function of β µ at fixed
T f π = β/N t = 0 . for various lattice sizes N t = 16 , , , , . Thecontinuum limit would be obtained by sending N t → ∞ . The N t -dependency of charge density and chargesusceptibility as function of µ at fixed κ = β/N t = T f π can be understood from the behavior of β µ n,n +1 ( β ) as function of β . With the expression for µ n,n +1 ( β ) from eq. (III.5), one finds the asymptoticbehavior of β µ n,n +1 ( β ) at small and large values of β to be given by: β µ n,n +1 ( β ) ≈ (cid:40) β log( n +1) β ) if β (cid:28) n +14 if β (cid:29) , (III.9)with a maximum separating the two regions. Clearly,for the curves in Figs. 4-5 to converge, the N t -values β μ 〈 n 〉 β / N t = N t = N t = N t = N t = N t = β μ β - N t ( 〈 n 〉 - 〈 n 〉 ) β / N t = N t = N t = N t = N t = N t = Figure 5. Same as Fig. 4 but at higher temperatures
T f π = β/N t = 1 . . have to be sufficiently large, so that the correspondingvalues of β ( N t ) = κ N t are all in the region β (cid:29) ,where β µ n,n +1 ( β ) is essentially independent of β .If the physical temperature is very low (i.e. if κ isvery small) then β ( N t ) can for small values of N t fall into the range β (cid:28) , where β µ n,n +1 ( β ) growsonly logarithmically instead of linearly with n . Asfunction of β µ , the successive transitions to higherand higher charge densities then happen for thesesmall systems at much lower values of β µ than forthe larger systems. This is the reason why in Fig. 4 thecharge density and susceptibility grow for N t = 16 much faster as function of β µ than for the largervalues of N t .Finally, it can also be understood, why the step-likebehavior of the charge density disappears when κ = β/N t is sufficiently large. To this end, we define the weight ratios , W n ,n ( µ, β, N t ) = (cid:18) I n ( β ) e µ n I n ( β ) e µ n (cid:19) N t = e N t sgn( n − n ) (cid:80) max( n ,n − n (cid:48) =min( n ,n ( µ − µ n (cid:48) ,n (cid:48) +1 ( β )) , (III.10)which tell us, as function of the parameters µ , β and N t , how strongly the term in the partition sum (III.2)with k = n is suppressed, compared to the term with k = n . Setting n = 0 and n = k , the partitionfunction (III.2) can then be written as: Z = ( I ( β )) N t ∞ (cid:88) k = −∞ W ,k ( µ, β, N t ) . (III.11)If we now plug into expression (III.10) for the weightratios, that according to (III.9) we have µ n,n +1 ( β ) ≈ (2 n + 1) / (4 β ) for β (cid:29) , expression (III.10) simpli-fies to: W ,k ( β µ, κ ) ≈ c ( β µ, κ ) e − ( k − β µ )22 κ , (III.12)with c ( β µ, κ ) = e (2 β µ )22 κ , and the partition function(III.11) reduces to a sum over the values of a shiftedGaussian with variance κ = β/N t at integer values.If κ is small, the Gaussian is strongly peaked aroundthe k -value that is closest to (2 β µ ) , so that there isalways only one k -value that significantly contributesto the partition sum (unless β µ is fine-tuned to be ex-actly in the middle between two successive k -values).If (2 β µ ) changes from being closest to one k -value tobeing closest to the next k -value, the peak moves veryabruptly from the old to the new dominant k , and theaverage k changes in a step-like manner as functionof β µ . This is not the case if κ is sufficiently large,as then the Gaussian is wide and also sub-dominant k -values contribute significantly to the partition sum,which allows the average k value to change muchmore smoothly. The value of κ at which the latter be-havior can be expected to set in, can be approximatedby the value for which the width of the σ interval ofthe Gaussian in (III.12) does no longer fit between twosuccessive k -values, i.e. if × (2 σ ) ≥ , which is thecase, if κ = σ ≥ / .This can be made more explicitly by noting that thesum in (III.11) reduces for β (cid:29) , using (III.12), to a Jacobi theta function : ϑ ( u, q ) = ∞ (cid:88) k = −∞ u k q k , (III.13)with q = e − / (2 κ ) and u = e β µ/κ . We can then useone of the Jacobi identities for ϑ ( u, q ) , namely: ϑ ( u (cid:48) ( u, q ) , q (cid:48) ( q )) = α ( u, q ) ϑ ( u, q ) , (III.14)with q (cid:48) ( q ) = e π / log( q ) = e − π κ , (III.15a) u (cid:48) ( u, q ) = e i π log( u ) / log( q ) = e − π i (2 β µ ) , (III.15b)and α ( u, q ) = (cid:113) − log( q ) π e log2( u )log( q ) = e − (2 β µ )22 κ √ π κ , (III.15c)to write the partition function (III.11) as: Z β (cid:29) = ( I ( κ N t )) N t ϑ ( u, q )= ( I ( κ N t )) N t ϑ ( u (cid:48) ( u, q ) , q (cid:48) ( q )) /α ( u, q )= ( I ( κ N t )) N t √ π κ e (2 β µ )22 κ · ϑ (cid:0) e − π i (2 β µ ) , e − π κ (cid:1) . (III.16)For sufficiently large κ , the theta function on the lastline converges towards unity as a small q suppressesall terms with k (cid:54) = 0 in the defining sum (III.13). Thepartition function then reduces to: Z ≈ ( I ( κ N t )) N t √ π κ e (2 β µ )22 κ , (III.17)and the corresponding charge density becomes: (cid:104) n (cid:105) = 12 N t ∂ log( Z ) ∂µ = 2 β µ , (III.18)with the linear behavior we observed in Fig. 3. B. Correlation function and mass spectrum
Next, we would like to investigate how the two-point function of the (0+1)-dimensional
O(2) -modelbehaves as function of µ . The general form of one-and two-point functions in the flux-variable formula-tion is derived in appendix A 2. Here we will just need (cid:10) φ − x φ + y (cid:11) = 1 Z ∂ Z∂s − x ∂s + y = Z − , +2 ( x, y ) Z , (III.19)with s ± x being per-site sources . For the partition func-tion (III.2) (or equivalently (III.11)), this two-pointfunction takes the simple form: (cid:10) φ − x φ + y (cid:11) ( µ, β, N t ) = ∞ (cid:80) k = −∞ W − s ( x,y )0 ,k ( µ, β, N t ) W s ( x,y )0 ,k +1 ( µ, β, N t )2 ∞ (cid:80) k = −∞ W ,k ( µ, β, N t ) , (III.20)with s ( x, y ) = (( y − x ) mod N t ) /N t , and where ( X mod Y ) with X ∈ Z and Y ∈ N is themodulo operation, that wraps X around the interval Note, that because of the continuum field φ ( x ) being dimension-less (the dimensionality is absorbed into f π ), the lattice fields φ ± x do not explicitly scale with the lattice spacing a . The local per sitesources , s ± x , do therefore also not explicitly scale with a ; only theglobal s ± do. This can be seen by noting that (cid:104) φ ± (cid:105) = V ∂Z∂s ± ,while (cid:104) φ ± x (cid:105) = ∂Z∂s ± x , where (cid:104) φ ± (cid:105) and (cid:104) φ ± x (cid:105) are both dimen-sionless and in the case of translation invariance, should even beequal, (cid:104) φ ± (cid:105) = (cid:104) φ ± x (cid:105) , ∀ x . { , . . . , Y − } . The W n ,n ( µ, β, N t ) are the weightratios introduced in (III.10). In order to avoid toolengthy expression we will often leave the dependencyof quantities on the parameters µ , β and N t implicit inthe following. Equation (III.20) simply states that if a φ − is inserted at site x and a φ + at site y , then, if thesource ˜ s is set to zero, ˜ s = 0 , the two sites have to beconnected by a continuous charged flux-line.For s ( x, y ) < / , it is useful to write (III.20) in theform: (cid:10) φ − x φ + y (cid:11) = ∞ (cid:80) k = −∞ W ,k W s ( x,y ) k,k +1 ∞ (cid:80) k = −∞ W ,k = ∞ (cid:80) k = −∞ W ,k e N t ( µ − µ k,k +1 ) s ( x,y ) ∞ (cid:80) k = −∞ W ,k , (III.21a)where on the second line, we made use of the lastline of (III.10) to write the W k,k +1 in terms of ex-ponentials, from which one can easily read off themasses of all the propagating modes. The restriction, s ( x, y ) < / , ensures that the largest term in the sumin the numerator of (III.21a) is the one with k = n if µ ∈ ( µ n − ,n ( β ) , µ n,n +1 ( β )] . Similarly, one can for s ( x, y ) > / write: (cid:10) φ − x φ + y (cid:11) = ∞ (cid:80) k = −∞ W ,k W − s ( x,y ) k,k − ∞ (cid:80) k = −∞ W ,k = ∞ (cid:80) k = −∞ W ,k e − N t ( µ − µ k − ,k ) s ( y,x ) ∞ (cid:80) k = −∞ W ,k , (III.21b)where the term with k = n in the sum in thenumerator of (III.21b) is this time largest if µ ∈ [ µ n − ,n ( β ) , µ n,n +1 ( β )) .From eqns. (III.21a) and (III.21b) we can see thatthe spectrum contains a massless mode whenever thechemical potential µ is set to one of the critical values µ n,n +1 , n ∈ Z .At fixed β and in the limit ( N t → ∞ ) , the φ ± -masses can also be extracted in the standard wayfrom the exponential decay of the connected two-pointfunction. To this end, we first note, that for ˜ s = 0 the U(1) -current is conserved and the φ ± -worldlines,which carry a U(1) -charge, therefore need to be con-tinuous. This means that (cid:104) φ − x φ + y (cid:105) is already a con-nected two-point function whose large distance behav-ior is dominated by the lightest mode and not by anydisconnected piece. For µ ∈ ( µ n − ,n ( β ) , µ n,n +1 ( β )] ,the φ + -mass is then obtained from (III.21a) with (cid:28) μ μ μ m ϕ + - mass ϕ - - mass Figure 6. The figure shows for the one-dimensional
O(2) -model with ˜ s = 0 and β = 0 . the zero-temperature φ ± -masses from eqn. (III.22a) and (III.22b) as functions of thechemical potential µ . t (cid:28) t ∗ by computing: m + ( µ, β ) = lim N t →∞ log (cid:18) (cid:10) φ − φ + t (cid:11)(cid:10) φ − φ + t +1 (cid:11) (cid:19) = − lim N t →∞ log( W n,n +1 ) /N t = 2 ( µ n,n +1 − µ ) , (III.22a)where t ∗ is given by the value of t for which theamplitude of the lightest forward-propagating modebecomes smaller than the amplitude of the lightestbackward-propagating mode (if µ = 0 , then t ∗ = N t / ). Similarly, using that (cid:104) φ +0 φ − t (cid:48) (cid:105) = (cid:104) φ − φ + N t − t (cid:48) (cid:105) ,the φ − -mass can for µ ∈ [ µ n − ,n ( β ) , µ n,n +1 ( β )) beobtained from (III.21b) with (cid:28) t (cid:48) (cid:28) N t − t ∗ as: m − ( µ, β ) = lim N t →∞ log (cid:18) (cid:10) φ − φ + N t − t (cid:48) (cid:11)(cid:10) φ − φ + N t − t (cid:48) − (cid:11) (cid:19) = − lim N t →∞ log( W n − ,n ) /N t = 2 ( µ − µ n − ,n ) , (III.22b)with t (cid:48) kept fixed while N t is sent to infinity.The masses (III.22a) and (III.22b) are shown in Fig. 6for β = 0 . .With the expressions for the φ ± -masses, we can nowalso write down an expression for t ∗ : requiring that e − m + ( µ,β ) t ∗ = e − m − ( µ,β ) ( N t − t ∗ ) and solving this for t ∗ , we find: t ∗ ( µ, β, N t ) = m − ( µ, β ) N t m + ( µ, β ) − m − ( µ, β )= ( µ − µ n − ,n ) N t µ n,n +1 − µ n − ,n , (III.23)where the n is such that µ ∈ ( µ n − ,n , µ n,n +1 ) .As the Euclidean two-point function (III.21) is asum of pure exponentials ∼ e − m t , a zero mode im-plies the presence of a constant piece in the two-point0function. In the vicinity of a critical point µ n,n +1 andfor sufficiently large N t , this constant piece can be ob-tained from the value of the two-point function (III.20)at t = t ∗ with t ∗ from (III.23). For µ slightly below µ n,n +1 , one finds: (cid:10) φ − φ + t ∗ (cid:11) ≈ W ,n W s ∗ n,n +1 + . . . W ,n + W ,n +1 + . . . ) ≈ e N t ( µ − µn,n +1)( µ − µn − ,n ) µn,n +1 − µn − ,n N t ( µ − µ n,n +1 ) ) , (III.24a)and for µ slightly above µ n,n +1 , correspondingly: (cid:10) φ − φ + t ∗ (cid:11) ≈ W ,n +1 W s ∗ n +1 ,n +2 + . . . W ,n + W ,n +1 + . . . ) ≈ e N t ( µ − µn +1 ,n +2)( µ − µn,n +1) µn +1 ,n +2 − µn,n +1 − N t ( µ − µ n,n +1 ) ) , (III.24b)where s ∗ = t ∗ ( µ, β, N t ) /N t and we made againused of (III.10). For sufficiently large N t , (III.24) as-sumes at a critical point µ = µ n,n +1 , n ∈ Z the value / and decays rapidly as µ moves away from µ n,n +1 .So far we have only looked at the behavior of thetwo-point function at fixed β , in which case, as wehave seen in the previous section, the limit ( N t → ∞ ) gives rise to the appearance of first-order transitionsat the very same values of the chemical potential, forwhich the zero modes and the constant piece appearin the two-point function (c.f. eq. (III.24)). Do thesezero modes and constant pieces survive when thelimit ( N t → ∞ ) is taken at fixed κ = β/N t = T f π instead of fixed β , in which case there are no firstorder transitions?As we have seen already in the previous section, thebehavior of the system as function of the chemical po-tential changes when we go from low to sufficientlyhigh temperatures. This should also be reflected in achange of behavior in the two-point function. To anal-yse the latter, we use again expression (III.12) for theweight ratios W n ,n ( β µ, κ ) at large β and write thetwo-point function as: (cid:10) φ − x φ + y (cid:11) ( β µ, κ ) = ∞ (cid:80) k = −∞ W ,k W s ( x,y ) k,k +1 ∞ (cid:80) k = −∞ W ,kβ (cid:29) = ∞ (cid:80) k = −∞ e − ( k − β µ )22 κ e − (2 k +1 − β µ ) s ( x,y )2 κ ∞ (cid:80) k = −∞ e − ( k − β µ )22 κ = ∞ (cid:80) k = −∞ e − ( k − β µ )22 κ e − β mk s ( x,y ) κ ∞ (cid:80) k = −∞ e − ( k − β µ )22 κ . (III.25) with mass terms β m k ( β µ, κ ) = 2 k + 1 − β µ ∀ k ∈ Z . (III.26)As long as the temperature is sufficiently small, i.e.for κ = β/N t = f π T < / , the Gaussian weights innumerator and denominator of (III.25) are sufficientlynarrow to give dominant weight to the terms for which k is closest to the value β µ . At a critical point β µ = β µ n,n +1 β (cid:29) = (2 n + 1) / , n ∈ Z , (III.27)the value of β µ is located precisely in the middlebetween two successive k -values. The Gaussianweights for the terms with k = n and k = n + 1 arethen equal, and the masses corresponding to thesetwo dominant modes assume the values β m n = 0 and β, m n +1 = 1 , meaning that the long-distancebehavior of the system is dominated by a zero mode.In Fig. 7 the N t -dependency of the rescaledmasses β m ± as functions of β µ is illustrated for κ = 0 . , corresponding to a low but non-zerotemperature. The spectrum in the uppermost panelwith N t = 1024 then corresponds to a system at β = κ N t ≈ . and looks therefore (up to therescaling of the axes by a factor of β ) identical tothe spectrum shown in Fig. 6. However, if now N t is increased while κ is kept fixed, β has to increasestoo and the critical values of β µ change according to(III.9): as long as β = κ N t (cid:28) , the critical valuesof β µ grow like β µ n,n +1 ( β ) ≈ β log (cid:0) n +1) β (cid:1) asfunction of increasing β = κ N t , but as soon as N t issufficiently large, so that β = κ N t (cid:29) , the criticalvalues of β µ converge towards their β -independentasymptotic values β µ n,n +1 ( β ) ≈ n +14 , n ∈ Z .The critical values then become uniformly spaced anthe mass-spectrum, which depends on the differencebetween successive critical values, becomes periodicin β µ , as illustrated in the lower-most panel of Fig. 7.At this point, a further increase of N t leaves thespectrum unchanged.For κ bigger than / , the Gaussian weights in(III.25) can no longer sufficiently project on the termsfor which k is closest to β µ and the long-distancebehavior of the system at a critical value of βµ will notnecessarily be dominated by a zero mode any more.By expressing the two-point function in terms of thetafunctions, we can write: (cid:10) φ − x φ + y (cid:11) ( β µ, κ ) β (cid:29) = ( u q ) s ( x,y ) ϑ ( u ( x, y ) , q )2 ϑ ( u, q )= ( u q ) s ( x,y ) α ( u, q ) ϑ ( u (cid:48) ( u ( x, y ) , q ) , q (cid:48) ( q ))2 α ( u ( x, y ) , q ) ϑ ( u (cid:48) ( u, q ) , q (cid:48) ( q ))= e − s ( x,y ) (1 − s ( x,y ))2 κ · ϑ (e − π i (2 β µ − s ( x,y )) , e − π κ )2 ϑ (e − π i (2 β µ ) , e − π κ ) , (III.28)1where q = e − / (2 κ ) , u = e β µ/κ , u ( x, y ) = u q s ( x,y ) and s ( x, y ) = (( y − x ) mod N t ) /N t , and on the sec-ond line, the transformations (III.15) were used. Forsufficiently large κ , the theta functions on the last lineof (III.28) converge towards unity and the two-pointfunction reduces therefore to: (cid:10) φ − x φ + y (cid:11) ( κ ) ≈
12 e − s ( x,y ) (1 − s ( x,y ))2 κ , (III.29)which is independent of β µ but depends only on thetemperature ∼ κ .To summarize: the expression (III.25) for the two-point function illustrates that zero modes will alwaysbe present in the large- N t limit when µ is set to oneof the critical points µ n,n +1 , n ∈ Z , regardless ofwhether κ is small or large. However, if κ , which setsthe temperature, is sufficiently large, the system willbe completely dominated by thermal modes and thezero-modes will no-longer give relevant contributionsto expectation values at the critical points. C. Explicit symmetry breaking
We have seen in the previous section that the
O(2) model, coupled to a chemical potential µ , can in (0+1)dimensions develop some sort of long-range orderwhen µ approaches certain critical values µ n,n +1 , n ∈ Z . In this section we would now like to checkwhether this long range order also gives rise to theformation of a non-zero condensate (cid:104) φ ± (cid:105) .To measure a condensate directly, one needs tobreak the global U(1) symmetry explicitly by setting ˜ s in (III.1) to a non-zero value. This is necessarybecause of the fact, that the partition function (III.1),although reformulated in terms of flux-variables, stillrepresents a path-integral that sums over all possibleconfigurations of the lattice field φ ± x , which means,that if the global symmetry is spontaneously broken,the path-integral still sums over all possible waysin which this spontaneous breaking can happen. Inexpectation values of observables like φ ± = √ e ± i θ ,which depend on the direction in which the symmetrygets broken, the contributions from configurations thatdiffer only by this direction would therefore exactlycancel each other. To avoid such cancellations, onehas to single out a particular direction in which thesymmetry should break, which is most easily doneby coupling φ ± x to a non-zero source s ± = ˜ s e ∓ i φs √ .The value of the partition function does not dependon the choice of the direction that is singled out andthe choice of the phase φ s of the source is thereforeirrelevant; only the magnitude ˜ s of the source matters.The value of φ s will only enter in expectation valuesof observables that are not invariant under the global U(1) symmetry. β μ β m N t = ϕ + ϕ - β μ β m N t =
65 536 ϕ + ϕ - β μ β m N t = ϕ + ϕ - Figure 7. The figures show for the one-dimensional
O(2) -model with ˜ s = 0 and κ = β/N t = 0 . the rescaled φ ± -masses, β m ± , as functions of β µ for different valuesof N t . With ˜ s > , the 1D partition function for the O(2) -model does not reduce to a single sum as in equation(III.2); the non-zero value of ˜ s breaks charge conserva-tion, so that k -variables on different links can now takeon different values, at the cost of producing monomersand thereby changing the site weight I p x (˜ s ) . One thenarrives at the following expression: Z = (cid:88) { k } (cid:89) x (cid:26) I k x ( β ) e µ k x · I ( k x − k x − ) (˜ s ) e i φ s ( k x − k x − ) (cid:27) = tr (cid:0) T N t (cid:1) , (III.30)where we have summed over the p -variables in orderto get rid of the delta-function constraints in (III.1),and after the second equality sign, N t is (as usual) the2temporal system size. The transfer matrix , T δ , can forexample be chosen as: ( T δ ) ab = I a ( β ) e µ a I ( a − b − δ ) (˜ s ) e i φ s ( a − b − δ ) | δ | / , (III.31)with δ ∈ {− , , } , depending on whether the site towhich the transfer matrix is associated contains an ex-ternal φ − , φ + or no external field (cf. the discussion inappendix A 2 and the appearance of Kronecker-deltasin the delta-function constraints for the one- and two-point partition functions).The one- and two-point functions can be expressed interms of the transfer matrix as: (cid:10) φ ± (cid:11) = tr (cid:0) T ± T N t − (cid:1) tr (cid:0) T N t (cid:1) , (III.32)and (cid:10) φ − φ + t (cid:11) = if t = 0 tr( T − T t − T T Nt − t − )tr( T Nt ) if t > . (III.33)In Fig. 8, we show (III.32) as function of µ when eval- μ 〈 ϕ 〉 N t = N t = N t = N t = N t = N t = Figure 8. The figure shows the φ ± -condensate as functionof the chemical potential µ for the (0+1)-dimensional O(2) -model with ˜ s = 0 . and β = 0 . , obtained from eq.(III.32) for various values of N t . The curves for N t = 1024 and N t = 4096 are essentially on top of each other. uated numerically with ˜ s = 0 . and β = 0 . forvarious values of N t . As can be seen: at the criticalpoints, µ = µ n,n +1 , n ∈ Z , the condensate, (cid:104) φ ± (cid:105) ,peaks and approaches for sufficiently large N t a finitevalue. A comparison of the value of the condensateand the magnitude of the constant piece of the two-point function (III.33) at these critical points furthershows, that: lim t →∞ (cid:10) φ − φ + t (cid:11) = (cid:10) φ − (cid:11)(cid:10) φ + (cid:11) . (III.34)However, comparing the numerically obtained resultfor this constant piece in the two point function for ˜ s > with the value we found above in (III.24) for ˜ s = 0 , reveals, that the value for ˜ s > is only abouthalf the value obtained with ˜ s = 0 .In order to better understand this behavior, we deter-mine analytically the leading contributions to the one-and two-point functions at the critical points µ n,n +1 , n ∈ Z .For µ = µ n,n +1 , the factors I k x ( β ) e µ k x in (III.30)are also for ˜ s > maximized if k x ∈ { n, n + 1 } . If ˜ s is not too large, we can then assume that the one-and two-point functions, (III.32) and (III.33), are wellapproximated by taking into account only the 2 by 2blocks of the transfer-matrix ( T δ ) ab from (III.31) forwhich a, b ∈ { n, n + 1 } .For the two-point function at time separation t onethen finds: (cid:10) φ − φ + t (cid:11) ( µ = µ n,n +1 , N t ) ≈ (cid:80) { k i ∈{ n,n +1 }} ∀ i (cid:0) T − (cid:1) k Nt − ,k (cid:0) T (cid:1) k ,k . . . (cid:0) T (cid:1) k t − ,k t (cid:0) T (cid:1) k t ,k t +1 . . . (cid:80) { k i ∈{ n,n +1 }} ∀ i (cid:0) T (cid:1) k Nt − ,k . . . (cid:0) T (cid:1) k Nt − ,k Nt − (III.31)(III.5) ≈ c ( t, I (˜ s ) , I (˜ s )) c ( N t − t, I (˜ s ) , I (˜ s ))4 c ( N t , I (˜ s ) , I (˜ s )) (III.36) = 18 (cid:18) (cid:0) ( Nt − t ) log( I s ) − I s ) I s )+ I s ) ) (cid:1) cosh (cid:0) Nt log( I s ) − I s ) I s )+ I s ) ) (cid:124) (cid:123)(cid:122) (cid:125) − (log( I (˜ s )+ d I s )d˜ s ) − log( I (˜ s ) − d I s )d˜ s )) ≈− d log( I s ))d˜ s = − ˜ s + O (˜ s ) (cid:1) (cid:19) ≈ (cid:18) (cid:0) ( N t / − t ) ˜ s (cid:1) cosh( N t ˜ s/ (cid:19) , (III.35)where the function c ( L, A, B ) is given by c ( L, A, B ) = (cid:98) L/ (cid:99) (cid:88) m =0 (cid:18) L m (cid:19) A L − m B m = 12 (cid:0) ( A + B ) L + ( A − B ) L (cid:1) , (III.36)and represents the partition function for a system of L ∈ N binary degrees of freedom, in which the twodistinct states have weights A and B , respectively, andwhere the states with weight B must always be occu-pied by an even number of degrees of freedom. The reason for the appearance of the function c ( L, A, B ) in (III.35) is explained in Figs. 9-10; in short: to eachsite of our (0+1)-dimensional lattice corresponds onebinary d.o.f. and the two possible states of the d.o.f.correspond to whether the two k -variables that touchat the site do ( ∼ state with weight A ) or do not ( ∼ statewith weight B ) have the same value.Along the same line, one finds the leading contri-butions to the one-point function at a critical point µ n,n +1 to be: (cid:10) φ ± (cid:11) ( µ = µ n,n +1 , N t ) ≈ (cid:80) { k i ∈{ n,n +1 }} ∀ i (cid:0) T ± (cid:1) k Nt − ,k (cid:0) T (cid:1) k ,k . . . (cid:0) T (cid:1) k Nt − ,k Nt − (cid:80) { k i ∈{ n,n +1 }} ∀ i (cid:0) T (cid:1) k Nt − ,k . . . (cid:0) T (cid:1) k Nt − ,k Nt − = c odd ( N t , I (˜ s ) , I (˜ s ))2 / c ( N t , I (˜ s ) , I (˜ s )) = − / tanh (cid:16) N t log( I (˜ s ) − I (˜ s ) I (˜ s )+ I (˜ s ) ) (cid:17) ≈ − / tanh( N t ˜ s/ , (III.37)where c ( L, A, B ) is again given by (III.36), and thenew function, c odd ( L, A, B ) = (cid:98) ( L − / (cid:99) (cid:88) m =0 (cid:18) L m + 1 (cid:19) A L − (2 m +1) B m +1 = 12 (cid:0) ( A + B ) L − ( A − B ) L (cid:1) , (III.38)has a similar meaning as c ( L, A, B ) , but the state withweight B has now to be occupied always by an oddinstead of even number of degrees of freedom.From (III.35) and (III.37) we can finally obtain anexpression for the connected two-point function at acritical point µ n,n +1 : (cid:10) φ − x φ + y (cid:11) conn. ( µ = µ n,n +1 , N t )= (cid:0)(cid:10) φ − x φ + y (cid:11) − (cid:10) φ − x (cid:11)(cid:10) φ + y (cid:11)(cid:1) ( µ = µ n,n +1 , N t ) ≈ (cid:32) cosh (cid:0) ( N t / − t ) ˜ s (cid:1) cosh( N t ˜ s/
2) + 1 − tanh ( N t ˜ s/ (cid:33) N t ˜ s (cid:29) ≈ cosh (cid:0) ( N t / − t ) ˜ s (cid:1) N t ˜ s/ , (III.39)where on the last line we used that tanh ( x ) quicklyapproaches unity for x > , so that the constantpiece in (cid:104) φ − x φ + y (cid:105) gets cancelled. The reason whythis cancellation happens only for sufficiently large N t ˜ s is, that the fields φ ± are, as mentioned earlier,not invariant under global U(1) -transformations and (cid:104) φ ± (cid:105) therefore receives cancelling contributions formdifferent U(1) -phases if N t ˜ s is not sufficiently largeto give dominant weight to just a single phase. Onlyif the (cid:104) φ ± (cid:105) are dominated by a single phase, the term (cid:104) φ + (cid:105)(cid:104) φ − (cid:105) in (III.39) can fully cancel the disconnecteddiagrams in (cid:104) φ − x φ + y (cid:105) .Now let us return to the question of why theconstant piece in the two-point function gets smallerwhen the source ˜ s changes from zero to a non-zerovalue. The first thing to note is, that for ˜ s > the con-nected two-point function (III.39) at a critical valueof µ does no-longer correspond to a zero-mode but toa mode of mass ˜ s . The cause of the constant pieces(III.24) we observed at critical µ for ˜ s = 0 is therefore4 xk n − nn + 1 n + 2 x x x x xk n − nn + 1 n + 2 x x x x Figure 9. The figure shows how the function (III.36) enters the denominator of (III.35). The x -axis represents the differentsites of the periodic (0+1)-dimensional lattice on which (III.30) is defined, and the y -axis display the values of the k -variablesbetween the sites. The chemical potential is set to the critical value µ = µ n,n +1 , so that k x = n ∀ x and k x = n + 1 ∀ x are equally likely and have maximum weight. However, as ˜ s > , the k -variables don’t have to be all the same, but canvary between the two values n and n + 1 , while other values are highly suppressed. Now, assume that we are initially in theconfiguration where k x = n ∀ x (left-hand panel). As all k -variables are equal, all site-weights in (III.30) assume the value I (˜ s ) . Next, we modify this configuration by picking some shift-points , e.g. { x , . . . , x } , at which the values of incomingand outgoing k -variable changes form n to n + 1 (upward shift-point) or from n + 1 back to n (downward shift-point). Asthe lattice is periodic, these shift-points have to appear in pairs. The modified configuration has lower weight than the originalone, as the site-weights for the shift-points { x , . . . , x } in (III.30) are reduced from I (˜ s ) to I (˜ s ) ; but, there are (cid:0) N t (cid:1) waysof picking the four shift-points, { x , . . . , x } , and therefore (cid:0) N t (cid:1) such modified configurations which are distinct but have thesame weight. Similarly, if the number of shit-points is not four, but more generally m , with m ∈ { , , . . . , (cid:98) N t / (cid:99)} , therewill be (cid:0) N t m (cid:1) configurations with a weight in which m site-weights have value I (˜ s ) instead of I (˜ s ) . The total weight ofall the modified configurations that can be obtained in this way from the config. with k x = n ∀ x (including the config. with k x = n ∀ x itself) is therefore: (cid:80) (cid:98) N t / (cid:99) m =0 (cid:0) N t m (cid:1) I N t − m (˜ s ) I m (˜ s ) . In the same way, one can modify the configuration with k x = n + 1 ∀ x (right-hand panel). One then gets the set of configurations which are the mirror-images of the ones obtainedfrom the config. with k x = n ∀ x , (i.e. the values n and n + 1 of the k -variables are interchanged), but as they are equallymany mirror-images, the total weight is again given by: (cid:80) (cid:98) N t / (cid:99) m =0 (cid:0) N t m (cid:1) I N t − m (˜ s ) I m (˜ s ) . xk n − nn + 1 n + 2 z z xk n − nn + 1 n + 2 x x x x z z Figure 10. The figure illustrates how the function (III.36) enters the enumerator of (III.35) and we proceed in a similar wayas in Fig. 9. The chemical potential is set to the critical value µ = µ n,n +1 , so that all k -variables have value n or n + 1 , andwe start with the maximum-weight configuration, displayed in left-hand panel, in which the external φ − at site z requiresthe incoming and outgoing k -variables to have values n and n + 1 , respectively, while the φ + at site z forces incomingand outgoing k -variables to have values n + 1 and n . As in Fig. 9, we can now pick an even number of lattice sites whichbecome shift-points at which the values of incoming and outgoing k -variables have to be n and n + 1 , respectively (upwardshift-point), or n + 1 and n (downward shift-point), unless the shift-point coincides with z or z , in which case incoming andoutgoing k -variable have to have the the same value instead of distinct ones. In the right-hand panel, we have again choosefour such shift-points, { x , . . . , x } , and obtain a modified configuration for which four site-weights are changed form I (˜ s ) to I (˜ s ) . But, in contrast to the situation in Fig. 9, the presence of the external fields on sites z and z , limits the numberof ways in which one can pick the four shift-points, { x , . . . , x } : z and z split the system into two parts and in orderto have k x ∈ { n, n + 1 } ∀ x , the number of shift-points must be even in both of these parts separately. Therefore, if weneglect the complications that arise at the boundaries between the two parts, the total weight of all configurations that canbe obtained by adding shift-points to the maximum weight configuration in the left-hand panel, is approximately given by: (cid:0) (cid:80) (cid:98) ∆ z/ (cid:99) m =0 (cid:0) ∆ z/ m (cid:1) I ∆ z/ − m (˜ s ) I m (˜ s ) (cid:1)(cid:0) (cid:80) (cid:98) ( N t − ∆ z ) / (cid:99) m =0 (cid:0) ( N t − ∆ z ) / m (cid:1) I ( N t − ∆ z ) / − m (˜ s ) I m (˜ s ) (cid:1) , where ∆ z = ( z − z ) . gone. The constant pieces appearing in (III.35) for ˜ s > are instead due to disconnected diagrams andcan be removed by subtracting (cid:104) φ − x (cid:105) (cid:10) φ + y (cid:11) in (III.39).By setting in (III.35) the source ˜ s to zero, the dis-connected diagrams are gone and at the same timethe massless modes reappear, so that we recover theresult from (III.24) for µ = µ n,n +1 . For ˜ s > , on the other hand, the minimum in the two-point functionat t = N t / changes smoothly as a function of N t from the value / for N t (cid:28) ˜ s − to the value / for N t (cid:29) ˜ s − .From the derivation of the result in (III.35), it canbe seen, that this decrease in the minimum of the5two-point function as function of N t when ˜ s > (remember that ˜ s still has to be small for our approxi-mation with the restricted transfer-matrix to be valid),is due to configurations with disconnected diagrams,which start to contribute to both, the partition functionas well as observables, but in different amounts. If wewrite (cid:104) φ − x φ + y (cid:105) = Z − , +2 ( x, y ) /Z (cf. appendix A 2),the number of configurations with disconnecteddiagrams, contributing to the partition function Z ,is always larger than the number of configurationswith disconnected diagrams, contributing to thetwo-point partition function Z − , +2 ( x, y ) , as in thelatter, the presence of the external fields at x and y puts additional constraints on the form of the possibledisconnected diagrams. The ratio Z − , +2 ( x, y ) /Z has therefore to decrease when ˜ s changes from zeroto a non-zero value. For small N t , the number ofpossible disconnected diagrams that can contributeto Z or Z − , +2 ( x, y ) , is relatively small, and becauseeach discontinuity leads to a suppression factor I (˜ s ) /I (˜ s ) in the weight of the correspondingconfiguration, disconnected configurations contributethen only marginally to the partition functions Z and Z − , +2 ( x, y ) . In this case, the fact, that there are lessdisconnected diagrams in Z − , +2 ( x, y ) than in Z isirrelevant. With increasing N t , the number of possibleconfigurations with disconnected diagrams growsrapidly for both, Z and Z − , +2 ( x, y ) , while the numberof fully connected configurations remains the same.For some sufficiently large N t , the configurations withdisconnected diagrams will therefore start to domi-nate in Z and Z − , +2 ( x, y ) over the fully connectedconfigurations, after which the fact, that the numberof disconnected diagrams that can contribute to Z ,is larger than the number of disconnected diagramsthat can contribute to Z − , +2 ( x, y ) , becomes relevantand at some point completely dominates the value of Z − , +2 ( x, y ) /Z .So far, we have for ˜ s > only discussed the N t -dependency of condensate and correlation function inthe case where β is kept fixed. However, as the com-bination σ := ˜ s N t ∼ J/T is dimensionless (cf.text below eq. (II.3) for case d = 1 ), the situationshould remain qualitatively unchanged if we keep in-stead of β , the quantities κ = β/N t = T f π and σ = ˜ s N t ∼ J/T fixed while N t is increased, pro-vided that κ and σ are sufficiently small, so that theapproximations, that lead to the expressions (III.35)and (III.37) for two- and one-point function, respec-tively, remain valid.For too large values of ˜ s , the coupling to the externalfield will dominate over the nearest-neighbour interac-tions and one expects that if the system develops longrange order, the condensate should approach the fol-lowing value: (cid:12)(cid:12)(cid:10) φ ± (cid:11)(cid:12)(cid:12) ≈ N t √ ∂∂ ˜ s log (cid:18) π (cid:90) − π d θ e N t ˜ s cos( θ − θ s ) (cid:19) = 1 √ I ( N t ˜ s ) I ( N t ˜ s ) ˜ sN t (cid:29) ≈ √ ≈ . . (III.40)It can be numerically verified that the full expression(III.32) reproduces this result, while our approximateexpression (III.37) clearly has to fail in doing so, as itcan apparently not grow larger than − / ≈ . .To improve the approximate formula (III.37), config-urations that include k -variables with values that aredifferent from n or n + 1 , would have to be included,whose large number an variety would quickly becomeunmanageable in a manual computation. IV. SUMMARY AND DISCUSSION
In Sec. III A we have seen, that the (0+1)-dimensional non-linear
O(2) -model at finite densityundergoes phase-transitions between different vacuawhen the chemical potential µ crosses the criticalvalues µ n,n +1 , n ∈ Z , given in (III.5). The differentvacua form a discrete set and can be labeled by theinteger-valued charge-density they carry. The twosubscripts n and n + 1 in µ n,n +1 refer to these chargedensities of the two vacua that are separated by thetransition at µ = µ n,n +1 .Sec. III B illustrated, that although the latticesystem is one-dimensional (it extends only in timedirection), the two-point function (cid:104) φ − x φ + y (cid:105) develops atthe transition points µ = µ n,n +1 , n ∈ Z a non-zeroconstant piece of magnitude / , which is due tozero-modes and indicates the development of somesort of long-range order in the system.In Sec. III C we then discussed the case ˜ s > : thenon-zero source removes all zero-modes and insteadallows configurations with disconnected diagrams tocontribute to the partition function and to expectationvalues of observables. These disconnected diagramsreplace the zero modes as cause of the constantpiece in the two-point function when µ approachesa critical value µ n,n +1 , n ∈ Z , and in addition giverise to a non-zero condensate (cid:104) φ ± (cid:105) . For sufficientlylarge system sizes, both, the constant piece in thetwo-point function, as well as the condensate developa pronounced peak when µ approaches one of thecritical values µ n,n +1 , n ∈ Z .These findings generalize to linear and non-linear O( N ) spin models with N ≥ ; in most of the equa-tions presented in Sec. III, the Bessel functions I k ( β ) would just need to be replaced by more complicatedlink-weight functions.6As mentioned earlier, in the solid state physicsinterpretation of the partition function for our (0+1)dimensional O(2) model, we consider the Euclideantime-direction as spatial dimension and let, insteadof /N t , the paramete β play the role of inversetemperatuere: β ∝ / ( k B T ) . The Euclidean actionthen turns into S = β H , with H being the Hamilto-nian of a corresponding one-dimensional solid statephysics O(2) -spin system. In this context, the abovementioned findings of long range order and a non-zero condensate in this 1D
O(2) spin system seemto conflict with the findings of Mermin-Wagner [1]and Wegner [5], that an O( N ) -symmetric spin systemwith finite-range interactions can in one or twodimensions not undergo spontaneous magnetization,and not develop long-range order. Why does ourmodel nevertheless show signs of long-range order ata discrete but infinite set of µ -values?We can come up with essentially three possiblereasons why the Mermin-Wagner theorem does notneed to apply in the present case:The first reason is, that in the solid state physicsinterpretation given above, the lattice spacing can beconsidered as physical, and the thermodynamic limitis therefore always obtained by sending N t to infinitywhile the lattice spacing a and the inverse temperature β are kept fixed. But, as we have seen in Secs. III Aand III B, if N t is sent to infinity at fixed β , then thesystem undergoes first order phase transitions at thevery same values of µ for which one can observe theappearance of long-range order.One could also argue, that the action of our modelis complex for µ (cid:54) = 0 when expressed in terms ofthe original spin variables, and it is not obvious, thatthe proof of the Mermin-Wagner theorem extends tothis case. In the solid state physics interpretation ofour O(2) model, the variables { θ x } x can for µ = 0 be considered as physical degrees of freedom, whichspecify the configuration of a set of classical spinson a 1D lattice. Each configuration has in this casea well-defined energy and Boltzmann weight. But,as soon as µ (cid:54) = 0 , the Hamiltonian is in generalcomplex and the θ -variables loose their direct phys-ical meaning, as configurations which are encodedin terms of physical degrees of freedom should takeon only real energy values. One could thereforeinfer, that the system can escape the Mermin-Wagnertheorem, because it is only the formulation in termsof these unphysical degrees of freedom that makes theHamiltonian look as if the theorem should apply. But,if one parametrizes the system also for µ (cid:54) = 0 in termsof physical degrees of freedom, these physical degreesof freedom (and the ways in which they interact) dono-longer satisfy the prerequisites for the theoremto apply: in the flux-variables representation of the O(2) partition function (III.1) in (0+1) dimensions, the interaction between the k -variables that live ondifferent links, becomes for example of infinite rangewhen ˜ s = 0 , due to the conservation of the U(1) -flux,which renders the Mermin-Wagner theorem inappli-cable [3, 4].Finally, a third argument for why the Mermin-Wagner theorem does not need to apply to our finitedensity
O(2) model, is motivated by the fact that thecharge density in our
O(2) system would in continu-ous Minkowski space be given by the time-componentof the conserved current j ν ∝ i (cid:0) ( ∂ ν φ − ( x )) φ + ( x ) − ( ∂ ν φ + ( x )) φ − ( x ) (cid:1) = ∂ ν θ ( x ) (up to a divergence-free vector field, which in (0+1) dimensions is just aconstant). One might therefore assume that the dif-ferent vacua of the corresponding Euclidean latticemodel, which carry different integer-valued chargedensities, differ by how often θ x wraps around the in-terval ( − π, π ] as x moves along a path that winds intime direction around the periodic lattice.This per lattice winding , N Ω , is the Euclidean timeintegral of the charge density n , and one can there-fore write: n = N Ω /N t , where N t is the temporal ex-tent of the periodic lattice. An integer charge density n therefore implies that the θ x variable had to wrap n · N t times around the interval ( − π, π ] as x wrapsonce around the periodic lattice in time-direction. Thismeans that θ x should undergo n full windings whiletraversing a single link. Such a per link winding can,of course, not be represented by the lattice field θ x , as ∆ θ x,d = ( θ x + (cid:98) d − θ x ) has to satisfy | ∆ θ x,d | < π if θ x ∈ ( − π, π ] ∀ x . But, we can for the moment con-sider the values θ x and θ x + (cid:98) d of the lattice field on ad-jacent sites of a link ( x, d ) that connects the sites x and x + (cid:98) d as boundary conditions for a 1d continuum field θ ( x ) that lives on the link ( x, d ) and interpolates be-tween the two boundary values. The number of wind-ings that this continuum field undergoes along the link ( x, d ) , will then have an impact on how strongly theaction for this continuum field changes when ∆ θ x,d is changed. So, a change in the winding number ofthe continuum field on the link ( x, d ) would imply achange in the interaction strength between the latticevariables θ x and θ x + (cid:98) d . As the continuum field will tryto behave classically, the winding number will growproportionally to the chemical potential (up to the factthat it can only change in discrete steps) and one cantherefore imagine that in the lattice formulation, theper link winding number of the link ( x, d ) is encodedin the combined values of µ and ∆ θ x,d .This suggests that the long-range order we observein our (0+1)-dimensional O(2) lattice system when-ever µ approaches one of the critical values µ n,n +1 from (III.5), with n ∈ Z , is not of the usual form,i.e. not related to the spontaneous breaking of a globalsymmetry and the formation of a homogeneous non-zero vacuum expectation value, but rather occurs be-cause the magnitude of the chemical potential is suchthat the corresponding preferred phase velocity ˙ θ ( x ) φ ± -fields is so high, that φ ( x ) undergoes mul-tiple windings as x moves along the distance of a sin-gle lattice spacing. In the field-theory context, this isclearly a lattice artefact, which occurs because the en-ergy scale set by µ exceeds the lattice cut-off scale. Inthe above mentioned solid state physics context, how-ever, where the one-dimensional lattice is spatial andthe chemical potential µ might play the role of a nega-tive resistance, the lattice spacing is physical and doesnot define an energy cut-off. The magnitude of µ istherefore in this case not restricted. V. ACKNOWLEDGEMENT
I would like to thank Philippe de Forcrand for pro-viding helpful comments on an earlier version of thismanuscript, which helped improving clarity.
Appendix A FLUX-VARIABLE FORMULATION OF O( N ) SPIN MODEL
In this appendix, we review, following [16–18],how the flux variable representation for the non-linear O( N ) spin model with a chemical potential and sourceterms is obtained. Starting point is the partition function for the action(II.4): Z = (cid:90) D (cid:2) φ (cid:3) exp (cid:32)(cid:88) x (cid:26) β d (cid:88) ν =1 (cid:0) φ x e µ τ δ ν,d φ x + (cid:98) ν + φ x e − µ τ δ ν,d φ x − (cid:98) ν (cid:1) + ( s · φ x ) (cid:27)(cid:33) , (A.1)with φ x ∈ S N − ⊂ R N . To carry out the dualization,we now write the exponential in (A.1) as a product of separate exponential factors for each individual termin the action: Z = (cid:90) D (cid:2) φ (cid:3) (cid:89) x (cid:26)(cid:18) d (cid:89) ν =1 exp (cid:0) β e µ δ ν,d φ − x φ + x + (cid:98) ν (cid:1) exp (cid:0) β e − µ δ ν,d φ − x + (cid:98) ν φ + x (cid:1) · (cid:18) N (cid:89) i =3 exp (cid:0) β φ ix φ ix + (cid:98) ν (cid:1)(cid:19)(cid:19) exp (cid:0) s + φ + x (cid:1) exp (cid:0) s − φ − x (cid:1) (cid:18) N (cid:89) i =3 exp (cid:0) s i φ ix (cid:1)(cid:19)(cid:27) , (A.2)where φ ± = √ ( φ ± i φ ) and s ± = √ ( s ∓ i s ) .Then we write these exponentials in terms of power-series: exp (cid:0) β e µδ ν,d φ − x φ + x + (cid:98) ν (cid:1) = (cid:88) η x,ν (cid:0) φ − x φ + x + (cid:98) ν β e µδ ν,d (cid:1) η x,ν η x,ν ! , (A.3a) exp (cid:0) β e − µδ ν,d φ + x φ − x + (cid:98) ν (cid:1) = (cid:88) ¯ η x,ν (cid:0) φ + x φ − x + (cid:98) ν β e − µδ ν,d (cid:1) ¯ η x,ν ¯ η x,ν ! , (A.3b) exp (cid:0) β φ ix φ ix + (cid:98) ν (cid:1) = (cid:88) χ ix,ν (cid:0) β φ ix φ ix + (cid:98) ν (cid:1) χ ix,ν χ ix,ν ! , (A.3c)and exp (cid:0) s + φ + x (cid:1) = (cid:88) m x (cid:0) s + φ + x (cid:1) m x m x ! , (A.3d) exp (cid:0) s − φ − x (cid:1) = (cid:88) ¯ m x (cid:0) s − φ − x (cid:1) ¯ m x ¯ m x ! , (A.3e) exp (cid:0) s i φ ix (cid:1) = (cid:88) n ix (cid:0) s i φ ix (cid:1) n ix n ix ! , (A.3f)8and after writing the φ x in polar form, i.e. φ + x = sin (cid:0) θ ( N − x (cid:1) · · · sin (cid:0) θ (2) x (cid:1) e i θ (1) x √ (A.4a) φ − x = sin (cid:0) θ ( N − x (cid:1) · · · sin (cid:0) θ (2) x (cid:1) e − i θ (1) x √ (A.4b) φ x = sin (cid:0) θ ( N − x (cid:1) · · · cos (cid:0) θ (2) x (cid:1) (A.4c)... φ N − x = sin (cid:0) θ ( N − x (cid:1) cos (cid:0) θ ( N − x (cid:1) (A.4d) φ Nx = cos (cid:0) θ ( N − x (cid:1) , (A.4e)and using the corresponding integration measure, D (cid:2) φ (cid:3) ∝ (cid:89) x (cid:18) N − (cid:89) i =2 (cid:0) sin (cid:0) θ ( i ) x (cid:1)(cid:1) i − (cid:19) d θ (1) x ∧ . . . ∧ d θ ( N − x , (A.5)the partition function (A.1) can be written as Z = (cid:88) { η, ¯ η,m, ¯ m } (cid:89) x (cid:26)(cid:18) (cid:89) ν (cid:0) β (cid:1) η x,ν +¯ η x,ν η x,ν ! ¯ η x,ν ! (cid:19) (cid:0) s + √ (cid:1) m x (cid:0) s − √ (cid:1) ¯ m x m x ! ¯ m x ! (cid:18) N (cid:89) i =3 (cid:18) (cid:89) ν β χ ix,ν χ ix,ν ! (cid:19) (cid:0) s i (cid:1) n ix n ix ! (cid:19) · e µ ( η x,d − ¯ η x,d ) π (cid:90) − π d θ (1) x e i θ (1) x (cid:0) m x − ¯ m x − (cid:80) ν ( η x,ν − ¯ η x,ν − ( η x − (cid:98) ν,ν − ¯ η x − (cid:98) ν,ν )) (cid:1) · (cid:18) N − (cid:89) j =2 π (cid:90) d θ ( j ) x (cid:0) cos (cid:0) θ ( j ) x (cid:1)(cid:1) (cid:80) ν (cid:0) χ j +1 x,ν + χ j +1 x − (cid:98) ν,ν (cid:1) · (cid:0) sin (cid:0) θ ( j ) x (cid:1)(cid:1) j − (cid:80) ν (cid:0) η x,ν +¯ η x,ν + η x − (cid:98) ν,ν +¯ η x − (cid:98) ν,ν + j (cid:80) i =3 (cid:0) χ ix,ν + χ ix − (cid:98) ν,ν (cid:1)(cid:1)(cid:19)(cid:27) . (A.6)To simplify the notation, we define new variables k x,ν ∈ Z and l x,ν ∈ N , so that: η x,ν − ¯ η x,ν = k x,ν , (A.7a) η x,ν + ¯ η x,ν = | k x,ν | + 2 l x,ν , (A.7b)and p x ∈ Z and q x ∈ N , so that: m x − ¯ m x = p x , (A.8a) m x + ¯ m x = | p x | + 2 q x , (A.8b)as well as on each site x the quantities A x = (cid:88) ν (cid:0)(cid:12)(cid:12) k x,ν (cid:12)(cid:12) + (cid:12)(cid:12) k x − (cid:98) ν,ν (cid:12)(cid:12) + 2 (cid:0) l x,ν + l x − (cid:98) ν,ν (cid:1)(cid:1) (A.9)and B ix = (cid:88) ν (cid:0) χ ix,ν + χ ix − (cid:98) ν,ν (cid:1) . (A.10) Now we carry out the angular integrals in (A.6), usingthat for M, N ∈ N , we have : π (cid:90) d θ sin M ( θ ) cos N ( θ ) =1 + ( − N (cid:0) M (cid:1) Γ (cid:0) N (cid:1) Γ (cid:0) M + N (cid:1) , (A.11)which yields: Compare (A.11) with the integral form of Eu-ler’s beta function, B ( m, n ) = Γ( m )Γ( n )Γ( m + n ) =2 (cid:82) π/ d θ cos m − ( θ ) sin n − ( θ ) , and use the symme- try properties of the trigonometric functions with respect toreflection at π/ . Z = (cid:88) { k,l,χ,p,q,n } (cid:89) x (cid:26)(cid:18) d (cid:89) ν =1 β | k x,ν | +2 l x,ν + (cid:80) Ni =3 χ ix,ν ( | k x,ν | + l x,ν )! l x,ν ! (cid:81) Ni =3 χ ix,ν ! (cid:19) · ( s + ) ( | p x | + p x )+ q x ( s − ) ( | p x |− p x )+ q x e µ k x,d ( | p x | + q x )! q x ! (cid:18) N (cid:89) i =3 ( s i ) n ix n ix ! (cid:19) δ (cid:0) p x − (cid:88) ν (cid:0) k x,ν − k x − (cid:98) ν,ν (cid:1)(cid:1) · W (cid:0) A x + | p x | + 2 q x , B x + n x , . . . , B Nx + n Nx (cid:1)(cid:27) , (A.12a)or, with s ± = ˜ s e ∓ i φs √ : Z = (cid:88) { k,l,χ,p,q,n } (cid:89) x (cid:26)(cid:18) d (cid:89) ν =1 β | k x,ν | +2 l x,ν + (cid:80) Ni =3 χ ix,ν ( | k x,ν | + l x,ν )! l x,ν ! (cid:81) Ni =3 χ ix,ν ! (cid:19) · ˜ s | p x | +2 q x e i φ s p x e µ k x,d ( | px | + 2 qx ) / ( | p x | + q x )! q x ! (cid:18) N (cid:89) i =3 ( s i ) n ix n ix ! (cid:19) δ (cid:0) p x − (cid:88) ν (cid:0) k x,ν − k x − (cid:98) ν,ν (cid:1)(cid:1) · W (cid:0) A x + | p x | + 2 q x , B x + n x , . . . , B Nx + n Nx (cid:1)(cid:27) . (A.12b)where δ ( x ) = (cid:40) if x = 00 else and W (cid:0) A, B , . . . , B N (cid:1) = Γ (cid:0) A (cid:1) N (cid:81) i =3 1+( − Bi Γ (cid:0) B i (cid:1) A / Γ (cid:0) N + A + (cid:80) Ni =3 B i (cid:1) . (A.13)Not that the Taylor expansions of the exponential func-tions in (A.3) always converge, so that the two expres-sions (A.12) are exact rewritings of the original latticepartition function (A.1). The one- and two-point functions (and in principleany other n-point function) can be obtained by pro-moting the sources s i , i ∈ { + , − , , . . . , N } to latticefields s ix and taking derivatives of the logarithm of thepartition function (A.12) with respect to the s ix on dif-ferent sites x : (cid:10) φ ix (cid:11) = ∂ log( Z ) ∂s ix = Z i ( x ) Z , (A.14) (cid:10) φ ix φ jy (cid:11) − (cid:10) φ ix (cid:11)(cid:10) φ jy (cid:11) = ∂ log( Z ) ∂s ix ∂s jy = Z ij ( x, y ) Z − Z i ( x ) Z Z j ( y ) Z , (A.15)where the expressions for the one- and two-point par-tition functions , Z i ( z ) = ∂Z∂s iz and Z ij ( z , z ) = ∂ Z∂s iz ∂s jz , (A.16)in terms of the flux- and monomer-variables are mosteasily obtained, by applying the derivatives to the orig-inal partition function (A.1) and then go again throughthe dualization steps described in the previous section(Sec. A 1). Each derivative of (A.1) with respect to asource s iz brings down an extra factor of φ iz , which,in the course of the dualization procedure translatesinto integer shifts in the arguments of the site weightsgiven by (A.13) and the delta function. For the one-point partition function one then finds: Z i ( z ) = (cid:88) { k,l,χ,p,q,n } (cid:89) x (cid:26)(cid:18) d (cid:89) ν =1 β | k x,ν | +2 l x,ν + (cid:80) Ni =3 χ ix,ν ( | k x,ν | + l x,ν )! l x,ν ! (cid:81) Ni =3 χ ix,ν ! (cid:19) · ( s + ) ( | p x | + p x )+ q x ( s − ) ( | p x |− p x )+ q x e µ k x,d ( | p x | + q x )! q x ! (cid:18) N (cid:89) i =3 ( s i ) n ix n ix ! (cid:19) · δ (cid:0) p x − (cid:0) δ i, + − δ i, − (cid:1) δ x,z − (cid:88) ν (cid:0) k x,ν − k x − (cid:98) ν,ν (cid:1)(cid:1) · W (cid:0) A x + | p x | + 2 q x + (cid:0) δ i, + + δ i, − (cid:1) δ x,z , B x + n x + δ i, δ x,z , . . . , B Nx + n Nx + δ i,N δ x,z (cid:1)(cid:27) , (A.17)and similarly, for the two-point partition function : Z ij ( z , z ) = (cid:88) { k,l,χ,p,q,n } (cid:89) x (cid:26)(cid:18) d (cid:89) ν =1 β | k x,ν | +2 l x,ν + (cid:80) Ni =3 χ ix,ν ( | k x,ν | + l x,ν )! l x,ν ! (cid:81) Ni =3 χ ix,ν ! (cid:19) · ( s + ) ( | p x | + p x )+ q x ( s − ) ( | p x |− p x )+ q x e µ k x,d ( | p x | + q x )! q x ! (cid:18) N (cid:89) i =3 ( s i ) n ix n ix ! (cid:19) · δ (cid:0) p x − (cid:0) δ i, + − δ i, − (cid:1) δ x,z − (cid:0) δ j, + − δ j, − (cid:1) δ x,z − (cid:88) ν (cid:0) k x,ν − k x − (cid:98) ν,ν (cid:1)(cid:1) · W (cid:0) A x + | p x | + 2 q x + (cid:0) δ i, + + δ i, − (cid:1) δ x,z + (cid:0) δ j, + + δ j, − (cid:1) δ x,z ,B x + n x + δ i, δ x,z + δ j, δ x,z , . . . , B Nx + n Nx + δ i,N δ x,z + δ j,N δ x,z (cid:1)(cid:27) . (A.18)The products of Kronecker deltas that are insertedin (A.17) and (A.18), can be interpreted as externalfields inserted into the system at particular positions.The Z i ( z ) and Z ij ( z , z ) represent therefore parti-tion functions for the system defined by Z , but inthe presence of external fields; receiving contributionsonly from configurations which are compatible withthe presence of the inserted external fields.In a similar way one can define arbitrary n -point par-tition functions Z i ··· i n n ( z , . . . , z n ) . But, it should benoted, that the one-point partition functions (A.17) canonly take on non-zero values if their correspondingsources s i are non-zero. This is due to the fact, that the world lines, which start or end at the inserted externalfield, also have to end somewhere, which is possibleonly if the system is able to produce an appropriatemonomer (which is the case only if the correspond-ing source s i is non-zero). This is just a manifestationof the fact that in order to have a non-zero conden-sate, the vacuum needs to have overlap with the field-operator. For the same reason also all other n -pointfunctions can be non-zero only if the inserted externalfields φ iz i come in pairs of particle and correspondinganti-particle (for i ∈ {±} , this means φ ± z φ ∓ z -pairs),or if the corresponding source s i is non-zero. [1] N. D. Mermin, H. Wagner, Absence of ferromagnetismor antiferromagnetism in one- and two-dimensionalisotropic Heisenberg models , Phys. Rev. Lett. 17, 1133(1966), DOI: 10.1103/PhysRevLett.17.1133.[2] N. N. Bogoliubov, Physik. Abhandl. Sowjetunion 6, 1,113, 229 (1962).[3] D. J. Thouless,
Long-range order in one-dimensional ising systems , Phys. Rev. 187, 732,DOI: 10.1103/PhysRev.187.732.[4] Y. Imry, Ann. Phys. 51, 1-27 (1969),DOI: 10.1016/0003-4916(69)90345-5.[5] F. Wegner,
Spin-ordering in a planar classicalHeisenberg model , Z. Physik 206, 465 (1967),DOI: 10.1007/BF01325702.[6] P. C. Hohenberg,
Existence of long-range order inone and two dimensions , Phys. Rev. 158, 383 (1967), DOI: 10.1103/PhysRev.158.383.[7] N. D. Mermin,
Crystalline order in two dimensions ,Phys. Rev. 176, 250 (1968), DOI: 10.1103/Phys-Rev.176.250; Errata: Phys. Rev. B 20, 4762 (1978),DOI: 10.1103/PhysRevB.20.4762; Phys. Rev. B 74,149902 (2006), DOI: 10.1103/PhysRevB.74.149902.[8] D. C. Hamilton,
Absence of long-range Overhauserspin-density waves in one or two dimensions , Phys.Rev. 157, 427 (1967), DOI: 10.1103/PhysRev.157.427.[9] M. A. Moore,
Additional evidence for a phase transi-tion in the plane-rotator and classical Heisenberg mod-els for two-dimensional lattices , Phys. Rev. Lett. 23,861 (1969), DOI: 10.1103/PhysRevLett.23.861.[10] V. L. Berezinsky,
Destruction of long range order inone-dimensional and two-dimensional systems havinga continuous symmetry group. I. Classical systems , Sov. Phys. JETP 32, 493 (1971).[11] V. L. Berezinsky,
Destruction of long range order inone-dimensional and two-dimensional systems havinga continuous symmetry group. II. Quantum systems ,Sov. Phys. JETP 34, 610 (1972).[12] R. Kishore, D. Sherrington,
On the non-existence ofmagnetic order in one and two dimensions , Phys.Lett. A, 42, 205-207 (1972), DOI: 10.1016/0375-9601(72)90862-6.[13] S. Coleman,
There are no Goldstone bosons in twodimensions , Commun. math. Phys. 31, 259 (1973),DOI: 10.1007/BF01646487.[14] J. M. Kosterlitz, D. J. Thouless,
Ordering, metastabil-ity and phase transitions in two-dimensional systems ,Journal of Physics C: Solid State Physics, 6(7), 1181(1973), DOI: 10.1088/0022-3719/6/7/010.[15] M. G. Endres,
Method for simulating O( N ) lat-tice models at finite density , Phys. Rev. D75, 065012 (2007), DOI: 10.1103/PhysRevD.75.065012, arXiv: hep-lat/0610029 .[16] F. Bruckmann, C. Gattringer, T. Kloiber,T. Sulejmanpasic, Dual lattice representa-tion for O( N ) and CP( N − models witha chemical potential , Phys. Lett. B749, 495(2015), DOI: 10.1016/j.physletb.2015.08.015, arXiv: 1507.04253 [hep-lat] .[17] T. Rindlisbacher, P. de Forcrand, Lattice simu-lation of the
SU(2) chiral model at zero andnon-zero pion density , PoS (LATTICE 2015) 171,DOI: 10.22323/1.251.0171, arXiv: 1512.05684 [hep-lat] .[18] T. Rindlisbacher,