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dcos θ tegral. However, the problem itself is more general dτ from integrating by parts, and we use standard i R β dτφ(τ)f(τ) than the toy model considered here and clearly per- identity for functional integrals φe− 0 = sists in more complicated models, including lattice Bose- D dcos θ δ(f), to get that cos θ must be constant (i.e., dτ = 0). Hubbard models. We use an exact method of calculating This δ-function allows us to doR the path integral over the partition function, first used for spin (with H = Sz) (cos θ), except for the initial value which we call x := by Cabra et al. [23], and demonstrate that the exact re- cosD θ(0). Taking all of this into account, the path integral sult differs from the correct partition function in the cases can then be written as of both normal-ordering of operators (as prescribed by 1 most textbooks) and when using Weyl ordering (i.e., it ∞ 2πiks(1 x) βH(x) ′ = dxe − − . (2) cannot be accounted for with the phase anomaly found Z 1 k= Z− by Solari and Kochetov [18, 20] and elaborated on by X−∞ Stone et al. [19]). The sum over k can be evaluated as a sum of delta func- We begin with the coherent state path integral for spin tions of the form δ(s(1 x) n) for all integers n. Since − − with the standard SU(2) algebra defined on the operators x is in the interval 1 to +1, only finitely many n con- − Sx,Sy,Sz with [Si,Sj] = iǫijkSk, and we define our tribute (n = 0 to n = 2s to be exact). We can rewrite Hilbert{ space} by taking the matrix representation of the the sum over n as a sum over m := s n and we get the − SU(2) group in (2s + 1)-by-(2s +1) matrices (s being the answer (dropping overall constants) spin of the system). Irrespective of the algebra, we can in s general define a Hermitian matrix H that acts on states βH(m/s) ′ = e− . (3) Z in our Hilbert space, and this will be our Hamiltonian. m= s Usually, H is polynomial of algebra generators. X− If s is the maximal state of Sz in our spin-s sys- Eq. (3) looks very promising, but H(m/s) is not the same | i n as m H m . First let us see where it does work. Take tem, then we can define spin-coherent states as = h | | i iφSz iθSy | i the simple Hamiltonian H = S , then n H n = s cos θ, e− e− s where (θ, φ) are coordinates on the z h | | i sphere S2 along| i the unit vector n (i.e., a point on and thus H(x)= sx. This immediately yields the standard Bloch sphere). These coherent states are s 2s+1 overcomplete such that 2 dn n n = 1 where βm 4π S H′ =S = e− , (4) n | i h | 2 Z z d = dφ d(cos θ) is the standard measure on S . Us- m= s R − ing this continuous, overcomplete basis, one can derive X the standard path integral for the partition function for and it is easily calculated (in operator language) that βH ′ = H=S . The two methods agree for the partic- spin from = tr e− in the standard way [1] discussed ZH=Sz Z z in the introduction:Z ular Hamiltonian H = Sz (the case considered by Cabra 2 et al. [23]). On the other hand, if we take H = Sz and β n 2 n 1 2 s = 1, we can evaluate Sz = 2 cos θ +1 ; from ′ = n(τ) exp dτ [ n(τ) ∂ n(τ) h | | i Z D − − h | τ i which we have Z ( Z0
1 + n(τ) H n(τ) ] . (1) H(x)= x2 +1 . (5) h | | i } 2 We call the partition function as given by the time- β β/ 2
Thus, H′ =S2 = 2e− + e− , but this conflicts with continuous path integral ′ in order to distinguish it from z Z β βH Z 2 = 2e− + 1 by more than just a multiplicative = tr e− since we will find that in general they may ZH=Sz Z constant. Thus, we have 2 = 2 for s = 1, and not agree. The path integral is over all closed paths (since H′ =S H=Sz Z z 6 Z it is the parititon function). The first term in the action in fact ′ 2 = H=S2 for all s> 1/2. ZH=Sz 6 Z z for Eq. (1), n ∂ n , is the Berry phase term and in (θ, φ) Importantly, the two methods agree for any Hamil- h | τ i coordinates n ∂τ n = si(1 cos θ)∂τ φ. tonian when s = 1/2. This comes from the fact that In order to− highlight h | i what− is wrong− with a na¨ıve use of any (diagonalized) Hamiltonian for a two state sytem Eq. (1), we employ the method of Cabra et al. [23]. We (s = 1/2) can be written as H = a + bSz (in fact 2 assume n(τ) H n(τ) = H(cos θ(τ)) for some function H = S = 1/4), and the above method gives ′ = h | | i z Z Z H(x) (this is true if and only if H is diagonal). This when H = a + bSz. 2 2 puts the φ dependence of the action solely in the Berry Also, if we let our Hamiltonian be H = Sz /s , then in phase term of the action. We then integrate the Berry the limit of s 1 Eq. (3) reproduces the correct result. phase term by parts; the boundary term is just ∆φ(1 This is a general≫ result for Hamiltonians that are finite − cos θ(0)) with ∆φ = φ(β) φ(0) = 2πk for any integer polynomials of Sz/s, and suggests that “semiclassically” k and cos θ(β) = cos θ(0) since− our paths are closed. We (i.e. s tends to infinity), we will still arrive at sensible must sum over the different topological sectors defined results. by the integer k (i.e., how many times φ wraps around Agreement can also be forced by considering H(x) = 2 the sphere). Thus, our only φ dependence is multiplying x instead of Eq. (5), but this corresponds to replacing Sz 3

µ U/2 µ 2µ U with Sz in the Hamiltoinian instead of just considering we have ′ 1+e − + , but 1+e +e − + h i 2 Z ∼ ··· Z ∼ H . In the H = Sz case, it is the difference between con- . With different asymptotics, and ′ are different h i 2 2 ··· Z UZ sidering S and S ; the latter gives correct results. expressions. Note that if we let µ µ+ in ′, that we z z → 2 Z There ish no ia priorih reasoni for this construction. will get same result. This substitution for µ corresponds To motivate looking for this same issue in a system to replacing n in Eq. (6) by n = z 2 when writing down h i | | with the Weyl-Heisenberg algebra (i.e., the harmonic our action (so instead of n2 , one gets n 2). oscillator algebra), it is known [24] that one can con- We now compare this toh thei semiclassicalh i result. Still tract u(2) (since we constructed our coherent states for considering Eq. (6), let change our algebra slightly to spins with su(2)) into the Weyl-Heisenberg algebra by incorporate a small parameter (akin to the standard ~ 1 → considering u(2) = span S ,S ,S ,S = u(1) su(2), 0 for normal semiclassics): h− , the representation index; { 0 x y z} ⊕ where we define [S0,Si] = 0. Then define the opera- see γ defined in [20]. We note here that different h’s 2 tors J0 := S0, J1,2(ǫ) := ǫSy,x, and J3(ǫ) := S0 + ǫ− Sz change the coherent states z in the following way: if to get the commutation relations [J ,J ] = iJ , z = 1 (x + iy), then x = q/c| i, y = p/d, and h = ~/(cd) 3 1,2 2,1 √2 2 ∓ [J1,J2] = iǫ J3 + iJ0, and [J0,Ji] = 0. If we let (and [a,a†]= h). We have used q and p as the standard ǫ 0, we recover− exactly the Weyl-Heisenberg algebra: → position and momentum for the harmonic oscillator. Up h4 = span 1,x,p,a†a with [x, p] = i,[a†a, x] = ip, until now we have been considering h = 1. { } 2 − [a†a,p] = ix. Observe that Sz is related to a†a in this We can write the propagator between two coherent contraction, so we might suspect that terms quadratic in states z and z using a Hubbard-Stratonovich trans- | ii | f i a†a might give problems like those found with the spin- formation and the propagator for the harmonic oscillator: coherent state path integral. iHT/h A Hamiltonian that uses the Weyl-Heisenberg alge- K(zf∗,zi; t)= zf e− zi (10) bra to construct its coherent states is the Bose-Hubbard h | | i iT 1 Φ + 1 iωT + i UhT model. For a single site, we can write = dωe h ω 2 8 , (11) r2πUh U Z H = µn + n(n 1), (6) where we have defined − 2 − i(ω+µ)T iT 2 1 2 2 where n = a†a and the a (a†) is the annihilation (cre- Φ = z∗z e + ω ( z + z ). (12) ω f i 2U − 2 | i| | f | ation) operator for the algebra [a,a†] = 1. The form n(n 1) = a†a†aa comes from the normal ordering re- In path integral notation we can write out the propaga- − quired from a path integral of the form tor [20]

∗ ∗ β z (T )=zf 2 1 2 ′ = z exp dτ (z∗z˙ z˙∗z) K(zf∗,zi; T )= z exp Φ[z,z∗]/h , (13) Z D − 2 − z(0)=z D { } Z ( Z0  Z I U µ z 2 + z 4 . (7) where Φ = Γ+ S, − | | 2 | |  1 2 2 Γ= z∗z(T )+ z∗(0)z z z , (14) We can solve this path integral with the same method 2 f I − | f | − | I | used to obtain Eq. (3) in the spin-coherent state path 1  T T  integral. Let z = √n eiθ, so that the measure becomes S = dt (zz˙∗ z∗z˙) i dt z H z . (15) 2 − − h | | i 2z = n θ and the action becomes S = dτ(inθ˙ Z0 Z0 D D D − µn + U n2). Integrating by parts on the nθ˙ term then Performing the standard semiclassical analysis and al- 2 R integrating over θ will fix n to be constant, and the gebra (see [20] and [19]) the semiclassical propagator boundary term willD fix n to be an integer. Since n is takes the form radial, it can only be positive so we directly obtain 1/2 2 1/2 iT 1 ∂ Φω − 2 Ksc(z∗,zi; T )= ∞ µnβ U n β f 2 ′ = e − 2 . (8) hU h ∂ω Z ω     n=0 X X 1 i exp Φω + (ω + µ)T i∆ , (16) But this differs from the partition function that we can × h 2 −   easily calculate in operator language: where the sum is over solutions to the consistency equa- ∂Φω i(ω+µ)T ∞ tion given by = 0 or ω = Uz z e , and µnβ U n(n 1)β ∂ω f∗ i = e − 2 − . (9) 1 − Z we have defined ∆ = (µ +2ω)T . The term ∆ comes n=0 2 X from the fixing of the fluctuation determinant anomaly We see that a similar problem to that of the spin coherent described in detail by Stone et al. [19] for the SU(2) case. state path integral here. To see it explicitly, for U 1, However, if we try to get Eq. (16) by using the method ≫ 4

n 2 n of steepest descent on Eq. (11) with h 0, we won’t (between time slices j and j + 1) j+1 Sz j , and we → n 2 n n 2hn | | i get the same result. This is because of what has been have Sz = 0, so j+1 Sz j can attain zero, shown by others [20, 21]: that the semiclassics will give but noth | for|− anyi paths thath are “close”| | i to each other (i.e. results consistent with the Weyl ordering of the Hamil- n n ) as the continuous time path integral assumes. j ≈ j+1 tonian (na¨ıvely ordering all a and a†’s symmetrically). As such, the discrete time path integral (before a continu- The usual normal ordered Hamiltonian takes the form ity assumption is imposed) can unambiguously give the U (inserting h’s) H = µn + 2 n(n h) while the Weyl correct results to a calculation. ordered Hamiltonian− takes the form− (up to a constant) To conclude, in the time-continuous formulation of U HW = µn + 2 n(n + h). If we derive Eq. (11) for HW , the path integral, neither the action suggested by Weyl- we will− find the the steepest descent exactly agrees with ordering nor the action constructed by normal ordering Eq. (16) just as expected [20, 21]. give the correct result when evaluating via path inte- Z While the semiclassical result is not a new one, it grals. shows that the path integral is not dealing with the Acknowledgements – We thank Michael Levin for stim- same Hamiltonian. Unfortunately, our exact calculation ulating conversations. This research was supported by the NSF CAREER award, DMR-0847224. suggests that the path integral is dealing with H′ = U 2 µn + 2 n while semiclassics suggests it is dealing with − U HW = µn + 2 n(n + 1). These two methods differ but both are− not the Hamiltonian under consideration. In the case of the Weyl ordered Hamiltonian, we can write [1] A. Altland and B. D. Simons, Condensed Matter Field our original Hamiltonian in Eq. (6) as H = HW Un Theory, 2nd ed. (Cambridge University Press, 2010). − which is Weyl ordered (up to a constant). This ordering [2] X. Wen, Quantum Field Theory of Many-body Systems, can be used to modify the path integral by an extra term reissue ed. (Oxford University Press, 2007). Un. This correction to the path integral suggested by [3] M. E. Peskin and D. V. Schroeder, An Introduction To Quantum Field Theory Weyl− ordering does not fix the exact calculation as can (Westview Press, 1995). [4] J. R. Klauder, in Path Integrals and their Applications be easily shown, but it does motivate an ad hoc correc- in Quantum, Statistical, and Solid State Physics, edited tion to the path integral to “fix” our exact calculation. by G. Papadopoulos and J. Devreese (Plenum Press, We use the following action (going back to h = 1): Antwerp, Belgium, 1977). [5] F. A. Berezin, Sov. Phys. Usp. 23, 763 (1980). [6] J. R. Klauder and B. Skagerstam, Coherent states: ap- U S = dt µ z 2 + z 2( z 2 1) . (17) plications in physics and mathematical physics (World − | | 2 | | | | − Scientific, 1985).   Z [7] R. J. Glauber, Phys. Rev. 131, 2766 (1963). [8] A. M. Perelomov, Commun. Math. Phys. 26, 222 (1972). This action is constructed by just changing the operator [9] R. Gilmore, Ann. Phys. 74, 391 (1972). n to a function z 2; while this gives correct results with | | [10] W. Zhang, D. H. Feng, and R. Gilmore, Rev. Mod. Phys. the method which gives Eq. (8), there is no a priori rea- 62, 867 (1990). son to suspect this of being the action. Similarly, if in [11] A. M. Perelomov, Generalized coherent states and their the spin-coherent state path integral, we replace the op- applications (Birkh¨auser, 1986). [12] J. R. Klauder, “The Feynman path integral: An histori- erator Sz with its expectation value n Sz n everywhere, h | | i cal slice,” (2003), arXiv:quant-ph/0303034v1. we will get the correct result. This means, in particular, 19 2 2 [13] M. Enz and R. Schilling, J. Phys. C , 1765 (1986). for H = Sz that instead of Sz in the spin-path integral 2 h i [14] K. Funahashi, T. Kashiwa, S. Sakoda, and K. Fujii, we have Sz . In general, if one substitutes the genera- J. Math. Phys. 36, 3232 (1995). h i tors of the coherent states in the Hamiltonian with their [15] K. Funahashi, T. Kashiwa, S. Nima, and S. Sakoda, expectation value, one obtains the correct result for Nucl. Phys. B 453, 508 (1995). with the methods used to derive Eq. (3) and Eq. (8). Z [16] V. I. Belinicher, C. Providencia, and J. da Providencia, J. Phys. A 30, 5633 (1997). Corrections aside, a simple way to see what has gone [17] J. Shibata and S. Takagi, Int. J. Mod. Phys. B 13, 107 wrong is to return to Eq. (5). This H(x) function can (1999). 2 28 not achieve the value 0, but H = Sz clearly has such an [18] H. G. Solari, J. Math. Phys. , 1097 (1987). eigenvalue. This is due to the fact that for higher dimen- [19] M. Stone, K. Park, and A. Garg, J. Math. Phys. 41, 8025 (2000). sional representations of SU(2) not every eigenvector of 31 S can be rotated into another with a standard SU(2) [20] E. A. Kochetov, J. Phys. A , 4473 (1998). z [21] M. Pletyukhov, J. Math. Phys. 45, 1859 (2004). rotation. On the other hand, the coherent states we used [22] A. Polkovnikov, Ann. Phys. 325, 1790 (2010). are a complete set for even higher dimensional represen- [23] D. C. Cabra, A. Dobry, A. Greco, and G. L. Rossini, J. tations, so in principle, we should not lose any informa- Phys. A 30, 2699 (1997). tion about the m = 0 state. Continuity in n seems to [24] R. Gilmore, Lie groups, physics, and geometry (Cam- be the culprit: H(x) came from a time discretized form bridge University Press, 2008).