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PT-Symmetric Matrix Quantum Mechanics ∗ Peter N. Meisinger and Michael C. Ogilvie Department of Physics, Washington University, St. Louis, MO 63130, USA (Dated: February 2, 2008) RecentlydevelopedmethodsforPT-symmetricmodelsareappliedtoquantum-mechanicalmatrix models. WeconsiderindetailthecaseofpotentialsoftheformV =−(g/Np/2−1)Tr(iM)pandshow how the calculation of all singlet wave functions can be reduced to solving a one-dimensional PT- symmetricmodel. Thelarge-N limitofthisclassofmodelsexists,andpropertiesofthelowest-lying singlet state can be computed using WKB. For p=3,4, the energy of thisstate for small values of N appearstoshowrapidconvergencetothelarge-N limit. Forthespecial caseofp=4,weextend recent work on the −gx4 potential to the matrix model: we show that the PT-symmetric matrix model is equivalent to a hermitian matrix model with a potential proportional to +(4g/N)TrΠ4. However, this hermitian equivalent model includes an anomaly term ~p2g/NTrΠ. In the large-N 7 limit, the anomaly term does not contribute at leading order to theproperties of singlet states. 0 0 2 n I. INTRODUCTION a J 3 Matrix models appear in many contexts in modern theoretical physics, with applications ranging from condensed 2 matter physics to string theory. Interest in the large-N limit of matrix models was strongly motivated by work on the large-N limit of QCD [1], but interest today is much wider. For example, Hermitian matrix quantum mechanics c 1 leads to a construction of two-dimensional quantum gravity coupled to c=1 matter [2]. v We will show below that the matrix techniques pioneered in [3] for Hermitian matrix quantum mechanics can be 7 extendedtoPT-symmetricmatrixquantummechanics,wherethematricesarenormalbutnotnecessarilyHermitian. 0 The large-N limit can then be taken in PT-symmetric matrix theories just as in the Hermitian case. Quantities of 2 1 interest such as the scaled ground state energy and scaled moments can be calculated using WKB methods. In the 0 special case of a quartic potential with the “wrong” sign, we prove using functional integration for all values of N 7 that the PT-symmetric model is equivalent to a hermitian matrix model with an anomaly, as in the one-component 0 case [4, 5]. Interestingly, the anomaly vanishes to leading order in the large-N limit. / h t - p II. FORMALISM e h The solution for all N of the quantum mechanics problem associated with the Euclidean Lagrangian : v 2 i 1 dM g X L= Tr + TrM4 (1) 2 dt N r (cid:18) (cid:19) a whereM isanN N HermitianmatrixwasfirstgivenbyBrezinetal.[3]. Thegroundstateψisasymmetricfunction × of the eigenvalues λ of M. The antisymmetric wave function φ defined by j φ(λ ,..,λ )= (λ λ ) ψ(λ ,..,λ ) (2) 1 N j k 1 N  −  j<k Y   satisfies the Schrodinger equation 1 ∂2 g + λ4 φ=N2E(0)φ (3) j "−2∂λ2j N j# X where E(0)is the ground state energy scaled for the large-N limit. This equation separates into N individual Schrodinger equations, one for each eigenvalue, and the antisymmetry of φ determines N2E(0) as the sum of the N lowest eigenvalues. ∗Electronicaddress: [email protected] 2 Here we solve the correspondingproblem where the potential term is PT-symmetric but not Hermitian. As shown by Bender and Boettcher [6], the one-variable problem may be solved by extending the coordinate variable into the complex plane. This implies that for PT-symmetric matrix problems, we must analytically continue the eigenvalues of M into the complex plane, and in general M will be normal rather than Hermitian. We consider the Euclidean Lagrangian 1 dM 2 g L= Tr Tr(iM)p (4) 2 dt − Np/2−1 (cid:18) (cid:19) with g >0. Making the substitution M UΛU+, with U unitary and Λ diagonal, we can write L as → 1 dλ 2 1 dH dH g L= j + (λ λ )2 (iλ )p (5) 2 dt 2 j − k dt dt − Np/2−1 j j (cid:18) (cid:19) j,k (cid:18) (cid:19)jk(cid:18) (cid:19)kj j X X X where dH dU = iU+ . (6) dt − dt In the analysis of conventional matrix models by Brezin et al., a variational argument shows that the ground state is a singlet, with no dependence on U. Because the λ ’s are in general complex for PT-symmetric theories, this j argument does not apply. However, in two cases we can prove that the ground state is indeed a singlet: for p = 2, which is trivial, and for p = 4, where the explicit equivalence with a hermitian matrix model proven below can be used. Henceforth, we will assume that the ground state is a singlet, but our results will apply in any case to the lowest-energy singlet state. We have now reduced the problem of finding the ground state to the problem of solving for the first N states of the single-variable Hamiltonian 1 g H = p2 (iλ)p. (7) 2 − Np/2−1 This Hamiltonian is PT-symmetric but in general not Hermitian. The case p = 2 is the simple harmonic oscillator. For p > 2, the Schrodinger equation associated with each eigenvalue may be continued into the complex plane as explained in [6]. We exclude the case p < 2, where PT symmetry is spontaneously broken and the eigenvalues of H are no longer real. III. GROUND STATE PROPERTIES AswithHermitianmatrixmodels. thegroundstateenergyisthesumofthefirstN eigenenergiesoftheHamiltonian H. Inthe largeN limit, this summay be calculatedusing WKB.A noveltyofWKB for PT-symmetricmodels is the extension of classical paths into the complex plane. This topic has been treated extensively in [6, 7]. We define the Fermi energy E as the energy of the N’th state F 1 N = dpdλθ[E H(p,λ)] (8) F 2π − Z where the path of integration must be a closed, classical path in the complex p λ plane. In order to construct the large-N limit, we perform the rescaling p √Np and λ √Nλ yielding − → → 1 H (p,λ)= p2 g(iλ)p (9) sc 2 − where the scaled Hamiltonian H is related to H by H =NH . We introduce a rescaled Fermi energy ǫ given by sc sc F E =Nǫ , which is implicitly defined by F F 1 1= dpdλθ[ǫ H (p,λ)]. (10) F sc 2π − Z After carrying out the integration over p, we have 1 1= dλ 2ǫ +2g(iλ)pθ[ǫ +g(iλ)p] (11) F F π Z q 3 wherethe contourofintegrationis takenalongapathbetweenthe turningpoints whicharetheanalyticcontinuation of the turning points at p=2. This equation determines ǫ as a function of g. F We define a scaled ground state energy E(0) by N−1 1 E(0) = E . (12) N N2 k k=0 X The WKB result for the sum of the energies less than E can be written as F N−1 N2 E = dpdλH (p,λ)θ[ǫ H (p,λ)] (13) k sc F sc 2π − k=0 Z X so that in the large-N limit E∞(0) is given by 1 E(0) = dpdλH (p,λ)θ[ǫ H (p,λ)] (14) ∞ sc F sc 2π − Z The integration over p is facilitated by using equation (10) to insert a factor of ǫ , giving F 1 E(0) =ǫ dpdλ[ǫ H (p,λ)]θ[ǫ H (p,λ)]. (15) ∞ F F sc F sc − 2π − − Z The integral over p then yields 1 E∞(0) =ǫF dλ[2ǫF +2g(iλ)p]3/2θ[ǫF +g(iλ)p]. (16) − 3π Z The turning points in the complex λ plane are 1/p ǫ λ− = F eiπ(3/2−1/p) (17) g (cid:18) (cid:19) 1/p ǫ λ = F e−iπ(1/2−1/p) (18) + g (cid:18) (cid:19) We integrate λ along a two-segment, straight-line path connecting the two turning points via the origin [6]. Solving equation (10) for ǫ , we find F 1 π p Γ(3/2+1/p) 2p p+2 ǫ = g2 , (19) F 2 sin(π/p)Γ(1+1/p) "(cid:16) (cid:17) (cid:18) (cid:19) # and solving (16) for the scaled ground state energy we have 1 p+2 p+2 π p Γ(3/2+1/p) 2p p+2 E(0) = ǫ = g2 . (20) ∞ F 3p+2 3p+2 2 sin(π/p)Γ(1+1/p) "(cid:16) (cid:17) (cid:18) (cid:19) # For p=2, this evaluates to E(0) = g/2 , in agreement with the explicit result for the harmonic oscillator. It is very interesting to compare the large-N result with results for finite N. The low-lying eigenvalues for the p Hamiltonian p2 (ix)p have been calculated by Bender and Boettcher in [6] for the cases p = 3 and p = 4; the case − p=2 is trivial. We can use their results by noting that the eigenvaluesof our Hamiltonian H are relatedto theirs by g2/(p+2) E = EBB. (21) j 2p/(p+2)N(p−2)/(p+2) j Results for p=3 and4 andsmall values of N arecomparedwith the large-N limit in Table 1. The energiesfor finite valuesofN rapidlyapproachtheN limit. Theapproachtothe limitappearsmonotonicinbothcases,butwith →∞ opposite sign. 4 N p=3 p=4 1 0.762852 0.930546 2 0.756058 0.935067 3 0.75486 0.935846 4 0.754443 0.936115 5 0.754251 0.936239 6 0.754147 0.936306 7 0.754084 0.936347 8 0.754043 0.936372 ∞ 0.753991 0.936458 TABLEI: The scaled ground state energy E(0) at g=1 for p=3 and p=4. N The expected value of TrM for large N is given by h i N−1 1 TrM = λ = dpdλλθ[E H(p,λ)]. (22) j F h i h i 2π − j=0 Z X Calculations of higher moments TrMn are carried out in the same manner. Upon rescaling, we find that TrM h i h i grows as N3/2, and the scaled expectation value is given by 1 1 µ= lim TrM = dpdλλθ[ǫ H (p,λ)] (23) N→∞N3/2 h i 2π F − sc Z which reduces to 1 µ= dλλ 2ǫ +2g(iλ)pθ[2ǫ +2g(iλ)p]. (24) F F π Z q Using the same two-segment straight line path as before, we find that 1 − 2 p+4 π p+2 π p+2 π Γ(3/2+1/p) p+2 Γ(1+2/p) µ= i sin cos . (25) − 2g p p Γ(1+1/p) Γ(3/2+2/p) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)(cid:20) (cid:21) For p=2, µ=0, as expected for a harmonic oscillator. For p>2, the expectation value µ is imaginary because λ j h i foreacheigenstateofthe reducedproblemis imaginary[6]. For p=3,µ= 0.52006i. For p=4,µ= 0.772539i. In − − the limit p , µ goes to -i. This behavior is easy to understand, because in this limit, the turning points become →∞ degenerate at i. − IV. SPECIAL CASE OF TrM4 For the case of a TrM4 interaction, we can explicitly exhibit the equivalence of the PT-symmetric matrix model with a conventionalHermitian quantum mechanicalsystem. As in the single-variable case, there is a parity-violating anomaly, in the form of an extra term in the Hermitian form of the Hamiltonian, proportional to ~. We show below that the anomaly term does not contribute at leading order in the large-N limit. The derivation of the equivalence closely follows the path integralderivationfor the single-variablecase [4, 5]. The Euclidean Lagrangianis 2 1 dM 1 g L= Tr + m2TrM2 TrM4 (26) 2 dt 2 − N (cid:18) (cid:19) and the path integral expression for the partition function is Z = [dM]exp dtL . (27) − Z (cid:26) Z (cid:27) 5 Motivated by the case of a single variable, we make the substitution M = 2i√1+iH (28) − where H is an Hermitian matrix. Because M and H are simultaneously diagonalizable, this transformation is tanta- mount to the relation λ = 2i 1+ih (29) j j − between the eigenvalues of M and the eigenvalues h of Hp. The change of variables induces a measure factor j [dH] [dM]= (30) Det[√1+iH] where the functional determinant depends only on the eigenvalues of H. The Lagrangianbecomes 1 (dH/dt)2 g L= Tr 2m2Tr(1+iH) 16 Tr(1+iH)2 (31) 2 1+iH − − N at the classical level. However, following [5], we note that in the matrix case the change of variables introduces an extra term in the potential of the form 2 1 d dh j ∆V = (32) 8 dh dλ j (cid:20) j (cid:18) j(cid:19)(cid:21) X which can be written as 1 1 1 1 ∆V = = Tr . (33) −32 1+ih −32 1+iH j j (cid:18) (cid:19) X The partition function is now [dH] 1 (dH/dt)2 16g 1 1 Z = exp dt Tr 2m2Tr(1+iH) Tr(1+iH)2 Tr (34) det √1+iH − 2 1+iH − − N − 32 1+iH Z (cid:26) Z (cid:20) (cid:18) (cid:19)(cid:21)(cid:27) We introduce(cid:2) a hermit(cid:3)ian matrix-valued field Π using the identity 2 1 1 iH˙ +1/4 = [dΠ]exp dtTr (1+iH) Π . (35) det √1+iH − 2 − 1+iH !  Z  Z  (cid:2) (cid:3)   Dropping and adding appropriate total derivatives and integrating by parts yields  g 1 1 Z = [dH][dΠ]exp dtTr 2m2(1+iH) 16 (1+iH)2+ (1+iH)Π2+Π˙(1+iH) Π (36) − − − N 2 − 4 Z (cid:26) Z (cid:20) (cid:21)(cid:27) The integration over H is Gaussian, and the shift H H +i gives → N 1 1 Z = [dΠ]exp dtTr Π˙2 2m2Π2+ Π4 Π . (37) − 64g − 4 − 4 Z (cid:26) Z (cid:20) (cid:18) (cid:19) (cid:21)(cid:27) After the rescaling Π 32g/NΠ we have finally → p 1 4g Z = [dΠ]exp dtTr Π˙2 2m2Π2 + Π4 2g/NΠ (38) − 2 − N − Z (cid:26) Z (cid:20) (cid:16) (cid:17) p (cid:21)(cid:27) We have now proven the equivalence of the PT-symmetric matrix model defined by 1 dM 2 1 g L= Tr + m2TrM2 TrM4 (39) 2 dt 2 − N (cid:18) (cid:19) 6 to the conventional quantum mechanics matrix model given by 2 1 dΠ 2g 4g L′ = Tr TrΠ m2TrΠ2+ TrΠ4. (40) 2 dt − N − N (cid:18) (cid:19) r This equivalenceimplies thatthe energyeigenvaluesofthe correspondingHamiltoniansarethe same. This couldalso be proven using the single-variable equivalence for the special case of singlet states, but the functional integral proof encompasses both singlet and non-singlet states at once. The equivalence of these two models also allows for an easy proof of the singlet nature of the ground state. Standard variational arguments show that the ground state of the Hermitian form is a singlet. The direct quantum mechanical equivalence of the single-variable case is then sufficient to prove that the ground state of the PT-symmetric form is also a singlet. As inthe single-variablecase,there is a linear termof order~ appearingin the LagrangianandHamiltonianof the Hermitian form of the model. This term represents a quantum mechanical anomaly special to the TrM4 model. To determine the fate of the anomaly in the large-N limit, we construct the scaled Hamiltonian of the Hermitian form in exactly the same way as for the PT-symmetric form. It is given by 1 1 H = p2 2gx m2x2+4gx4, (41) sc 2 − N − p indicating that the effect of the anomaly is absent in leading order of the large-N expansion. One easily checks for the m=0 case that the Hermitian formwithout the linear term reproduces the PT-symmetricprediction for E∞(0) at p=4. Acknowledgments The authors would like to thank Carl M. Bender for useful discussions, and Stefan Boettcher for providing data used in constructing Table 1. We gratefully acknowledge the support of the U.S. Department of Energy. [1] G. ’t Hooft, Nucl. Phys.B 72, 461 (1974). [2] V. A.Kazakov and A. A.Migdal, Nucl.Phys. B 311, 171 (1988). [3] E. Brezin, C. Itzykson, G. Parisi and J. B. Zuber,Commun. Math. Phys.59, 35 (1978). [4] C. M. Bender, D. C. Brody, J. H. Chen, H. F. Jones, K. A. Milton and M. C. Ogilvie, Phys. Rev. D 74, 025016 (2006) [arXiv:hep-th/0605066]. [5] H. F. Jones, J. Mateo and R. J. Rivers, Phys.Rev.D 74, 125022 (2006) [arXiv:hep-th/0610245]. [6] C. M. Bender and S. Boettcher, Phys. Rev.Lett. 80, 5243 (1998) [arXiv:physics/9712001]. [7] C. M. Bender, S. Boettcher and P. Meisinger, J. Math. Phys. 40, 2201 (1999) [arXiv:quant-ph/9809072].