The fermion loop formulation naturally separates partition functions into
their canonical sectors. Here we discuss various strategies to make use of this
for supersymmetric SU(N) Yang-Mills quantum mechanics obtained from dimensional
reduction in various dimensions and present numerical results for the separate
canonical sectors with fixed fermion numbers. We comment on potential problems
due to the sign of the contributions from the fermions and due to flat
directions.Comment: 7 pages, 3 figure

Bergner, Georg

Liu, Hang

Wenger, Urs

English

arXiv

Urs Wenger

Georg Bergner

Hang Liu

Crossref

Canonical simulations of supersymmetric SU(N) Yang-Mills quantum mechanics

The fermion loop formulation naturally separates partition functions into their canonical sectors. Here we discuss various strategies to make use of this for supersymmetric SU(N) Yang-Mills quantum mechanics obtained from dimensional reduction in various dimensions and present numerical results for the separate canonical sectors with fixed fermion numbers. We comment on potential problems due to the sign of the contributions from the fermions and due to flat directions

Bern Open Repository and Information System (BORIS)

PoS(LATTICE 2015)241Canonical simulations of supersymmetric SU(N)Yang-Mills quantum mechanicsGeorg Bergner, Hang Liu and Urs Wenger∗Albert Einstein Center for Fundamental PhysicsInstitute for Theoretical PhysicsUniversity of BernSidlerstrasse 5CH–3012 BernSwitzerlandE-mail: bergner@itp.unibe.ch, liu@itp.unibe.ch, wenger@itp.unibe.chThe fermion loop formulation naturally separates partition functions into their canonical sectors.Here we discuss various strategies to make use of this for supersymmetric SU(N) Yang-Millsquantum mechanics obtained from dimensional reduction in various dimensions and present nu-merical results for the separate canonical sectors with fixed fermion numbers. We comment onpotential problems due to the sign of the contributions from the fermions and due to flat directions.The 33rd International Symposium on Lattice Field Theory14 -18 July 2015Kobe International Conference Center, Kobe, Japan∗Speaker.c© Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). http://pos.sissa.it/source: https://doi.org/10.7892/boris.86072 | downloaded: 13.3.2017PoS(LATTICE 2015)241Canonical simulations of SYM quantum mechanics Urs Wenger1. IntroductionSince long it is suspected that SU(N) gauge theories can be regarded as the low energy ef-fective theory of N D-branes in specific parameter regimes. In this way, dimensionally reducedlarge-N supersymmetric Yang-Mills (SYM) gauge theories might provide a nonperturbative for-mulation of the string/M-theory which describes the dynamics of the D-branes. In particular, theconnection between black p-branes and SYM gauge theories in d = (p+1) dimensions allows tostudy black hole thermodynamics through the corresponding strongly coupled gauge theory. Apartfrom providing tests of the gauge/gravity duality, SYM quantum mechanics is interesting per se.There are various intriguing expectations concerning the behaviour of the theory in specific sectorsof fixed fermion number [1, 2]. In particular, while certain canonical sectors only have a discreteenergy spectrum, some sectors allow for the discrete spectrum to be immersed in a continuous onereaching down to zero energy. These continuous spectra can presumably be associated with theso-called flat directions of the potential which are present in the classical theory, but may or maynot survive the quantisation of the theory.In these proceedings we report on our ongoing effort to perform nonperturbative calculationsin SYM quantum mechanics with gauge group SU(N). Here we concentrate on N = 4 SYMquantum mechanics which is obtained from dimensional reduction of the N = 1 SYM gaugetheory in d = 4 dimensions by compactifying the three spatial dimensions. To define the theorywe employ the lattice regularisation in Euclidean time proposed in [3, 4]. The bosonisation of thetheory on the lattice by means of the fermion loop formulation [5, 6, 7] decomposes the fermioncontributions into fermion sectors with fixed fermion number. Our recent progress in understandingthe algebraic structure of the fermion loop formulation allows the explicit construction of transfermatrices [8]. The transfer matrices in turn provide the starting point for the construction of localfermion update algorithms which allow to directly simulate fixed canonical sectors of the theory.In the following we summarise the derivation of the transfer matrices for generic fermion num-ber sectors, recapitulating our results from [8], and show first results for some simple observablessuch as the Polyakov loop, the moduli of the bosonic fields and the fermion action from simulationsin fixed canonical sectors of the theory.2. Lattice regularisation and canonical sectorsWe directly start withN = 4 SYM quantum mechanics obtained from dimensional reductionofN = 1 SYM in d = 4 dimensions down to d = 1 dimension. The action in Euclidean time canbe written asS =1g2∫ β0dt Tr{(DtXi)2− 12[Xi,X j]2+ψDtψ−ψσi [Xi,ψ]}(2.1)where Dt = ∂t − i[A(t), ·] denotes the covariant derivative using the time component of the SU(N)gauge field A(t), while the spatial components become the bosonic fields Xi(t) with i = 1,2,3. Theanticommuting 2-component complex fermion fields ψ(t), ψ(t) interact with the bosonic fields Xithrough a Yukawa-type interaction involving the three Pauli-matrices σi. We note that all the fieldsare in the adjoint representation of SU(N). The discretisation of the Lagrangian on a time lattice2PoS(LATTICE 2015)241Canonical simulations of SYM quantum mechanics Urs Wengerwith Lt points is straightforward and yields for the bosonic partSB =1g2Lt−1∑t=0Tr{DtXi(t)DtXi(t)− 12 [Xi(t),X j(t)]2}(2.2)where DtXi(t) =U(t)Xi(t +1)U†(t)−Xi(t) is the covariant forward derivative and U(t) is an ele-ment of the gauge group SU(N). For the fermionic part of the action we introduce a Wilson termwith Wilson parameter r = 1 in order to avoid fermion doubling. With this choice of r the masslessWilson Dirac operator in one dimension involves just the forward derivative and one obtainsSF =1g2Lt−1∑t=0Tr{ψ(t)Dtψ(t)−ψ(t)σi [Xi(t),ψ(t)]} (2.3)for the fermion action. More specifically, we haveSF =12g2Lt−1∑t=0[−ψaα(t)W abαβ (t)e+µψbβ (t+1)+ψaα(t)Φacαβ (t)ψcβ (t)]≡ ψDp,a[U,Xi;µ]ψ (2.4)where W (t) denote the real adjoint gauge link matricesW abαβ (t) = 2(σ0)αβ ⊗Tr{T aU(t)T bU(t)†} . (2.5)Note that we have introduced a chemical potential µ in the standard way [9]. The subscripts p,afor the Dirac matrix D denote periodic or antiperiodic boundary conditions, respectively, for thefermionic fields. The Yukawa interaction matrices Φ(t) are nmaxf ×nmaxf with nmaxf = 2(N2−1) andreadΦacαβ (t) = (σ0)αβ ⊗δ ac−2(σi)αβ ⊗Tr{T a[Xi(t),T c]} . (2.6)Two remarks are in order. Firstly, all supersymmetry breaking terms apart from the onesintroduced by the discretisation in eq. (2.2) and (2.4) are forbidden by the gauge symmetry. Hence,supersymmetry is expected to be automatically restored in the continuum limit without any finetuning [3, 4]. Secondly, the Wilson term breaks the time reversal symmetry, or equivalently theparticle-hole exchange symmetry. This reflects itself in the fact that the action in eq. (2.4) onlyallows forward propagating fermions. As a consequence, the exchange symmetry between therelated fermion sectors with n f and nmaxf −n f becomes exact only in the continuum limit.Let us now derive exact expressions for the fermionic contributions to the partition functionof the theory for a given fixed gauge and boson field background – the canonical determinants. Inquantum mechanics the lattice regulated determinant of the Dirac matrix can readily be calculated,and one obtainsdetDp,a[U,Xi;µ] = det[T ∓ e+µLt ] with T ≡ Lt−1∏t=0Φ(t)W (t) . (2.7)This essentially corresponds to the dimensionally reduced determinant for Wilson fermions derivedin [10, 11] except that here the dimensional reduction is from the full matrix to a nmaxf × nmaxf’flavour’ matrix. It is now easy to get the canonical determinants from the fugacity expansiondetDp,a[U,Xi;µ] =2(N2−1)∑n f=0(∓eµLt )n f detDn f [U,Xi] (2.8)3PoS(LATTICE 2015)241Canonical simulations of SYM quantum mechanics Urs Wengerwhich identifies the canonical determinants as the coefficients of the characteristic polynomial.These coefficients can be expressed in terms of the elementary symmetric functions Sk of order kof the eigenvalues {τi, i = 1, . . . ,nmaxf } of T ,Sk(T )≡ Sk({τi}) = ∑1≤i1<···<ik≤nmaxfk∏j=1τi j , (2.9)and one eventually obtainsdetDn f [U,Xi] = Snmaxf −n f (T ) . (2.10)So the crucial object for the calculation of the canonical determinants is the product T of thematrices Φ(t) and W (t) which in fact is a product of transfer matrices [8], as was suspected alreadyin [10].3. Transfer matrices for the canonical sectorsThe explicit construction of the fermion transfer matrices for each fermion sector is most easilydone via the fermion loop formulation [8] which in essence is an exact (hopping) expansion of thefermionic Boltzmann factor to all orders. In this formulation, the contributions of the fermionsto the partition function are obtained by summing over all possible closed oriented fermion loopswhich are forward propagating in time for any given gauge and boson field background. Theloop configuration space naturally separates into subspaces characterised by the number of forwardpropagating fermions n f . The transfer matrix elements are explicitly given in terms of the cofactorsC(Φ) and the complementary minors M(W ),(TΦn f)AB= CABAA(Φ) = (−1)p(A,B) detΦABAA , (3.1)(TWn f)AB= MAB(W ) = detW AB , (3.2)where A,B are sets of indices A,B⊆ {1, . . . ,nmaxf } of order n f and p(A,B) = ∑i∈A i+∑ j∈B j. ΦABAAdenotes the matrix obtained from Φ by deleting the rows with indices from B and the columnswith indices from A, while W AB denotes the matrix obtained from W by picking only the rows withindices from A and columns with indices from B. The size of the transfer matrices is given by thenumber of such sets for a given n f and corresponds to the number of forward propagating fermionstates Nstates = nmaxf !/(nmaxf −n f )! ·n f !. The fermion contribution to the partition function in sectorn f is then simply given bydetDn f [U,Xi] = Tr[Lt−1∏t=0TΦn f (t) ·TWn f (t)](3.3)and one can use the Cauchy-Binet formula and some further algebra [8] to show that[Lt−1∏t=0TΦn f (t) ·TWn f (t)]AB= (−1)p(A,B) detT AAAB =CAAAB(T ) , (3.4)4PoS(LATTICE 2015)241Canonical simulations of SYM quantum mechanics Urs Wenger0 0.5 1 1.5 2β0.30.40.50.60.70.80.91Pnf=16nf=15nf=  1nf=  00 0.5 1 1.5 2β0.30.40.50.60.70.80.91Pnf=16nf=15nf=14nf=13nf=12nf=11Figure 1: The modulus P of the Polyakov loop as a function of β for SU(3) on a Lt = 5 lattice in variouscanonical sectors.hence the canonical determinant is simply given by the sum over the principal minors of order n fof T denoted by En f (T ),detDn f [U,Xi] =∑BdetT ABAB ≡ En f (T ) . (3.5)Finally, it is easy to show that En f (T ) = Snmaxf −n f (T ) which proves the equivalence between therepresentation using the transfer matrices and the one in eq. (2.10).Some remarks are in order. Firstly, we note that the matrix T describes the dimensionallyreduced effective action for the Polyakov loop coupled to the bosonic fields Xi. Secondly, ourresult for the canonical determinants in principle allows for local fermion update algorithms, but inpractice only the sectors with n f = 0 and n f = nmaxf can be implemented straightforwardly, whilein other sectors algorithms along the lines in [12] can be employed. Thirdly, the construction of thetransfer matrices and the calculation of the canonical determinants in terms of those is applicableto QCD, since the algebraic structures of the theories are the same.4. Canonical simulationsHere we present our first results from simulations of the system employing the gauge groupSU(N) with N = 3 and nmaxf = 16 directly in the various canonical sectors1. First we note thatthe canonical determinants are real because the eigenvalues τi of T are real or come in complexconjugate pairs. Furthermore, for the sectors n f = 0 and n f = nmaxf (quenched) one can prove thatthe canonical determinants are positive. In these two sectors we update the bosonic degrees offreedom using a local Metropolis algorithm. In the other sectors we use Metropolis updates basedon eq. (3.5) and currently simulate only in the configuration space with positive determinants.In the following we show results on a lattice with temporal extent Lt = 5. We measure themoduli of the Polyakov loop and the scalar field defined byP =∣∣∣∣Tr∏tU(t)∣∣∣∣ , R2 ≡ |X |2 = Xai Xai . (4.1)1Note also the recent effort using the RHMC algorithm for SU(2) in [13].5PoS(LATTICE 2015)241Canonical simulations of SYM quantum mechanics Urs Wenger0 0.5 1 1.5 2β00.10.20.30.40.5R2nf=16nf=15nf=  1nf=  00 0.5 1 1.5 2β00.10.20.30.40.5R2nf=16nf=15nf=14nf=13nf=12nf=11Figure 2: The square of the modulus of the scalar field R2 as a function of β for SU(3) on a Lt = 5 lattice invarious canonical sectors.We note that in some sectors the simulations become unstable and R2 grows without bound. Webelieve that this is because the flat directions become unstable due to lattice artefacts, and we expectthe behaviour to disappear towards the continuum limit. In the left panel of Figure 1 we show P asa function of the temporal extent of the system parametrised by β for the sectors n f = 0 and 16, andfor n f = 1 and 15. Each pair should be degenerate in the continuum and we see that this is indeedthe case for β . 0.6, while for larger values of β lattice artefacts lift the degeneracy. The (physical)differences between the sectors is illustrated in the right panel of Figure 1 where we show P in thesectors n f = 16 down to 11.Next, in Figure 2 we show the square of the modulus of the scalar field R2 as a function of βfor the same combinations of sectors as in Figure 1. Again we find that the degeneracy between then f = 0 and 16, n f = 1 and 15, and so on, is lifted by lattice artefacts towards large values of β .Finally, in Figure 3 we show the fermionic action SF = 〈ln det Dn f 〉n f as a function of β invarious canonical sectors. We find that the degeneracy of this observable between the mirror sectorsbecomes better and better towards β → 0, suggesting that reweighting between the mirror sectorscould become feasible in that limit, or more generally towards the continuum limit.0 0.5 1 1.5 2β-8-404812S Fnf=16nf=15nf=14nf=13nf=  3nf=  2nf=  1nf=  0Figure 3: The fermionic action as a function of β for SU(3) on a Lt = 5 lattice in various canonical sectors.6PoS(LATTICE 2015)241Canonical simulations of SYM quantum mechanics Urs Wenger5. Summary and outlookIn this contribution we summarise the derivation of explicit transfer matrices forN = 4 SYMquantum mechanics with generic gauge group SU(N) discretised on a time lattice. The transfermatrices are defined separately in each canonical sector with fixed fermion number n f and formthe basis for canonical simulations of the theory. One caveat is that in those sectors where thecanonical determinants are not positive definite, the local Metropolis algorithm is currently notvery efficient and only samples configurations with positive determinants.Several paths are now open for further investigation. From an algorithmic viewpoint, it is in-teresting to examine the systematics of reweighting ensembles of configurations from one fermionsector to another, or from simulations at finite (imaginary) chemical potential. Concerning thephysics of the model, it is interesting to calculate correlation functions and energy spectra in thevarious canonical sectors. The investigation of the phase transition in the large-N limit of theN = 16 SYM quantum mechanics is most useful for a further understanding of the thermodynam-ics of certain black holes. The results in these proceedings are a first step towards these goals.References[1] J. Wosiek, Spectra of supersymmetric Yang-Mills quantum mechanics, Nucl.Phys. B644 (2002)85–112, [hep-th/0203116].[2] M. Campostrini and J. Wosiek, High precision study of the structure of D=4 supersymmetricYang-Mills quantum mechanics, Nucl.Phys. B703 (2004) 454–498, [hep-th/0407021].[3] S. Catterall and T. Wiseman, Towards lattice simulation of the gauge theory duals to black holes andhot strings, JHEP 0712 (2007) 104, [arXiv:0706.3518].[4] S. Catterall and T. Wiseman, Black hole thermodynamics from simulations of lattice Yang-Millstheory, Phys.Rev. D78 (2008) 041502, [arXiv:0803.4273].[5] D. Baumgartner and U. Wenger, Supersymmetric quantum mechanics on the lattice: I. Loopformulation, Nucl. Phys. B894 (2015) 223–253, [arXiv:1412.5393].[6] D. Baumgartner and U. Wenger, Supersymmetric quantum mechanics on the lattice: II. Exact results,Nucl. Phys. B897 (2015) 39–76, [arXiv:1503.0523].[7] D. Baumgartner and U. Wenger, Supersymmetric quantum mechanics on the lattice: III. Simulationsand algorithms, Nucl. Phys. B899 (2015) 375–394, [arXiv:1505.0739].[8] K. Steinhauer and U. Wenger, Loop formulation of supersymmetric Yang-Mills quantum mechanics,JHEP 12 (2014) 044, [arXiv:1410.0235].[9] P. Hasenfratz and F. Karsch, Chemical potential on the lattice, Phys.Lett. B125 (1983) 308.[10] A. Alexandru and U. Wenger, QCD at non-zero density and canonical partition functions with Wilsonfermions, Phys.Rev. D83 (2011) 034502, [arXiv:1009.2197].[11] K. Nagata and A. Nakamura, Wilson Fermion Determinant in Lattice QCD, Phys. Rev. D82 (2010)094027, [arXiv:1009.2149].[12] U. Wenger, Efficient simulation of relativistic fermions via vertex models, Phys.Rev. D80 (2009)071503, [arXiv:0812.3565].[13] Z. Ambrozin´ski and P. Korcyl, Numerical investigations of Supersymmetric Yang-Mills QuantumMechanics with 4 supercharges, PoS LATTICE2014 (2015) 253, [arXiv:1411.6904].7

http://boris.unibe.ch/86072/1/LATTICE%202015_241.pdf

Canonical simulations of supersymmetric SU(N) Yang-Mills quantum mechanics

Abstract

Similar works

Full text

Available Versions

Crossref

Bern Open Repository and Information System (BORIS)