QUANTUM STATISTICS And QUANTUM GRAVITY David Mermin
MERMIN-WAGNER THEOREM and DLR EQUATIONS inQUANTUM STATISTICS and QUANTUM GRAVITYMark Kelbert, Swansea/Sao PauloJoint work with Yurii Suhov, Cambridge/Sao PauloDavid Mermin0 s foot. In his book ‘About Time’ (2005):”Henceforth, by 1 foot we shall mean the distance light travels in a nanosecond.A foot, if you will, is a light-nanosecond (and a nanosecond, even more nicely,can be viewed as a light foot). If it offends you to redefine the foot then youmay define 0.299792458 meters to be 1 phoot, and think ”phoot” whenever youread ”foot”.”A nanosecond 10 9 second 1000 picosecond1 foot 1 light-nanosecond 0.3 mThe cycle time for radio frequency 1 GHz (1 109 hertz) 1 light-nanosecondThe Mermin-Peres magic square. The product of 9 numbers 1 should beeither 1 or 1. This means that it is not possible to construct a 3 3 table withentries 1 and 1 such that the product of the elements in each row equals 1and the product of elements in each column equals 1. But it is nearly possibleto do so with the richer algebraic structure based on Pauli matrices:σx I σx σzI σzσx σxσy σyσz σzI σx σz σxσz Iwith Pauli matrices σx , σy , σz01100i-i0100-11. Gibbs measures and Dobrushin-Lanford-Ruelle (DLR) equationThe definition of a Gibbs random field on a lattice requires some terminology:The lattice: A countable set L Zd .The single-spin space: A probability space (S, S, λ).The configuration space: (Ω, F), where Ω S L and F S L .The set L of all finite subsets of L.Given a configuration ω Ω and a subset Λ L, the restriction of ω toΛ is ωΛ (ω(t))t Λ . If Λ1 Λ2 and Λ1 Λ2 L, then the configurationωΛ1 ωΛ2 is the configuration whose restrictions to Λ1 and Λ2 are ωΛ1 and ωΛ2 ,1
respectively.For each subset Λ L, FΛ is the σ algebra generated by the family offunctions (σ(t))t Λ , where σ(t)(ω) ω(t). This σ algebra is just the algebraof cylinder sets on the lattice.The potential: A family Φ (ΦA )A L of functions ΦA : Ω R such thatfor each AP L, ΦA is FA measurable. For all Λ L and ω Ω, the seriesHΛΦ (ω) A L,A Λ6 ΦA (ω) exists.The Hamiltonian in Λ L with boundary conditions ω̄, for the potential Φ,is defined byHΛΦ (ω ω̄) HΛΦ (ωΛ ω̄Λc ),(1.1)cwhere Λ L \ Λ.The partition function in Λ L with boundary conditions ω̄ and inversetemperature β R (for the potential Φ and λ) is defined byZΦZΛ (ω̄) λΛ (dω) exp( βHΛΦ (ω ω̄)).(1.2)QHere λΛ (dω) is the product measure t Λ λ(dω(t)).A potential Φ is λ admissible if ZΛΦ (ω̄) for all Λ L, ω̄ Ω and β 0.A probability measure µ on (Ω, F) is a Gibbs measure for a λ admissiblepotential Φ if it satisfies the Dobrushin-Lanford-Ruelle (DLR) equationsZZΦ 1µ(dω̄)ZΛ (ω̄)λΛ (dω) exp( βHΛΦ (ω ω̄))1A (ωΛ ω̄Λc ) µ(A),(1.3)for all A F and Λ L. We say that a phase transition is observed if thesolution of (1.3) is not unique.2. Classical case: Heisenberg modelWerner Heisenberg studied a model of classical statistical mechanics on ad-dimensional lattice Zd with spins of the unit length si R3 , si 1, eachone placed on a lattice node.This was a prelude to Heisenberg quantum model with formal HamiltonianXĤ J x σjx σjx0 J y σjy σjy0 J z σjz σjz0(2.1) j j 0 1Formal Hamiltonian of classical model:XXH φij (si , sj ) i,j i j 1with a coupling Jij between neighboring spins.Invariant with respect of rotation group O(3).2Jij hsi , sj i(2.2)
Mermin-Wagner principle: if d 1, 2 an externalmagnetic field cannot destroy this symmetryOriginal proof is quite involved. A short proof is possible based on Gibbs Bogolyubov inequality: if H H 0 λH and a free energy F (λ) β 1 ln Tr e βH2then ddλF2 0. λ.3. Schlosman’s rotators on Z2 :A phase transition with spontaneous breaking of discrete symmetry andpreservation of continuous symmetry.φij (x, x0 ) β cos(x x0 ) if i j 2,φij (x, x0 ) β cos2 (x x0 ) if i j 1.(3.1)0 in all other cases.Discrete symmetry is a rotation of all spins on even sub-lattice by π.When β the model exhibits a phase transition destroying the discretesymmetry but preserving the continuous symmetry.4. Quantum Hamiltonian on a graphFormal Hamiltonian on frustrated 2D lattice Γ:iX1h XH j J(d(j, j 0 ))V (xj , xj 0 )20j Γ(4.1)j6 jAssume V (gx, gx0 ) V (x, x0 ), g G compact Lie group, and V (x0 , x00 ) , x V (x0 , x00 ) , x0 x00 V (x0 , x00 ) V̄and a reasonable decay of J(d), say J(d) d 3 .3(4.2)
5. Quantum gravity: Lorentzian triangulationsRandom Lorentzian triangulations are parametrized by critical size-biasedGalton-Watson (GW) treeswith the offspring numbers {kt , t 1, 2, . . .} conditional upon non-extinction.Select one particle from kt 1 and generate its offspring family with MGF f 0 (x),i.e. p̃k kpk . All other particles have the same offspring law as in the classicalGW process.The sized-biased critical branching processes have been studied by RussellLions (Indiana University), Robin Pemantle (UPenn) and Yurval Peres(Microsoft Research).Lemma 1. Let the offspring distribution has a finite second moment. Thena.s.1kt Ct ln 2 t(5.1)4
6. Quantum Gibbs statesin a finite volume Λ Γ are linear positive normalizedfunctionals on the C Qalgebra BΛ of bounded operators in space HΛ L2 (Mi , ν):i ΛϕΛ (A) trHΛ RΛ Awhere (6.1) exp βHΛwith Ξβ,Λ trHΛ exp βHΛ .RΛ Ξβ,Λ(6.2)Restriction to a finite volume Λ0 Λ : 0Λ0ϕΛΛ (A0 ) trHΛ0 RΛ A0 , A B(Λ0 )where(6.3)0ΛRΛ trHΛ\Λ0 RΛ .0ΛIn a similar way define operators RΛ xΓ0 \Λ(6.4)with boundary conditions xΓ0 \Λ .0ΛTheorem 1. For all given β (0, ) and a finite Λ0 Γ, operators RΛforma compact sequence in the trace-norm topology in HΛ0 as Λ % Γ. Furthermore, given any family of (finite or infinite) sets Γ0 Γ0 (Λ) Γ and boundaryΛ0conditions (i.e. particle configurations) xΓ0 \Λ , operators RΛ xalso form aΓ0 \Λcompact sequence in the trace-norm topology.on 00Λis a positive definite opMoreover, any limiting point, RΛ , for RΛ xΓ0 \Λerator of trace one which possesses the following invariance property: g G,00UΛ0 (g) 1 RΛ UΛ0 (g) RΛ .(6.5)The proof is based on the followingLemma 2. Let ρn (x, y) be a sequence of kernels defining positive-definite operators Rn of trace class and with trace 1 in a Hilbert space L2 (M, ν) whereν(M ) . Suppose there exists the following limit, uniform in x, y M :lim ρn (x, y) ρ(x, y),n which defines a positive-definite trace-class operator R of trace 1. Thenlim kRn Rktr 0n where kAktr tr AA 1/2.5
7. Symmetries in the Hubbard modelBrian Flowers, Baron Flowers FRS (1924-2010), was a British physicist, he waseducated in Swansea at the Bishop Gore School. Rector of Imperial College(1973-1985), VC of University of London (1985-1990). While VC of the University of London, he became known for making extensive notes during committeemeetings. People thought that maybe he didnt trust the minutes. Later, whenhis textbook ”An introduction to numerical methods in C ” came out, it allbecame clear.Walter Marshall FRS (1932-1996) from Cardiff gained a PhD under RudolfPeierls. He succeeded Brian Flowers as Head of Theoretical Physics Division atAERE (Atomic Energy Research Establishment) Harwell.John Hubbard (1931-1980) from London was the Head of the Theoretical PhysicsGroup at AERE. He left the UK for the US in 1976, following Marshall’s promotion to director of the AERE and a subsequent major reform of its facilities in Harwell. He joined Brown University and the IBM Laboratories in SanJose, California, where his research focused on the study of critical phenomena: phase transitions near which universal behaviour, independent of materialspecific properties, is observed.When asked what the book ”The Many-Body Problem” was about, declaredthat it was a murder mystery.LWith vertex i Γ associate a bosonic Fock-Hilbert space H Hk , Hk kddLsym(M),M R/Zd-dimensionaltorus.TheactionofGislifted to2unitary operator UΛ (g):UΛ (g)φ(x Λ ) φ(g 1 x Λ ), (g, x) G M gx M.Formal Hamiltonian (with boundary conditions)hPPPP(x)(HΛ φ)(x Λ ) 12 j Λ x x (j) j j Λ x x (j) U (1) (x)PPP 12 j Λ x,x0 x (j) 1(x 6 x0 )U (2) (x, x0 ) 21 j,j 0 Λ 1(j 6 j 0 )J(d(j, j 0 )) iPP0 0x x (j),j 0 x (j 0 ) V (x, x ) φ(xΛ ) j,j 0 1(]x (j) 1, ]x (j ) κ) RP0ν(dy)[φ(x (j,x) (j ,y) (j)) φ(x (j))].x x (j)(7.1)Assume thatzeΘ 1(7.2) V̂ ) andwhere Θ κβ(Ū (1) κŪ (2) κJ(1)hXi supJ(l)J(d(j 0 , j))1(d(j 0 , j) l) .j 0 Γj ΓTheorem 2. Assume that all potentials are invariant under continuous groupG and satisfy some addtional conditions. Then for all β, z satisfying (7.2) all6
states corresponding the Hamiltonian (7.1) are G invariant : A BΛ0 andg Gϕ(A) ϕ(UΛ 1(g)AUΛ0 (g))08. Bose gas in R2The simplest example of breaking translational symmetry is the wettingtransition in 2D Ising model for β . Let n be a unit vector, and Dn bethe straight line line through the origin with normal n. Denote by Dn the lengthof the segment Dn [ 1, 1]2 , and define the following boundary conditionηn sign(x, n).The surface tension in the direction n is defined byηnZ1ln Λ l .l lDnZΛlτ (n, β) lim(8.1)We prove that the translational invariance is preserved if β β0 .The local Hamiltoniann(Hn,Λ φn )(xn1 )1X ( j φn )(xn1 ) 2 j 1XV (x(j) x(j 0 ))(xn1 )(8.2)1 j j 0 nwhich acts on functions φn Lsym(Λnr ) where Λnr stands for the set of n points2configurations in Λ with a hard core of radius r . Define a partition functionΞβ,n (Λ) trLsymGβ,n,Λ , Gβ,n,Λ exp [ βHn,Λ ](Λnr )2(8.3)positive-definite trace-class operator in Lsym(Λnr ). Similar define Gβ,n,Λ x(Λ)c2cfor a boundary condition x(Λ) . Next, for a given fugacity z 0 defineXGΛ x(Λ)c z n Gβ,n,Λ x(Λ)c ,n 0Ξ(Λ x(Λ)c ) Xz n Ξβ,n (Λ x(Λ)c ) trH(Λ) GΛ x(Λ)c .n 0Now the density matrix (for simplicity select x(Λ)c )Rβ,Λ 1Gβ,ΛΞβ (Λ)(8.4)defines the Gibbs state, i.e., a linear positive normalized functional ϕz,β,Λ onthe C algebra of bounded operators A B(Λ): ϕz,β,Λ trH(Λ) ARβ.Λ , A B(Λ).(8.5)7
Λ0Λ0Λ0are integral operReduced density matrices (RDMs) RΛ, RΛ x(Λc ) and Rators, sayZ Λ0RΛφΛ (x0 ) FΛΛ0 (x0 , y0 )φΛ (y0 )dy0 .(8.6)Cr (Λ)Next, we define the limiting density matrices as Λ % R2 . Here and below weassume that z, β satisfy the condition ρ̄ z exp 4β V̄ Rd /r0d 1.(8.7)Here V̄ max[0, V (r), r0 r R], R stands for the radius of potential,r0 stands for the radius of the hard core. However, it is valid z (0, 1) if thetwo-body potential V 0. We also assume that V̄ (2) max[ V 00 (r) , r0 r R] .Λ0Theorem 3. The family FΛ x(Λc ) (x0 , y0 ) is compact in the space of continuous0functions C (Cr (Λ0 ) Cr (Λ0 )). Any limiting point F Λ0 (x0 , y0 ) determines apositive-definite operator RΛ0 of trace 1. Consider a pair of limit-points alonga sequence Λ(l) % R2Λ0Λ0Λ0F Λ1 lim FΛ(l) x(Λ(l) lim FΛ(l) x(Λ(l)c), Fc).l (8.8)l Then for any Λ1 Λ0 the compatibility property holds:Rβ,Λ1 trH(Λ0 \Λ1 ) Rβ,Λ0 .(8.9)We establish the translation invariance of bose-gas:Theorem 4. Under condiitons formulated belowϕ(A) ϕ(S(s)A), A B(Λ0 )(8.10)Rβ,S(s)Λ0 U Λ0 (s)Rβ,Λ0 U S(s)Λ0 ( s)(8.11)S(s)A U S(s)Λ0 ( s)AU Λ0 (s) B(S(s)Λ0 ).(8.12)orHereFeynman-Kac representation in a cube ΛΛ0FΛ x(Λc ) (x0 , y0 ) ZW (x0 ,y0 ) Px0 ,y0 (dΩ 0 )z K(Ω0 ) αΛ (Ω 0 )χΛ0 (Ω Λ0 )ΞΛ0 ,Ω0 [Λ \ Λ0 x(Λc )].Ξ[Λ x(Λc )]Pwhere K(Ω 0 ) ω Ω k(ω ), αΛ (Ω 0 ) and χΛ0 (Ω Λ0 ) indicates that the paths0are always inside Λ but outside Λ0 at moments β.8
9. Method: Fröhlich-Pfister argumentproves that a.a. quenched Gibbs measures generated by U are G invariant.The basis is the following property of specifications (conditional probabilities)of Gibbs measuresγ {γΛ (ω ω̄) Φ1e βHΛ (ω ω̄) }.ZΛΦ (ω̄)Lemma 3. (H.-O. Georgii: Gibbs Measures and Phase Transitions)Let for any cylindrical set A a, b 0 and Λ Zd withaγΛ (g 1 A .) bγΛ (gA .) γΛ (A .).(9.1)Then g preserve any measure µ Gibbs(γ).We illustrate the method for a classical model on Lorenzian triangulation.Let Tr be the union of the first r layers of the Lorentzian tree T . Let G be ad dimensional torus. Identify g with the vector θ and define a gauged actionon the layer j, r 1 j n:gn θn jX11Q(n r) t j 1 r t ln twhereQ(n r) n rXt 21.t ln tLet Et,t 1 kt kt 1 be the number of edges between levels j and j 1.Thenφ Xhv,v 0 in rX Et,t 1 θ 2(gn (v) gn (v )) 0ln ln(n r) t 2 t2 (ln t)202(9.2)as n . The series in (9.2) converges due to Lemma 1.A tuned-shift argument:Lemma 4. Let µ be a FK-DLR measure, and an event D is localized in Λ0 .Then measure µ is S(s) invariant if and only ifµ(S(s)D) µ(S( s)D) 2µ(D) 0.Proof Let τ S(s). Thenµ(τ k 1 D) µ(τ k 1 D) 2µ(τ k D).9(9.2)
The sequence {µ(τ k D)} is convex and bounded. Hence, it has to be constant,in particular µ(τ 1 D) µ(D).10. Feynman-Kac formula for density matrix kernelThe idea goes back to Jean Ginibre, Paris-Sud 11 University. exp βHΛ φ (xΛ ) ZYv(dy(j))Kβ,Λ (xΛ , yΛ )φ(yΛ ).(10.1)M Λ j ΛThe integral kernel Kβ,Λ (xΛ , yΛ ) admits a Feynman–Kac (FK) integral representationZ PβxΛ ,yΛ (dω Λ ) exp hΛ (ω Λ ) .(10.2)Kβ,Λ (xΛ , yΛ ) βWxΛ ,yΛDefine an energy for a path configuration ω Λ {ωj , j Λ} over Λ,X0hj,j (ω j , ω j 0 )hΛ (ω Λ ) (j,j 0 ) Λ Λ0where hj,j (ω j , ω j 0 ) represents an integral along trajectories ω j and ω j 0 :j,j 0h0Zβ(ω j , ω j 0 ) J(d(j, j )) dτ V ω j (τ ), ω j 0 (τ ) .00hj,j (ω j , ω j 0 ) yields the ‘energy of interaction’ between trajectories ω j and ω j 0 ,and hΛ (ω Λ ) equals the ‘full potential energy’ of the path configuration ω Λ .11. Breaking of continuous symmetry Theorem 5. Take Γ Z2 with distance d(j, j 0 ) max j1 j10 , j2 j20 .Take M S 1 G where S 1 R/Z is a unit circle, with a standard metricρ(x, x0 ) min x x0 , 1 x x0 and the group operation of addition mod1. Assume that the two-body potentials J(d(j, j 0 )) and V (x, x0 ), j, j 0 Z2 ,x, x0 S 1 , are of the formJ(d(j, j 0 )) 1, j j 0 1, 0, j j 0 6 1,V (x, x0 ) cos 2π(x x0 ), ρ(x, x0 ) θ, , ρ(x, x0 ) θ,10
where θ (0, 1/4). Then, β (0, ), a measure µe µeβ which is not S 1 invariant. Consequently, the corresponding state ϕe ϕeeis not S 1 -invariant.µTheorem 5: a sketch of the proof:Consider a sequence of ‘cooled’ boundary conditions x and a measure µ which is a limiting point of Gibbs measures in Λn . If µ is not rotation invariant,we are done. Otherwise, select an arc α (x 1/200, x 1/200). So, theweight of α 1/99 for n large enough. Next, given η (0, 1] consider a sequenceof boundary conditionsx̃j,η x j1 ηθ, j (j1 , j2 ).For η 1 unique compatible configuration inside the box. Hence, n η (0, 1) such that the weight of α 2/3. Any limiting point of this sequenceis not rotation invariant.11
12. Dobrushin-Lanford-Ruelle (DLR) equations in quantum statisticsThe standard approach to phase transitions in quantum statistics are KMS(Kubo-Martin-Schwinger) states. In Heisenberg picture ατ (A) : eiHτ Ae iHτhατ (A)Biβ Tr[Rατ (A)B] Tr[RBατ iβ ] hBατ iβ (A)iβ .(12.1)RHS and LHS of (12.1) are the boundary values of an analytic function of z. Ifthere is a phase transition or spontaneous symmetry breaking, the KMS stateis not unique. In the case of the density matrix R with positive elements wedevelop a simpler approach based on the DLR equations. DefinepΛ (xΛ , ω Λ ) 1exp hΛ (ω Λ )ΞΛ(12.2)Given Λ0 Λ consider the partially integrated probability densityZΛ00pΛ (ω ) : dνΛ\Λ0 (ω Λ\Λ0 )pΛ (ω 0 ω Λ\Λ0 )WΛ\Λ0where ω 0 ω Λ\Λ0 stands for concatenation of two loop configurations.00DLR equations: set Λ0 such that Λ0 Λ0 Λ, the density pΛΛ (ω )obeysZΛ00pΛ (ω ) dνΛ\Λ0 (ω Λ\Λ0 )WΛ\Λ0Λ\Λ0 pΛ(ω Λ\Λ0 )ΞΛ0 \Λ0 (ω 0 , ω Λ\Λ0 ).ΞΛ0 (ω Λ\Λ0 )Define the class of infinite-volume Gibbs states G(β) satisfying DLR equations.Theorem 6. For all β (0, ), the sequence of Gibbs states ϕΛ(n) contains asubsequence ϕΛ(nk ) such that finite Λ0 Γ and A0 BΛ0 , we have:lim ϕΛ(nk ) (A0 ) ϕ(A0 )k where state ϕ G(β). Consequently, class G(β) is non-empty.Theorem 7. For all β (0, ) and finite Λ0 Γ, any Gibbs state ϕ G(β) isG invariant.12
References[1] M. Kelbert, Y. Suhov. A quantum MerminWagner theorem for quantumrotators on two-dimensional graphs. Journ. Math. Phys., 54, No 3, 2013http://dx.doi.org/10.1063/1.4790885[2] M. Kelbert, Y. Suhov. A quantum Mermin–Wagner theorem for a generalized Hubbard model on a 2D graph. Advances Math. Phys., V.2013, ID637375, 2013arXiv; 1210.8344 [math-ph][3] M. Kelbert, Y. Suhov, A. Yambartsev. A Mermin-Wagner theorem forGibbs states on Lorentzian triangulations. Journ. Stat. Phys., 150 (2013),671–677[4] M. Kelbert, Y. Suhov, A. Yambartsev. A Mermin-Wagner theorem onLorentzian triangulations with quantum spins, Brazilian Journ. Prob., 2013arXiv:1211.5446v1 [math-ph][5] Y. Suhov, M. Kelbert. FK-DLR states of a quantum bose-gas.arXiv:1304.0782 [math-ph][6] Y. Suhov, M. Kelbert, I. Stuhl. Shift-invariance of FK-DLR states of a 2Dquantum bose-gasArXiv:1304.4177 [math-ph]13
MERMIN-WAGNER THEOREM and DLR EQUATIONS in QUANTUM STATISTICS and QUANTUM GRAVITY Mark Kelbert, Swansea/Sao Paulo Joint work with Yurii Suhov, Cambridge/Sao Paulo David Mermin0s foot. In his book ‘About Time’ (2005): "Henceforth, by 1 foot
An indirect way of observing quantum gravity e ects is via the gauge / gravity corre-spondence, which relates quantum eld theories and quantum gravity. 1.3 Approaches to quantum gravity List of the largest existing research programmes. Semiclassical gravity Energy-momentum-t
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