JHEP01(2017)010 Springer January 3, 2017 October 22, 2016 : : December 27, 2016 spins interacting : 2 N 10.1007/JHEP01(2017)010 Received Published Accepted doi: , ), hyperoctahedral, symmetry. Z N, ( O Published for SISSA by [email protected] , partition function admits a saddle point in which the symmetry N . 3 ). We further demonstrate that this ‘matrix saddle’ correctly computes 1608.05090 N ( The Authors. O c 1/N Expansion, Gauge-gravity correspondence

We introduce a transverse field Ising model with order , [email protected] matrix quantum mechanics. At the critical point, the low energy excitations are observables at weak and strong coupling. The matrix saddle undergoes a contin- N N Stanford, CA 94305-4060, U.S.A. E-mail: [email protected] Department of Physics, Stanford University, Open Access Article funded by SCOAP ArXiv ePrint: eigenvalue distribution becomes disconnected.large The criticalwaves excitations propagating in are an described emergent 1 by + 1 dimensional spacetime. Keywords: via a nonlocal quartic interaction.We The show model that has the an large is enhanced to large uous quantum phase transition at intermediate couplings. At the transition the matrix Abstract: Sean A. Hartnoll, Liza Huijse and Edward A. Mazenc Matrix quantum mechanics from qubits JHEP01(2017)010 ]). 13 16 1 15 17 6 18 28 29 30 22 9 4 20 4 – 1 – 4 v v 26 28 10 8 10 18 26 23 12 1 24 B.2 Moments from the spin system: first order A.1 Dirac quantization A.2 Rotation to eigenvalue basis B.1 Moments from the spin system: zeroth order 6.3 Microscopic description6.4 of the excitations Non-singlet modes 5.3 Behavior and5.4 matching at large Existence of the topological quantum phase transition 6.1 Collective excitations6.2 of the eigenvalue Solution distribution to the perturbation equations 4.3 Collective field Hamiltonian 5.1 Behavior and5.2 matching at small Phase transition to a disconnected eigenvalue distribution 4.1 Matrix quantum4.2 mechanics Hamiltonian Eigenvalue Hamiltonian 1 Introduction The phrase ‘Itof from discrete Qubit’ qubits, expresses isRecent the the developments on intuition correct several that fronts framework havetheoretic for universal emphasized ideas the thinking quantum in power about characterizing computing, of physical physical quantum systems. information reality Firstly, (e.g. phases [ of matter with the same B Perturbative matching with the spin system 7 Discussion A Constrained quantization details 6 Low energy excitations 5 Ground state 3 Map to a constrained4 matrix quantum mechanics Collective field Hamiltonian Contents 1 Introduction 2 A nonlocal transverse-field Ising model JHEP01(2017)010 – ]. ]. 7 17 13 – 5

Spacetime matrix. Low energy N spins. Conventional local 2 N below. The starting point is 1

]. Secondly, the ground state of a ]. Instead of a quantum field theory, 4 , 14 3 ], and this order is quantified through the 2 ]. These three ideas are intimately related: ]. The low energy dynamics is then described 12 – 2 – 23 – – 9 ]. ] suggests that the emergence of spacetime itself 8 21 26 – Matrix quantum mechanics ] to a fully fledged quantum mechanical correspondence. 24 , 16 20 , matrix quantum mechanics. To achieve this, we show that a matrix quantum mechanics for a single real symmetric ma- 15 N N saddle our model undergoes a continuous quantum phase transition. At

N Ising spins . A certain nonlocal transverse field Ising model will be shown to admit a continuum limit We proceed to solve the matrix quantum mechanics using standard techniques [ In this paper we give a completely explicit realization of the emergence of two dimen- ]. At the quantum critical point, non-singlet modes are decoupled and we can focus on 20 the gapless singlet (eigenvalue) sectorby [ the propagation ofspacetime a is collective closely field related intwo to an dimensional that string emergent of theory 1+1 [ the dimensional target spacetime. space This dynamics of the ‘tachyon’ field in the critical point, ato continuum limit be is described possible bytrix. and large the This critical generalizes excitations asystems will and previously matrix be established models shown connection [ between nonlocal classical spin logic we follow containsa two particular steps, nonlocal as transverse-field illustrated Isingtransverse-field in model Ising figure with models order admitwhich they continuum are limits described close bywe quantum to wish field to quantum theories engineer [ critical aparticular points large large in provide the ‘architecture ofemergent spacetime’ spacetime [ requires anentanglement emergent structure (‘quantum locality order’) that of the will underlying microscopic necessarily quantum be state [ reflected insional the spacetime (‘It’) from a system of a large but finite number of interacting qubits. The entanglement in the quantumquantum state system of with the local system dynamicsof [ is short distance strongly entanglement, constrained reflected byThirdly, in a the an necessary Ryu-Takayanagi area accumulation formula law [ inrequires the a entanglement large entropy [ amount of microscopic entanglement in ‘stringy’ degrees of freedom that given by a matrixexcitations quantum of this mechanics matrix for quantumsemiclassical mechanics a are eigenvalue single distribution. described by real, collective symmetric excitations of large an emergent symmetries can have differing quantum order [ Figure 1 JHEP01(2017)010 ] , ). 15 N = 0 ( QCP 4 (1.2) (1.1) v T O ). ). The → 1.1 ]. 1.1 4 v 15 ground state ]. We instead N 15 . below). Numerical ] — at present we do not 0 in [ and spatial coordinate 2 z DA 31 S c 0. The classical limit of metastable saddle would physics at small and large > T > z CD N S 4 v N matrix worth of spins, with z BC ) of this low energy singlet sector S N ) symmetry is enhanced to 1.2 z AB , in which the large × Z S h . saddle point of the partition function. N N, , the low energy physics can be mapped ∼ =1 ( diverges logarithmically as 2 Our results combined with those in [ O N = 0 N L 1 X φ QCP 4 2 q v ∂ A,B,C,D ) model is known to exhibit a structural glassy 2 – 3 – = 4 c ∞ N v 4 v − = + below for the ‘matrix saddle’. The connectivity of ¨ φ ) in an emergent 1+1 dimensional spacetime, obeying 4 ) favors certain ‘antiferromagnetic crystalline’ ordered 2 v ) symmetry (the hyperoctahedral group). The classical x AB Z t, q S 1.1 ( ) invariant ‘matrix saddle’ undergoes a continuous topolog- φ N, =1 ( N N ( X O A,B O = 0 regime with the new transverse field term in ( h − T = , was previously studied at temperatures H → ∞ 4 v . The focus of this paper, however, will not be on symmetry breaking. Instead, 4 v ]. ]. The symmetries of the model are restored by either thermal fluctuations or 27 15 At the quantum critical point shown in figure We will argue that the At any nonzero temperature or transverse field, the symmetry breaking dynamics in The model we study involves a symmetric A similar phase transition occurs in Ising models on planar random graphs [ 1 will be defined below. The emergent local dynamics ( (of single-matrix quantum mechanics) from atruly spin interesting spacetime system physics is from a qubits.does small step In not towards particular, have obtaining the enough low degrees energy singlet of sector freedom tosee describe a direct black relation holes between or the random other graph manifestations model and ours. The waves propagate inleading to a the box existence whose ofq gapless length critical excitations. The speed lead to the phasethe diagram eigenvalue of distribution figure istransition shown to in the finite blue. temperature transition The present red in line theto classical the connects model dynamics our of [ quantum a local phase field be a good enoughsaddle starting [ point; after all, our entire universeical is quantum likely phase such transition a ateigenvalue metastable some distribution becomes disconnected. we will describe theThis physics saddle of is a singledThe specific out saddle large by point the willcoupling. fact be that seen We the have toparticular not capture at proven the that the correct it criticalperspective large of remains coupling realizing the where an explicit dominant a emergent saddle quantum spacetime, at a phase large all transition couplings, occurs. in From the the model is subtle.phase The below classical ( a criticalstudy temperature will presumably (shown be schematicallyand needed finite in to figure see if a quantum spin glass phase survives at work in themodel quantum is invariant under an quartic Ising spin interaction instates ( [ quantum fluctuations induced by the transverse field term in the Hamiltonian ( Hamiltonian We consider the ‘antiferromagnetic’this case with model, the coupling JHEP01(2017)010 , 4 13 ) as a (2.1) 1.1 characterizes the and a disconnected 6 4 v

4

) symmetry. In section we introduce the nonlocal h/v N x, y, z . ( 2 O =

Quantum disordered shows that the Ising model partition 3 Quantum critical point . . . N , i touches on connections with discrete gauge . In both of these limits, the matrix saddle 7 5 = 1 – 4 – . At the quantum critical point the eigenvalue distribution ‘matrix saddle’ of the transverse-field Ising model ( 4 v N , in section 4 , A, B

v

]. It does, however, exhibit emergent local dynamics in an i BA 28 S ‘matrix saddle’ with an emergent =

N and coupling i

Thermally disordered AB T collective field Hamiltonian describing the eigenvalue dynamics of the 0 S

N 0 T Glassy regime . Phase diagram of the large ]. It thereby provides the simplest realization of emergent spacetime dynamics. The remainder of the paper proceeds as follows. In section 30 , 2 A nonlocal transverse-field IsingThe model model will be built out of a symmetric matrix worth of spin-half operators. That is tum critical point, gapless excitationsspacetime propagating are locally found. in The an conclusion intheories, emergent section 1+1 the dimensional possibility offermionic realizing realizations fast of scrambling qubits, dynamics and without two quenched dimensional string disorder, theory. corresponding matrix quantum mechanics.tonian The is ground shown state to of haveeigenvalue a the distribution connected collective at eigenvalue field distribution large Hamil- atis small shown to captureuous the quantum ground phase state transition oflow separating the energy these spins. excitations two This about regimes. section the Section also matrix describes quantum the mechanics contin- ground state. At the quan- 29 transverse-field Ising model in morefunction detail. admits a Section large we derive the large function of temperature becomes disconnected. The critical dynamics is described by matrix quantumof mechanics. stringy entanglement, cf. [ emergent spacetime, with the associated accumulation of short distance entanglement [ Figure 2 JHEP01(2017)010 ) 1 in  N are ) is ( V (2.3) (2.2) Z O V ]. The N, 17 ( limit can O N of a given column i ) interactions. n z 1. The objective is to ). The potential . In disordered phases S ( 1 and the remainder to  N 2 2.1  V Z of resolutions of the identity and and perform a Suzuki-Trotter M . i . N ) i σ z S | to equal ). The role of the transverse field . That is, the spin is flipped along S 12 i indices of ( | ( AB n 2.2 will appear in the Hamiltonian, they V ,N σ . operators. That is, the terms in has eigenvalues i 21 = z A, B ··· 1 entries). However the group S βH ) above. However, the universal properties is therefore a nonlocal interaction. These i + tr , S i AB −  σ | S e 1.1 V x AB = 1 and z . S AB – 5 – i S i 12 = Tr z AB S symmetry. The symmetries are generated by acting A,B X as in ( S Z h N 2 V − Z = , for each x ij ). This can be seen as follows. The columns of matrices in H ). symmetry given by permuting rows and simultaneously the S Z i N ( 6= N N, N j ) enjoys a O S ( Q O = 1. This requires one of the 2.2 2 i ≡ n i i Q ]. Here we are interested in the full-blown quantum dynamics induced by P . Thus we write (Tr is a trace over the Hilbert space) 16 , β 15 ) are orthonormal with integer components. The components Z The dynamics is conveniently encoded in the quantum partition function at inverse The Hamiltonian ( We will study Hamiltonians of the following form N, ( We now carry out a seriesthe of partition standard function. steps in That moving is, towardsin we a introduce path terms a integral large of description number of the simultaneous eigenstates symmetry at this point (theentries easiest is way generically to not seemanifestly another this a matrix is subgroup with that of a rotation of a matrixtemperature with therefore satisfy be zero. This holdsIt for is all columns, clear which thatthe must symmetries such furthermore generated be a by orthogonal matrix theof to semi-direct then the each product describes model, other. of these ‘signed symmetries permutations’, are which unbroken. are We precisely can emphasize that there is no an entire row and correspondingThere column, with is the furthermore spin on an corresponding the diagonal columns. being flipped The twice. hyperoctahedral full group symmetry groupO generated by these symmetries is the studied in [ the first, transverse field,term term is in to the ‘quantum Hamiltonian disorder’ the ( ground state createdon by the the state tr with the Hamiltonian is agiven function by of matrix the multiplication matrixinteraction of of terms the all commute withwill each work other. with Where a we microscopic wish quartic of to the give critical concrete point results, wemultiplication we will structure describe is do present. not The depend statistical on physics of the some potential, models so in long this as class the was matrix Here and elsewhere, tr refers to a trace over the write down a Hamiltonian forbe these represented spins by a such bosonic thatcondition matrix the that quantum dynamics the mechanics in of matrix the be theLet type symmetric large first may us not solved emphasize in be [ essential, thatare but while is the technically both same convenient. operator e.g. and both act on the same spin We will normalize the operators so that each JHEP01(2017)010 ) . 2.1 (3.2)      , 2 to impose
are increas- (Φ) +1 AB m 1. In the final n AB m µ F (3.1) 2  σ 2 1  ε i − − (Φ) 2 ) F εh +1 (2.4) AB 1 m  m AB m 1 σ σ Φ ) can be exchanged for | (  −  x AB 2.4 2 β
M A,B X AB m ) onto that of a matrix Φ of itself is an auxiliary variable, iµ e 2 ≡ AB m K e 0 limit of the imaginary time ε h S 2.4 Φ AB m → ) +  1 + β/M | δ dµ ε m m σ ≡ , ε ( σ AB m AB m i ε o h as follows: Φ Φ V +1 d d Y A,B tr m ) AB m σ   | m Y Y σ 0 discontinuous paths contribute to the path ( εH m,A,B m,A,B =1 – 6 – o M − X V m e Z Z → tr | ε ε m − ε ). Such continuum limits are subtle in spin systems. = − in the partition function ( σ τ e    h ) = ( limit. ). The quantity σ =1 =1 ( AB m M M AB Y Y εh m exp m N σ F Φ 2 2 2 2 N N N N independent spins only before the symmetry constraint ( log( } → } } } 1 1 1 1 2 × N X AB m X X X 2 ∈{ ∈{ ∈{ ∈{ M ε/ M M M σ σ σ σ ) means that as = . In both of these steps, we have dropped numerical prefactors (2’s ··· ··· ··· ··· F e 2.4 K 2 2 2 2 N N N N } } } } 1 in ( 1 1 1 X X X X e K ∈{ ∈{ ∈{ ∈{ 1 1 1 1 σ σ σ σ in ’s) that will only contribute to an unimportant overall normalization of the partition = = ≈ ε π The continuum limit corresponds to taking the The sums over spin states Z especially powerful in the large discretization, so that Φ The integral. With discontinuous paths, one is not even guaranteed that the Riemann integral for any function and function. In the second step wethe furthermore constraints. introduced Lagrange multipliers While the two steps above are rather trivial, this reformulation will be integrals over symmetric matrices of bosons Φ as opposed to quantumingly field relevant theory), rather than highernot irrelevant. order manifestly interactions be Furthermore, such a oursteps as symmetry continuous carefully. breaking Φ phase transition. transition For will these reasons, we go through the In this section wecontinuous map bosonic the fields. Ising spinunderstood Some partition mapping of function of ( the spinsparticularities onto steps in bosons we our in go case local through because Ising in are models. quantum familiar However, mechanics from there (with are the no some well- spatial dimension, i.e. is imposed. The expansionline of the we exponent have in set thenot second part line of requires the microscopic model. 3 Map to a constrained matrix quantum mechanics In these sums there are decomposition. Thus JHEP01(2017)010 . h . (3.6) (3.5) (3.3) (3.4) #)  1 limit will coupling. . In such 4 − v N AB 2
˜ . This integral µδ is more serious . i µ ε . AB
− 1 ), can be expected Φ ··· iµ −  3.4 − ). In general, we must + 2 AA 2 AB = ) side, typical excitations 2.4 Φ µ  4 v 2 Φ AB 2 AB A X dτ µ d . X i ) invariant.  µ + N 2 (Φ) ) becomes ( 1 + ˜ h ! V 0
O 2.4 2 saddle points that are invariant under tr N K ), with the above caveats in mind, the (Φ) − N V + dτ 3.4 coupling, to three but not four orders in 2 2 + 4 Z Φ  v 2 ) becomes Φ  → dτ tr d 2.4 – 7 – ) Φ µ  dτ d m  2 → K (Φ 2  K V 1 →

tr − 2 tr 2 ) "
X +1 ε dτ AB m β Φ 2 Φ 0 ε Z , and the higher derivative terms are generically important. The  − . Such states will be seen to exist at the gapless quantum critical h − h m are to be determined in the spirit of effective field theory, by matching AB ( ) invariant (i.e. it does not have the form of a matrix multiplication). µ (Φ  . It is not yet a matrix quantum mechanics, however, as the final term N ,... . It is consistent to look for large e 2 ( 0 exp µ K E AB X µ symmetry from the full partition function, corresponding to permuting rows O µ i D N K,K Φ S D ], as we discuss in more detail below. First imagine integrating out the matrix Φ Z ) is not ) to obtain an effective theory for the matrix of Lagrange multipliers 15 A genuine matrix quantum mechanics is obtained as follows. Here we take inspiration Keeping the leading time derivative term in ( = 3.5 3.5 Z this symmetry. These are matricesmultipliers where are all equal off-diagonal and termsa all in saddle diagonal the the terms matrix constraint are of becomes also Lagrange equal: from [ in ( inherits an and columns of The partition function ismatrices Φ now and a quantumin mechanical ( path integralThis over is two expected symmetric giventhe that continuum the limit), steps and so the far original have model been was exact not (up to subtleties with taking continuum limit of the partition function ( point described below. Therefore,continuum it limit is theory, keeping only the closeto lowest to correctly order capture the derivative the term quantum low in criticalis energy ( point physics. also where We sufficient will the in seeperturbation the that theory the limiting for leading cases the order ground of kinetic state weak term energy, and to leading order at strong Higher order time derivatives comeAway from with the inverse quantum powers critical of point,have the on energies the microscopic disordered of energy (small order scale coefficients with the microscopics. This expansionwith can be energies truncated ∆ to the leading power for excitations for the remaining finite differenceallow term higher derivative in terms the in final the exponent replacement: of ( be powerful enough to favoras sufficiently well-behaved is paths, familiar, we the put first this term concern aside in and, the final exponent of ( The presence of jagged and even discontinuous paths at any fixed nonzero exists. In the expectation (and we will later see this explicitly) that the large JHEP01(2017)010 . ) 2 2 ) N ) as C 2.2 (4.1) (3.7) ~n · 3.7 ). This A ~n constraints ( ) is written . 4.1 N 1.1 A,C ). saddle point is !) P out of order = . 3.7 N µN N ] show that indeed, ]. These ideas were  z DA
− S 15 Φ 33 , z CD − (Φ) S 32 ], who noted the existence T V z BC saddles captured by ( 15 Φ S + 2 ν N z AB ]) with a quartic interaction. Φ S − 14 µ constraints are relaxed to
+ 2 N 2 and can be dropped to leading order N  − A,B,C,D N Φ P Φ dτ d : T A  Φ ) symmetry of the original Hamiltonian ( ~n 2 Z K µ – 8 – of Lagrange multipliers in order to impose the

N, − ( tr ). This leads us to an exercise in the quantization , but the components are unconstrained. Our model may by relaxing the individual normalization constraints O AB N (Φ) ν dτ 3.7 N = V β quantum rotor model (cf. [ 0 ). Thirdly, we change variables to a collective field and 2 − | Z N ) below. Secondly we diagonalize the matrix Φ and obtain A ˙ Φ − ~n However, in the more established cases, interactions are local | 4.12 T ( 4.9 2 ˙ Φ ). -dimensional ‘rotors’, 2 K N N exp ( ,  µ O N ) and ( D Φ 4.8 = tr constrained matrix quantum mechanics described by ( D L N Z = is a single field, not a matrix. We will refer to large matrix ]also considered an intermediate case where the order µ Z The ‘softening’ of spins at large 15 [ 2 the eigenvalue Hamiltonian ( (rather than one constraint).as The a quartic quartic interaction ‘matrix of Each multiplication’ Ising rotor spin is interactioncorrespondingly then of be normalized ( related as to a large We have introduced aconstraint new that matrix Φ beobtain the symmetric. quantum mechanical Therewill Hamiltonian be will and equations be ( constraints three following steps from ( in this section. Firstly we To obtain the groundsponding state to wavefunction, the we partitionof first function constrained ( need systems. the The Schr¨odingerequation Lagrangian corre- is good enough for our purposessolve of the realizing large an emergent spacetime. We therefore proceed to 4 Collective field Hamiltonian the effective spin can takein becomes nonlocal less models. constrained. However, Itoutside the is numerical of unclear Monte-Carlo the results that in low thisclassical [ intuition temperature spin holds system glassy at regime,saddle all the in temperatures. matrix our quantum saddle Thecoupling. case evidence correctly will As for describes be noted the the restricted in dominance to the of perturbation introduction, the theory a matrix at potentially weak metastable and large strong has a long history, goingapplied back to the to classical the version classicalof of ‘spherical the our ‘matrix nonlocal model’ saddle’ Ising [ above. modeland by hence [ the softening oftype renormalization spins group is flow. an As intuitive spins process are that locally occurs grouped during together, the a range spin-blocking of values Here ‘matrix saddles’. In these saddles,has the been enhanced to (that is to say, thevariables, spins and along hence the can diagonal be ofThe neglected the partition — matrix function correspond we then will to becomes see some concrete evidence for this later). The second of these terms is subleading at large JHEP01(2017)010 ). 3 4.8 (4.6) (4.7) (4.8) (4.9) (4.2) (4.3) (4.4) (4.5) . . ). The advantage (Φ) ) are automatically CD V Φ 4.22 ) ]. 4.4 AB + tr 34 A . Φ  2 2 — lead to the Hamiltonian 1
. N . ¯ MN ∂ , A ∂ · . , − Φ = 0 = 0
  MN T T 2 Φ BC Π) = 0 1 Φ BA Π δ ∂ (Φ) N T are not manifestly Hermitian. Making them Φ − − V X MN ∂ − AD ) and constraints ( H Φ δ Π ¯ ∂ tr(Φ + ). The Hamiltonian then becomes, as a CD · ) becomes the commutator (with an extra + Π + Φ and 4.26 limit can explicitly be treated in the saddle T Φ 4.6 2 AB 4.5 4 ∂ , i Π BD Φ – 9 – ) — see appendix N CD 2 δ N , , ∂ − K 1 N 2 2 + 4.1  AC N . We wish to find the ground state of this Hamilto- δ = 0 = 0  N 1 2 = 2 + CD

T AB 1 2 ≡ 2 ∂ T Φ Φ N i = tr T N = ). − − − AB AB Φ H − ∂ S Φ Φ 4.8 = ), as well as the second constraint in ( ¯ ∂ T tr Dirac · AB Φ } ), the operators Π 4.3 CD ¯ ∂ P  Π 4.8 CD tr ≡ Π K 1 , 2 T · − AB ) and ( S Φ = 4.6 { = 1 and introduced the symmetric derivative H ~ , as usual). This means that the momentum operator must be represented as i Upon quantization, the Dirac bracket ( In equations ( 3 Use of the Dirac bracket has ensured these constraints commute with the Hamiltonianexplicitly ( Hermitian (i.e. by addingconstant the shift hermitian in conjugate the and dividing Hamiltonian by ( two) simply leads to an overall Here we defined nian, which must be solved together with the operator identities The second constraint in ( satisfied once thedifferential momentum operator, is given by ( We have set by Dirac rather than Poisson brackets. In this case (see appendix factor of The first linefurthermore constrains to the have components nothe of momentum sphere). perpendicular Φ As to to usual the with lie sphere a on (and constrained hence Hamiltonian a system, to high the remain dynamics dimensional on is sphere determined and with Π the momentum conjugate to Φ, together with the constraints of this last formulationpoint is approximation. that I.e. the the collective large field is4.1 the ‘master field’ Matrix [ quantumStandard mechanics manipulations Hamiltonian starting from ( obtain the final collective field Hamiltonian ( JHEP01(2017)010 , , ) N i λ ( (4.17) (4.15) (4.16) (4.10) (4.11) (4.12) (4.13) (4.14) ]. The V . It will 24 O i X H + ), as it should. : ]). At large   O 2 20 4.12 ! i ∂ i λ i X , ) is introduced as

ij 2 will increase the energy. The 1 ∂ λ, t Ω . N ( ij ∂ . ρ ij − δ ij i )) i , ∂ Ω t λ ∂ ) ( ∂ i j i . . λ = 2 λ λ . ) O T 2 j ij i − − λ H O X i Λ N λ · 1 λ ( 4 − + ( = i δ λ − 2 i dO λ i

) = H π →  ) λ ) λ . 4.20 0 iρ λ ) ( ∂ ( ) . ρ λ λ ) 0 ( 2 π ( 0 , λ ), so that λ ˆ λ . λ π π λ ( ) = ) 0 0 − λ H i∂ λ N P λ − λ, t dλλ ( ∂ ( ( λ 0 2 iρ √ λ ( H π λ Z X ) ) can be expressed in terms of ) = 0 0 − ) iδ = iρ 0 λ λ λ ) ( λ ∂ ( = λ ( + 0 ρ π ( 4.12 0 ρ  ) 0 π ) terms identified in the sentences below ( i λλ 2 c λ λ dλ ) λ , t dλ 0 ∂ 0 – 11 – and N λ i 2 λ − Z ˆ ρ , λ ( ( Z ) = ρ λ 0 − λ 2 ( N i , π 0 ( scaling ) 2 ) ) = N √ dλ λ X λ . ( N ( λ N, = λ, t dλ δ π Z ∂ H . This means that all the remaining terms are of the same ( λ 1 + ρ 2 ρ ρ ˆ π ∂  Z 2  ) ) = N i ) N λ λ λ ˆ ( − V N/K ( ( ) is not manifestly Hermitian because there is a nontrivial mea- − = π π ) = λ = π λ λ ∂ i dλρ ( 4.20  V ∂ λ∂ is chosen to remove the linear-in-momenta terms from ( X Z λ K 1 i K compared the third term. The last term in the second line is similarly 2 ∂ 4 X 2

2 in the second term in the first line of ( + N ) / λ ( Hamiltonian then becomes to satisfy N dλρ ) using the chain rule. Thus X λ Z ( ]. The shift = The Hamiltonian ( There are now two constraints 18 iδ/δρ H We have dropped the subleadingThe in general large solution to this equation (using both of the constraints in ( sure factor in the wavefunction normalization.wavefunction, which This amounts factor to can shifting bein removed the [ by momenta rescaling the we need subleading. We willground drop state of these interest, twoorder terms and henceforth. must be kept.terms We it We will can must see also scale see as shortly that that for in the potential the to compete with the other These two constraints require the From this scaling weis can subleading immediately in see that the second term in the first line of ( The factor of 1 arises together with taking the principal value of the integral in ( Here the Hilbert transform The large The partial derivatives in− the Hamiltonian ( We define the canonical conjugate momenta to be JHEP01(2017)010 ) (5.1) (5.2) (5.3) (5.4) 4.26 (4.26) , while 2 ) we are N . 2.2 Hamiltonian 2 K N N limit. It is also 32 ! − ) N λ 2 ( N ) that does the job of V 2 λ c ). From the Hamilto- ( + +  X 2 . ) . 4.22 N 3 λ 1 ) ( 2 c K λ H ( N + ρ 32 1 4 . The final collective field Hamil-  − dλρ 2 N 2 λ = 0 without loss of generality (in fact, ) above — concerning the continuum . 2 ) + Z ) to be stable. The model (  c c 4 λ ) , 2 ). Classically the momentum vanishes in ( λ 3.4 3 λ 4 − π 5.1 λ 4 ( π ( to impose the constraints. We also used the 1 N v λ π c ρ = ∂ Kv λ 2 is kept fixed in the large ) c 2 − λ ) 4 = λ∂ ) = – 12 – ( ) v ) λ 4 λ π ( λ λ ˆ ( v ( λ and ( H ∂ V V 1 ρ  ) c ) to simplify these terms. Various terms that arise dλρ λ + ( K 1 2 2 ) Z ). ]) that 4.22 λ ). This semiclassical approach holds because, in the collec- 

dλρ ( 19 ) 5.1 ρ 2 λ Z ( = 0. Therefore we must minimize 2 4.22 K 1 π π KN ’s are subleading at large 24 ]. The coupling dλρ 2 π  15 ) Z − λ ( = ’s and H dλρ ρ 0, in order for the minimum of ( Z ) of course commutes with the constraints ( > , the ground state is found by classically minimizing the Hamiltonian ( explicitly Hermitian, and so we can set 4 ) we can now characterize the ground state as well as the low energy collective N ] = v is an arbitrary constant. We only need to find one 4.26 ρ H [ c 4.26 E Following the discussion around equation ( We will specialize at this point to the quartic potential can be shown to drop out of the final results in any case). The large convenient to introduce which will play the role of a dimensionless couplinglimit in of the the discrete continuum time theory. derivative — the collective field Hamiltonian (obtained via matrix We take studying then corresponds to amodel quantum considered disordering of in the [ quartic nonlocal classical Ising It is trivial now to minimize ( We introduced Lagrange multipliers identity (that we learnt from [ At large subject to the constraints ( tive field path integral, typicalthere configurations is have now action a (andthe single energy) ground field of state, order degree so of that freedom tonian ( nian ( eigenvalue excitations. 5 Ground state We have used thein constraints commuting ( making c is now where JHEP01(2017)010 ) .  (5.6) (5.7) (5.8) (5.5) 4.26 ··· becom- + 2 a 2 4 ˆ v ) 2 n + . n  = 0 distribution is a 2 + 3) N λ 4 n 4 v v 1)(10+8 + ˆ — are precisely reproduced + 2)( 2 − 2 ]. This requires the parameter 4 a n . n o v ( ( ), although we will not need the n ,   ··· Nλ 2 5.1 √ N + 128 λ ··· , 2 4 ) can be written o in ( + − v + 4 2 o 2 } 4 5.1 to give ˆ v Nλ 2 λ v 4 2ˆ  √ , c 1) v 1 √ 4 − c s + 1088ˆ [ − v { in terms of the couplings. The integrals can be 48 4  + 2 n ∈ – 13 – v ( 2 a − n λ n N λ 16ˆ 2 ) — i.e. up to order 16 1 − − √ and n 2 o 4 − 2 o 5.8 λ λ λ 1 = = 2 )  depend on  λ θ o a ( inside the integral. Note that the ˆ } λ N 4 , a 8 π ) v o dλρ + 2) n √ λ n { Z ) at all couplings. This will allow us to achieve two things. First, )Γ( ) = ) determine +1 2 Γ(2 n λ 1 n 5.3 ( ? N Γ( 4.22 ρ ) in small ˆ = = 5.5 i n 2 ) of potential. must of course be real and nonnegative everywhere. ρ 5.3 tr Φ h 0. The parameters An important class of observables are the single trace moments of the eigenvalue dis- The distribution that minimizes the energy ( +1 1 n > . This will be important for us to argue that indeed there is a continuous quantum 2 4 N These observables directly characterizethat the the full three eigenvalue terms distribution. in We equation have ( verified Wigner semicircle and the integrals that arise are elementary. tribution, which correspond microscopicallya to weak traces coupling of expansion powers one of obtains the matrix of spins. In It is easy todistribution obtain ( the perturbative solution to high orders. One can simply expand the 5.1 Behavior andThe matching constraints at ( small performed explicitly in an expansion at small ˆ single connected component, with range a explicit relation. We willing discuss negative, shortly in the whichdensity the phase eigenvalue transition distribution associated becomes with disconnected. The eigenvalue We have assumed that the parameters are such that the eigenvalue distribution is only a capture the critical behaviorfield of the ground matrix state saddle.v also Second, correctly we will reproducesphase see that transition that of the at collective intermediate theuniversal coupling eigenvalue spin in dynamics model at the a matrix at continuouschoice saddle. quantum ( small critical Finally, point and we is independent note large of that the the a quantum critical point. Ourthe presentation in ground the state remainder and willwith be the as the low follows. potential energy We excitations ( willwe find of will the collective be fieldpoint. able Hamiltonian These ( to are isolate universal singular the properties singular of the behavior critical and point and critical so excitations should correctly at the critical quantum mechanics) is only guaranteed to correctly describe the low energy excitations at JHEP01(2017)010 . ) B 3.4 (5.9) (5.11) (5.10) ], which 15 , the moments 4 4 v in the ground state . 3 4 v ). This matching is one , the ground state energy ··· 4 4 v + 5.8 4 4 v 14848ˆ − 3 4 . v . . These mismatches between bosons and h 1 3 16 4 1 v 16 − + 256ˆ = 2 4 – 14 – v ) does not match that of the spin system. The spin 8ˆ Λ = K ) is in fact sufficient to describe the matrix saddle − 3.4 5.10 4 . We have obtained the spin answer both numerically v ) to be 4 4 v 5.1 15360ˆ = Λ + 2ˆ − 2 0 below to argue that the matrix saddle undergoes a quantum phase E N K 5.4 ) from standard quantum mechanical perturbation theory in the microscopic by explicit microscopic computations in the spin system. The matching with 5.10 n The crucial difference between the quantum and classical models is the need in the Despite the above agreements, we have found that at order One can also obtain the ground state energy in a weak coupling expansion. The ground , one needs to deal with the discrete time derivative in the bosonic description. An al- 4 positive outcome from thethe matching leading to time low derivative orders termto in in these perturbation ( low theory. orders.important We Knowing in learn the section that behavior oftransition the at matrix intermediate couplings. saddle at weak coupling will be above, this limit is onlytation justified of close the mismatch, to then, av is quantum that critical beyond point. theternative The first possibility simplest few is interpre- orders that the inthe continuum perturbation limit matrix theory is saddle in not is the notdominant source saddle the of to dominant disagreement, low but saddle orders that in in general, perturbation yet theory. happens In to either case, coincide there with is the an important computed above will alsospins are disagree in at contrast to order we high have temperature verified perturbation agrees in through the to classical fourth model order of and [ quantum probably model to all to orders. take a continuum limit in time. As discussed around equation ( of the matrix quantum mechanics in ( system answer is instead and analytically. Directly related to the mismatch in energy at order not worried about operatorterm ordering aside, ambiguities that we appear haveenergy in reproduced ( Dirac the quantization. remaining This spin terms system. up This to matching order fixes Here we have includedhave an not kept undetermined track overall of constant the Λ. overall normalization of This the is partition present function and because also we we have problem that is solvedof by our the main matrix results, quantum wesomewhat mechanics have technical. put in the ( entire computation in an appendixstate only energy because is it evaluated is from ( microscopics fixes the coefficient of the matrix saddle kinetic term to be The zeroth and firstPerturbative computation order of the spin moments computations of the of spin system the translate moments into a are combinatorial given in appendix for all JHEP01(2017)010 ) 5.3 (5.14) (5.15) (5.12) (5.13) : ] and be- QCP 4 + ˆ v . . → 4 Nλ 3 ˆ v 4 √ v . ,  2 − N λ ··· Nλ − + . Illustrative eigenvalue √ 2 +  . λ 4 5 ˆ v N . The integrals appearing in ). Inserting the distribution −   5.5 2 − . QCP 4 λ 1 v QCP 4 − ˆ v = ˆ  2 1776 . 4 N 2 λ v 0 .  5 2 log ≈ one finds that the leading non-analyticity s . The constraints may be solved and the 2 4  r ˆ + 4 v π ˆ v 500 9  = − 2 − − = < λ – 15 – λ , the most important phenomenon that occurs is 1 4 QCP o v − − QCP 4 λ v QCP = 0 in the solution ( 4 2 QCP 4 ˆ v ) above, the critical coupling is found to be N λ v a  < λ = ˆ 2   , the eigenvalue distribution is disconnected. Thus the 4.22 π 4 2 4 ˆ v v N λ 500 27 − ) takes the form ], it is this singular contribution that truly captures the emer- = 2 + 0 ], with 0 24 5.1 λ − E 3 4 3  ˆ v d ) can be evaluated analytically in terms of elliptic functions. Solving Nλ θ d ]. The divergence above describes the approach to the critical point 2 √ . This is consistent with our expectation above that the effects of these N K 40 8 − π N 4.22 N , , √ + 39 degrees of freedom should be subleading at large 2 Nλ ) = N λ √ ( ? − = 0 into the constraints ( ρ a Past the critical value of ˆ The singular behavior of the eigenvalue distribution at the critical point leads to a In obtaining the ground state energy of the spin system to fourth order in perturbation out of The support oftween the [ distribution has twoenergy components, evaluated in between a [ distributions similar just way above to and what below the was critical done coupling in are section shown in figure (‘off-diagonal’) excitations are only parametrically gapped close to thesolution singular that region. minimizes ( connection point, where there iserties a divergent are density of determined states. universally,experienced independent This by is of the why the the eigenvalues. critical As forma prop- in of discretized the the worldsheet emergence of external [ twogence potential dimensional of ( string local theory spacetime from excitations. We will recall in the following section that non-singlet A third order continuousdistributions transition [ is characteristicfrom of the topological connected side. changes The in singular eigenvalue contribution comes from eigenvalues close to the dis- weak non-analyticity in the groundthe state constraints energy ( at ˆ the constraints in an expansioncauses about a ˆ divergence in the third derivative of the ground state energy as ˆ At the critical point the width of the distribution is Moving to larger values ofa the topological coupling phase ˆ transition inbution the becomes eigenvalue distribution. disconnected Namely, when thewith eigenvalue distri- theory, we have found thatleading virtual order states in involving theN diagonal spins never contribute at 5.2 Phase transition to a disconnected eigenvalue distribution JHEP01(2017)010 (5.16) (5.17) (5.18) (5.19) ]. This 15 term) here 4 v 2. The critical . 1.5 . = 0 4 ··· v + . 4 . 1.0 / . 1 ··· )  4 1 = 1 ) + v 4 n N 2 v 2(2ˆ λ √ ) = 0.5 λ − 0 ( λ E ( 2 = 1 + δ dλρ 1 N + N λ Z ) + 0.0 N 4 +1 ⇒ 1 n v √ – 16 – , λ N tends to + , λ = ··· ( ) and solving the constraints, one finds that at large -0.5 i δ ··· + n  → ∞ 2 + 4 4 / 5.15 2 4 N 1 v v ) tr Φ 4 h 1 = ˆ v ) = 15, while the disconnected distribution has ˆ -1.0 . 0 associated with quantum fluctuations has dropped out of this +1 λ 1 n ( 2(2ˆ E ρ 2 K = 0 N − K N 4 v = 1 177. -1.5 . 0 − λ ≈ 0.2 0.1 0.0 0.5 0.4 0.3 ]. Note that the energies have been shifted by one in figure 1 of [ QCP 4 N v ρ 15 . Illustrative eigenvalue distributions on each side of the transition. The connected the support of the eigenvalue distribution tends towards two narrow strips with The ground state energy as → ∞ 4 ˆ Indeed, the energy scale formula at leading order,exactly as reproduces we the should ground expect. statefound The energy in of leading [ the order matrix (linear phase in of the classical spin model tends towards the sum of delta functions The moments of the distribution are therefore all given by Using the disconnected solution ( v There are no quantum fluctuations of the spins in this limit, as the eigenvalue distribution distribution shown has ˆ coupling is ˆ 5.3 Behavior and matching at large Figure 3 JHEP01(2017)010 ). It (5.21) (5.22) (5.20) 5.17 above. of classical Ising ] is also relevant = 0. The ground i 5.2 35 h ordered | between 50 and 100 and found . ) agree with those of the ground N ! 1 5.18 = 2, take 1 − limit by setting k − . , 2 k , also in agreement with the matrices N 2 = 0) model exhibits glassy physics and 4 ! limit, although we do not know how to integer, the exact ground states can be M N M h 1 − α v k √ → ∞ N −  ) of the bosonic partition function correctly 1 1 = 4 1 1 1 v − − k k 3.7

2 2 ord – 17 – , with E M M = k limit taken in the matrix quantum mechanics did 2 2 ) and moments ( 1

N turns out to be a little subtle. N M = 2 14 or slightly lower. This is consistent with the energy = . α 5.19 ] — the discussion in section 8 of [ , the classical ( 1 k N 2 N 15 ≈ ]. M α 15 , this seems to indicate that classical states with energy close to limit of the matrix quantum mechanics described by ( = 1. These states are then in agreement with the matrix quantum k ). In these states the matrix of Ising spins (taken to be valued in α microscopic spin system. one then has . Determining = 2 α 5.19 k → ∞ N 4 → ∞ = 2 v 4 ). Given that the large v N 5.18 We have seen that the matrix saddle ( However, it was noted in [ For generic large but finite It is convenient to discuss the classical topological quantum phase transition of the sort described in section 1) is such that all the rows are mutually orthogonal. Symmetric orthogonal matrices are 5.4 Existence ofLet the us topological explicitly quantum make phaseN the transition argument that the matrix saddle undergoes acaptures continuous large the ground state to several orders in perturbation theory about the free limit answer ( not presuppose the crystalline states alwaysfind exist them. in the large These matrices have allfound eigenvalues in equal to the immediately follows that the spin moments agree with the matrix quantum mechanics For general — that for thefound special and values these have of mechanics result (  easily constructed iteratively as follows. Firstly for so Monte-Carlo computations are unable(classical) to find Monte-Carlo the simulations true for ground variousthat state. values states of We always have performed exist with of glassy states reported in [ of these ordered states must be of the form for some constant how both the ground state energystate ( of the states of this modelspins. will be Individually these some will specific all matrix necessarily break configurations the symmetries of the model. The energy provides another check on our computations. More importantly however, we now describe JHEP01(2017)010 ) 3.6 (6.6) (6.1) (6.2) (6.3) (6.4) (6.5) = 0 and 4 v . 2 K above. ) to simplify the N 32 . . 5.2 ? 5.2 − ρ ). Away from these   2 ) ) λ 5.3 λ , ( ( ) 0 V δρ λ 2 ( + 4 π π  . 0 2 (section , λ 0 + ) 2 ) ∂ λ 2 ) N 0 λλ ( ∞ )] . ρ λ, t λ ) ( − 1 4 = λ, λ )( 2 ) ( 0 δρ 3 4 π λ, t λ ) P ( v 0 ) ∂δπ 0 + λ − ) + ? δπ ( λ suggests that the higher derivative terms in 2 λ ( P ρ λ ( ) may be important. However, these terms are )] ( ρ [( ? 0 δ λ – 18 – ρ 5.1  ) satisfies the properties of a projection operator. 3.4 0 )( dλ ) ) symmetry implies that a collective field descrip- ) symmetry. Such terms would lead to additional λ ) = ) can be written in the more transparent form N ( Z 0 ) = ) = 0 + N ( ? λ, λ ( P ∂π ( O O [( λ, t λ, t P λ, λ 4.26 ) = ( (  dλρ ( λ ρ π P )( K Z 1 2 K 1  2 ) P ∂π ) has been used as well as the expression ( ( λ , so there must be a topological phase transition at intermediate = ( . These will be shown to be described by waves propagating in an ∞ 4.22 (2) ) means that dλρ = ) and in the classical limit H QCP 4 4 Z v v 4.22 5.1 = H denotes the projector evaluated on the background solution ? P The collective eigenvalue excitations are found by considering small perturbations The conclusion above pertains to the matrix saddle, independently of additional physics = 0 (section 4 Here The linearized perturbations are controlled by the quadratic Hamiltonian The constraint ( around the ground state solution where The collective field Hamiltonian ( Here the constraint ( Hilbert transform. We have introduced the projection of the momentum We can now characterizecritical the gapless coupling excitations atemergent and 1+1 close dimensional to spacetime. the continuous quantum 6.1 Collective excitations of the eigenvalue distribution such as quantum glassiness and orderingphase that may diagram. dominate regions of the zero temperature 6 Low energy excitations tion will exist atdisconnected all at couplings.couplings. The eigenvalue Because distribution the isleast eigenvalue true connected distribution for at at perturbativebe corrections a continuous to given and the in coupling free the is and universality unique class classical of (this limits), that is the described transition at in will section limits, the mismatch describedthe in expansion section of the discretestill derivative consistent ( with ancontributions emergent to perturbation theoryfor but the do Lagrange not multipliers. lead The to a breakdown of the ansatz ( v JHEP01(2017)010 (6.7) (6.8) (6.9) ) = 0, (6.12) (6.10) (6.13) (6.14) (6.15) (6.11) L = 1 ma- − ( c φ , with L . )) to . L L )) 0 ), due to the presence of − − ( q : ( φ , } 2 φ 6.10
), which can be written λ − 2 ( . 0 ) ) q φ, π φ L ∂ . 4.22 { q ( ∂ )) ) φ . run from ) = 0 ( q . λ ( q φ ( q 2 2 ( φ , π dλ + ( ? π q λ φ ρ 2 . ). The new ingredient is the remaining ( ∂ 2 π ) r ) q ), such that )) φ ) = 0 2 π Z ∂ λ λ q r = L ( ( ( 2 6.10 ( ¯ ? ? 2 q P π r = ) in the most natural way, so that 1 1 ρ ρ φ ( N λ ? 1 ( 4 ρ can be written as φdq − q dq 6.8 = q – 19 – q dq − L ∂ ) = L = = − ) L dq dλ 0 dq ∂ Z Z Z q − δρ L δπ L ( 2 π = λ − K − π φ ∂ 4 Z q L r ( 2 ≡ δ = = Q (2) ) = 0 H δρ dλ q, q ], and in particular one sees an emergent 1+1 dimensional free ( Z ¯ P ] there are two things we can do to simplify the quadratic Hamilto- 37 , 0 = 37 36 , 36 ). Firstly, the change of variables to 6.6 The equations above are quite similar to the well-known results for the The change of variables above has the additional benefit of turning one of the con- Following [ This constraint commutes withthe the projection quadratic operator, Hamiltonian as ( which it the should. emergent collective Thisthe eigenvalue is dynamics equations a is of nonlocal motion local, constraint. and we characterize proceed To the see to linear explicitly the modes. solve extent to trix model, e.g.field [ dynamics in theconstraint, quadratic the Hamiltonian linearization ( of the second constraint in ( Choosing the constant of integration fromone ( therefore has the boundary condition Then we can write the constraint (that the total number of eigenvalues does not change) as straints into a boundary condition. Let the variable where the projector in the new variables The quadratic Hamiltonian then takes a more conventional form nian ( Secondly, the canonical transformation to new variables JHEP01(2017)010 ) 6.19 . (6.24) (6.16) (6.17) (6.18) (6.20) (6.21) (6.22) (6.23) (6.19) ) = 0, ω L ( ). Thus φ 5.9 , rescale the , L ,  ), that contains 0 via ( 1 . dq ) will then be the h   6.19 )) from the expression. 0 2 1 q ω 6.21 ( L 2  λ ( O 0 , n q ∂ + ) L 0 )) q 2 . In this limit, the solution that q n c π , ( ω − c ; to see the factor of 2
= ) = 0 is then ) q ) ω λ 2 ( ( L n q λ ( ω . ( φ ωL/c q 6.19 c ∂  . The condition ( and will quantize the frequencies sin . , 1 γ ∂ ω o = L q h L , ω 0 K − π  − /φ 2 ) Z o dq ωL/h φ φ , O o L  γ π 2 q = )) γ – 20 – 0 ) is to be fixed by imposing that the solution obey ∂ = ω + c ), which in turn depends on q q c + ( q o o K K 2 π π ( ) γ φ 2 2 λ n L 6.20 ( ω 0 q = = + c ∂ ˙ φ q φ ) sin 0 ˙ ( via ( π t q ω n − c iω K ), which eliminates all other factors of − q sin ( e , o o , integration by parts, together with careful treatment of boundary o ω ) will fix the ratio φ φ 6.19 ω ), and performing similar integrations by parts, one finds that at  iγ sin — shows that the integral in ( 6.15 ) at all times. These equations can be solved. Looking for solutions iωt iωt ) = in ( L q 6.15 − − 2 − ¯ e e q / t, q Z 1 L | 6.15 ( = = is related to  φ ) of large = λ γ φ c q − 6.21 λ (which depends on time but not γ ∼ | ω ) = 0, one has ) The above solution simplifies when L λ − ( ( ? The previous two equationsthe together effects imply of that the the nonlocalfurther second constraint, satisfies term drops the out in remaining at ( boundary large condition This is the usual cancellationin in the a constraint rapidly ( oscillatinglarge integral. Using this result and ( the limit ( terms — in particular, usingρ the fact that near the endpoints of the eigenvalue distribution which we will nowsolution, show the is frequency also appears in thecoordinate the limit combination in whichRecall a local that dispersion iswe recovered. can In think the instatement terms that of a large the number combination of microsopic spins participate in the collective mode. In The overall normalization isand unimportant. the The constraint remaining ( boundary condition, where the constraint ( with a definite frequencyφ of oscillation, and imposing part of the boundary conditions, The Hamiltonian equations of motion can be cast in the following form where 6.2 Solution to the perturbation equations JHEP01(2017)010 ) 5.12 (6.28) (6.29) (6.25) (6.26) (6.27) in the . This ) above. h n . 1.2 4  ˆ v ω − . The result is ) this is seen to  5.5 QCP 4 QCP 4 ) is solved because ˆ v v ) can also be satisfied , h/L = ˆ → 6.15 + v 4 Z 6.15 v ), ∈ as ˆ , 5.9 diverges. From the definition ) is compatible with the exci- L , including order one values, as . L n ··· ) 6.21 , n v + ) in ∆ 1 correspond to odd values of c π − , 1 h/ 0 6.27 n  1) 4 ω L ˆ v n h . = 0 2 ) as well as ( − ) and constraint ( log( at the quantum critical point. This is done 2 − φ 2 n π 2 q L π ) are dropped. They are part of the universal 5 4 ) is odd. These low-lying excited states allow ∂ 6.20 (2 6.14 ) = 0. This, finally, is our sought-after emergent 2 2 24 – 21 – QCP c 4 λ ). The L = 3.4 = ˆ → ∞ ( v ( at some point. In the solution ( in ( q ≤ n −  0 φ n ∂ L c λ ω ¨ ω φ 6.19 log = ) = min. h small this means that the integral will be dominated by 5 6 L E will diverge logarithmically if the background eigenvalue a − ∆ = ( L ) and ( whereas φ L q , ω = 0. Using the critical values of the parameters given in ( 6.18 ) and using the growth ( ) are only a subset of the very low energy modes. While beyond λ = 0 = 0. For o 6.28 6.28 a γ no longer needs to be large. The constraint ( n ) vanishes linearly in near the critical point, the condition ( ] is even in λ ( , using the definition of ? = 1 in ( /c h ), it is clear that ) ρ n ), but L = 0 when → ∞ ), one obtains λ 6.12 + L integer. These are just the solutions to the free wave equation 6.24 q ( 5.13 n 0 n The excitations ( A certain class of critical states can be found for all As The energy carried by the collective modes can be written in units of the microscopic of ( ω n on the lowest excitation close to the quantum phasethe transition, objectives of ∆ this study, it shouldat be least possible numerically, to and solve the therebyby constraint obtain equations the the exactly, or matrix full quantum spectrum mechanics. of excitations that are captured sin[ us to bound theby closure putting of the gapthe as bound by setting in the solution givenω by ( critical excitations. Itmically is as clear the from critical theseas point modes follows. is that approached. the A energy more gap precise is bound closingwe can now logarith- be explain. put The on boundary conditions the ( gap as the quantum critical point is approached. tations having small energymeans relative that to these the excitations barein are microscopic which consistently the captured scale: higher by the derivative terms matrix in quantum mechanics ( distribution occur at the contribution close to and ( energy scale As the system isof tuned L to in the ( critical point, the length with the boundary conditions spacetime locality from the spin system. This is the wave equation we quoted in ( with JHEP01(2017)010 . Ac- (6.30) (6.35) (6.32) (6.33) (6.34) (6.31) spins). diagonal k . ), we see |←i n ), one finds to 2.2 = 4 ··· direction: 5.5 n + x i |→i predicted by the 1 n . 4 v  N,N . 3 it is therefore sufficient = 8 ,
··· 23 diagonal spins costs energy + (1) ··· n , h/L 3 , E m +  12 | 4  is an order one number, and in 3 4 . h v v . + ). Microscopically speaking, the ω L  . . − |←i |→i BA 2 ··· ) / S 6.25 = = m + 393216ˆ i are highly degenerate. At first order in +1) + 216 + , but because of the symmetry constraint, N 2 − 51 ( h |←i |←i n , has a satisfying interpretation. It is the  , 2 4 hn N x z 4 v (2 h ⊗ σ σ v hn 35 h , using the background solution ( off-diagonal and ,  4 = 4 spins. To get v – 22 – n 23 |→i = 4 = 2 34 n , (0) n + 3200ˆ = − 12 E Pauli matrix such that (0) (0) n n | 4 i |→i |←i 4 v 0 E x E h v + ψ = = σ | 32ˆ , we must also flip ∆ i ) then lead to the excitation energies ) at small ˆ − 41 ), flipping AB , |→i |→i 4 + 8 1 S 6.26 ‘rectangles’ of off-diagonal spins (i.e. a total of z x 2.2  34 6.12 2 k σ σ , n π , the degeneracy is partially lifted. Let us focus on states that off-diagonal spins in the free theory or alternatively 2 23 4 = , = v n ) are as follows. These states are given by a linear superposition of n L 12 scaling. Comparing with the microscopic Hamiltonian ( | E h = 1 this would be ∆ = 6.31 N k i is the eigenvector of the ) is roughly the energy to flip ) we were not careful to treat the dynamics of the diagonal spins correctly. One set 6.26 |→i The excited states with energy ∆ Evaluating the integral ( = 0), the ground state of the spin system has all spins pointing in the 3.6 excited singlet | 4 v of states that reproduce thecollective degeneracy-splitting field energy in of ( ∆ all possible ways of flipping For instance, for if we flip an off-diagonal spin perturbation theory in do not primarily involvein flipping ( diagonal spins, as in the mapping to the matrix saddle Excited states then correspond tocording flipping to a the certain Hamiltonian number ( of spins from The energy difference for a single flipped spin is 2 Here energy cost of flipping spins (or some( combination thereof). To see this, first note that, with no interactions The modes with frequencies ( The first term in this expansion, ∆ excitations will beintuition particular for superpositions what these of superpositions flipped are spins. from weak coupling. We can get some limited At any fixed distance fromparticular the has critical no point, thethat length ( to flip a large butby order one the number emergent of spins. 1+1 These dimensional are free the scalar collective excitations field described ( 6.3 Microscopic description of the excitations JHEP01(2017)010 (6.36) (6.37) ), with rectangles 6.35 k have been flipped other different states nN . . 4 ) = v A, B, C, D v N 8 ) above that this is the region ∆ = kN h/ i 6.27 51 , log( 35 ∼ , ) . Therefore, this is the same region where 23 /a , different other edges. The resulting restricted 12 → ∞ | – 23 – log(1 L N δH | ∼ 41 , ). This state is invariant under permuting rows and 34 ] , h 6.32 non-sing. 23 rectangles can mix with 4 23 – , E k — mixes two different rectangles of spins that share one edge ∆ 21 12 h ) whose spectrum can be found exactly in principle. At generic ) will become highly dressed at order one couplings. Nonethe- . for these states as follows. The quartic interaction Hamiltonian admit an eigenstate given by the excited singlet state ( n δH ). The modes become heavy if the corresponding eigenvalues are 4 n 4.16 v 4 6.35 v δH denotes a state in which the four spins 4.15 = 8 i = 8 (1) n ]. ]), recall from the computation around ( E (1) n 38 23 E ). Given a background eigenvalue distribution, this is a quadratic Hamiltonian rectangles. This is because one can choose any of the 4 edges of the ) — let’s call it A, B, C, D 4.15 | k An alternative approach to isolating the singlet dynamics would be to realize an emer- Components of the matrix Ω are associated with pairs of eigenvalues, as in for instance States such as ( 1.1 gent gauged matrix quantum mechanics,This in may which be the non-singlet possible statesfield. by are appropriately projected coupling The out. the decouplingemergent spin of gauge system symmetry the to in a non-singlet the dynamical modes low magnetic energy at theory. the critical point is equivalent to an While the decoupling only occursbution close (cf. to [ the disconnectionresponsible point for of the the growth eigenvalue ofthe distri- the gapless length singlet excitations propagate locally. closely spaced. Upondensity tuning of to states the closetightly to quantum spaced the critical eigenvalues in point point, this where region therenon-singlet and the a excitations is eigenvalue divergence as distribution in a the [ splits. divergence energy of in This the leads corresponding the to for the matrix Ωpoints in in parameter ( space there isexcitations no [ hierarchy between the energy of singlet and non-singet the Hamiltonian ( critical behavior. 6.4 Non-singlet modes Non-singlet modes ofnian ( the matrix quantum mechanics are described by the Hamilto- to mix, and amatrix given elements edge of can mixeigenvalue ∆ with less, they give some feel for the types of spin excitations involved in the emergent with a factor of 8. For example: Furthermore, a state with with relative to thecolumns ground and is state therefore a ( singlet state.describing singlet That is excitations. good, We as can theleads collective see field to that should ∆ indeed the be splittingin of ( the degenerate state indeed Here JHEP01(2017)010 N ]. ]. In this 40 50 , 39 gauge theories N ]. Therefore, quantum 49 , 41 ]. 48 matrix quantum mechanics by a N ], involves quenched random interac- 2 edges (if the spins along the diagonal / 46 – +1) 42 N ( ]. Matrix interactions may remove the need for – 24 – N is then precisely the gauge symmetry that acts 48 2 , 2 spins live on the edges. The maximally connected symmetries. From this point of view, the topological / 47 gauge theory on a highly connected graph in which N 2 + 1) Z N ( ] considered the classical version of the system we have stud- N 15 ) defines a 1.1 it has been pointed out to us by Xiaoliang Qi and Zhao Yang that the the paper [ bosonic matrix quantum mechanics can also emerge as the low energy descrip- spins, but only order 2 N vertices, while the symmetry described in section N N 2 Z An exciting possibility is that gauge theories on complete graphs may give a general may be a relation between glassiness and fastFermions: scrambling [ tion of a many-body fermioniccase, quantum the ‘qubit’ mechanics nature with of nonlocal thefermions. interactions underlying Hilbert The [ space fact is that due fermions to at the Grassmann different statistics sites of anticommute, however, means that even tractable models for fast scrambling.this The behavior, currently best the understood Sachdev-Ye-Kitaevtions. microscopic model model Another for [ model withfield quenched disorder Sherrington-Kirkpatrick that model may [ scramblequenched quickly is disorder the in transverse- achieving fast scrambling, as they did for glassiness. Indeed, there ied because it is aThe solvable matrix model multiplication of structure adynamics. of structural the glass, interactions that Another was is,sociated sufficient feature without to quenched with of disorder. obtain ‘fast matrix frustrated scrambling’spin interactions quantum systems is chaotic that lead dynamics that to [ they emergent matrix are quantum expected mechanics might to be be expected to as- give structure, such as blackof holes with spacetime Bekenstein-Hawking leading entropy andtheoretic, to hyperoctahedral stringy Ryu-Takayanagi structure substructure entanglement. on its ownspacetime is It dynamics. enough to may connect also directly with be emergent thatScrambling: the gauge microscopic framework for realizingsymmetries pin emergent the matrix indices quantumlead on mechanics, to the matrix as spin multiplication the interactionssystems in gauge in a that pairs, continuum allow limit. in anThis a This emergent way may will matrix that be quantum be can helpful mechanics necessary for naturally with finding to more qubit than obtain one a matrix. higher dimensional emergent spacetime with richer nature of the graph allows forare this order gauge theoretic interpretation, despite thetransition fact we that have there encountered isin a which cousin the of familiar eigenvalue distribution transitions in of large Wilson loops becomes disconnected [ spin Hamiltonian ( vertices are all connected to eachcan other be by neglected, i.e. eachThe vertex is not connectedat to the itself, then this is a complete graph). This paper has constructedfinite a dimensional spin ‘regularization’ Hamiltonian. of The large structureto we other have built physical has systems, suggestive connections as we now describe. Gauge theory: 7 Discussion JHEP01(2017)010 ] 50 (7.3) (7.1) (7.2) ].  ) 53 λ , ( 52 V i ] shows that it can be ) λ ( 54 ]) of transverse field spin − P 51 ]. It is natural to ask whether − ) . 26 λ , – 2 ( ) +  λ 24 ), becomes i P ( , . ) h ρ matrix of fermions. In these construc- λ 20 5.2 1 π 2 ( π CB 2 − N +  P ~σψ ) i ) − λ ( λ ) AC ( π ¯ λ ψ 3 − λ ( ), using ( – 25 – ∂ 2 P + = is a large P − h 4.26 ) ) = AB AC λ λ ~ S ( ( ψ dλλ 3 +  P P Z h  2 1 πK 6 1  KN 2 π dλ 8 ], or new classes of models altogether. ) that arises from an underlying spin system admits a string theoretic Z − the unconstrained single matrix quantum mechanics gives a nonpertur- 55 ] (that paper focuses on SU(2) invariant Heisenberg-like interactions) may = 4.9 50 H that is otherwise present in two dimensional string theory [ ].  are the Pauli matrices and P 51 ~σ A modern perspective on single matrix quantum mechanics [ We are especially gratefulproject to to Brian get Swingle for going.Basu, some Erez It helpful Berg, is John commentsAndreas also Cardy, that Sumit a allowed Karch, Das, this pleasure Eduardo Steve to Fradkin,Sachdev, Kivelson, Jeff acknowledge Steve Harvey, useful Shenker, Igor Shamit Julian discussions Klebanov, Kachru, Sonner, with David Natalie Pallab Tong Paquette, and Zhao Xiaoliang Yang. Qi, Subir considered can be developedspacetime. for These could more be complicated spinas regularizations BFSS models of theory matrix with [ quantum an mechanics theories emergent such dynamical Acknowledgments modes considered as the simplest versionthe of AdS/CT a correspondence. holographic However, description there ofA is spacetime natural no in question dynamical the gravity for spirit in future of this work simple is model. whether qubit constructions along the lines we have We see that the final term due to the constraint breaks the decoupling of the two chiral then the collective field Hamiltonian ( space [ String theory: bative description of two dimensional stringthe theory constraint [ ( interpretation. We can make one preliminary comment in this regard. If we define models, via the change of variables Here tions there is typically also an emergent gauge symmetry acting on the fermion Hilbert space (cf. the Jordan-Wignerpaper, transform). the Similar low to energy theare boson models valued matrix in we quantum have aconsidered obtained mechanical compact in in wavefunctions space. [ this obtained In inpossibly fact, [ describe the ‘fractionalized’ general spin class liquid of phases quantum (see mechanical models e.g. [ simple-looking Hamiltonians have a complicated representation as a matrix on the Hilbert JHEP01(2017)010 (A.8) (A.3) (A.4) (A.5) (A.6) (A.7) (A.1) (A.2) , = 0 . . .  AB 
˙ ν
∂L , Φ ∂ = 0 Φ , − BA ≡ − ) BA T Φ T N Φ Φ , AB − Φ ν − − , ν Π . ν Φ AB i = 0 T is a matrix’s worth of multipliers. AB AB 1 − ∂g ) + ∂q ’s are zero at all times: χ Φ
AB i 1 = Φ ν N , AB N χ tr(Φ ≡ ∂f AB 1 Π ν ∂p − u − − AB = 0 ≡ − Φ AB 2 ∂H = Φ ˙ i T ∂ν µ AB X T ∂ AB 1 ∂L ∂g − ∂p (Φ Φ + ∂µ i ∂H µ ≡ 0 1 = µ ∂f − χ , χ ∂q ), i.e. µ } – 26 – 0 1 T − Π u = i , χ X } 6.12 (Φ) + + ,H L T ) = 0 (Φ) V = 0 H AB 1 , V N − } ≡ ,H µ χ 0 1 = − − { Π AB Π + χ AB ˙ ˙ T f, g { T Φ Φ Φ ˙ = Φ ≡ { T T H Π K 0 1 = ˙ Φ AB χ 0 1 AB 1 K 1 = 2 Π 2 K χ χ tr(Φ in the main text.   4 AB AB X ˙ ∂L ≡ − Φ 0 = ˙ 0 = ˙ ∂ = = tr 0 2 = tr χ ≡ L H is a single Lagrange multiplier, whereas AB ’s are arbitrary at this point. 1 µ Π u For consistency we now have to require that the It follows that we find the secondary constraints: where we used the usual Poisson bracket: where the imply that the most general Hamiltonian takes the form However the primary constraints The momenta conjugate to the fields are clearly and hence the Hamiltonian na¨ıve is The starting point is the Lagrangian ( Recall that This appendix gives someis details performed regarding in the section quantization of a constrained systemA.1 that Dirac quantization A Constrained quantization details JHEP01(2017)010 = AB 3 (A.9) (A.17) (A.12) (A.13) (A.14) (A.15) (A.16) (A.10) (A.11) χ and 0 = ˙ } T , . ,H AB 2 0 3 χ , χ = 0 , . {
} AB 2 u BA = , AB 1 , g BA j u 0 3 Π , . Π χ BA χ AB X , = { − , Π − 0 1 ij + }  = 0 u T C 0 2 AB BA AB } T χ i Π = Φ 0 2 Π) = 0 (Φ) Π ,H Π . u } T ≡ V } T − AB f, χ X AB j 1 + K { ν Π AB 3 2 { , χ tr(Φ K i Π + ij = AB 1 µ, H X χ T = − { χ } { Π T = = – 27 – AB 1 = AB } − , , K 1 } , χ u µ ν ,H 2 ij T f, g  AB 2 = 0 = 0 M AB X ,H 0 = ˙ 0 = ˙ χ
0 2 T { ≡ { + χ Φ N Π) = 0 = tr ⇒ ⇒ 0 1 { = T χ − − H 0 1 = Dirac u Φ Φ 0 2 AB 2 = 0 = 0 } , respectively, and does not introduce any further constraints. ’s are zero at all times. This gives T χ χ 2 + µ 2tr(Φ Φ χ AB 2 AB f, g H u ν { . These free parameters indicate a gauge freedom and can be fixed 1 0 = ˙ 0 = ˙ tr ≡ − = u 0 3 T and χ 0 2 H u , the inverse of the Poisson bracket matrix of the constraints: 1 − fixes M } = T C ’s since they do not commute. Instead, imposing 0 = ˙ ,H 2 To ensure that observables remain on the constrained surface, the dynamics and com- At this stage the dynamics is still not fully determined because we still have the χ AB 3 χ where These are the results quoted in the main text. mutators must be evaluated using the Dirac bracket: together with the constraints In the end we find that the system can be described by the Hamiltonian: { unknown parameters by the convenient gauge conditions: and thus further secondary constraints These constraints cannot bethe added to the Hamiltonian without changing the dynamics of the Hamiltonian: and now require that the Since these constraints commute with the primary constraints, we can also add them to JHEP01(2017)010 ) , . It 5.8 ), to (B.3) (B.1) (B.2) ij δ CB (A.23) (A.18) (A.19) (A.20) (A.21) (A.22) i 4.8 ) λ O . ( = δ AB ij AC µ , ν Λ) . = = ) to be solved is T 0 4 CD O 1.1 AB 4 . Φ . + ( , where Λ direction: AB AB using textbook quantum and hence this limit will ∂ z DA , χ O , χ x Φ 4 Φ CB h S ). The reader may find it Λ v ∂ 2 Π) T O 1 BA ) means that there are only , T N z CD C 1.1 O Π AB ) S . 2.1 Φ − λ − ( ) z BC δ 2tr(Φ . AB S . − |←i X |→i AB 2 BC AC / − δ iB ) = = z = AB T O S = = Π +1) AD O δ Ai N 0 3 AB ( ) |←i ∂ |←i Φ AB 3 + T , with the result N x z + ( ∂ X ⊗ σ σ O M A,B,C,D BD i CB δ is equivalent to large AB , χ – 28 – ) 4 = ( |→i ) N v Φ 4 ∂λ O AC v ∂ N = i δ i + Pauli matrix such that , χ AB (Λ ( i |→i |←i λ − 0 Φ 1 2 x ∂λ x BA ψ = = AC ∂ σ AB Φ | ) = Φ S T X ABi ), the following manipulation is important. T ) in the main text. − = O |→i |→i ( A,B X i 4.5 z 4.12 x tr(Φ δ , although the identification ( Dirac h AB σ ∂ σ } − N ∂λ = − i CD λ = = = Φ AB Π 0 2 ) i , H X AB 2 O AB Λ T Φ { O , χ ( = 0, the ground state has all spins pointing in the δ , χ AB 4 µ ν v 2 spins. This model can be analyzed perturbatively in is the eigenvector of the ) = / = Π = Π AB 0 1 |→i +1) χ When For ease of reference, recall that the transverse-field Ising model ( The symmetric matrix Φ can be diagonalized via Φ = AB 1 (Φ N δ χ ( Here All indices run fromN 1 to mechanical perturbation theory. Small describe quantum disordered spins. instructive to see how a combinatorial structure arises from the spin system. B Perturbative matching withIn the this spin appendix we system match thewith matrix perturbative quantum mechanics computations expansions of in the the moments ( microscopic model ( which implies that Using this result we find the eigenvalue Hamiltonian ( follows that This is the result quoted in ( A.2 Rotation toIn eigenvalue going basis from the Hamiltonian acting on wavefunctions of the matrix Φ, equation ( It is straightforward to compute and invert The eight sets of constraints are JHEP01(2017)010 . n B (B.7) (B.8) (B.9) (B.4) (B.5) (B.6) ≡ (B.10) . When 0 . B ] N i spins twice 2 , − we only have n n +1 2 . n ) N z 1 1 N A S A n n z B z B 1 = S ]tr[( S 2 n n n B − i B n 2 ...B n z . A ) 2 z A z i , , and take any of the other B , where we defined X 2 1 1 ) S , − n B n ...S -direction. To leading order in AB i 1 ...S 2 N . 2 tr[( 0 x I 2 ( A 1 B 2 = ψ -direction. It follows that only the B , . . . , n 2 | − O 2 − x z i A  A z A 2 ) n n + 2) S + ...n I S 2 = 1 z 2 X B n ) 2 1 n n i A S =2 z 2 A − 1 i n I 1 S , something nice happens. There are two z B ( z )Γ( B 1 2Γ(2 . It turns out that at large S  ,...n z AB n S X − 1 1 i ] + with 1 i ≡ tr B 2 B =2 0 Γ( B 1 B i 1 | 1 − 1 ψ 0 − z A – 29 – | n ...S i z A A + +1 = 2 ψ ] S ) n S h A B 2 n 1 n 2 z 2 − N z ) ! B = S B = n 1 z ...B 2 S i 1 − 1 = 2 S = 1. We thus have to count how many of these terms i n B tr[( B 2 B n 2 NI n we have just shown that and 1 z 2 AB X ) N tr[( I B i S | δ ...A tr Φ N 0 A and i z = 2 A AB 1 h 1 ψ A i S A = 2 n z B h 1 = 2 A S A I = 1 1 δ = ( =  ), with all spins pointing in the A n z 1 AB ...n 2 z S ) BA A that flip any given spin an even number of times can contribute to , that is terms such as: X B.2 z =1 n S i  S N n (

z ...B 2 AB  flips a spin that is pointing along the n 2 ) S z B z tr X 1 ): S ...B S ( = 0 ground state we can evaluate the moments at zeroth order in perturbation 1 . In terms of the microscopic spin variables  AB B 4 4 5.8 given in ( n v v X i , this expectation value is obtained by solving the following combinatorial problem: 0 ...A 1 ψ N | A We consider the first operator, which acts on spin The above recursion relationsaddle is in solved ( by precisely the expression found from the matrix This result leads to a recursion relation. Let us define then at leading order in large That is, the cases wherespin the as second, the first fourth, operator. sixth,we . The do . . other , this, cases or i.e. give 2n-thtypes subleading enforce operator of contributions acts at terms large on that the arise,those same those in in which they which the are identified separated. spins This are leads adjacent to in the trace and operators and force it toto also consider the act cases on spin where we used there are in large the operator terms in tr the expectation value indominate the at ground large state. Of these terms, those that flip From the theory in with B.1 Moments from the spin system: zeroth order JHEP01(2017)010 to the (B.15) (B.12) (B.13) (B.14) (B.11) x 4 − m 2 − , k 2 ) 4 ··· − . k n ( i 2 in the unperturbed 1 0 I k ] + i A ψ 4 ). m | E k ) 2 spins twice, and the T 2 | z 1 I − k A S − B.1 δH 2 4 0 | I z n A E and k tr[( S h ) i ! k,m i ). That is, the operators in the 3 0 X 4 k k ψ 3 | − ( | ) depending on whether the four n ] 4 B.5 2 m N n A 2 2 T ) − + , 4 B.11 z k ] that flip 2 4 . A S . . We need to compute: n k 3 − − as in ( 4 2 have only 2 or 4 spins flipped and thus ] flips back these same 2 or 4 spins and 2 AA j z k k A ) v ) 2 n 2 2 i ··· z N S z tr[( T 2 I . − − ) ) | k S S n | n 2 z 0 m ( 2 2 k I ] + ψ S I I ( 4 h = 0. ) = k ≡ 3 ) m 0 2 k – 30 – z 2 A ) ( I ψ I /h T k S A k 6= ) for ( 4 3 X 2 k k T v A A I are the energies of X 2 tr[( j T = 2 z 2 A k n n B.10 I i 2 S 3 E 1 N 2 spins twice. In terms of the discussion of the previous ) I 1 ψ | − k + X 1) ] are the first and second terms in ( ( j,k,m 4 n n 2 and 4 2 − − A + 2 ) 0 n n T z N n δH 2 n 2 E ( 2 I S A n n 1 on the ground state is to flip 2 or 4 spins from the plus 3 N 2 + z A and tr[( 1 S | N

0 0 ground state with 4 δH H ψ = is non-zero only when tr[( . Thus h ] = . N ...A i i 1 X 0 1 n k 2 A | ψ H ) | ] z n ) are contiguous or not. Taking combinatorial factors into account we obtain: S 2 ) z direction. It follows that the states tr[( S B.12 x tr[( The action of | we only have to consider the terms in tr[( 0 s in ( s are the ones that flip the ψ The last step used the expression ( T subsection, we see that to leading order There are then four types of terms contributing to ( where and where we pair up theT operators within each N remaining 4 spins once.terms that In can particular, be we written find as follows: that the leading contributions come from Hamiltonian, minus h when it flips all other spins that it acts on an even number of times. At leading order in Here we have used theto standard the perturbation state, theory formula for the first order correction combinatorial problem. In particular, the ‘matrixdisordered saddle’ large indeed captures the fully quantum B.2 Moments from theWe can spin now system: include first the first order order correction in This agreement shows explicitly that the matrix quantum mechanics is solving the correct JHEP01(2017)010 ]. , , Phys. (B.16) (B.17) SPIRE , IN ][ (2006) 96 ··· + ). 2 ), namely i | k 0 B.17 5.9 ψ E | ]. hep-th/9303048 ). ··· − [ ]. 0 δH + 5.8 Phys. Rev. Lett. | ]. E 2 , k correction to the moment is SPIRE | h N 4 IN h 2 SPIRE 4 v 2 , J. Barrow et al. eds., Cambridge ][ v k IN SPIRE X − (1993) 666 IN ][ 4 4 . ][ N N v v 71 4 A quantum source of entropy for black . h n n ]. v 2 2 h moments in ( n I I 2 1 2 4 I ]. 16 1) 1) v 1) − − = + 2 + 2 – 31 – − n n Area laws for the entanglement entropy — A review + 2 n n ( ( hep-th/9401070 K SPIRE n n n [ n ( IN hep-th/0603001 3 3 ]. n [ ), the first order in [ 2 2 N N − Phys. Rev. Lett. arXiv:0808.3773 , [ = = = ), which permits any use, distribution and reproduction in B.11 Science & ultimate reality cond-mat/0510613 SPIRE i [ Holographic derivation of entanglement entropy from AdS/CFT Black hole entropy in canonical quantum gravity and superstring IN 1 Topological entanglement entropy Detecting topological order in a ground state wave function ψ , in (1994) 2700 ][ | (1986) 373 ) in ( n 2 ) , Cambridge University Press, Cambridge U.K. (2004). (2010) 277 (2006) 181602 z D 50 B.15 S CC-BY 4.0 D 34 82 96 tr( This article is distributed under the terms of the Creative Commons | Entropy and area (2006) 110405 0 It from qubit ψ Quantum field theory of many-body systems: from the origin of sound to an origin 96 h 2 hep-th/0510092 Phys. Rev. ), provided that we make the identification quoted in ( [ Phys. Rev. , , 5.8 Rev. Mod. Phys. Phys. Rev. Lett. theory Rev. Lett. holes of light and electrons 110404 University Press, Cambridge U.K. (2004). We have similarly reproduced the order Using the result ( S. Ryu and T. Takayanagi, L. Susskind and J. Uglum, L. Bombelli, R.K. Koul, J. Lee and R.D. Sorkin, M. Srednicki, J. Eisert, M. Cramer and M.B. Plenio, X.G. Wen, A. Kitaev and J. Preskill, M. Levin and X.-G. Wen, D. Deutsch, dependence (all the single trace moments, effectively the entire eigenvalue distribution) [8] [9] [5] [6] [7] [2] [3] [4] [1] References Open Access. Attribution License ( any medium, provided the original author(s) and source are credited. Thus the matrixproblem quantum even mechanics upon the ground (perturbative)n state inclusion of is the quartic solving spinhas the interactions. been The correct reproduced entire combinatorial from matching a single effective coupling in ( This expression agrees precisely withtion the in leading ( order matrix quantum mechanics correc- JHEP01(2017)010 ] , Nucl. , June Phys. Phys. , , , JHEP ] , ]. ]. matrix model , in the ]. SPIRE SPIRE arXiv:1205.3855 = 1 ]. IN [ IN c Class. Quant. Grav. ][ ]. ][ , SPIRE Commun. Math. Phys. hep-th/9403137 SPIRE IN , [ [ string theory IN matrices from a nonlocal SPIRE ]. ]. ]. Nucl. Phys. Proc. Suppl. ][ IN D , N 2 ][ ]. SPIRE (2012) 1230013 SPIRE SPIRE ]. Black hole thermodynamics and IN IN IN [ Matrix models as solvable glass Large- Black hole non-formation in the (1994) 3987 hep-th/0012228 SPIRE ][ ][ ]. limit in symplectic matrix models [ arXiv:1412.1092 IN ]. A 27 [ SPIRE Planar diagrams N ][ gravity and hep-th/9108019 IN D 50 [ Quantum corrections to holographic , ]. SPIRE D ]. 2 SPIRE arXiv:1307.2892 IN [ [ IN ]. (1993) 809 cond-mat/9407086 ][ – 32 – SPIRE [ ]. SPIRE , Cambridge University Press, Cambridge U.K. (2011). IN hep-th/9411028 IN (2015) 195009 A 8 hep-th/0503112 hep-th/0411174 ][ On the large- Phys. Rev. , SPIRE ][ [ [ , IN ]. 32 SPIRE Lectures on (2013) 074 (1980) 511 ][ Int. J. Mod. Phys. IN Vortices and the nonsinglet sector of the , arXiv:1504.07985 Entanglement entropy in two dimensional string theory hep-th/9304011 [ One-dimensional string theory with vortices as the upside down 11 [ On the architecture of spacetime geometry (1995) 1012 Two-dimensional unoriented strings and matrix models SPIRE The quantum collective field method and its application to the IN B 165 74 (2006) 039 (2005) 078 hep-th/9501090 [ [ JHEP ]. 09 01 , (1991) 459 hep-th/9511214 Theoretical Advanced Study Institute in Elementary Particle Physics arXiv:1212.5183 Int. J. Mod. Phys. [ Long strings in two dimensional string theory and non-singlets in the [ , SPIRE String theory in two-dimensions hep-th/0310195 (2015) 121602 JHEP JHEP IN What is string theory? [ (1983) 307 Nucl. Phys. , , Class. Quant. Grav. [ De Sitter musings B 354 Quantum phase transitions , , (1995) 6901 115 Geometric entropy of nonrelativistic fermions and two-dimensional strings Degrees of freedom in two-dimensional string theory ]. ]. Phys. Rev. Lett. (1996) 224 , B 215 D 51 (2004) 002 (1978) 35 (2014) 214002 SPIRE SPIRE IN IN matrix model Rev. 45BC 1–26, Boulder, U.S.A. (1992), [ matrix model proceedings of the 06 Nucl. Phys. matrix oscillator planar limit Phys. models spin system 59 Rev. Lett. information loss in two-dimensions [ 31 entanglement entropy S.R. Das, S.R. Das, J. Polchinski, D. Anninos, J.L. Karczmarek, J.M. Maldacena and A. Strominger, J.M. Maldacena, I.R. Klebanov, P.H. Ginsparg and G.W. Moore, D.J. Gross and I.R. Klebanov, D. Boulatov and V. Kazakov, A. Jevicki and B. Sakita, I. Andric, A. Jevicki and H. Levine, J. Gomis and A. Kapustin, D. Anninos, S.A. Hartnoll, L. Huijse and V.L. Martin, E. Br´ezin,C. Itzykson, G. Parisi and J.B. Zuber, S.A. Hartnoll and E. Mazenc, S. Sachdev, L.F. Cugliandolo, J. Kurchan, G. Parisi and F. Ritort, E. Bianchi and R.C. Myers, T. Faulkner, A. Lewkowycz and J. Maldacena, T.M. Fiola, J. Preskill, A. Strominger and S.P. Trivedi, [29] [30] [26] [27] [28] [23] [24] [25] [21] [22] [18] [19] [20] [16] [17] [13] [14] [15] [11] [12] [10] JHEP01(2017)010 , , 04 D 94 176 ] B 39 (2016) Phys. Lett. 07 strings , JHEP (1952) 821 (1981) 277 lattice gauge , = 1 86 N invariant (2016) 106 Phys. Rev. ) JHEP D B 98 Phys. Rev. Phys. Rev. , , N , 08 , ]. arXiv:1607.01801 SU( , arXiv:0808.2096 Recent developments in [ Phys. Rev. JHEP SPIRE , , IN Phys. Lett. , in ][ , (1994) 7647 (2008) 065 A 27 symmetric quantum dynamical ) 10 N ]. Sherrington-Kirkpatrick model in a transverse SU( A bound on chaos J. Phys. ]. ]. JHEP , ]. , SPIRE cond-mat/9212030 ]. IN – 33 – [ [ SPIRE SPIRE ]. Replica field theory for deterministic models. II. A SPIRE IN IN [ Bi-local holography in the SYK model IN SPIRE ][ ]. [ IN The spectrum in the Sachdev-Ye-Kitaev model Remarks on the Sachdev-Ye-Kitaev model SPIRE ][ ]. Planar limit for IN (1993) 3339 Planar limit of the singlet spectrum for [ (1980) 1103 SPIRE Fast scramblers ]. Possible third order phase transition in the large- IN , (2015). The Spherical Model of a Ferromagnet 70 String field theory and physical interpretation of (1980) 446 21 ][ SPIRE Gapless spin fluid ground state in a random, quantum Heisenberg (1990) 1639 IN SPIRE expansion in atomic and particle physics phase transition in a class of exactly soluble model lattice gauge theories ]. ][ ]. IN D 21 [ ∞ (1980) 403 A 5 /N Interferometric approach to probing fast scrambling 1 = arXiv:1604.07818 Ising model on a dynamical planar random lattice: exact solution , G. ’t Hooft ed., Springer, Germany (1980). [ Spherical model as the limit of infinite spin dimensionality SPIRE N arXiv:1601.06768 IN B 93 The KITP Lectures [ [ J. Math. Phys. ]. ]. ]. ]. Phys. Rev. Lett. , (1986) 140 Phys. Rev. , , arXiv:1603.06246 [ SPIRE SPIRE SPIRE SPIRE IN IN arXiv:1503.01409 IN cond-mat/9406074 IN (1989) 11828. [ [ 007 (2016) 106002 field: Absence of replica symmetry breaking due to quantum fluctuations (2016) 001 Phys. Lett. [ magnet systems theory quantum hamiltonians by the[ quantum collective field method Mod. Phys. Lett. gauge theories non-random spin glass with[ glassy behaviour A 119 [ (1968) 718 N.Y. Yao et al., Y. Sekino and L. Susskind, J. Maldacena and D. Stanford, P. Ray, B.K. Chakrabarti and A. Chakrabarti, A. Kitaev, J. Polchinski and V. Rosenhaus, A. Jevicki, K. Suzuki and J. Yoon, J. Maldacena, S.H. Shenker and D. Stanford, S. Sachdev and J. Ye, G. Marchesini and E. Onofri, D.J. Gross and E. Witten, S.R. Wadia, M. Mondello and E. Onofri, S.R. Das and A. Jevicki, E. Witten, E. Marinari, G. Parisi and F. Ritort, T.H. Berlin and M. Kac, H.E. Stanley, V.A. Kazakov, [48] [49] [46] [47] [43] [44] [45] [41] [42] [38] [39] [40] [36] [37] [34] [35] [32] [33] [31] JHEP01(2017)010 , 12 04 (1991) ]. JHEP , SPIRE JHEP B 362 , IN [ ]. (1991) 3 matrix reloaded Nucl. Phys. , SPIRE = 1 IN c B 359 ][ M theory as a matrix model: a Nucl. Phys. = 1, hep-th/9610043 ]. c [ – 34 – ]. ]. Grassmann matrix quantum mechanics -dimensional string theory for SPIRE = 1 IN Strings from tachyons: the SPIRE SPIRE (1 + 1) ][ S IN IN [ (1997) 5112 ][ D 55 (1991) 2664 Classical limit of ]. hep-th/0304224 arXiv:1512.03803 B 44 Phys. Rev. [ [ , Mean-field theory of spin-liquid states with finite energy gap and topological orders SPIRE IN [ (2003) 054 conjecture (2016) 138 Phys. Rev. 125 D.J. Gross and I.R. Klebanov, J. McGreevy and H.L. Verlinde, T. Banks, W. Fischler, S.H. Shenker and L. Susskind, X. Wen, J. Polchinski, D. Anninos, F. Denef and R. Monten, [53] [54] [55] [51] [52] [50]