Quantum Jamming: Critical Properties of a Quantum Mechanical Perceptron
Claudia Artiaco, Federico Balducci, Giorgio Parisi, Antonello Scardicchio
QQuantum Jamming: Critical Properties of a Quantum Mechanical Perceptron
Claudia Artiaco,
1, 2, 3
Federico Balducci,
1, 2, 3
Giorgio Parisi,
4, 5, 6 and Antonello Scardicchio
2, 3 SISSA, via Bonomea 265, 34136, Trieste, Italy The Abdus Salam International Center for Theoretical Physics, Strada Costiera 11, 34151, Trieste, Italy INFN Sezione di Trieste, Via Valerio 2, 34127 Trieste, Italy Dipartimento di Fisica, Sapienza Universit`a di Roma, P.le A. Moro 2, 00185 Roma, Italy INFN, Sezione di Roma I, P.le A. Moro 2, 00185 Roma, Italy Nanotec-CNR, UOS Rome, Sapienza Universit`a di Roma, P.le A. Moro 2, 00185, Roma, Italy
In this Letter, we analyze the quantum dynamics of the perceptron model: a particle is constrainedon a N -dimensional sphere, with N → ∞ , and subjected to a set of randomly placed hard-wallpotentials. This model has several applications, ranging from learning protocols to the effectivedescription of the dynamics of an ensemble of infinite-dimensional hard spheres in Euclidean space.We find that the jamming transition with quantum dynamics shows critical exponents different fromthe classical case. We also find that the quantum jamming transition, unlike the typical quantumcritical points, is not confined to the zero-temperature axis, and the classical results are recoveredonly at T = ∞ . Our findings have implications for the theory of glasses at ultra-low temperaturesand for the study of quantum machine-learning algorithms. Introduction.—
Constraint Satisfaction Problems [1](CSPs), born in computer science, have taken up a promi-nent role also in statistical mechanics. Methods from thetheory of disordered systems have been proposed to shedlight on the possible origin of their computational dif-ficulty [2–6] and inspire efficient algorithms [7] to solvethem. While problems defined in terms of discrete vari-ables map naturally to spin glasses, CSPs with continu-ous variables have shown a deep connection with struc-tural glasses [8–13].A notable example of CSP with continuous variablesis the sphere packing problem [14, 15]. Sphere systemshave gained plenty of attention among the glass physicscommunity, and their jamming transition has been in-corporated into the framework of glassy theory [11]. Inthis context, the perceptron, another CSP with con-tinuous variables, has been recognized as the simplestmean-field model presenting a jamming transition in thesame universality class of high-dimensional sphere sys-tems [14–17]. Exactly solvable models have always playedan important role in increasing our understanding of thephysics of glasses, both qualitatively and quantitatively.Furthermore, the perceptron has several applications inlearning protocols [18–20] and constitutes the buildingblock of deep neural networks.Recently, partly motivated by the technologicalprogress in quantum computation [21], many authorshave been looking at ways to use quantum dynamics tospeed up the solution of the classical problems. In thecase of discrete variable CSPs, a growing body of litera-ture has investigated the impact of quantum dynamics onthe spin-glass transition [22–27], and it has been foundthat disordered quantum systems display a plethora ofnew phenomena, such as Many-Body Localization (MBL)[28–38]. The study of CSPs with continuous variablesendowed with quantum dynamics, surprisingly, has notreceived the same kind of attention so far, but it promises to be equally far-reaching. For instance, in view of theconnection to structural glasses, it might provide cluesfor the anomalous (i.e. non-Debye) behavior of thermo-dynamic quantities in glasses at ultra-low temperatures.These phenomena, such as C V ( T ∼ ∼ T [39–41], areindeed naturally explained in terms of quantum mechan-ics [42, 43], but no firm results or solvable toy modelsexist (see for example [44] for a critical view).The purpose of this Letter is to address, for the firsttime, the jamming transition deep in the quantum regime[45] through the perceptron model. We show that quan-tum mechanical effects change the nature of the criti-cal phase radically. We find that, for any (cid:126) (cid:54) = 0, thecritical exponents are different from the classical onesand independent of the temperature. We also find that C V ( T ∼ ∼ e − ∆ /T at small T , while at higher tem-peratures the specific-heat has a power-law behavior.Remarkably, the latter result, valid in the deep quan-tum regime, resembles the semiclassical results of Refs.[46, 47], connecting the physics on the two sides of thejamming transition [58]. Model.—
The perceptron model can be formulated asa particle living on a N -dimensional sphere, subjected toa set of randomly placed obstacles. The vector X repre-sents the position of the particle on the sphere ( X = N ),and the obstacles are represented by the N -dimensionalvectors ξ µ = ( ξ µ , . . . , ξ µN ), where µ = 1 , . . . , M = αN and ξ µi are i.i.d. Gaussian random variables with zeromean and unit variance. For each obstacle, one definesthe constraint h µ ( X ) = 1 √ N ξ µ · X − σ > , (1)and the cost function is V = (cid:80) Mµ =1 v ( h µ ( X )). We are in-terested in the hard-wall potential case in which v ( h ) = 0if h > v ( h ) = ∞ if h <
0; hence, all the constraintsmust be satisfied (see Fig. 1). The limits
N, M → ∞ are a r X i v : . [ c ond - m a t . d i s - nn ] D ec FIG. 1: Finite dimensional representation of the perceptronmodel at σ = 0, N = 3, M = 4. Each constraint is repre-sented by a plane passing through the origin, and cuts thesphere in half. The particle can move in the allowed (lightblue) region. The jamming transition is reached when thenumber of obstacles is such that (with probability 1 in the N, M → ∞ limit) there is no light blue volume left anymore. taken, eventually, keeping α ≡ M/N finite.The classical system (recovered for (cid:126) = 0) is indepen-dent of the temperature and presents two phases, de-termined by whether there is or there is not any vol-ume left by the intersection of the M constraints. Morespecifically, one has to consider the limit of the set W ≡ (cid:84) Mµ =1 { X ∈ R N : X = N ∧ h µ ( X ) > } as N, M → ∞ : in the satisfiable (SAT) phase, a position X for the particle satisfying all the constraints can befound with probability one. In the unsatisfiable (UN-SAT) phase, instead, W becomes empty and the CSPproblem has no solution. The sharp SAT-UNSAT tran-sition is induced by increasing the constraint density α up to α c ( σ ).The features of the SAT-UNSAT transition depend on σ [15]. For σ >
0, the constraints { h µ > } force the par-ticle X to be closer than some distance to each obstacle;thus, the allowed region is convex. The free energy hasa single minimum and the replica-symmetric (RS) solu-tion is everywhere stable. On the contrary, when σ < σ = 0, at the border ofthe RS stable region, for which the jamming point cor-reponds to α c (0) = 2. In this way, we can reach thejamming point within the RS ansatz, but capturing thephysics of the glassy phase ( σ <
0) .The model is quantized by imposing the canonical com-mutation relations [ ˆ X i , ˆ P j ] = i (cid:126) δ ij . The Hamiltonianreads ˆ H = ˆ P m + M (cid:88) µ =1 v ( h µ ( ˆ X )) . (2) Methods.—
We wish to compute the quenched disorder average of the free energy F = − β − ln Z , Z = Tr( e − βH ).Following the lengthy but straightforward procedure in[46, 48], which introduces replicas a, b = 1 , ..., p (witheventually p → q = N − (cid:104) X a ( t ) · X b ( s ) (cid:105) (for a (cid:54) = b and any t, s ), the correlation function of a single replica G ( t − s ) = N − (cid:104) X a ( t ) · X a ( s ) (cid:105) − q , and a Lagrange mul-tiplier µ to enforce the spherical constraint. G ( t ) , q, and µ need to be found self-consistently. To this purpose, itis convenient to introduce a one-dimensional, β (cid:126) -periodicauxiliary process with the same autocorrelation function (cid:104)•(cid:105) r = 1 Z (cid:73) Dr e − (cid:82)(cid:82) β (cid:126) dtβ (cid:126) dsβ (cid:126) r ( t ) G − ( t − s ) r ( s ) • (3)( Z is a suitable normalization). Then, the quenched freeenergy in the RS approximation, per dimension and perreplica, reads [59] − βf = 12 (cid:88) n ∈ Z ln G n + q G − βm (cid:88) n ∈ Z ω n G n − βµ (cid:104) (cid:88) n ∈ Z G n − (1 − q ) (cid:105) + αγ q (cid:63) ln (cid:104) e − β (cid:82) β (cid:126) dtβ (cid:126) v ( r ( t )+ h ) (cid:105) r , (4)where we denoted the Fourier transform by • ( ω ) ≡ (cid:90) β (cid:126) dtβ (cid:126) e − iωt • ( t ) , • n ≡ • ( ω n ) (5) ω n ≡ πn/β (cid:126) being the Matsubara frequencies, and γ q (cid:63) • ( h ) ≡ (cid:82) ∞−∞ dh √ πq e − h / q • ( h ). It is also convenient todefine the self-energyΣ( ω ) = β − G − ( ω ) − mω − µ, (6)and fix Σ(0) = 0.As said before, the extremization of (4) with respectto G n , q, µ gives rise to a set of self-consistency equations(see [48]). To solve them we have implemented an iter-ative method, together with a Montecarlo sampling forthe calculation of (cid:104)•(cid:105) r , (for an analog calculation in theSK model see [49–52]). Results.—
As a first result, we obtain the value of theorder parameter q as a function of α, β ; it is plottedin Fig. 2 against the classical counterpart q cl ( α ), ob-tained at (cid:126) = 0 [15]. Unlike the quantum case, q cl ( α )is independent of the temperature and goes to 1, for α → α c = 2, with the critical exponent κ cl = 1 (validfor σ ≥
0, while for σ < κ cl = 1 . ... [15]): (1 − q cl ( α )) (cid:39) (2 − α ). The value of q for (cid:126) > β = β = β = β = α q FIG. 2: Edwards-Anderson order parameter as a functionof the constraint density α for various temperatures. Frombottom to top: infinite temperature classical dynamics (red)to finite temperature quantum dynamics ( β = 2 , , O ( α ), β = ∞ results are shown as dashed black lines (whilethe horizontal black line is a reference for the value q = 1).Notice how, as soon as α (cid:38) , the temperature dependence of q is effectively lost (it is ∼ e − cβ/ (2 − α ) ). S = = = = - - - - - - - - - - [ - α ] Log [ - q ] / S κ FIG. 3: Edwards-Anderson order parameter close to the crit-ical point α = 2. From top to bottom, increasing the numberof Trotter slices S = 4 , , ,
32 for sufficiently large β , theslope increases. For reference, the classical value of the slope(from (1 − q ) ∼ (2 − α )) is shown as the diagonal dashedblack line. In the inset are shown the values of the slope withtheir errors, and its extrapolation to S → ∞ to the value κ = 2 . ± .
1, quoted in the text. boundary conditions on the walls. Moreover, it becomesmore concentrated the larger the aspect ratio of the bil-liard, namely if one of the sides is larger than the oth-ers. Quantitatively, one finds q > q cl already at low-est order in α . Indeed, from the self-consistency equa-tions [48], q = α (cid:104) r (cid:105) v ( h =0) + O ( α ) where the average (cid:104)•(cid:105) v ( h =0) , when β → ∞ , indicates the expectation valueover the ground state of a harmonic oscillator with a wallin the origin. This problem is easily solved and one finds q = π α + O ( α ), to be compared with q cl = π α + O ( α ). Fig. 2 shows that the quantum order parameter q de-pends on the temperature T = 1 /β for α (cid:46)
1, andthen, increasing α , becomes independent of T through acrossover. From the classical calculation, we expect thatthe typical linear size of the allowed region for the par-ticle on the sphere vanishes as (cid:96) ∼ √ − q cl ∼ √ − α for α →
2. Thus, as soon as the energy gap to thefirst excited state becomes larger than the temperature,i.e. roughly when (cid:126) m (1 − q cl ) ∼ (cid:126) m (2 − α ) (cid:38) T , the quan-tum dynamics is effectively at zero temperature and theorder parameter q becomes independent of T . More-over, in the following we will show that the gap, deepin the quantum regime, grows even faster than (2 − α ) − when α →
2. Since the quantum dynamics recovers theclassical dynamics only when the de Broglie wavelength λ T ∼ (cid:126) / √ mT (cid:28) (cid:96) , on approaching jamming quantummechanics dominates. Hence, for any T, (cid:126) , m , as α → T → ∞ before α → κ regulating the re-lation (1 − q ) ∼ (2 − α ) κ in the quantum regime canbe extracted by looking at the low-temperature, large- α data. As usual, a sufficiently large number of Trotterslices S must be taken, and it increases as α →
2, makingthe numerical simulations more demanding. However,fortunately, the asymptotic region is reached already at α (cid:38)
1. The data in Fig. 3 clearly show that the criti-cal exponent of the quantum theory is not the classicalone, κ cl = 1, and it departs more and more from it asthe number of Trotter slices is increased. We have per-formed a log-log fit to extract such critical exponent, ina region of α ∈ [1 , . S → ∞ , we find κ = 2 . ± . κ > α → G − n = βm ( ω n + (cid:126) / m ) / (1 − q ), we were able tosolve explicitly the self-consistency equations for β → ∞ ,finding κ = 3 /
2. The value κ (cid:39) ω ) is considered.The internal energy per degree of freedom u (see[46, 48]) is independent of β , like q , already at α (cid:38) u ∼ (cid:126) m (2 − α ) for α →
2. This can be again interpreted in terms ofreduced volume and uncertainty principle, and confirmsthe previous result κ (cid:39) ∼ (cid:126) m (1 − q ) for α →
2. This im-plies that, if we focus on frequencies ω (cid:28) ∆ / (cid:126) , or times t (cid:29) (cid:126) / ∆, there is no dynamics. In order to see somedynamical behavior one should consider G ( ω (cid:38) ∆ / (cid:126) ).As shown in Fig. 4, at these large frequencies the formof the self-energy Σ( ω ) changes significantly. Indeed, atany α <
2, the self-energy is an analytic function of ω in a neighborhood of the origin ω = 0 (inset of Fig. 4).As α →
2, this behavior becomes extended to increas-ing values of ω . At larger frequencies, however, Σ( ω )develops a linear behavior. Moreover, for any α < ω →∞ Σ( ω ) = 0, as can be seen from its definition[48]. Performing a log-log fit, we find that the con-stant contribution to the autocorrelation function scalesas βµ ∼ (1 − q ) δ where δ (cid:39) − .
9. From a quadratic fitof Σ( ω ) at small ω , the coefficient of the quadratic termresults instead almost independent of (1 − q ).The behavior of Σ( ω ) defines the effective dynamicsof the theory, and its analytical properties around theorigin determine the low-temperature behavior of ther-modynamical observables. Both the analyticity of Σ( ω )around ω = 0 and the independence of β of all the ob-servables, including the internal energy u , show that thespecific heat is non-analytic in T when α →
2. Moreprecisely, our findings show that C V ( T ∼ ∼ e − ∆ /T ,due to the presence of the gap. However, since at not-so-small ω it holds Σ( ω ) ∼ | ω | , the specific heat presentsa power-law behavior at high enough temperatures, i.e. C V ( T > T cutoff ) ∼ T γ . Since ∆ → ∞ as α → T cutoff → ∞ too.The linear dispersion Σ( ω ) ∼ | ω | , observed in the crit-ical regime, reminds us of the result of [46], where theauthors perform a semiclassical analysis to investigatethe UNSAT phase with soft potentials. In [46], they sent (cid:126) → (cid:126) /T kept fixed, while in our study (cid:126) is keptfinite. They found the linear dispersion Σ( ω ) ∼ | ω | in aneighborhood of the origin ω = 0, implying a power-lawbehavior of C V ( T ) at small T near the jamming point.The similarities between the two results are surprising,since the regimes considered are different, and suggestthat the linear dispersion Σ( ω ) ∼ | ω | might be a univer-sal feature of quantum models near jamming. Conclusions.—
We have investigated the quantum per-ceptron with hard-wall potentials as a model for jam-ming. We have studied the replicated, quenched free en-ergy in the RS approximation, finding a quantum crit-ical point corresponding to the classical jamming point α c = 2 at σ = 0. Usually, quantum critical points areconfined and influence the physics around T = 0 [53].We instead find that the quantum jamming critical pointexists for any temperature , and the classical results arerecovered only by taking T → ∞ before α → α c : it isthe classical critical point to be confined to T = ∞ . Wefind quantum critical exponents different from the clas-sical ones, and an exponentially small C V ( T ) at small T . The dispersion relation G ( ω ) − ∼ | ω | for frequen- S = = =
16 S =
320 200 400 600 80005000100001500020000 ω Σ ( ω ) FIG. 4: Self-energy Σ( ω ) at α = 1 . β = 1 / as a function ofthe Matsubara frequency ω , for increasing number of Trotterslices (accessing higher and higher frequencies). We see thatΣ( ω ) develops a linear ω behavior (black, dashed line) forintermediate ω ’s, while retaining its analyticity in terms of ω around the origin for any q < ω ) at small ω ’s for α = 1 . β = 8. cies higher than the gap, but not asymptotically large,implies a power-law specific-heat for T > T cutoff , where T cutoff diverges at the critical point. This shows a sur-prising connection of our findings with the ones of thesemiclassical analysis in [46], where a different region ofparameters was considered, that deserves to be furtherinvestigated.An appealing extension of this work would be to con-sider soft potentials, having a finite v (cid:48) ≡ ∂v/∂r | r =0 [60],as in the case of structural glasses. Employing soft po-tentials, it is possible to access the UNSAT phase deepin the quantum regime. We do expect that the quantumjamming transition will turn into a crossover (like theclassical one does) but the same phenomenology outlinedin this paper should be observed as far as the change inthe potential on length scales O ((1 − q ) / ) is large withrespect to the gap ∆ ∼ (1 − q ) − . This means that for(1 − q ) (cid:38) ( v (cid:48) ) − / , or α (cid:46) − c ( v (cid:48) ) − / , the physics isdominated by the hard-wall quantum jamming criticalpoint. The robustness with temperature of the quan-tum critical point, shown in our results, implies that thequantum character of the system even with soft poten-tials cannot be neglected. Therefore, it suggests thatthe standard approaches used to study glassy systems atultra-low temperatures, which add quantum effects ontop of the classical landscape [54–56], might be inade-quate.Another interesting extension of this study would beto move to the regions with σ (cid:54) = 0. The case σ > σ <
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In particular, the thermodynamic observables, likespecific heat and order parameter, are divergence-free.[60] A similar reasoning applies for the case v (cid:48) = 0 but ∂ k v/∂r k | r =0 (cid:54) = 0 with k > Supplementary MaterialDerivation of the self-consistency equations
We need to compute the quenched disorder average of the free energy F = − β − ln Tr( e − βH ), with H given byEq. (2) (main text). Introducing the imaginary time t , the Lagrange multiplier λ associated to the constraint X = N and p replicas, one can find F as a function of the overlap matrix Q ab ( t, s ) = (cid:104) X a ( t ) · X b ( s ) (cid:105) /N, (S7)where Q ab ( t, s ) periodic in t and s with period β (cid:126) and a, b = 1 , ..., p are replica indices. The quenched free energy f ,per dimension N and per replica p can be written as: − βpf = 12 ln det ˆ Q ( t, s ) + m (cid:126) (cid:88) a (cid:90) β (cid:126) dt ∂ s Q aa ( t, s ) | s = t − m (cid:126) (cid:88) a (cid:90) β (cid:126) dt λ a ( t )( Q aa ( t, t ) −
1) + α ln ζ, (S8)where ζ = exp (cid:18) (cid:88) a,b (cid:90) (cid:90) β (cid:126) dtβ (cid:126) dsβ (cid:126) Q ab ( t, s ) δ δr a ( t ) δr b ( s ) (cid:19) · exp (cid:18) − (cid:126) (cid:88) c (cid:90) β (cid:126) dt v ( r c ( t )) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) r c ( t )=0 . (S9)The RS ansatz for the saddle point is: Q ab ( t, s ) RS = [ q d ( t − s ) − q ] δ ab + q (S10)where q d ( t ) − q is the autocorrelation function of a replica, while the off-diagonal order parameter q is the analog ofthe Edwards-Anderson order parameter: It is the overlap of two different replicas. As usual, one shall send p → Q .We need to find the saddle point with respect to variations of Q , namely of q d ( t ) and q , and µ ≡ mλ . To do this,it is convenient to define G ( t − s ) ≡ q d ( t − s ) − q. (S11)From the β (cid:126) -periodicity in imaginary time, we can consider as variables the countable set of Fourier components of G ( t ), i.e. { G n } n ∈ Z . Then, the quenched free energy in the RS approximation, per dimension N and replica p , is(Eq. (4), main text) − βf = 12 (cid:88) n ∈ Z ln G n + q G − βm (cid:88) n ∈ Z ω n G n − βµ (cid:104) (cid:88) n ∈ Z G n − (1 − q ) (cid:105) + αγ q (cid:63) ln (cid:104) e − β (cid:82) β (cid:126) dtβ (cid:126) v ( r ( t )+ h ) (cid:105) r , (S12)where ω n ≡ πn/β (cid:126) are the Matsubara frequencies, γ q (cid:63) • ( h ) ≡ (cid:82) ∞−∞ dh √ πq e − h / q • ( h ) and (cid:104)•(cid:105) r = 1 Z (cid:73) Dr e − (cid:82)(cid:82) β (cid:126) dtβ (cid:126) dsβ (cid:126) r ( t ) G − ( t − s ) r ( s ) • . (S13)The saddle-point equations for the parameters G n , µ and q are G − n = βmω n + βµ + β Σ n (S14) (cid:88) n G n = 1 − q (S15) q = αγ q (cid:63) (cid:104) r (cid:105) v (S16)where we defined the self-energy Σ n ≡ α (cid:0) G − n − G − n γ q (cid:63) ( (cid:104) r ∗ n r n (cid:105) v − δ n (cid:104) r (cid:105) v ) (cid:1) /β, (S17)with (cid:104)•(cid:105) v = (cid:104) e − β (cid:82) β (cid:126) dtβ (cid:126) v ( r ( t )+ h ) •(cid:105) r (cid:104) e − β (cid:82) β (cid:126) dtβ (cid:126) v ( r ( t )+ h ) (cid:105) r . (S18)These equations define self-consistently the dynamics of the auxiliary random process r ( t ). Iterative solution of the self-consistency equations
The self-consistency equations (S14)-(S16) can be solved with an iterative method, together with a Montecarlosampling. The parameters G n , µ and q can be initialized with arbitrary values. However, numerically it is convenientto proceed in a stepwise manner, from smaller to higher α ’s. The algorithm is composed by three steps.The first step is to compute the self-energy Σ n and use it to update the autocorrelation function G n , iteratively.However, the computation of the self-energy involves the averages (cid:104)•(cid:105) v . To evaluate them, we use a Path IntegralMontecarlo (PIMC) simulating the dynamics of the β (cid:126) -periodic process r ( t ), in the potential generated by G − ( t − s )and v ( r ( t ) + h ), as sketched in Fig. 5. The former, when Σ n ≡
0, contains a kinetic term ( mω n /
2) plus a harmonicpotential ( µ/ r ( t ) > − h . When Σ n (cid:54) = 0 both contributions (kineticand potential) change, and the dynamics of r ( t ) becomes non-trivial. FIG. 5: Sketch of the Path Integral Montecarlo (PIMC) used to simulate the dynamics of the β (cid:126) -periodic process r ( t ) ( r (0) = r ( β (cid:126) )) in the potential generated by G − ( t − s ) and v ( r ( t ) + h ). The PIMC consists in proposing a move r ( t s ) → r ( t s ) + δ ( t s )for every time step t s , which is accepted or rejected according to the Metropolis algorithm. We improved this simple schemeusing both the method of images and the rigid movement of the time chain r ( t s ), as discussed in the text. Numerically, it is convenient to consider the period of the process as β , reabsorbing (cid:126) in the mass m → m/ (cid:126) .Moreover, the period has to be discretized: The number of Trotter slices is S = β/a , where a is the time-sliceamplitude, and, setting β = 2 L and a = 2 − K , it holds S = 2 L + K . In this way, we can define a discrete Fouriertransform f n = S (cid:80) S − s =0 f ( t s ) e iω n t s where ω n = 2 πn/β with n ∈ [0 , S − β the set { ω n } becomesdenser, while decreasing a one can access higher frequencies.The PIMC algorithm consists in proposing a move r ( t s ) → r ( t s ) + δ ( t s ) for every time step t s , which is accepted orrejected according to the Metropolis algorithm with weight given by G − and v . However, the presence of the hard-wall potential makes the convergence of the Montecarlo very demanding, and it is not sufficient to reject the attemptedmoves with r < − h to have a good numerical protocol. Thus, we implemented an improved Montecarlo samplingwhich exploits the method of images. We modified the free particle kinetic term of the Hamiltonian ( mω n / P ( r , | r , β ) of the free particle, we used P ( r , | r , β ) − P (Im( r ) , | r , β )where Im( r ) = − r − h is the image of r when the wall is in − h . Another expedient we adopted is to add a movewhich translates rigidly the time chain r ( t s ), i.e. r ( t s ) → r ( t s ) + δ with δ independent of t s .The presence of the convolution γ q (cid:63) • ( h ) in the definition of Σ n (Eq. (S17)) implies the evaluation of (cid:104)•(cid:105) v for manypositions − h of the wall. We approximate this Gaussian integral with the Gauss-Hermite quadrature, always with,at least, 10 sample points. This first step of the iterative method stops when G n is converged for every n within afixed tolerance (we fixed the relative difference between G old n and G new n to be < . G n verifies the identity in Eq. (S15). If it does so, we can go to thethird step; otherwise, µ is changed via the bisection method and the first step is performed again.The third step consists in computing the r.h.s. of Eq. (S16) with the converged G n and µ and check if the identityin Eq. (S16) is verified. If it is so, we have found the parameters which solve the self-consistency equations; if not, q is changed and one has to repeat all the procedure from the first step. Exponent κ = 3 / in the quadratic approximation Setting G − n = βm ( ω n + (cid:126) / m ) / (1 − q ) as in the text, the spherical constraint (Eq. (S15)) is automatically satisfiedup to exponentially small corrections, and the values of m and q can be fixed by Eqs. (S14) and (S16). Note that thereis an equation of the form (S14) for every n ∈ Z , yielding a deeply overcomplete set of constraints for our ansatz, butwe restrict to the n = 0 case only.It is convenient to set x ≡ r/ √ − q and H ≡ h/ √ − q , so that Eq. (S16) becomes q (1 − q ) / = α (cid:90) dH √ πq e − (1 − q ) H q (cid:104) ψ ( H )0 | x | ψ ( H )0 (cid:105) , (S19)where the reduced Schr¨odinger problem to solve is − dψ ( H ) k dx + 18 x ψ ( H ) k = E ( H ) k ψ ( H ) k , ψ ( H ) k ( H ) = 0 . (S20)Self-consistently we will show that only the ground-state contribution matters (i.e. k = 0). With this in mind we haveemployed the one-parameter variational wavefunction ψ ( H ) ( x ; L ) = 1 √ Z ( x − H ) θ ( x − H ) e − x / L , (S21)with an appropriate normalization Z , for which the energy reads E ( H ) ( L ) = 1 + L L φ ( H/ √ L )( H + 3 L ) − HLφ ( H/ √ L )( H + L ) − HL (S22)where φ ( y ) ≡ √ πe y Erfc( y ) , Erfc being the complementary error function. The equation dE ( H ) /dL = 0 can besolved separately in the regions H (cid:29) L , | H/L | (cid:28) H (cid:28) L by using suitable expansions. Remembering that q → H ∼ (cid:112) q/ (1 − q ) → ∞ ) we see that the important region is H (cid:29) L , andself-consistently we obtain H/L (cid:29)
1. We find (cid:104) ψ ( H )0 | x | ψ ( H )0 (cid:105) (cid:39) H + 3 / H − / + O ( H − / ) and by inserting it in Eq.(S19) we arrive at q = α (cid:20) (1 − q ) ξ (cid:18) q − q (cid:19) + q (cid:21) (S23)with ξ ( λ ) = (cid:90) ∞ dH √ πλ e − H / λ (cid:20) (6 H ) / / + · · · (cid:21) = 3 / Γ(5 / √ π / λ / + · · · . (S24)Eq. (S23) can now be solved for q , yielding κ = 3 / q = 1 − √ π / (2 − α ) /
24 Γ(5 / / . (S25)The same scaling has been observed by solving the Schr¨odinger equation (S20) numerically, discretizing the x -axisand employing imaginary-time evolution to find the ground state.Knowing q as a function of α , we can now solve the n = 0 case of Eq. (S14) with the same technique. It reads m = βγ q/ (1 − q ) (cid:63) (cid:104) ψ ( H )0 | x | ψ ( H )0 (cid:105) conn . (S26)By means of the same variational ansatz we find that the connected average in the equation above is 3 / H − / θ ( H )+ · · · , and finally m = β / Γ(1 / / (cid:18) − qq (cid:19) / . (S27)Thus we see that, as q → β/m → ∞ and our approximation to take only the ground state becomes more and morereliable. Internal energy
The (regularized) internal energy per degree of freedom is u = 12 β (cid:88) n ∈ Z µ + Σ n mω n + µ + Σ n , (S28)as derived in [46].We find that u is independent of β , like q , already at α (cid:38)
1, but it strongly depends on the number of Trotterslices S . Extrapolating the data for S → ∞ we obtain the result in Fig. 6, which shows a divergence of the energyas α →
2. This is again interpreted in terms of reduced volume and uncertainty principle. In particular, we observethat u ∼ (cid:126) m (2 − α ) with good accuracy for α →
2, in a region where the dependence on β is lost. This confirms theresult κ (cid:39)
2, obtained from the behavior of q (Fig. 3, main text). α u α = α = α = α = α = α = / S u s FIG. 6: Internal energy u as a function of the density of constraints α . The dashed line is a fit of the form u = A (2 − α ) − κ (1 + B (2 − α )+ C (2 − α ) ) with κ = 2 . q . This confirms u ∼ (1 − q ) − ∼ (2 − α ) − asdiscussed in the main text. In the inset one can see, from bottom to top for α = 0 . , . , . , , . , . , .