Replicated liquid theory in $1+\infty$ dimensions

TL;DR

This paper develops an exact replicated liquid theory for spatially varying glasses in high transverse dimensions, providing insights into diverging lengths and spatial profiles near glass transitions.

Contribution

It introduces a novel space-dependent free-energy functional for glasses in infinite transverse dimensions, enabling analysis of spatial variations and transition behaviors.

Findings

Exponents match previous mean-field models.

Predicts non-trivial spatial profiles of the glass order parameter.

Identifies diverging lengths near glass transitions.

Abstract

We develop a replicated liquid theory for structural glasses which exhibit spatial variation of physical quantities along one axis, say $z$-axis. The theory becomes exact with infinite transverse dimension $d-1 \to \infty$. It provides an exact free-energy functional with space-dependent glass order parameter $\Delta_{ab}(z)$. As a first application of the scheme, we study diverging lengths associated with dynamic/static glass transitions of hardspheres with/without confining cavity. The exponents agree with those obtained in previous studies on related mean-field models. Moreover, it predicts a non-trivial spatial profile of the glass order parameter $\Delta_{ab}(z)$ within the cavity which exhibits a scaling feature approaching the dynamical glass transition.

Equations

$$H=\sum _{i=1}^N\frac{\text{quantity}|{\mathbf{p}}_i{|}^2}{2m}+\sum _{i<j}v(r_{ij})+\sum _{i=1}^{N}U(z_i)$$

$$v(r)=\left\{\begin{array}[]{ll}\infty&(r\leq D)\\ 0&(r>D)\end{array}\right.$$

$$\rho (z)=\sum _{i=1}^N⟨\delta (z-z_i)⟩$$

$$\frac{-\beta F[\rho ]}{S}$$

$$f(r^2)=e^{-\beta v(r)}-1$$

$$r_{12}^2=(z_1-z_2{)}^2+\sum _{\mu =1}^{d-1}(x_{\mu 1}-x_{\mu 2}{)}^2.$$

$$\frac{N}{S}={\int }_{-\infty }^{\infty }dz\rho (z)$$

$$0$$

$${\mathbf{x}}_i^a=({\mathbf{x}}_i{)}_{\mathrm{c}}+{\mathbf{u}}_i^a$$

$$({\mathbf{x}}_i{)}_{\mathrm{c}}=((x_i^1{)}_{\mathrm{c}},(x_i^2{)}_{\mathrm{c}},\dots ,(x_i^{d-1}{)}_{\mathrm{c}},(z_i{)}_{\mathrm{c}})=\frac{1}{m}\sum _{a=1}^m{\mathbf{x}}_i^a$$

$${\Delta }_{ab}(z)={\alpha }_{aa}(z)+{\alpha }_{bb}(z)-2{\alpha }_{ab}(z){\alpha }_{ab}(z)\rho (z)=\frac{d}{D^2}\frac{1}{S}\sum _{i=1}^N⟨{\mathbf{u}}_i^a\cdot {\mathbf{u}}_i^b\delta (z-(z_i{)}_{\mathrm{c}})⟩$$

$$z=\frac{D}{\sqrt{d}}\hat{z}$$

$$\frac{-\beta F_m[\rho ,{\alpha }_{ab}]}{S}\frac{\sqrt{d}}{D}$$

$$={\int }_{-\infty }^{\infty }d\hat{z}\rho (\hat{z})\text{quantity}1-\ln (\rho (\hat{z}){\lambda }_{\mathrm{th}}^d)+d\ln m+\frac{(m-1)d}{2}\ln \frac{2\pi e(D/{\lambda }_{\mathrm{th}}{)}^2}{d^2}+\frac{d}{2}\ln \det \text{quantity}((\alpha (\hat{z}){)}^{m,m})+(-\beta U)(\hat{z})$$

$$+\frac{d}{2}\frac{{\Omega }_d{D}^d}{d\sqrt{2\pi }}{\int }_{-\infty }^{\infty }d\hat{z}\rho (\hat{z}){\int }_{-\infty }^{\infty }d{\hat{z}}^{\prime }\rho ({\hat{z}}^{\prime })e^{-\frac{(\hat{z}-{\hat{z}}^{\prime }{)}^2}{2}}\text{quantity}(-{\mathcal{F}}_{\mathrm{int}}({\Delta }_{ab}(\hat{z},{\hat{z}}^{\prime }))).$$

$$-{\mathcal{F}}_{\mathrm{int}}({\Delta }_{ab}(\hat{z},{\hat{z}}^{\prime }))={\int }_{-\infty }^{\infty }d\xi e^{\xi }{e}^{-\frac{1}{2}{\sum }_{ab}{\Delta }_{ab}(\hat{z},{\hat{z}}^{\prime }){\partial }_{{\xi }_a}{\partial }_{{\xi }_b}}{\text{quantity}[\prod _a{e}^{-\beta v\text{quantity}(D^2\text{quantity}(1+\frac{{\xi }_a}{d}{)}^2)}-1]|}_{\text{quantity}{\xi }_a=\xi }.$$

$${\Delta }_{ab}({\hat{z}}_1,{\hat{z}}_2)=\frac{{\Delta }_a({\hat{z}}_1)+{\Delta }_b({\hat{z}}_2)}{2}$$

$$0$$

$$\frac{N}{S}=\frac{D}{\sqrt{d}}{\int }_{-\infty }^{\infty }d\hat{z}\rho (\hat{z})$$

$$\phi (\hat{z})=\rho (\hat{z}){\Omega }_d\text{quantity}(\frac{D}{2}{)}^d$$

$$\hat{\phi }(\hat{z})\equiv \frac{2^d\phi (\hat{z})}{d}=\frac{\rho (\hat{z}){\Omega }_d{D}^d}{d}.$$

$${\alpha }_{ab}(\hat{z})=(m{\delta }_{ab}-1)\alpha (\hat{z})$$

$$\Delta (\hat{z})=2m\alpha (\hat{z}){\Delta }_{ab}(\hat{z})=\Delta (\hat{z})(1-{\delta }_{ab})$$

$$\frac{-\beta F_m^{1RSB}\text{quantity}[\text{quantity}\Delta (\hat{z})]}{S}\frac{\sqrt{d}}{D}\frac{{\Omega }_d{D}^d}{d}=\int d\hat{z}\hat{\phi }(\hat{z})\text{quantity}1-\ln (\hat{\phi }(\hat{z})(d/{\Omega }_d)({\lambda }_{\mathrm{th}}/D{)}^d)+d\ln m+\frac{(m-1)d}{2}\ln \frac{2\pi e{D}^2}{d^2{\lambda }_{\mathrm{th}}^2}+(-\beta m{\hat{U}}_0(\hat{z})),$$

$$+$$

$$-{\mathcal{F}}_{\mathrm{int}}(\Delta )={\int }_{-\infty }^{\infty }d\xi e^{\xi -\frac{1}{2}\Delta (\hat{z},{\hat{z}}^{\prime })}\text{quantity}[g^m\text{quantity}(\xi ,\Delta (\hat{z},{\hat{z}}^{\prime }))-1]$$

$$g\text{quantity}(\xi ,\Delta )=e^{\frac{1}{2}\Delta {\partial }_{\xi }^2}{e}^{-\beta v(D^2(1+\xi /d{)}^2)}=\int \mathcal{D}w{e}^{-\beta v\text{quantity}(D^2\text{quantity}(1+\frac{\xi +\sqrt{\Delta }w}{d}{)}^2)}$$

$$\frac{1}{\Delta (\hat{z})}$$

$$f(\xi ,\Delta )=-\ln g(\xi ,\Delta )$$

$$\delta \Delta (\hat{z})=\Delta (\hat{z})-\Delta .$$

$$-M({\hat{z}}_1,{\hat{z}}_2)$$

Full text

1. I Introduction

2. II Model

3. III Construction of an inhomogeneous replicated liquid theory

4. III.1 Inhomogeneous liquid theory

5. III.2 Inhomogeneous replicated liquid theory

6. III.3 One Step RSB solution

7. IV Diverging length scales at glass transitions

8. IV.1 Spatial correlation of glassy fluctuations around the dynamical transition

9. IV.2 Glass transitions within cavities

10. IV.2.1 Hard-spheres

11. IV.2.2 Cavity size dependence of the dynamical transition

12. IV.2.3 Spatial profile of the glass order parameter around the dynamical glass transition

13. IV.2.4 Correlation length extracted from the fluctuation around the saddle point in the bulk system

14. IV.2.5 Summary of the behavior of the system close to the dynamical transition point

15. IV.2.6 Cavity size dependence of the Kauzmann transition

16. V Conclusion and outlook

17. A Liquid theory in $1+(d-1)$ dimensions with $d-1\gg 1$

18. A.1 Basic setup

19. A.2 Large dimensional limit

20. A.3 Compression

21. B Replicated liquid theory in $1+(d-1)$ dimensions with $d-1\gg 1$

22. B.1 Basic setup

23. B.2 Compression

24. C One step RSB solution

25. C.1 1RSB ansatz

26. C.2 Free energy

27. C.3 Equation of states

28. C.4 Uniform system

29. C.5 Dynamical transition in uniform system

30. C.6 Longitudinal Hessian

31. C.7 Chemical potential and pressure

32. C.8 Complexity

33. D Cavity system

34. E Hard-sphere system

35. E.1 Basics

36. E.2 Kauzmann transition in cavity

37. F Useful Formulas

38. F.1 Asymptotic behavior of the error function

39. F.2 The function of $f(L)$

40. G Collection of some useful formulas

41. G.1 Proof of $\Omega_{d-1}=\sqrt{d/2\pi}\Omega_{d}$

42. G.2 Integration of Gaussians

43. H Numerical details to solve the integral equation Eq. (203)

I Introduction

Recently the exact mean-field theory for super-cooled, glass forming liquids which becomes exact in the large dimensional limit $d\to\infty$ was established kurchan2012exact ; kurchan2013exact ; charbonneau2014exact ; charbonneau2014fractal ; YZ14 ; rainone2015following ; 1130848328245474560 ; urbani2023gardner

which is based on the density functional theory of liquids hansen1990theory and the replica method developed in the studies of disordered systems MPV87 . This is a significant theoretical achievement since the key notions of glass physics kirkpatrick1987stable ; kirkpatrick1987connections ; angell2000relaxation ; cavagna2009supercooled ; berthier2011theoretical conceived by preceding experimental, numerical and theoretical studies such as the dynamical glass transition, thermodynamic glass transition (Kauzmann transition), jamming and yielding were firmly established and understood in a unified manner in a single theoretical framework based on 1st principles. The central object is the glass order parameter $\Delta_{ab}$ which parameterize the relative mean-squared displacements among replicated liquids $a=1,2,\ldots$. It remains to be clarified to what extent the features established in the large dimensional limit, such as the very existence of the Kauzmann transition, remain valid in finite dimensional real systems.

An obvious drawback of $d\to\infty$ theories is that, by its construction, it cannot describe any spatial variation or fluctuation of physical quantities. Actually glasses forming liquids and glasses are known to exhibit various interesting spatial heterogeneities such as the dynamical heterogeneity observed in the supercooled liquid state karmakar2014growing ; biroli2008thermodynamic ; bouchaud2004adam , the isostatic length which diverges approaching jamming wyart2005rigidity and formation of shear-bands approaching yielding or fracture maloney2006amorphous . Aiming to capture these spatial heterogeneities theoretically, we develop an exact mean-field theory for $1+\infty$ dimensional system so that we become able to describe the spatial variation and fluctuation of physical properties along one spatial axis, say $z$-axis. The main object is the space dependent glass order parameter $\Delta_{ab}(z)$. Our approach is related but somewhat different from the usual Ginzburg-Landau (GL) type, field theoretic descriptions lubensky2023renormalization . While such GL approaches will become useful at sufficiently long-wave lengths, our theory is derived microscopically so that it is precise also at the particle scales. We believe it will become particularly useful in situations like jamming and yielding where accurate microscopic descriptions at the scale of particles are indispensable.

The purpose of this paper is twofold. First we develop a generic replicated liquid theory in $1+(d-1)$ dimensions which becomes exact in $d-1\to\infty$ limit. Second we test the scheme by analyzing the length scales which diverge approaching the dynamic/static glass transitions using hard-spheres as the simplest glass forming system.

The analysis of the diverging length scales is done in two setups. First setup is an in an infinitely large system $-\infty<z<\infty$. There we analyze the spatial correlation of the thermal fluctuations of the glass order parameter. We find a length scale which diverges approaching the dynamical glass transition. It was originally predicted by a study on a mean-field spin-glass model kirkpatrick1987stable whose phenomenology closely resemble that of structural glasses kirkpatrick1987connections . Later the inhomogeneous MCT (mode coupling theory) biroli2006inhomogeneous obtained the length scale from a 1st principle. The 2nd setup is a cavity system of finite depth $L$ containing the hard-spheres. This setup allows one to study the correlation lengths, called as the point-to-set lengths in the literature biroli2008thermodynamic ; bouchaud2004adam , which diverge approaching the two glass transitions: the dynamical transition already mentioned above and the static glass transition or the Kauzmann transition. We capture the correlation lengths through the following two features. One is the finite size effect on the dynamic/static length glass transitions: we study how the glass transition points are affected by the finiteness of the cavity size $L$. The other is the spatial variation of the glass order parameter $\Delta_{ab}(z)$ viewed as a function of the distance from the cavity wall. It turned out to be very similar to the behavior of the order parameter associated with the surface critical phenomena binder1972phase ; kikuchi1985monte1 . Our results reconfirm the critical exponents obtained in previous theoretical studies based on mean-field spin-glass model kirkpatrick1987stable , inhomogeneous MCT biroli2006inhomogeneous , a Kac glass model franz2007analytic and a one-dimensional chain of discrete cells containing a hard-sphere liquid ikeda2015one . Our analysis can be viewed as a thermodynamic (static) counterpart of the inhomogeneous MCT biroli2006inhomogeneous and a continuous limit of the chain model ikeda2015one . Our work is also motivated in part by the replica theory of supervised learning by a prototypical deep neural network which revealed a non-trivial spatial profile of the glass order parameter yoshino2020complex .

II Model

We consider an assembly of particles $i=1,2,\ldots,N$ of mass $m$ contained in a cylinder of cross-section area $S$ as shown in Fig. 1. The coordinates of the particles are given by ${\bf x}_{i}=(x_{1i},x_{2i},\ldots,x_{d-1i},z_{i})$ while their momentum are ${\bf p}_{i}$. The particles are interacting with each other through a two-body interaction potential $v(r_{ij})$ with $r_{ij}=\quantity|{\bf x}_{i}-{\bf x}_{j}|$. Then the Hamiltonian is given by.

[TABLE]

The last term represents a potential which can be used, for example, to confine the particles within a finite region along the z-axis. We are interested in static macroscopic properties of the system in equilibrium at temperature $T$ or inverse temperature $\beta=1/k_{\rm B}T$ with $k_{\rm B}$ being the Boltzmann constant.

As a specific model system we will consider hard-spheres (HS) with diameter $D$ with the interaction potential given by,

[TABLE]

Thus the Boltzmann factor associated with the HS potential becomes $\exp(-\beta v(r))=\theta(r)$ where $\theta(r)$ is the Heaviside step function.

III Construction of an inhomogeneous replicated liquid theory

In this section, we discuss the construction of an inhomogeneous replicated liquid theory. To this end we first construct a density functional theory for simple liquids in $1+(d-1)$ dim space which becomes exact in $d-1\to\infty$ limit. We obtain an exact form of the free-energy functional expressed in terms of density profile $\rho(z)$ which is allowed to vary along the $z$-axis. Next, we replicate the system and construct an inhomogeneous replicated liquid theory with the free-energy expressed exactly in terms of the density profile $\rho(z)$ and space dependent glass order parameter $\Delta_{ab}(z)$. Then we derive the self-consistency equation which determines the density profile and the glass order parameter. We show how various thermodynamic quantities including chemical potential, pressure, and structural entropy (complexity) can be computed. The details of the derivations are shown in appendix A, B and C.

III.1 Inhomogeneous liquid theory

We assume that the system is uniform within each cross-section of the cylinder and the density varies only along the $z$ axis (see Fig. 1). Then we naturally introduce a microscopic density profile,

[TABLE]

where $\langle\ldots\rangle$ is the thermal average. Although the density functional theory for liquids is well established hansen1990theory , we show in appendix A.1 the derivation of the density functional expressions for the sake of clarity and pedagogical reasons. The free-energy of the system as a functional of the density profile $\rho(z)$ is obtained as,

[TABLE]

where $\lambda_{\rm th}=h/\sqrt{2\pi mk_{\rm B}T}$ is the thermal de Broglie wave length and

[TABLE]

is the Mayer function and

[TABLE]

In the large dimensional limit $d-1\to\infty$ contributions from higher orders in the Mayer expansion becomes negligible FP99 ; 1130848328245474560 and the above expression becomes exact.

The equilibrium density profile $\rho(z)$ is the one which minimizes the free-energy functional $F[\rho]$ under the constraint,

[TABLE]

It can be obtained by solving

[TABLE]

Here we introduced a Lagrange parameter $\beta\mu$ where $\mu$ can be interpreted as the chemical potential. The above equation yields $\mu$ which must be adjusted to satisfy Eq. (7).

In appendix A.2 we explain a rescaled length scale (see Eq. (12)) and a rescaled density (see Eq. (19)) which we will use below. There we also present the free-energy functional and the chemical potential expressed in terms of these rescaled quantities. In appendix A.3 we consider compression of the system to obtain an expression of pressure.

III.2 Inhomogeneous replicated liquid theory

Next let us consider inhomogeneous replicated liquid theory aiming to extend the conventional theory for the homogeneous system 1130848328245474560 . We considers system of $m$-replicas $a=1,2,\ldots,m$ all obey the same Hamiltonian Eq. (1). The basic idea of the cloned or replicated liquid theory mezard1999first is to consider the possibility of spontaneous formation of a ’molecular liquid state’ with molecules made of ’replicas’. To describe such a molecular liquid state, it is convenient to decompose the coordinate $\bm{x}_{i}^{a}$ of particle $i$ in the $a$ replica as,

[TABLE]

with centers of the ’molecules’ $i=1,2,\ldots,N$ located at

[TABLE]

The 2nd term in the r.h.s of Eq. (9) represents thermal fluctuations within the molecules. We introduce the space dependent glass order parameters as,

[TABLE]

where $a$,$b$ are indices for replicas $a,b=1,2,\ldots,m$. Formation of a molecular liquid state is signaled by the presence of finite glass order parameter $0<\Delta_{ab}<\infty$. Physically this means emergence of a glassy state while $\Delta_{ab}=\infty$ in a purely liquid state and $\Delta_{ab}=0$ in a jammed state.

It has been realized that the relevant length scale for the fluctuation of the inter-particle distance is $O(1/\sqrt{d})$ in the large dimensional limit $d\to\infty$ kurchan2012exact ; 1130848328245474560 (see appendix A.2). This naturally leads us to introduce a scaled coordinate $\hat{z}$ as,

[TABLE]

Here $D$ is the microscopic length scale which characterize the interactions, which is the diameter of spheres in the case of hard-spheres.

As we explain in the appendix B.1, the free energy functional of the replicated system is obtained exactly in the large dimensional limit $d-1\to\infty$ using the glass order parameter introduced above as,

[TABLE]

Here, $-\mathcal{F}_{\rm{int}}$ which represents the replicated Mayer function, is defined as

[TABLE]

We also introduced a short-hand notation,

[TABLE]

The equilibrium density profile $\rho(\hat{z})$ is the one which minimizes the free-energy under the constraint Eq. (17) (see below). The density profile $\rho(\hat{z})$ and glass order parameters $\Delta_{ab}(\hat{z})$ are obtained by solving

[TABLE]

The 1st equation yields the chemical potential $\mu$ which must be adjusted to satisfy Eq. (7) which reads as

[TABLE]

In the case of hard-sphere like systems, $D$ is the radius of particles. For those cases it is convenient to introduce the volume fraction $\varphi(\hat{z})$,

[TABLE]

where $\Omega_{d}$ is the volume of $d$-dimensional unit sphere, In order to study glassy states in the large dimensional limit it becomes convenient to introduce a scaled volume fraction 1130848328245474560 (see appendix A.2),

[TABLE]

In appendix A.2 we also report an expression of the chemical potential, and the free-energy functional for the bulk (uniform system) for references. In appendix B.2 we consider compression of the system and obtain an expression of the pressue.

III.3 One Step RSB solution

The simplest ansatz for the matrix form of the glass order parameter defined in Eq. (11) is

[TABLE]

or

[TABLE]

which are symmetric under permutations of replicas $a=1,2,\ldots,m$. This symmetry is the so-called replica symmetry. However we call these ansatz as one step RSB (1RSB) ansatz in the present paper for the reasons we explain in appendix C and B.1. This ansatz is known to be valid at densities lower than the so called Gardner transition density kurchan2013exact ; urbani2023gardner in the uniform system.

Using this ansatz we can evaluate thermodynamic quantities as we explain in appendix C. The free-energy is obtained as

[TABLE]

with the replicated Mayer function given by,

[TABLE]

where

[TABLE]

Here we used the identity Eq. (223) and a short-hand notation Eq. (224), $\int\mathcal{D}z\ldots=\int^{\infty}_{-\infty}\frac{dz}{\sqrt{2\pi}}e^{-\frac{z^{2}}{2}}\ldots$.

By taking a functional derivative of the free energy by $\Delta(\hat{z})$, we obtain the self-consistent equation for $\Delta(\hat{z})$,.

[TABLE]

where

[TABLE]

IV Diverging length scales at glass transitions

Now let us test our theoretical framework analyzing diverging length scales at dynamic/static glass transitions of hard-spheres in two different setups.

IV.1 Spatial correlation of glassy fluctuations around the dynamical transition

In the first setup we consider an infinitely large system $-\infty<z<\infty$ and examine the spatial correlation function of the fluctuation of the glass order parameter around the equilibrium value $\Delta$,

[TABLE]

This is done by analyzing the Hessian matrix. As explained in appendix C.6 we obtain the Hessian matrix as

[TABLE]

with

[TABLE]

By performing Fourier transform we find

[TABLE]

with

[TABLE]

From the above result we immediately find the spatial correlation function the local mean-squared displacements Eq. (27) as,

[TABLE]

with the correlation length $\xi$ given by

[TABLE]

Now let us focus on the dynamical transition at $\hat{\varphi}_{\rm d}$. As we recall in appendix C.5, the dynamical transition takes place at the spinodal point where the saddle point solution disappears kirkpatrick1987stable . Approaching $\hat{\varphi}_{\rm d}$ we find the correlation length behaves,

[TABLE]

with

[TABLE]

which measures the distance to the critical point $\hat{\varphi}_{\rm d}$. Here we used the fact that $M_{0}$ is nothing but the Hessian of the bulk system which scales as

[TABLE]

close to the dynamical transition density $\hat{\varphi}_{\rm d}(m)$ (See Eq. (192)) This scaling reflects the fact that the dynamical transition takes place at a spinodal point binder1987theory ; kirkpatrick1987stable . On the other hand $M_{2}$ is essentially a constant close to the critical point.

IV.2 Glass transitions within cavities

Now we turn to our 2nd setup which is a cavity system. The techinical details are reported in appendix D. It is prepared as follows. We consider again an infinitely large system $-\infty<\hat{z}<\infty$ with uniform density $\hat{\varphi}(\hat{z})=\hat{\varphi}$. Suppose that the entire system is in the liquid state. Then we freeze-out the system setting $\Delta(\hat{z})=0$ everywhere except for the ’cavity’ region $0<\hat{z}<\hat{L}_{\rm cav}$. The free energy of such a cavity system within the 1RSB anasatz is given by

[TABLE]

where $\int_{\rm{ex-cav}}d\hat{z}$ is the integral outside the cavity

[TABLE]

Then with the free energy given above, we obtain the self-consistent equation for the order parameter in the cavity as

[TABLE]

This equation must be solved under the condition that $\Delta(\hat{z})=0$ outside the cavity, i. e $\hat{z}<0$ or $\hat{L}_{\rm cav}<\hat{z}$ (see Eq. (203)). Since we will consider volume fractions lower than that of the Kauzmann transition, we fix the parameter $m$ as $m=1$ 1130848328245474560 in the following.

IV.2.1 Hard-spheres

We specifically analyzed the case of hard-spheres. The technical details are reported in appendix E. The function $g(\xi,\Delta)$ which appear in $\mathcal{F}_{\rm int}(\Delta)$ defined in Eq. (23) is obtained for the hard-spheres as,

[TABLE]

where $\Theta(x)=\quantity(1+\erf(x))/2$ and $\erf(x)$ is the error function.

First we analyzed the bulk (uniform) system solving the saddle point equation Eq. (25) with $\hat{\varphi}(\hat{z})=\hat{\varphi}$. Looking for the density at which the saddle point equation (with $m=1$) disappears, we obtain the dynamical transition density of the bulk system as

[TABLE]

We solve the saddle point equation numerically. For the details of the numerical procecure see appendix H. In Fig.2 panel (a),(b) we show a representative spatial profile of the glass order parameter of the hard-spheres in the cavity. The dotted lines represent $\Delta=\Delta_{\rm bulk}$ obtained in the bulk system ($\hat{L}_{\rm cav}=\infty$). We can see that by moving far from the edge of the cavity the value of the order parameter becomes close to that of the bulk system. Closer to the edge of the cavity, the order parameter becomes smaller meaning that the system is more constrained there due to the frozen region outside the cavity.

IV.2.2 Cavity size dependence of the dynamical transition

In Fig.2 panel (c),(d) we show the variation of the order parameter with respect to the changes of the volume fraction $\hat{\varphi}$ of the hard-spheres in the cavity. In the case $\hat{L}_{\rm cav}=40$ we found the solution disappears at $\hat{\varphi}\sim 4.8$ while the solution disappears at $\hat{\varphi}\sim 4.7$ for $\hat{L}_{\rm cav}=10$. This suggest the dynamical transition density $\hat{\varphi}_{\rm d}$ depends on the cavity size $\hat{L}_{\rm cav}$ such that the $\hat{\varphi}_{\rm d}(\hat{L}_{\rm cav})$ becomes smaller decreasing $\hat{L}_{\rm cav}$. This means the system in a smaller cavity is more strongly constrained due to the frozen region outside the cavity so that the glass state remains up to lower volume fractions.

We obtained the dynamical transition density $\hat{\varphi}_{d}(\hat{L}_{\rm cav})$, where the solution disappears, for various cavity sizes $\hat{L}_{\rm cav}$. From this result we defined the point-to-set (PS) length biroli2008thermodynamic bouchaud2004adam of the dynamical transition $\xi^{\rm PS}_{d}(\hat{\varphi})$ as

[TABLE]

treating $\hat{L}_{\rm cav}$ as a running parameter. As shown in Fig.3 panel (c), $\xi^{\rm PS}_{d}(\hat{\varphi})$ obeys well the anticipated scaling

[TABLE]

IV.2.3 Spatial profile of the glass order parameter around the dynamical glass transition

We have seen that by moving far from the edge of the cavity the value of the order parameter $\hat{\Delta}(\hat{z})$ becomes close to that of the bulk system. To characterize such a spatial profile of the order parameter let us introduce

[TABLE]

where $\Delta_{\rm bulk}$ is the value of the order parameter of the bulk system $\hat{L}_{\rm cav}=\infty$. As shown in Fig. 3 (a) it exhibits an exponential decay as a function of $\hat{z}$ , by choosing a large enough $\hat{L}_{\rm cav}$, such that it can be well fitted by

[TABLE]

with a characteristic length scale $\xi_{\rm d}^{\rm profile}$.

In Fig. 3 (a) we find clear $\hat{L}_{\rm cav}$ dependence. In smaller $\hat{L}_{\rm cav}$ systems, the exponential decay saturates at smaller $\hat{z}$. It is natural to expect that the saturation will disappear in the limit $\hat{L}_{\rm cav}/\xi_{\rm d}^{\rm profile}\to\infty$. The red dots in Fig. 3 (c) are $\xi_{\rm d}^{\rm profile}$ obtained analyzing the system with $\hat{L}_{\rm cav}=2000$. We find $\xi_{\rm d}^{\rm profile}$ clearly grows approaching the dynamical transition density $\hat{\varphi}_{\rm d}$ similarly to $\xi_{\rm d}^{\rm hessian}$ and $\xi_{\rm d}^{\rm PS}$.

Close to the critical point $\delta\varphi\sim 0$, it is natural to expect that the spatial profile of the order parameter, namely $\Delta(\hat{z},\delta\varphi)$ as a function of $\hat{z}$ exhibits a universal scaling feature. To see this, it is convenient to use the following decomposition,

[TABLE]

where we introduced a dimensionless function,

[TABLE]

where $\Delta_{\rm d}-\Delta_{\rm bulk}\sim\delta\varphi^{1/2}$ close to the critical point (see Eq. (178)). It is natural to expect that $f(\hat{z},\delta\varphi)$ becomes a universal function of $\hat{z}/\xi_{\rm d}^{\rm profile}$. Note also that by definition

[TABLE]

In fact Fig.3 (c) implies the scaling holds in $\delta\varphi\to 0$ limit.

One can expect that $\xi_{\rm d}^{\rm profile}$ agrees with $\xi_{\rm d}^{\rm hessian}$ based on the following argument. In our setup the region outside the cavity consists of frozen, immobile spheres. Let us imagine to replace them with spheres confined around the equilibrium reference points by some Hookian springs. Indeed the field $\epsilon(\hat{z})$ which is introduced through Eq. (109) as the field conjugated to the order parameter $q_{ab}(\hat{z})$ and thus $\Delta(\hat{z})$ (see Eq. (137) and Eq. (139)) exactly serves as such a Hookian spring. At the level of linear response, the change of the density profile $\delta\Delta(\hat{z})$ due to a change in the strength of the Hookian spring $\delta\epsilon(\hat{z}^{\prime})$ is related to the spatial correlation function of the spontaneous fluctuation of the mean-squared displacement $\delta\Delta(\hat{z})$ as,

[TABLE]

This implies introduction of Hookian springs of strength $\epsilon$ in the region $-\infty<\hat{z}^{\prime}<0$ on an initially homogeneous system amounts to inducing a change of the density profile at distance $\hat{z}$ from the wall as,

[TABLE]

where we used Eq. (32). To recover the original setup in which the system is completely frozen outside the cavity we have to consider $\epsilon\to\infty$ which goes beyond the simple linear-response argument. Nonetheless the above argument hints the relation $\xi_{\rm d}^{\rm profile}=\xi_{\rm d}^{\rm hessian}$.

IV.2.4 Correlation length extracted from the fluctuation around the saddle point in the bulk system

As we discussed in sec IV.1, characteristic length scale associated with the dynamical transition can also be obtained by analyzing fluctuations around the saddle point. For the hard spheres we obtained the key parameters

[TABLE]

as explained in Appendix C.6. In Fig. 3 (c) we also display $\xi_{\rm d}^{\rm hessian}$ we obtained using $M_{0}$ and $M_{2}$ in Eq. (34). Remarkably $\xi_{\rm d}^{\rm hessian}$ perfectly matches with the characteristic length $\xi_{\rm d}^{\rm profile}$.

IV.2.5 Summary of the behavior of the system close to the dynamical transition point

To summarize, we found three length scales i) the point-to-set length $\xi_{\rm d}^{\rm PS}$, ii) the correlation length of thermal fluctuation $\xi_{\rm d}^{\rm hessian}$ and iii) the characteristic length of the spatial profile of the glass order parameter in the cavity $\xi_{\rm d}^{\rm profile}$ all scales as $\delta\varphi^{-1/4}$ approaching the dynamical glass transition point $\hat{\varphi}_{\rm d}$. Our result confirms that the point-to-set length is proportional to the correlation length.

In the present paper we only considered the intra-state thermal fluctuation reflected in the longitudinal mode of the Hessian matrix for simplicity. More generally one has to consider inter-state and sample-to-sample fluctuations reflected in the replicon and anomalous modes all of which contribute the so called $\chi_{4}$ franz2013static ; franz2011field ; PhysRevE.106.024605 .

As far as we are aware of, the fact that the spatial profile of the glass order parameter $\Delta(\hat{z})$ also reflect a correlation length $\xi_{\rm d}^{\rm profile}$ has not been appreciated in the context of glass physics (see however berthier2012static ). As shown in Fig.3 (c), we find $\xi_{\rm d}^{\rm profile}=\xi_{\rm d}^{\rm hessian}$. The situation appears very similar to the surface critical phenomena, for example of ferromagnets, where one observes that the spatial profile of the order parameter reflects the spatial correlation length of spontaneous thermal fluctuations binder1972phase ; kikuchi1985monte1 ; okabe1986general .

IV.2.6 Cavity size dependence of the Kauzmann transition

Finally let us focus on the Kauzmann transition in the cavity system. The complexity (structural entropy) $\Sigma$ can be computed using the free-energy Eq. (37) as explained in appendix E.2.

We find

[TABLE]

with

[TABLE]

The Kauzmann transition density of the cavity system obtained at $\Sigma=0$ is

[TABLE]

where $\hat{\varphi}_{K,\rm{bulk}}$ is the Kauzmann transition density for the bulk system $\hat{L}_{\rm cav}=\infty$. Thus in cavity systems the Kauzmann transition occurs at lower densities than in bulk systems and the transition density increases increasing the cavity size $\hat{L}_{\rm cav}$. Physically this can be understood again as the consequence of the constraint imposed by the frozen system outside the cavity.

From this result we defined the point-to-set (PS) length of the static transition $\xi^{\rm PS}_{\rm s}(\hat{\varphi})$ as

[TABLE]

treating $\hat{L}_{\rm cav}$ as a running parameter. We find

[TABLE]

V Conclusion and outlook

To conclude we constructed a framework of the inhomogeneous replicated liquid theory that can treat glasses whose physical properties evolve along a one-dimensional axis. The theory becomes exact in the limit of infinite transverse dimensions. Extension of the theory to describe $2$, $3$ dimensional inhomogeneities, with $\infty$ dimensional transverse space, is straight forward. The dynamical version of our theory which extends the dynamical mean-field theory for the bulk system maimbourg2016solution can also be constructed straightfowradly which we will report elsewhere.

In the present paper we successfully applied our scheme to analyze diverging length scales at dynamic/static glass transitions. In particular we implemented a cavity setup to measure the point-to-set length biroli2008thermodynamic bouchaud2004adam in the replicated liquid theory. Our results reconfirm the critical exponents obtained in previous theoretical studies. Relevance of the static length scales in glassy dynamics may be questioned as it remains very small when the dynamics become extremely slow PhysRevX.12.041028 . In the real glassy dynamics other features such as avalanche like process which facilitates the relaxation and enhance the dynamical heterogeneity may play important roles. However a recent work report that in deeply annealed glass the point-to-set length reasonably describes a dynamically observed characteristic length shiraishi2024characterizing .

The obvious limitation of our theory is that it is still very strongly ’mean-field’ and miss many subtle features in real finite dimensional liquids. In particular the presence/absence of the ’dynamic glass transition’ (or even the ideal ’static glass transition’) in realistic systems cannot be addressed. However we believe it will be useful as a tool to study properties of the solid, glassy states at low temperatures/ high densities, which have sufficiently long life-times, from 1st principles. It is straightforward to adapt the glass state following rainone2015following scheme in our setup. It will be particularly interesting to apply such a scheme to study the emergence of spatial inhomogeneities in amorphous solids under compression, shear e.t.c. approaching yielding and jamming jin2017exploring ; jin2018stability ; jin2021jamming ; pan2023review . Both yielding and jamming require precise microscopic descriptions as demonstrated by the $d\to\infty$ theories charbonneau2014exact ; rainone2015following . However the characteristic spatial heterogeneities wyart2005rigidity ; maloney2006amorphous associated with them have not been studid by 1st principle theories. Our $1+\infty$ dimensional theory will be a promising tool to investigate these problems.

Acknowledgements.

We thank Yuki Rea Hamano, Yusuke Hara, Atsushi Ikeda, Harkuni Ikeda, Macoto Kikuchi, Hyonggi Kim, Kang Kim, Norifumi Maruyama, Kota Mitsumoto Hideyuki Mizuno, Misaki Ozawa, Yuki Takaha, Touta Yoshida, Akira Warita and Francesco Zamponi for useful discussions. This work was supported by KAKENHI (No. 21K18146) (No. 20H00128) from MEXT, Japan and JST SPRING, Grant Number JPMJSP2138.

Appendix A Liquid theory in $1+(d-1)$ dimensions with $d-1\gg 1$

A.1 Basic setup

Here we develop a density functional expression of the free-energy of the system of $N$ particles given by the Hamiltonian Eq. (1). Let us write the number density (per unit length), which varies along $z$ as,

[TABLE]

and introduce an identity,

[TABLE]

where ${\cal D}\hat{\rho}(z)$ is a functional integration over $\hat{\rho}(z)$ and $\delta(...)$ is a functional delta. In the 2nd equation we introduced an integral representation of the functional delta introducing a function $\phi(z)$ which can be related to the so called intrinsic chemical potential hansen1990theory .

Then the free-energy of the system can be written as

[TABLE]

Inserting ’1’ in the last equation, using the identity Eq. (58), and performing some interchanges of the order of integrals one finds that the free-energy $F$ can be expressed formally as,

[TABLE]

with a functional $F[\hat{\rho}]$ defined as,

[TABLE]

Here the functional $G[\phi]$ is defined as,

[TABLE]

In Eq. (62) we also introduced the Mayer functionhansen1990theory ,

[TABLE]

and the ’activity’,

[TABLE]

In Eq. (62) we also introduced a parameter $\lambda$ to organize the Mayer expansion discussed below. It will be put back to $1$ after organizing the expansion.

Physically the functionals $F[\rho]$ and $G[\phi]$ introduced above can be regarded as a free-energy functionals. We anticipate that the functional integration in Eq. (61) can be done by the saddle point method for $N\gg 1$ which yields,

[TABLE]

with the saddle point $\phi^{*}(z)=\phi^{*}[\hat{\rho}](z)$ is determined by

[TABLE]

Thus Eq. (65) is a Legendre transformation associated with the Laplace transformation Eq. (61).

Now we evaluate $G[\phi]$ treating the effect of interactions perturbatively, i. e. Mayer expansion.

[TABLE]

Here $-\beta G_{0}[\phi]$ is the free-energy of the ideal gas which is obtained using the Stirling’s formula $\ln N!\sim N\ln N-N$,

[TABLE]

with $S=\int d^{d-1}x$ being the surface area of the cross-section (See Fig. 1). We also introduced

[TABLE]

which is the thermal average took within the non-interacting system.

The outline of the analysis goes as follows. We consider series expansions,

[TABLE]

The series for $G$ can be determined by analyzing Eq. (67). Using the result in Eq. (66) the series for $\phi^{*}$ can be obtained. Finally using these in Eq. (70) we will obtain the series for $F$. Up to 1st order one can find,

[TABLE]

To work out $\phi_{0}^{*}$ explicitly we use Eq. (68) in Eq. (66) and find,

[TABLE]

Using this in Eq. (71) we find the ideal-gas part or the entropic part of the free-energy as,

[TABLE]

Now let us consider the effect of interactions. From Eq. (67) we find,

[TABLE]

Using this in Eq. (72) we find,

[TABLE]

Collecting the above results we obtain up to 1st order in the Mayer-expansion,

[TABLE]

where we introduced

[TABLE]

which is the number density field per unit volume. Note that $\rho(z)$ must be normalized such that,

[TABLE]

Finally the thermodynamic free-energy $F$ can be obtained through Eq. (60) where the functional integral can be evaluated by the saddle point method for $N\gg 1$,

[TABLE]

Here we introduced a Lagrange multiplier $\beta\mu$ to impose the normalization condition Eq. (79) where $\mu$ can be regarded as the chemical potential.

As is well known hansen1990theory , the contributions from higher order terms in the Mayer expansion into the free-energy $F[\rho]$ can be represented by one-particle irreducible diagrams. It has been shown that they become negligible in the large dimensional limit $d\to\infty$ frisch1985classical ; PhysRevA.36.2422 ; 1130848328245474560 .

A.2 Large dimensional limit

Here we derive an expression of the free-energy functional useful in $d-1\gg 1$ limit. We consider two-body potential $v(r)$ characterized by a microscopic length scale $D$ such that it becomes a function of $\xi$ defined as,

[TABLE]

in $d\to\infty$ limit 1130848328245474560 . Then we can write

[TABLE]

where $\Omega_{d}$ is the volume of $d$-dimensional unit sphere. We also introduced a scaled coordinate $\hat{z}$ such that

[TABLE]

Finally we obtain

[TABLE]

using $\Omega_{d-1}=\sqrt{d/2\pi}\Omega_{d}$. Here we introduced

[TABLE]

The equilibrium density profile $\rho^{*}(z)$ is obtained as Eq. (80) using the chemical potential,

[TABLE]

where the chemical potential $\mu$ should be chosen such that

[TABLE]

becomes satisfied.

In the case of hard-sphere like systems, $D$ is the radius of particles. For those cases it is convenient to introduce the volume fraction $\varphi(\hat{z})$,

[TABLE]

In order to study glassy sates in the large dimensional limit it is convenient to introduce a scaled volume fraction 1130848328245474560 ,

[TABLE]

Using this the free-energy can be expressed as

[TABLE]

where we omitted a constant $N(\Omega_{d}D^{d}/d)\ln((\lambda_{\rm th}/D)^{d}(d/\Omega_{d}))$. Here we also introduced a parametrization of the external potential,

[TABLE]

which is convenient to consider $d\to\infty$ limit. Similarly the chemical potential can be expressed as

[TABLE]

This expression implies that the external potential can be designed to realize any desired density profile $\hat{\varphi}(z)=O(1)$.

A.3 Compression

Let us discuss compression (or decompression) of our system. To this end we parameterize the changes of the volume as

[TABLE]

Thus we are compressing for $\eta>0$ and decompressing for $\eta<0$. A change of the volume amounts to a change of the boundary condition. By writing the original coordinate system as ${x^{\prime}_{1},x^{\prime}_{2},\ldots,x^{\prime}_{d-1},z^{\prime}}$, we can introduce a new coordinate system ${x_{1},x_{2},\ldots,x_{d-1},z}$ with $x_{\mu}=x^{\prime}_{\mu}(1+\eta/d)$ for $\mu=1,2,3,\ldots,d-1$ and $z_{\mu}=z^{\prime}_{\mu}(1+\eta/d)$. With the new coordinate system, the boundary condition is brought back to the original one.

Then the expression of the free-energy Eq. (60) becomes

[TABLE]

Then the free-energy functional Eq. (84) becomes

[TABLE]

We can compute the pressure as

[TABLE]

using $V\frac{\partial}{\partial V}=-\frac{\partial}{\partial\eta}$. The reduced pressure is obtained as,

[TABLE]

with $\rho=N/V$. We have $N=(SD/\sqrt{d})\int d\hat{z}\rho(\hat{z})$ (see Eq. (87)). In the last equation we performed an integration by parts.

In the 2nd term of Eq. (97) we find a contribution to the pressure due to the external potential. Disregarding the latter, the reduced pressure verifies

[TABLE]

which is equivalent to the thermodynamic relation $PV=\mu N-F$.

Appendix B Replicated liquid theory in $1+(d-1)$ dimensions with $d-1\gg 1$

B.1 Basic setup

Now we turn to derive the free-energy expression for the glassy states. The free-energy $F$ of the system is related to the logarithm of the partition function which can be expressed in terms of a replicated system,

[TABLE]

where the partition function of the replicated systems $a=1,2,\ldots,n$ is given by,

[TABLE]

In the last equation, anticipating the spontaneous formation of the molecular liquid state, we divided the $n$ replicas into $n/m$ subgroups ${\cal C}=1,2,\ldots,n/m$ each of which consists of $m$ replicas. Group ${\cal C}=1$ consists of replicas $a=1,2,\ldots,m$, ${\cal C}=2$ consists of replicas $a=m+1,m+2,\ldots,2m$ and so on. Thus we can write,

[TABLE]

It has been established in $d\to\infty$ 1130848328245474560 that the parameter $m$ should be set as follows. In the genuine liquid phase at high enough temperatures/small enough densities, $m=1$. At low enough temperatures/large enough densities beyond the so called Kauzmann transition (static glass transition) ideal glass phase can emerge where $m$ should be chosen such that the complexity remains [math] Mo95 . There is an intriguing intermediate temperature/density bounded by the so called dynamical glass transition and the Kauzmann transition. There are a large number of glassy metastable states emerge but the system is categorized still as a liquid $m=1$ in the thermodynamic sense. The information of the glassy metastable states is contained in the so called Franz-Parisi’s potential FP95 which is a term proportional to $1-m$ within the replicated free-energy $-\beta mF$ Eq. (101).

The partition function associated with the group of $m$ replicas reads as,

[TABLE]

In the last equation, again anticipating the spontaneous formation of the molecular liquid state, we introduced new coordinates for spatial integrations writing the coordinate $\bm{x}_{i}^{a}$ of particle $i$ in the $a$ replica as,

[TABLE]

This representation is convenient when the replicated liquid becomes a ’molecular liquid’ with ’molecules’ $i=1,2,\ldots,N$ whose centers of mass are located at

[TABLE]

The 2nd term in the r.h.s of Eq. (103) represents the thermal fluctuations within the molecules. Note that a sum rule

[TABLE]

must hold for the relative coordinates. Note also that there are $(N!)^{m-1}$ different permutations of the particles labels by $i=1,2,\ldots,N$ and $a=1,2,\ldots,m$ to form such molecules and that the system (Hamiltonian) is invariant under these permutations.

In the following we will use the molecular coordinate Eq. (103) but drop the subscript $c$ for the center of mass to lighten the notations. Let us write the number density (per unit length) field of the ’molecules’, which varies along $z$ as,

[TABLE]

and introduce an identity,

[TABLE]

Let us introduce the space dependent glass order parameter $q_{ab}(z)$ as,

[TABLE]

where $\langle\ldots\rangle$ is the appropriate thermal average. Finiteness of it means formation of a molecular liquid state, i. e. a glass state. On the contrary, in a genuine liquid state, such molecules should be dissociated so that this parameter diverges. Based on this observation let us introduce another identity,

[TABLE]

Note that

[TABLE]

because of the sum rule Eq. (105). Thus we find $q_{am}=-\sum_{b=1}^{m-1}q_{ab}$, $q_{mb}=-\sum_{a=1}^{m-1}q_{ab}$ and $q_{mm}=\sum_{b=1}^{m-1}\sum_{b=1}^{m-1}q_{ab}$.

Using these we can write,

[TABLE]

where we defined

[TABLE]

with

[TABLE]

Note that $F_{m}[\hat{\rho},q]$ and $G_{m}[\phi,\epsilon]$ are related to each other by the Legendre transform. The functional integrations in Eq. (112) can be done by the saddle point method for $N\gg 1$ which yield,

[TABLE]

with the saddle point $(\phi^{*}(z),\epsilon^{*}(z))$ is determined by,

[TABLE]

In Eq. (113), similarly to Eq. (62), we introduced a parameter $\lambda$ to organize the Mayer expansion. There we also introduced the replicated Mayer function,

[TABLE]

and the non-interacting part of the free-energy or the free-energy of ideal-gas made of the ’molecules’,

[TABLE]

with

[TABLE]

where we introduced the ’activity’,

[TABLE]

In Eq. (113) we also introduced,

[TABLE]

which is the thermal average took within the non-interacting system.

Let us note that we replaced $U(z+(u^{d})_{a})$ by $U(z)$ dropping the correction terms due to $u^{d}_{a}$. As we will see below $|{\bf u}|^{2}\sim O(1/d))$ so that the correction terms can be neglected in $d\to\infty$ limit.

Again we treat the effect of interactions perturbatively assuming the parameter $\lambda$ defined in Eq. (113) is small. Similarly to Eq. (65) we consider series expansions,

[TABLE]

in terms of $\lambda$. We find,

[TABLE]

To work out $\phi_{0}^{*}$ and $\epsilon^{*}_{ab}$ we use Eq. (118) and Eq. (119) in Eq. (116) and find,

[TABLE]

using this in Eq. (123) we find the ideal gas part (entropic) part of the free-energy,

[TABLE]

We also used Eq. (78) which reads $\rho(z)=\hat{\rho}(z)/S$.

Now let us consider the effect of interactions as we did in the previous section. From Eq. (113) we find,

[TABLE]

Using this in the last equation of Eq. (122) we find,

[TABLE]

where we defined,

[TABLE]

which is the thermal average of fluctuations within a molecule and the average over the solid angle $\Omega$ associated with the displacement vector ${\bf r}_{12,\perp}$,

[TABLE]

Now

[TABLE]

Here we drooped $((u^{d}_{1})^{a}-(u^{d}_{2})^{a})(z_{1}-z_{2})$ anticipating $|{\bf u}|^{2}\sim O(1/d)$.

Now writing

[TABLE]

we find

[TABLE]

Let us pause for a moment to investigate the averaging over the solid angle $\Omega$ of the unit vector $\hat{r}_{12,\perp}={\bf r}_{12,\perp}/r_{12,\perp}={\bf y}/\sqrt{d-1}$ defined in Eq. (131). We can notice that the average over the solid angle $\Omega$ can be done assuming the vector $\hat{r}_{12,\perp}$ obey a Gaussian distribution in $d\to\infty$ limit,

[TABLE]

This can be seen by writing

[TABLE]

In $d-1\to\infty$ limit the integration over $\kappa$ can be done (formally) by the saddle point method so that different $y_{\mu}$’s can actually be regarded as independent Gaussian random variables with zero mean and unit variance. Based on this observation we find,

[TABLE]

In the last equation we introduced $\alpha_{ab}$ such that

[TABLE]

In order to have sensible results in $d\to\infty$ limit we consider $\alpha_{ab}\sim O(1)$ which means $\langle u^{2}\rangle\sim O(1/d)$. With this scaling we could drop higher order terms that appear in the cumulant expansion of $\ln\langle\ldots\rangle_{\epsilon}$. Note also that $\langle{\bf u_{\perp}}_{a}\cdot{\bf u_{\perp}}_{b}\rangle_{\epsilon}=q_{ab}$ dropping $1/d^{2}$ correction.

Using this back in Eq. (134) we find,

[TABLE]

Here we used Eq. (81) $r_{12,\perp}=D(1+\xi/(d-1))$, $\frac{\partial}{\partial y_{a}}=(d-1)/(2D^{2})\frac{\partial}{\partial\xi_{a}}$ and the identity $f(x+a)=e^{a\frac{d}{dx}}f(x)$ for a generic function $f(x)$. We introduced

[TABLE]

assuming

[TABLE]

for $a=1,2,\ldots,m$. We also used Eq. (83) which reads $z=(D/\sqrt{d})\hat{z}$. Then using these expressions back in Eq. (129) and using

[TABLE]

we find

[TABLE]

In the last equation we performed integrations over parts with respect to $\xi$ to eliminate the term $\alpha_{d}...$ in Eq. (129). We also used Eq. (78) which reads $\rho(\hat{z})=\hat{\rho}(z)/S$.

On the other hand the entropic part of the free-energy Eq. (127) can be rewritten as,

[TABLE]

Again we used Eq. (83) which reads $z=(D/\sqrt{d})\hat{z}$.

To wrap up the results we find

[TABLE]

with

[TABLE]

where $\rho^{*}(\hat{z})$ and $q_{ab}^{*}(\hat{z})$ are solutions of the saddle point equations,

[TABLE]

The 1st equation yields the chemical potential,

[TABLE]

Using the scaled volume fraction $\hat{\varphi}$ defined in Eq. (89) we can write

[TABLE]

Similarly the chemical potential can be expressed as,

[TABLE]

It is useful to recall the special case of uniform density profile $\hat{\varphi}(\hat{z})=\hat{\varphi}$ and spatially uniform glass order parameter $\Delta_{ab}(\hat{z})=\Delta_{ab}$. In this case, the free-energy becomes,

[TABLE]

Here we have switched off the external potential $U$.

B.2 Compression

Now let us extend the analysis in sec. A.3 to discuss compression (or decompression) of our system in the glassy states. In the replicated system we may consider compressing each replica differently using $\eta_{a}$ ($a=1,2,\ldots,m$).

Then the expression of the free-energy Eq. (101) becomes

[TABLE]

Then the free-energy functional Eq. (145) becomes

[TABLE]

We can compute the pressure assuming uniform deformation $\eta_{a}=\eta$ $(a=1,2,\ldots,m)$ and using Eq. (96). Here we omitted the contribution from the external potential. The reduced pressure is obtained as,

[TABLE]

with $\rho=N/V$. We have $N=(SD/\sqrt{d})\int d\hat{z}\rho(\hat{z})$ (see Eq. (87)). It can be seen that it verifies

[TABLE]

which is equivalent to the thermodynamic relation $PV=\mu N-F$.

Appendix C One step RSB solution

C.1 1RSB ansatz

As we described at the beginning of sec B.1 we are considering ’molecular liquid’ made of $m$ replicas. For the space dependent order parameter $\alpha_{ab}(\hat{z})$ with $a=1,2,\ldots,m$ and $b=1,2,\ldots,m$, we consider the simplest ansatz,

[TABLE]

This ansatz reflects the symmetry of the system under permutations of the replicas $1,2,\ldots,m$. This is the so called replica symmetry so that this ansatz may be called as a replica symmetric (RS) ansatz 1130848328245474560 . In the present paper we prefer to call this ansatz as a one-step replica symmetry broken (1RSB) ansatz because we are considering a realization of a molecular liquid state where the replica symmetry involving all replicas $1,2,...,n$ is reduced down to that within a molecule $1,2,\ldots,m$ as we described in sec B.1.

Because of the sum rule Eq. (110) and Eq. (137) we find $\alpha_{d}(\hat{z})$ becomes $\alpha_{d}(\hat{z})=(m-1)\alpha(\hat{z})$. The we can rewrite the ansatz as,

[TABLE]

Equivalently using $\Delta_{ab}(\hat{z})$ defined in Eq. (139) the ansatz can be written also as,

[TABLE]

with $\Delta(\hat{z})=2(\alpha_{d}(\hat{z})+\alpha(\hat{z}))=2m\alpha(\hat{z})$.

C.2 Free energy

Now let us evaluate the free-energy Eq. (145) using the 1RSB ansatz. In the entropic part of the free energy we find,

[TABLE]

In the interaction part of the free-energy we find,

[TABLE]

where

[TABLE]

We have also introduced

[TABLE]

To derive the above equations we used the ideitities Eq. (222), Eq. (223) and the short-hand notation Eq. (224).

To sum up, we obtain the free energy within the 1RSB ansatz as,

[TABLE]

Or equivalently

[TABLE]

In the last equation the term ’const’ means contributions independent of $\Delta(\hat{z})$ and we introduced,

[TABLE]

which is the so called Franz-Parisi’s potential.

C.3 Equation of states

The integration over $q_{ab}$ Eq. (111) can be done by the saddle point method. The saddle point is found by solving,

[TABLE]

which yields a self-consistent equation of $\Delta(\hat{z})$,

[TABLE]

where we introduced

[TABLE]

C.4 Uniform system

Let us recall the special case of uniform density profile $\hat{\varphi}(\hat{z})=\hat{\varphi}$ and spatially uniform glass order parameter $\Delta(\hat{z})=\Delta$. In this case, the free-energy Eq. (C.2) becomes, switching off the external potential $U$,

[TABLE]

with $g(\xi,\Delta)$ defined in Eq. (161) and the term ’const’ representing contributions independent of $\Delta$ and

[TABLE]

is the Franz-Parisi’s potential.

On the other hand the equation of state Eq. (C.3) becomes for the uniform glass state,

[TABLE]

Taking another derivative we find the Hessian of the uniform system

[TABLE]

with

[TABLE]

C.5 Dynamical transition in uniform system

Solution to the equation of state Eq. (170) has been studied in detail in previous works. For instance in the the case of hard-spheres one finds non-trivial solutions $\infty>\Delta>0$ for high enough densities,

[TABLE]

where $\hat{\varphi}_{\rm d}(m)$ is the so called dynamical transition density. The dynamical transition point can be considered as a sort of a spinodal point: the non-trivial solution associated with a local minimum of the free-energy, which exists at higher densities, disappears there.

Close to the dynamical transition point we may expand the Franz-Parisi’s potential as

[TABLE]

with

[TABLE]

At the saddle point the 1st derivative must vanish (the equation of state Eq. (170) ),

[TABLE]

and the 2nd derivative (Hessian) is obtained as,

[TABLE]

Considering Eq. (176) at $\hat{\varphi}=\hat{\varphi}_{\rm d}$ we find $A_{0}=0$. We also note that 2nd derivative must vanish at $\hat{\varphi}=\hat{\varphi}_{\rm d}$ since it is a spinodal point as stated above, which implies $B_{0}=0$. Using these observations in Eq. (176) we find

[TABLE]

Then this implies

[TABLE]

at the saddle point close to the dynamical transition density $\hat{\varphi}_{\rm d}$.

C.6 Longitudinal Hessian

In order to study the stability of the solutions to the saddle point equation we have to examine the Hessian matrix. In the present paper we limit ourselves to what is called as ’longitudinal mode’,

[TABLE]

with $X(\Delta)$ defined in Eq. (172).

Then the free-energy around a glass state characterized with $\Delta^{*}(\hat{z})$ (which verifies the equation of state Eq. (C.3)) can be expanded as

[TABLE]

with

[TABLE]

For simplicity let us consider a glass state with spatially uniform density profile $\hat{\varphi}(\hat{z})=\hat{\varphi}$ and spatially uniform glass order parameter $\Delta(\hat{z})=\Delta$. In this case the longitudinal Hessian becomes translationally invariant,

[TABLE]

Introducing Fourier transforms as

[TABLE]

we find the integral in the 2nd term in the r.h.s of Eq. (181) becomes

[TABLE]

with

[TABLE]

Thus we find

[TABLE]

with

[TABLE]

Using these results we find the spatial correlation function of the fluctuation of the glass order parameter as,

[TABLE]

with the correlation length $\xi_{\rm d}^{\rm hessian}$ given by

[TABLE]

Note that $M_{0}$ is nothing but the Hessian of the bulk system (see Eq. (171)). Thus we must have

[TABLE]

We have also expected a scaling feature of $M_{0}$ approaching the dynamical transition point as Eq. (179) which reads as

[TABLE]

with $\delta\hat{\varphi}$ defined in Eq. (35) which measures the distance to the critical point. This implies

[TABLE]

Performing numerical analysis in the hard-sphere system for the case $m=1$, we indeed find

[TABLE]

as shown in Fig. 4. Here $M_{2}=\frac{1}{2\Delta_{\rm d}^{2}}$ follows using $M_{0}=0$ which holds at the critical point in Eq. (188) with $m=1$.

C.7 Chemical potential and pressure

In the present paper we will not investigate the ideal glass phase beyond the Kauzmann transition where $0<m<1$. As long as $m=1$, the chemical potential Eq. (149) and the (reduced) pressure Eq. (153) become the same as those of the liquid given by Eq. (92) and Eq. (97). We will consider compression on the glassy metastable state performing state following rainone2015following , which amount to compress $m-1$ subset of replicas (see sec B.2) in a subsequent work.

C.8 Complexity

Let us analyze the complexity using the 1RSB free-energy Eq. (162) or Eq. (163). To this end we have to examine first more closely the constant contribution to the entropic part of the free-energy. We find

[TABLE]

Here we used $\Omega_{d}=\pi^{\frac{d}{2}}/\Gamma(1+d/2)$ and $\Gamma(1+z)\sim\sqrt{2\pi z}\quantity(z/e)^{z}$ ($z\gg 1$). Using this we find

[TABLE]

Using the above expression we find the 1RSB free energy Eq. (163) as,

[TABLE]

The complexity $\Sigma$ per molecule can be derived as $\Sigma^{*}=-m^{2}\partial_{m}\quantity(-\frac{\beta F_{m}^{\rm 1RSB}}{mN})$ 1130848328245474560 . We have $N=(SD/\sqrt{d})\int d\hat{z}\rho(\hat{z})=(SD/\sqrt{d})\frac{d}{\Omega_{d}D^{d}}\int d\hat{z}\hat{\varphi}(\hat{z})$ (see Eq. (87) and Eq. (89)). Thus we obtain the complexity per molecule as,

[TABLE]

Appendix D Cavity system

Here we consider a cavity system of size $\hat{L}_{\rm{cav}}$ defined in the region $0<\hat{z}<\hat{L}_{\rm{cav}}$. The glass order parameter outside the cavity is set to zero

[TABLE]

The density is set to be uniform $\hat{\varphi}(\hat{z})=\hat{\varphi}$ across the whole system inside and outside the cavity.

In the cavity system the free-energy Eq. (148) becomes,

[TABLE]

where

[TABLE]

Note that the integrand in the double integral is symmetric with respect to the exchange of $\hat{z}_{1}$ and $\hat{z}_{2}$ so that we can replace the double integral by

[TABLE]

We see that in the last expression the first term represents the interaction between particles inside the cavity and the second term represents that between particles inside and outside the cavity. From the above expression we obtain the self-consistent equation for the glass order parameter in the cavity.

Assuming the 1RSB solution within the cavity, the saddle point equation Eq. (C.3) becomes modified as,

[TABLE]

Similarly the complexity Eq. (198) becomes,

[TABLE]

Appendix E Hard-sphere system

Here we collect some details for the hard-sphere system.

E.1 Basics

We consider hard-spheres whose interaction potential $v(r)$is given by

[TABLE]

where $D$ is the diameter with $\theta(x)$ being the Heaviside step function. Using the scaled coordinate $\xi$ Eq. (81) we find,

[TABLE]

Then the factor ${\cal F}$ defined by Eq. (85) becomes

[TABLE]

The function $g(\xi,\Delta)$ defined in Eq. (161) becomes

[TABLE]

where

[TABLE]

and $\erf(x)$ is the error function. Similarly, $f^{\prime}(\xi,\Delta)=-\partial_{\xi}\ln{g(\xi,\Delta)}$is,

[TABLE]

where we introduced

[TABLE]

See Sec F.1 for more details.

We obtain the self-consistent equation of order parameter $\Delta(\hat{z})$ in hard-sphere system as,

[TABLE]

where we introduced,

[TABLE]

Let us consider here a very large density regime where $\Delta(\hat{z})$ will become very small. In such a regime we find

[TABLE]

using this is in Eq. (212) we find

[TABLE]

This is consistent with

[TABLE]

which is known for the bulk system parisi2010mean .

E.2 Kauzmann transition in cavity

Let us consider the complexity in the cavity system Eq. (204) of the hard-sphere system at high densities where $\Delta(\hat{z})\sim 0$. We find

[TABLE]

where we introduced

[TABLE]

and

[TABLE]

To derive the above results we also used Eq. (202), Eq. (236) and Eq. (240).

Therefore, the Kauzmann transition density $\hat{\varphi}_{K}(\hat{L}_{\rm{cav}})$ of the cavity system at which the complexity vanishes is obtained as,

[TABLE]

where $\hat{\varphi}_{K,\rm{bulk}}$ is the Kauzmann transition density for the bulk system $\hat{L}_{\rm cav}=\infty$. Thus in cavity systems the Kauzmann transition occurs at lower densities than in bulk systems and the transition density increases increasing the cavity size $\hat{L}_{\rm cav}$.

We obtain the PS length at the Kauzmann transition,

[TABLE]

Thus, the exponent of the PS length at the Kauzmann transition is $-1$.

Appendix F Useful Formulas

The following formula can be proved by taking the direct derivative.

[TABLE]

The following formula can be proved with the Taylor expansion $f(h+\delta)=\sum_{n=0}^{\infty}\frac{\delta^{n}}{n!}\partial_{h}^{n}f(h)$.

[TABLE]

where we introduced a short-hand notation,

[TABLE]

F.1 Asymptotic behavior of the error function

The err function $\erf(x)$ is an odd function,

[TABLE]

The behavior of the error function at $x\to\infty$ is

[TABLE]

This can be proved as follows.

[TABLE]

With $\partial_{x}e^{-x^{2}}=-2xe^{-x^{2}}$, for the second term,

[TABLE]

Using the above equation, the behavior of $\Theta(x)$ at $x\to\infty$ is,

[TABLE]

And,

[TABLE]

behaves asymptotically like

[TABLE]

When performing numerical calculations using $\Theta(x)$ or $r(x)$, you can use these asymptotic expressions.

F.2 The function of $f(L)$

We derive

[TABLE]

For the err function $\erf(x)$,

[TABLE]

With this,we can derive

[TABLE]

In addition, the asymptotic behavior of

[TABLE]

in $x\to\infty$ is

[TABLE]

from Eq.226.

Appendix G Collection of some useful formulas

G.1 Proof of $\Omega_{d-1}=\sqrt{d/2\pi}\Omega_{d}$

The $\Omega_{d}$ is the volume of a hyper-sphere of radius $1$ in $d$-dimension,

[TABLE]

Here, $\Gamma$ if gamma function. With the Stirling’s approximation,

[TABLE]

For sufficiently large $d$,

[TABLE]

because

[TABLE]

Thus we get

[TABLE]

G.2 Integration of Gaussians

The integral of a Gaussian is as follows.

[TABLE]

[TABLE]

Appendix H Numerical details to solve the integral equation Eq. (203)

We solved the integral equation Eq. (203) to obtain the space dependent order parameter $\Delta(\hat{z})$ for the system of hard-spheres in the cavity as follows.

The cavity region $0<\hat{z}<\hat{L}_{\rm cav}$ is divided into segments of width $d\hat{z}=0.1$ for each of which we evaluate $\Delta(\hat{z})$ as follows. 2. 2.

The integrations over $\hat{z}^{\prime}$ and $\xi$ are done using the simple Euler method with $d\hat{z}^{\prime}=d\xi=0.1$. For the evaluation of the function $g(\xi,\Delta)$ we used the expression Eq. (208) with Eq. (209). For the evaluation of the function $f^{\prime}(\xi,\Delta)$, we used Eq. (210), Eq. (211) and Eq. (235). 3. 3.

Let us call the initial guess for $\Delta(\hat{z})$ (see 6) as $\Delta(\hat{z};t=0)$. 4. 4.

We evaluate the r.h.s of Eq. (203) using $\Delta(\hat{z};t)$ to obtain $1/\Delta(\hat{z};t+1)$. 5. 5.

The procedure 4 is repeated for the iteration steps $t=0,1,2,\ldots$ until the order parameters converge. 6. 6.

For the initial value of $\Delta(\hat{z},t=0)$, we used the solution of the self-consistent equation Eq. (170) for the bulk system with $m=1$. Eq. (170) is also solved iteratively until the solution converges.