Quantum phase transitions without thermodynamic limits
Abstract
A new microcanonical equilibrium state is introduced for quantum systems with finite-dimensional state spaces. Equilibrium is characterized by a uniform distribution on a level surface of the expectation value of the Hamiltonian. The distinguishing feature of the proposed equilibrium state is that the corresponding density of states is a continuous function of the energy, and hence thermodynamic functions are well defined for finite quantum systems. The density of states, however, is not in general an analytic function. It is demonstrated that generic quantum systems therefore exhibit second-order phase transitions at finite temperatures.
1. Introduction
The derivation of phase transitions in quantum statistical mechanics typically requires the introduction of a thermodynamic limit, in which the number of degrees of freedom of the system approaches infinity. This limit is needed because the free energy of a finite system is analytic in the temperature. However, phase transitions are associated with the breakdown of the analyticity of thermodynamic functions such as the free energy. Hence, in the canonical framework, the thermodynamic limit is required to generate phase transitions. Although the existence of this limit has been shown for various systems (Glimm & Jaffe 1987; Ruelle 1999), the procedure can hardly be regarded as providing a satisfactory description of critical phenomena. In any event, the fact is that phase transitions are known now to occur in finite quantum systems (Gross 2001). For example, the spherically symmetric cluster of 139 sodium atoms exhibits a solid-to-liquid phase transition at approximately 267 K, a temperature that is some 104 K greater than the melting point of bulk sodium (Schmidt et al. 1997). The canonical framework is inadequate as a basis for the understanding of such phenomena.
One can consider, alternatively, the possibility of a derivation based on the microcanonical ensemble. The usual construction of this ensemble (Kittel 1958; Landau & Lifshitz 1980; Huang 1987) is to define the entropy by setting S=kB ln nE, where nE is the number of energy levels in a small interval [E, E+ΔE], degenerate energy levels being counted with the appropriate multiplicity. The temperature is then obtained from the thermodynamic relation T dS=dE. This approach, however, is not well formulated because (a) it relies on the introduction of an arbitrary energy band ΔE, and (b) the entropy is a discontinuous function of the energy. To resolve these difficulties, a scheme for taking the thermodynamic limit in the microcanonical framework was introduced in Griffiths (1965). For finite quantum systems, however, the difficulties have remained unresolved, and it is an open issue how best to address the problem of phase transitions for such systems.
The purpose of this paper is to demonstrate the following: (i) if the microcanonical density of states is defined in terms of the relative volume, in the space of pure quantum states, occupied by the states associated with a given energy expectation E, then the entropy of a finite-dimensional quantum system is a continuous function of E, and the temperature of the system is well defined; and (ii) the density of states so obtained is in general not analytic, and thus for generic quantum systems predicts the existence of second-order phase transitions, without the consideration of thermodynamic limits.
It is remarkable in this connection that similar types of second-order transitions have been observed recently for classical spin systems, for which the associated configuration space possesses a non-trivial topological structure (Casetti et al. 2003; Kastner & Schnetz 2006).
The paper is organized as follows. We begin with the analysis of an idealized quantum gas to motivate the introduction of a new microcanonical distribution. This leads to a natural definition of the density of states Ω(E). Unlike the number of microstates nE, the microcanonical density Ω(E) is continuous in E. As a consequence, we are able to determine the energy, temperature, and specific heat of elementary quantum systems, and work out their properties. In particular, we demonstrate that in the case of an ideal gas of quantum particles, each particle being described by a finite-dimensional state space, the system exhibits a second-order phase transition where the specific heat decreases abruptly.
2. Ideal gas model
Let us consider a system that consists of a large number N of identical quantum particles (for simplicity, we ignore issues associated with spin-statistics). We write 
for the Hamiltonian of the composite system, and 
(i=1, 2, …, N) for the Hamiltonians of the individual constituents of the system. The interactions between the constituents are assumed to be weak, and hence to a good approximation we have
If the system as a whole is in isolation, then for equilibrium we demand that the total energy of the composite system should be fixed at some value Etotal. In other words, we have 
. It follows that 
. Now consider the result of a hypothetical measurement of the energy of one of the constituents. In equilibrium, owing to the effects of the weak interactions, the state of each constituent should be such that, on average, the result of an energy measurement should be the same. That is to say, in equilibrium, the state of each constituent should be such that the expectation value of the energy is the same. Therefore, writing E=N−1Etotal for the specific energy, we conclude that in equilibrium the gas has the property that 
for i=1, 2, …, N. In other words, the state of each constituent must lie on the energy surface 
E in the pure state manifold for that constituent. Since N is large, this will ensure that the uncertainty in the total energy of the composite system, as a fraction of the expectation of the total energy, is vanishingly small. Indeed, it follows from the Chebyshev inequality that
For convenience, we can describe the distribution of the various constituent pure states, on their respective energy surfaces, as if we were considering a probability measure on the energy surface 
E of a single constituent. In reality, we have a large number of approximately independent constituents; but owing to the fact that the respective state spaces are isomorphic we can represent the behaviour of the aggregate system with the specification of a probability distribution on the energy surface of a single ‘representative’ constituent.
The point of view that we are taking can thus be summarized as follows. In classical mechanics, the phase space of an ideal gas in a finite container has a huge dimension; and if one had perfect knowledge of the initial conditions, then the subsequent dynamics of the gas would correspond to a dynamical trajectory through this phase space. In practice, we have only limited access to information about the gas, and as a consequence, the best we can do is to characterize the classical gas as a whole by the use of a probability distribution on the phase space of a single constituent of the gas. Likewise, although the state space of a quantum gas is given by the tensor product of a very large number of Hilbert spaces, in the ideal gas approximation the state of the system as a whole, to the extent that this can be determined, can be represented as a probability distribution on the space of pure states associated with a single constituent. The associated density matrix can then be worked out by use of a standard construction.
3. Microcanonical equilibrium
In equilibrium, the relevant probability distribution is uniform on the energy surface, since the equilibrium distribution should maximize an appropriate entropy functional on the set of possible probability distributions on 
E. From a physical point of view, we can argue that the constituents of the gas approach an equilibrium as follows: on the one hand, weak exchanges of energy result in all the states settling on or close to the energy surface; on the other hand, the interactions will induce an effectively random perturbation in the Schrödinger dynamics of each constituent, causing it to undergo a random motion on 
E that in the long run induces uniformity in the distribution on 
E. We conclude that the equilibrium configuration of a quantum gas is represented by a uniform measure on an energy surface of a representative constituent of the gas.
The theory of the quantum microcanonical equilibrium state presented here is analogous in many respects to the well-known symplectic formulation of the classical microcanonical ensemble described, for example, in Khinchin (1949), Ehrenfest & Ehrenfest (1959) and Thompson (1972). There is, however, a subtle difference. Classically, the uncertainty in the energy is fully characterized by the statistical distribution over the phase space, and for a microcanonical distribution with support on a level surface of the Hamiltonian the energy uncertainty vanishes. Quantum mechanically, however, although the statistical contribution to the energy variance vanishes, there remains an additional purely quantum-mechanical contribution. Hence, although the energy uncertainty for the composite system is negligible for large N, the energy uncertainties of the constituents will not in general vanish. An expression for energy uncertainty associated with a typical constituent will be given in equation (8.3) below.
4. Density of states
To describe the equilibrium represented by a uniform distribution on the energy surface 
E, it is convenient to use the symplectic formulation of quantum mechanics. Let 
denote the Hilbert space of states associated with a constituent. We assume that the dimension of 
is n+1. The space of rays through the origin of 
is a manifold Γ equipped with a metric and a symplectic structure. The expectation of the Hamiltonian along a given ray of 
then defines a Hamiltonian function
, λ∈
\0. The Schrödinger evolution on 
is a symplectic flow on Γ, and hence we may regard Γ as the quantum phase space. Specifically, Γ is given by the complex projective space 
n, regarded as a real manifold, endowed with the Fubini–Study metric. Our approach to quantum statistical mechanics thus unifies two independent lines of enquiry, each of which has attracted attention in recent years: the first of these is the ‘geometric’ or ‘dynamical systems’ approach to quantum mechanics, which takes the symplectic structure of the space of pure states as its starting point (Kibble 1979; Anandan & Aharonov 1990; Ashtekar & Schilling 1998; Benvegnù et al. 2004); and the second of these is the probabilistic approach to the foundations of quantum statistical mechanics, in which the space of probability distributions on the space of pure states plays a primary role (Jaynes 1957; Khinchin 1960; Brody & Hughston 1998; Goldstein et al. 2006; Jona-Lasinio & Presilla 2006).The level surface 
E in Γ is defined by H(ψ)=E. The entropy associated with the corresponding microcanonical distribution is
Consider a large system composed of two independent parts, each in a state of equilibrium. Each subsystem is thus described by a microcanonical state with support on the Segré variety corresponding to disentangled subsystem states. Let us write Ω1(E1) and Ω2(E2) for the associated state densities, where E1 and E2 are the initial specific energies of the two systems. Now imagine that the two systems interact weakly for a period of time, during which energy is exchanged, following which the systems become independent again, each in a state of equilibrium. As a consequence of the interaction, the state densities of the systems will now be given by expressions of the form Ω1(E1+ϵ/N1) and Ω2(E2−ϵ/N2), for some value of the total exchanged energy ϵ, where N1 and N2 represent the number of constituents belonging, respectively, to each of the two systems. The value of ϵ can be determined by the requirement that the total entropy
5. Phase transitions for finite quantum systems
The quantum microcanonical ensemble introduced here is applicable to any isolated finite-dimensional quantum system for which the ideal gas approximation is valid. The volume integral in equation (4.3) can be calculated by lifting the integration from Γ to 
and imposing the constraint that the norm of |ψ〉 is unity. Then we can write
is the volume element of 
. Making use of the standard Fourier integral representation for the delta function, and diagonalizing the Hamiltonian, we find that equation (5.1) reduces to a series of Gaussian integrals (see Brody et al. (2005) for further details). Performing the ψ-integration, we then eventually obtain the following integral representation for the density of states:
. Carrying out the integration, we find that the density of states is given by
{A} denotes the indicator function (
{A}=1 if A is true, and 0 otherwise). In equation (5.3), we let m denote the number of distinct eigenvalues Ej (j=1, 2, …, m), and δj the multiplicity associated with the energy Ej. Thus, 
. In the non-degenerate case, for which δj=1 for j=1, 2, …, m, there is some further simplification in formula (5.3), and we obtain
6. Non-degenerate spectra
In the case of a Hamiltonian with a non-degenerate spectrum of the form Ek=ϵ(k−1), for k=1, 2, …, n+1, where ϵ is a fixed unit of energy, the density of states (5.4) reduces to
Figure 1 Density of states Ω(E) for non-degenerate (n+1)-level systems for n=3, 4, 5, with energy eigenvalues Ek=k (k=0, …, n). Ω(E) is n−2 times differentiable in E.
The structure of the space of pure states in quantum mechanics is intricate, even for relatively elementary systems. In particular, as the value of the energy changes, the topological structure of the energy surface undergoes a transition at each eigenvalue (Brody & Hughston 2001). For example, in the case of a non-degenerate three-level system, the topology of the energy surface changes according to 
, as the energy is raised from Emin to Emax (
denotes a two-plane compactified into a two-sphere at a point corresponding to the intermediate eigenstate). These structural changes in the energy surfaces induce a corresponding non-trivial behaviour in the thermodynamic functions.
As an illustration, we consider a four-level system and compute the specific heat as a function of temperature. The result is shown in figure 2, where we observe that the specific heat drops abruptly from 2kB to kB/2 at the critical temperature Tc defined by kBTc=ϵ/2. Therefore, this system exhibits a second-order phase transition, in this case at the critical energy Ec=ϵ. This example shows that the relationships between phase transitions and topology discovered recently in classical statistical mechanics (Franzosi & Pettini 2004; Angelani et al. 2005; Kastner 2004, 2006) carry over to the quantum domain where, arguably, they may play an even more basic role. Figure 2 Specific heat for a non-degenerate four-level system (dotted line, n=3, Ej=0, 1, 2, 3), a four-level system having a degenerate first excited state (dashed line, n=4, Ej=0, 1, 1, 2, 3), and a quantum Ising chain (solid line, J=1/4, B=1). In the quantum Ising chain, we have C(T)∼(T−Tc)−2 away from Tc, whereas in the vicinity of Tc we have C(T)∼(T−Tc)−13 for T>Tc. (We set kB=1 here.)
For a system with a larger number of non-degenerate eigenstates, the specific heat also increases abruptly as T is reduced. In this case, the specific heat is continuous, and the discontinuity is in a higher-order derivative of the energy. For a system with n+1 non-degenerate energy eigenvalues, the (n−1)-th derivative of the energy with respect to the temperature has a discontinuity. The phenomenon of a continuous phase transition is generic and is also observed if the eigenvalue spacing is not uniform.
7. Degenerate spectra
In a system with a degenerate spectrum, the effects of the phase transition can be enhanced. In particular, the volume of 
E increases more rapidly as E approaches the first energy level from below, if this level is degenerate. This leads to a more abrupt drop in the specific heat (figure 2).
As an example, we consider a quantum Lenz–Ising ferromagnetic chain with three spins. The Hamiltonian is
denotes the third Pauli matrix for spin k, and J and B are constants. We have in mind a gas of weakly interacting molecules, each modelled by a strongly interacting quantum Ising chain. The eigenvalues of the Hamiltonian are E1=−3J−3B, E2,3,4=−J−B, E5,6,7=J+B, and E8=−3J+3B. As the temperature is reduced, the specific heat rises rapidly in the vicinity of the critical point
8. Density matrix and energy uncertainty
Finally, we show the existence of a natural energy band associated with the quantum microcanonical distribution. The microcanonical density matrix for the energy E is given by
denotes the projection operator onto the state 
corresponding to the point ψ∈Γ. The squared energy uncertainty is
denotes the uniform average of the energy eigenvalues. To check that ΔH vanishes at E=Emax, we note that the first moment of Ω(E) is given by the integral of H(ψ) over Γ. Hence, by use of a trace identity obtained in Gibbons (1992) we have
It remains to be seen whether the new quantum microcanonical distribution can be put to the test in some definitive way, and in particular whether the phase transitions it predicts actually correspond to observable phenomena.
D.C.B. acknowledges support from the Royal Society. The authors thank M. Kastner and T. W. B. Kibble for helpful comments.
