is also required when using free boundary conditions, but with an applied field . If the applied field is small enough that it can be treated as a perturbation to the system in zero-field, then the magnetic susceptibility can be estimated. This is done by using the eigenstates computed by the transfer matrix approach and computing the energy shift with second-order perturbation theory, then comparing with the free-energy expansion . One finds where is the Curie constant (a value typically associated with the susceptibility in magnetic materials). This expression is also true for the one-dimensional Ising model, with the replacement . Two dimensions The two-dimensional XY model with nearest-neighbor interactions is an example of a two-dimensional system with continuous symmetry that does not have long-range order as required by the Mermin–Wagner theorem. Likewise, there is not a conventional phase transition present that would be associated with symmetry breaking. However, as will be discussed later, the system does show signs of a transition from a disordered high-temperature state to a quasi-ordered state below some critical temperature, called the Kosterlitz-Thouless transition. In the case of a discrete lattice of spins, the two-dimensional XY model can be evaluated using the transfer matrix approach, reducing the model to an eigenvalue problem and utilizing the largest eigenvalue from the transfer matrix. Though the exact solution is intractable, it is possible to use certain approximations to get estimates for the critical temperature which occurs at low temperatures. For example, Mattis (1984) used an approximation to this model to estimate a critical temperature of the system as The 2D XY model has also been studied in great detail using Monte Carlo simulations, for example with the Metropolis algorithm. These can be used to compute thermodynamic quantities like the system energy, specific heat, magnetization, etc., over a range of temperatures and time-scales. In the Monte Carlo simulation, each spin is associated to a continuously-varying angle (often, it can be discretized into finitely-many angles, like in the related Potts model, for ease of computation. However, this is not a requirement.) At each time step the Metropolis algorithm chooses one spin at random and rotates its angle by some random increment . This change in angle causes a change in the energy of the system, which can be positive or negative. If negative, the algorithm accepts the change in angle; if positive, the configuration is accepted with probability , the