Anderson’s orthogonality catastrophe in one dimension induced by a magnetic field
According to Anderson’s orthogonality catastrophe, the overlap of the -particle ground states of a free Fermi gas with and without an (electric) potential decays in the thermodynamic limit. For the finite one-dimensional system various boundary conditions are employed. Unlike the usual setup the perturbation is introduced by a magnetic (vector) potential. Although such a magnetic field can be gauged away in one spatial dimension there is a significant and interesting effect on the overlap caused by the phases. We study the leading asymptotics of the overlap of the two ground states and the two-term asymptotics of the difference of the ground-state energies. In the case of periodic boundary conditions our main result on the overlap is based upon a well-known asymptotic expansion by Fisher and Hartwig on Toeplitz determinants with a discontinuous symbol. In the case of Dirichlet boundary conditions no such result is known to us and we only provide an upper bound on the overlap, presumably of the right asymptotic order.
Keywords: Fermi gas, Toeplitz matrix, Hilbert matrix, Szegő limit theorem, Fisher–Hartwig conjecture
In 1967, P.W. Anderson  discovered that in the limit the ground state of a free gas of non-interacting fermions is orthogonal to the ground state of the same system but perturbed by an external potential. The asymptotic behaviour of the overlap of the two ground states is of the order with a constant which depends on scattering parameters of the potential. 30 years later, I. Affleck  related Anderson’s orthogonality catastrophe (AOC) to an asymptotic expansion of the difference of the ground-state energies of the free and the perturbed system. Assuming that the latter is of the asymptotic form Affleck argued that . Here, the leading term, , is called Fumi term and the coefficient, , of the next-to-leading term is called finite-size energy.
In this paper, we study AOC and the difference of the ground-state energies for a one-dimensional system which is not perturbed by an external potential but rather by an external magnetic field given by a magnetic (vector) potential . We confine the particles to the interval , impose periodic or Dirichlet boundary conditions, and consider the thermodynamic limit where the particle density tends to a fixed finite value . In , quantum fluctuations of the current in a metallic loop were studied. This led to a logarithmically divergent term similar to the orthogonality catastrophe in the present paper.
It is well-known that in one spatial dimension the Hamiltonian with magnetic field can be mapped to a free Hamiltonian (with the same or slightly modified eigenvalues as the free problem) by a gauge transformation. Note that while the ground-state energy is changed little (or not at all) by the magnetic field, the effect on the overlap of the ground states is surprisingly strong. Therefore, we cannot expect that Affleck’s identification of the AOC exponent with the finite-size energy holds.
The main part of this paper is dedicated to the asymptotic analysis of the overlap of the ground state of the free Hamiltonian and the one with a magnetic field. For periodic boundary conditions, the leading order of the asymptotic expansion of the overlap of the two ground states is with a finite constant . A precise formulation of this result is given in Theorem 2.3. Here, is half of the integral of (except for additive multiples of ). This behaviour is remindful of the one for potentials. In Theorem 2.5 we give an asymptotic upper bound of this overlap, namely with a finite constant . This second result holds for periodic as well as Dirichlet boundary conditions. It is noteworthy that the obtained exponent does not depend on the specific form of the magnetic potential but only on its integral which is related to the magnetic flux.
For the proof of both theorems we rewrite the inner product of the two ground states as a Toeplitz determinant with an -dependent symbol. Due to this -dependence, classical Szegő limit theorems including those for Fisher–Hartwig symbols cannot be applied directly. The crucial step of our proofs is that this determinant is factored into a determinant, which is asymptotically of order 1, times a Toeplitz determinant with a discontinuous -dependent symbol. This discontinuous symbol in turn can be interpreted as corresponding to an effective flux which looks like the one induced by Dirac’s -function as a magnetic potential. In the case of periodic boundary conditions the leading -asymptotics of this particular Toeplitz determinant with discontinuous symbol is given by a Toeplitz determinant with an -independent symbol of the Fisher–Hartwig class. This exact Toeplitz determinant has been analysed by Fisher and Hartwig in their 1968 paper, which they used to bring forward a more general conjecture on the asymptotics for a certain kind of discontinuous and singular symbols . Using their result we find the leading asymptotics of the overlap of the respective ground states. An upper bound is determined by using the inequality , where the exponent on the right-hand side is the so-called Anderson integral. We study this Anderson integral to obtain the asymptotic upper bound for periodic boundary conditions. For Dirichlet boundary conditions we choose a different approach which requires the asymptotic analysis of a determinant of a Toeplitz plus a Hilbert matrix.
In the following section we specify the quantum mechanical model and fix the notation. As a first simple result we obtain the Fumi term and the finite-size energy in Subsection 2.2. The precise formulation of our main results on Anderson’s orthogonality catastrophe, Theorems 2.3 and 2.5, is given in Subsection 2.3. In Section 3 we prove these two theorems. For this purpose we introduce the notion of generalised Toeplitz matrices. In Lemma 3.1 we give a decomposition of the overlap of the ground states into a bounded part and a Toeplitz determinant with discontinuous symbol as mentioned above. This lemma is proved in Section 4. In Subsection 3.1 we show Theorem 2.3. Subsections 3.2 is devoted to the proof of Theorem 2.5.
Finally, we comment on the mathematical work in this field all of which is rather recent. The papers [4, 5, 6] focus on AOC (caused by potentials and not magnetic fields) and prove upper bounds on the overlap which are all in agreement with Anderson’s bounds. In  and , the finite-size energy and its relation to the coefficient in AOC is studied in a one-dimensional Fermi system. Yet, we do not know whether Anderson’s prediction about the decay rate of the overlap is sharp. However, in the model of this paper we are able to prove the sharp decay rate. But we also notice that Affleck’s correspondence between the coefficient and the finite-size correction does not hold here. This should not diminish the value of this fascinating relation in the case of a potential.
2 Model and results
2.1 The Hamiltonian for a single particle
We consider a single, spinless particle confined to the interval . The Hilbert space is , equipped with the usual inner product given by . We study the system for periodic and for Dirichlet boundary conditions where arguments and results for the latter can be taken over to Neumann boundary conditions (or more general Robin boundary conditions). We denote the eigenfunctions, eigenvalues etc. for different boundary conditions by the same respective symbols. It will become clear from the context which boundary conditions are used.
Periodic boundary conditions
The Hamiltonian of a single free particle on this interval is with periodic boundary conditions. The energy eigenvalues denoted by and the corresponding eigenstates are determined by
Then we have, normalised to one in -norm, and , . Note that all eigenvalues except for are twofold degenerate and the eigenvalue 0 is non-degenerate.
Let , the Sobolev space of integrable and continuously differentiable functions with integrable derivatives. We denote the multiplication operator given by by as well. In order to have a well-defined operator on satisfying periodic boundary conditions, we additionally assume . The Hamiltonian of the system with the external magnetic (vector) potential is . The spectrum is determined by the eigenvalue problem
This eigenvalue problem can be solved using the gauge transformation where is given by
The quantity is called magnetic flux and represents the phase shift due to the magnetic field. Since and , the corresponding eigenvalues and eigenfunctions are easily determined: and , . On the one hand, all eigenvalues except for 0, which is non-degenerate, are degenerate with multiplicity two if is an integral multiple of and, if is a half-integral multiple of , the eigenvalue 0 is also degenerate with multiplicity two. On the other hand, in any other case all eigenvalues are non-degenerate.
Dirichlet boundary conditions
respectively. Moreover, we only assume , but not that . The free and the magnetically perturbed Hamiltonian have the same spectrum, with . This is easily shown using the gauge transformation where is defined as above. The corresponding normalised eigenfunctions are given by
In contrast with the periodic case, all eigenvalues are non-degenerate and is not an eigenvalue.
One could regard as the magnetic field (whatever the physical interpretation of a magnetic field in one dimension is). Due to the condition that is integrable, has to decay to zero at infinity which, for instance, rules out the case of a constant magnetic field. Moreover, the integral of has to vanish. In the periodic case the additional restrictions and for any occur.
We want to remind the reader that the gauge transformation maps the Hamiltonian with magnetic field, Eq. (3), to a free Hamiltonian (with the same or slightly modified eigenvalues as the free problem (1)). However, the eigenfunctions depend on this gauge. Therefore the effect of the magnetic potential on the overlap of the ground states can be expected to be more drastic than the one on the difference of the ground-state energies.
2.2 non-interacting fermions and their ground-state energy
Periodic boundary conditions
Now we consider a system of non-interacting spinless fermions on the interval with periodic boundary conditions. The corresponding -particle Fock space is the -fold antisymmetric tensor product of the one-particle Hilbert space, . The energy of this system is represented by the Hamiltonian . Note that for we recover the free Hamiltonian. The ground-state energy as well as corresponding eigenfunctions of these two fermion system can be calculated from the eigenvalues and -functions of the one-particle system as derived in Subsection 2.1. The magnetic flux at position is decomposed as
with and . With this notation the ground-state energy of is . The set is for odd and for even . Here and henceforth we set for even and for odd . Note that the ground-state energy is twofold degenerate in the cases and and non-degenerate otherwise. Then the difference of the ground-state energies is equal to for odd and to for even .
Therefore the sequence has two limit points and the asymptotic behaviour differs for the subsequences with even and odd values of , respectively. We find
as with for a fixed limit density . Here the parameter is given by where is chosen appropriately. Thus .
Note that the leading term of the asymptotic expansion, which is of order 1 and called Fumi term, is zero. However, the next term in this expansion which is of order , the finite-size energy, is, in general, non-zero for finite values of .
Moreover, we find that the ground state of the Hamiltonians is the Slater determinant Note that .
Dirichlet boundary conditions
Analogously to the periodic case we define the -particle Hamiltonians and with Dirichlet boundary conditions. The corresponding ground-state energies are . They are independent of and therefore the difference of the ground-state energies vanishes for all particle numbers . The ground state of the perturbed system is the Slater determinant .
2.3 Anderson’s orthogonality catastrophe
We mainly study the asymptotic behaviour of the overlap of the ground states of the free and the perturbed system. In the case of periodic boundary conditions we can determine the asymptotics of the overlap rather explicitly:
Theorem 2.3 (Anderson’s orthogonality catastrophe, periodic boundary conditions).
Let and . Assume , , , at least for a subsequence of ’s, with . Then there exists a constant such that the overlap of the two -fermion ground states satisfies
in the thermodynamic limit with . In particular, the overlap vanishes in the considered limit for . For the overlap is equal to 1.
The proof of this theorem is the main task of this paper and is carried out in Section 3. For our proof it is crucial that the matrix on the left-hand side of Eq. (7) is a Toeplitz matrix. The leading term of the asymptotics results from a Szegő–type limit theorem for a Toeplitz determinant with a discontinuous symbol as given by Fisher and Hartwig .
Levitov, Lee and Lesovik derived a similar result to Eq. (7) in a time-dependent setup [3, 10]. They consider a loop, i. e. an interval of length with periodic boundary conditions, in a fixed external potential. By a short pulse a magnetic flux is induced which leads to a time-dependent vector potential in the Hamiltonian. The overlap of the ground states of the system long before the pulse and long afterwards are compared where the thermodynamic limit is taken at fixed Fermi energy.
For reasons that will become clear soon, we, moreover, show a certain upper bound to the overlap of the ground states for both boundary conditions. The bound is obtained with the inequality and calculating the leading term of the so-called Anderson integral. For Dirichlet boundary conditions the direct calculation of the Anderson integral is omitted in favour of another approach using a property of a Hilbert matrix appearing in the asymptotic analysis of the determinant.
Theorem 2.5 (Upper bound for Anderson’s orthogonality catastrophe, periodic and Dirichlet boundary conditions).
Let and . With the same assumptions as in Theorem 2.3, an upper bound for the asymptotics of the overlap is given by
with a constant as with .
The proof of this theorem is given in Subsection 3.2.
Up to the constant , the negative exponent of the exact asymptotics in Eq. (7) and the negative exponent of the upper bound in Ineq. (8) satisfy “Anderson’s rule”. So they are essentially the sine respectively the arc sine of each other. Note that the negative exponent is sharp and cannot be improved.
When considering the overlap for Dirichlet boundary conditions, one is, in leading order, led to the asymptotic analysis of
Here is a Hilbert matrix with , , and a projection on an -dimensional subspace. In order to determine this asymptotics, we would need a Szegő limit theorem for the determinant of the sum of the Toeplitz matrix and the square of the Hilbert matrix . We are only aware of asymptotic results for the sum of a Toeplitz and a Hankel matrix requiring a certain relation between the two symbols which, however, do not apply here. It seems reasonable that the overlap is independent of boundary conditions and in particular the same as for periodic boundary conditions.
If we would allow a magnetic field with non-vanishing (finite) integral then from the start we can only allow Dirichlet boundary conditions. The logarithm of the overlap is now to leading order of the form . It seems that the next-to-leading term is of the order and that it is of the same form (possibly up to a constant) as above if we replace by in the above asymptotics for vanishing total magnetic field .
3 Asymptotic analysis of Toeplitz determinants
Let be a complex-valued (essentially bounded) function. The operator of multiplication by is denoted by as well. A generalised Toeplitz matrix with symbol is defined as the -matrix
where is an orthonormal basis of the underlying Hilbert space. The corresponding determinant is referred to as the (generalised) Toeplitz determinant. Here the generalisation is with respect to the choice of the basis. In our context we choose the eigenbasis of the corresponding free Hamiltonian. In the case of periodic boundary conditions this basis consists of plane waves and the matrix elements are the Fourier coefficients of . In particular, they only depend on the difference of the indices and we recover a Toeplitz matrix in the usual sense. For Dirichlet boundary conditions, the emerging matrix is a sum of a usual Toeplitz and a Hankel matrix. For a deeper study and an overview of results for Toeplitz and related matrices see, for instance, [11, 12].
Hereinafter, we study symbols of the form . In the case of periodic boundary conditions is given by
with and as defined in Eqs. (5) and (6), respectively. The function has a jump of height at the boundary . This becomes particularly simple if we take, at least informally, the discontinuous magnetic potential . More precisely, we set
which is reminiscent of an integral of .
As , has a jump discontinuity of height
at the boundary , but also at with height .
For Dirichlet boundary conditions we do not have the term . Hence we set
Here the jump discontinuities at 0 and at the boundary are of the same height .
Now we have . Note that the symbol of this Toeplitz determinant depends on the parameter , which also tends to infinity, unlike in classical Szegő limit theorems. To circumvent this problem is factorised into the Toeplitz determinant with the symbol and a bounded term as the main step in the proof of the theorems. This Toeplitz determinant in turn can be rewritten as a usual Toeplitz determinant with a Fisher–Hartwig symbol times a remainder term of order 1. More precisely, we use the following lemma.
For the system with periodic or Dirichlet boundary conditions assume , , and for some . Then we have
for any and . In general, depends on and . Moreover, there are positive constants , and such that
The proof of this lemma is postponed to Section 4. It is based on a factorisation of the determinant and a decomposition of some of the resulting matrices.
For simplicity we henceforth assume that the quotient is kept at a fixed finite value when we take the thermodynamic limit. However, this condition may be relaxed to and the following statements still hold true.
3.1 Proof of Theorem 2.3
Proof of Theorem 2.3.
where . This can be verified by a straightforward computation using Eq. (6). With the abbreviation we obtain the -matrix
This is a Toeplitz matrix with the symbol which is discontinuous at zero and at the boundary, but has no singularities.
If is compactly supported, is constant for sufficiently large values of and the constant value is . If we relax the condition that has compact support, the -dependence of the symbol has to be taken into account. However, with and , has to decay faster than at infinity. Therefore as . Thus the first error term decays faster than . Consequently, we may assume the magnetic potential to be compactly supported from now on. In this special case there is an such that is constant (i.e. does not depend on ) and is equal to for every .
By the above arguments, the parameter can be considered to have the constant value for sufficiently large . The Toeplitz determinant we obtain this way has already been considered by Fisher and Hartwig in 1968  as an example where the conjecture they brought forward in the same paper, now called Fisher–Hartwig conjecture, holds. They explicitly calculate the leading asymptotic term in the limit using Cauchy’s determinant formula. So we find
as with . Note that the constant depends on and is given explicitly in . Related results on the Fisher–Hartwig conjecture can be found in [13, 14, 15]. Merging the above results and Lemma 3.1 we obtain the assertion of Theorem 2.3. ∎
In his study on Anderson’s orthogonality catastrophe , P. Lloyd constructs a model unitary transformation between the eigenfunctions of the free Hamiltonian and the perturbed Hamiltonian . The matrix elements [16, Eq. (A.2.4)]of this unitary transformation with respect to the free eigenfunctions are of the same form as in (12) with replaced by the phase factor of the potential, which allowed him to justify the exponent in Anderson’s orthogonality catastrophe.
3.2 Proof of Theorem 2.5
Periodic boundary conditions and the Anderson integral
We first show Theorem 2.5 for periodic boundary conditions. To this end we study the asymptotic behaviour of the corresponding Anderson integral. Again, it suffices to study the asymptotics of the Toeplitz determinant, , with the discontinuous symbol . For simplicity we assume and to be odd in the following and set . For the area of summation has to be shifted by to the left and for even the term with has to be removed from the sum. With the two projections we have
The inequality , which holds for trace class operators , yields
where is the Anderson integral. Since , the individual summands of the Anderson integral only depend on the difference of the indices . Rescaling and renaming the summation variables we obtain
On the one hand, the second and the fourth sum on the right-hand side are monotonously decaying at least as for large . On the other hand, the first and the third sum are diverging for and their respective asymptotics are . Hence the leading term of the asymptotics of the Anderson integral is as with . Together with Lemma 3.1 the above derivation yields
thus finishing the proof of Theorem 2.5 for periodic boundary conditions.
Dirichlet boundary conditions and the Hilbert matrix
For the Dirichlet case we use a different way to prove the respective assertion of Theorem 2.5. This approach has the advantage that we have equalities until the very last step where the only estimate of this proof comes into play. It shows a possible way to tackle the problem of the exact asymptotics of the overlap as already mentioned in Remark 2.7.
For simplicity we consider the particle number to be even and set throughout this subsection. For odd particle numbers some of the following formulae are slightly more complicated, but the general method works as well. In order to further simplify we have a closer look at the matrix elements that appear in this determinant. By the trigonometric addition theorems we find
We obtain for , for and for odd . Merging these results yields
In symbolic notation the resulting determinant has the form
where we interchanged columns and rows for the last identity. Due to the particular structure of the matrix elements we obtain
Here the two -matrices and have the entries and , respectively. The determinant is further reduced to
using the -periodicity of and with . The matrix is defined by . Using the elementary formula we obtain the decomposition . Here the matrices are given by , , and . Introducing the operators , and the orthogonal projection we obtain
Note that is self-adjoint, . In order to estimate the determinant we need the trace norms of these operators. To begin with,
Analogously, we obtain
for large . The mixed terms are estimated via the Cauchy–Schwarz inequality:
Since gives the leading term for large