Formalism for the solution of quadratic Hamiltonians with large cosine terms
We consider quantum Hamiltonians of the form where is a quadratic function of position and momentum variables and the ’s are linear in these variables. We allow and to be completely general with only two restrictions: we require that (1) the ’s are linearly independent and (2) is an integer multiple of for all so that the different cosine terms commute with one another. Our main result is a recipe for solving these Hamiltonians and obtaining their exact low energy spectrum in the limit . This recipe involves constructing creation and annihilation operators and is similar in spirit to the procedure for diagonalizing quadratic Hamiltonians. In addition to our exact solution in the infinite limit, we also discuss how to analyze these systems when is large but finite. Our results are relevant to a number of different physical systems, but one of the most natural applications is to understanding the effects of electron scattering on quantum Hall edge modes. To demonstrate this application, we use our formalism to solve a toy model for a fractional quantum spin Hall edge with different types of impurities.
In this paper we study a general class of quantum Hamiltonians which are relevant to a number of different physical systems. The Hamiltonians we consider are defined on a dimensional phase space with . They take the form
where is a quadratic function of the phase space variables , and is linear in these variables. The ’s can be chosen arbitrarily except for two restrictions:
are linearly independent.
is an integer multiple of for all .
Here, the significance of the second condition is that it guarantees that the cosine terms commute with one another: for all .
For small , we can straightforwardly analyze these Hamiltonians by treating the cosine terms as perturbations to . But how can we study these systems when is large? The most obvious approach is to expand around , just as in the small case we expand around . But in order to make such an expansion, we first need to be able to solve these Hamiltonians exactly in the infinite limit. The purpose of this paper is to describe a systematic procedure for obtaining such a solution, at least at low energies.
The basic idea underlying our solution is that when is very large, the cosine terms act as constraints at low energies. Thus, the low energy spectrum of can be described by an effective Hamiltonian defined within a constrained Hilbert space . This effective Hamiltonian is quadratic in since is quadratic and the constraints are linear in these variables. We can therefore diagonalize and in this way determine the low energy properties of .
Our main result is a general recipe for finding the exact low energy spectrum of in the limit . This recipe consists of two steps and is only slightly more complicated than what is required to solve a conventional quadratic Hamiltonian. The first step involves finding creation and annihilation operators for the low energy effective Hamiltonian (Eq. 11). The second step of the recipe involves finding integer linear combinations of the ’s that have simple commutation relations with one another. In practice, this step amounts to finding a change of basis that puts a particular integer skew-symmetric matrix into canonical form (Eq. 14). Once these two steps are completed, the low energy spectrum can be written down immediately (Eq. 19).
In addition to our exact solution in the infinite limit, we also discuss how to analyze these systems when is large but finite. In particular, we show that in the finite case, we need to add small (non-quadratic) corrections to the effective Hamiltonian in order to reproduce the low energy physics of . One of our key results is a discussion of the general form of these finite corrections, and how they scale with .
Our results are useful because there are a number of physical systems where one needs to understand the effect of cosine-like interactions on a quadratic Hamiltonian. An important class of examples are the edges of Abelian fractional quantum Hall (FQH) states. Previously it has been argued that a general Abelian FQH edge can be modeled as collection of chiral Luttinger liquids with Hamiltonian Wen (1990); Frohlich and Kerler (1991); Wen (1995, 2007)
Here , with each component describing a different (bosonized) edge mode while is a matrix that describes the velocities and density-density interactions between the different edge modes. The commutation relations for the operators are where is a symmetric, integer matrix which is determined by the bulk FQH state.
The above Hamiltonian is quadratic and hence exactly soluble, but in many cases it is unrealistic because it describes an edge in which electrons do not scatter between the different edge modes. In order to incorporate scattering into the model, we need to add terms to the Hamiltonian of the form
where is a -component integer vector that describes the number of electrons scattered from each edge mode Kane and Fisher (1995). However, it is usually difficult to analyze the effect of these cosine terms beyond the small limit where perturbative techniques are applicable. (An important exception is when is a null vector Haldane (1995), i.e. : in this case, the fate of the edge modes can be determined by mapping the system onto a Sine-Gordon model Wang and Levin (2013)).
Our results can provide insight into this class of systems because they allow us to construct exactly soluble toy models that capture the effect of electron scattering at the edge. Such toy models can be obtained by replacing the above continuum scattering term by a collection of discrete impurity scatterers, , and then taking the limit . It is not hard to see that the latter cosine terms obey conditions (1) and (2) from above, so we can solve the resulting models exactly using our general recipe. Importantly, this construction is valid for any choice of , whether or not is a null vector.
Although the application to FQH edge states is one of the most interesting aspects of our results, our focus in this paper is on the general formalism rather than the implications for specific physical systems. Therefore, we will only present a few simple examples involving a fractional quantum spin Hall edge with different types of impurities. The primary purpose of these examples is to demonstrate how our formalism works rather than to obtain novel results.
We now discuss the relationship with previous work. One paper that explores some related ideas is Ref. Gottesman et al., 2001. In that paper, Gottesman, Kitaev, and Preskill discussed Hamiltonians similar to (1) for the case where the operators do not commute, i.e. . They showed that these Hamiltonians can have degenerate ground states and proposed using these degenerate states to realize qubits in continuous variable quantum systems.
Another line of research that has connections to the present work involves the problem of understanding constraints in quantum mechanics. In particular, a number of previous works have studied the problem of a quantum particle that is constrained to move on a surface by a strong confining potential Jensen and Koppe (1971); Da Costa (1981). This problem is similar in spirit to one we study here, particularly for the special case where : in that case, if we identify as position coordinates , then the Hamiltonian (1) can be thought of as describing a particle that is constrained to move on a periodic array of hyperplanes.
Our proposal to apply our formalism to FQH edge states also has connections to the previous literature. In particular, it has long been known that the problem of an impurity in a non-chiral Luttinger liquid has a simple exact solution in the limit of infinitely strong backscattering Kane and Fisher (1992a, b); d. C. Chamon et al. (1996); Chamon et al. (1995). The infinite backscattering limit for a single impurity has also been studied for more complicated Luttinger liquid systems Chklovskii and Halperin (1998); Chamon and Fradkin (1997); Ponomarenko and Averin (2007); Ganeshan et al. (2012). The advantage of our approach to these systems is that our methods allow us to study not just single impurities but also multiple coherently coupled impurities, and to obtain the full quantum dynamics not just transport properties.
The paper is organized as follows. In section II we summarize our formalism and main results. In section III we illustrate our formalism with some examples involving fractional quantum spin Hall edges with impurities. We discuss directions for future work in the conclusion. The appendices contain the general derivation of our formalism as well as other technical results.
Ii Summary of results
ii.1 Low energy effective theory
Our first result is that we derive an effective theory that describes the low energy spectrum of
in the limit . This effective theory consists of an effective Hamiltonian and an effective Hilbert space . Conveniently, we find a simple algebraic expression for and that holds in the most general case. Specifically, the effective Hamiltonian is given by
where the operators are defined by
and where is an matrix defined by
This effective Hamiltonian is defined on an effective Hilbert space , which is a subspace of the original Hilbert space , and consists of all states that satisfy
A few remarks about these formulas: first, notice that and are matrices of -numbers since is quadratic and the ’s are linear combinations of ’s and ’s. Also notice that the operators are linear functions of . These observations imply that the effective Hamiltonian is always quadratic. Another important point is that the operators are conjugate to the ’s:
This means that we can think of the ’s as generalized momentum operators. Finally, notice that
The significance of this last equation is that it shows that the Hamiltonian can be naturally defined within the above Hilbert space (5).
We can motivate this effective theory as follows. First, it is natural to expect that the lowest energy states in the limit are those that minimize the cosine terms. This leads to the effective Hilbert space given in Eq. 5. Second, it is natural to expect that the dynamics in the directions freezes out at low energies. Hence, the terms that generate this dynamics, namely , should be removed from the effective Hamiltonian. This leads to Eq. 2. Of course this line of reasoning is just an intuitive picture; for a formal derivation of the effective theory, we refer the reader to appendix A.
At what energy scale is the above effective theory valid? We show that correctly reproduces the energy spectrum of for energies less than where is the maximum eigenvalue of . One implication of this result is that our effective theory is only valid if is non-degenerate: if were degenerate than would have an infinitely large eigenvalue, which would mean that there would be no energy scale below which our theory is valid. Physically, the reason that our effective theory breaks down when is degenerate is that in this case, the dynamics in the directions does not completely freeze out at low energies.
To see an example of these results, consider a one dimensional harmonic oscillator with a cosine term:
In this case, we have and . If we substitute these expressions into Eq. 2, a little algebra gives
As for the effective Hilbert space, Eq. 5 tells us that consists of position eigenstates
If we now diagonalize the effective Hamiltonian within the effective Hilbert space, we obtain eigenstates with energies . Our basic claim is that these eigenstates and energies should match the low energy spectrum of in the limit. In appendix A.1.1, we analyze this example in detail and we confirm that this claim is correct (up to a constant shift in the energy spectrum).
To see another illustrative example, consider a one dimensional harmonic oscillator with two cosine terms,
where is a positive integer. This example is fundamentally different from the previous one because the arguments of the cosine do not commute: . This property leads to some new features, such as degeneracies in the low energy spectrum. To find the effective theory in this case, we note that and , . With a little algebra, Eq. 2 gives
As for the effective Hilbert space, Eq. 5 tells us that consists of all states satisfying
One can check that there are linearly independent states obeying the above conditions; hence if we diagonalize the effective Hamiltonian within the effective Hilbert space, we obtain exactly degenerate eigenstates with energy . The prediction of our formalism is therefore that has a -fold ground state degeneracy in the limit. In appendix A.1.2, we analyze this example and confirm this prediction.
ii.2 Diagonalizing the effective theory
We now move on to discuss our second result, which is a recipe for diagonalizing the effective Hamiltonian . Note that this diagonalization procedure is unnecessary for the two examples discussed above, since is very simple in these cases. However, in general, is a complicated quadratic Hamiltonian which is defined within a complicated Hilbert space , so diagonalization is an important issue. In fact, in practice, the results in this section are more useful than those in the previous section because we will see that we can diagonalize without explicitly evaluating the expression in Eq. 2.
Our recipe for diagonalizing has three steps. The first step is to find creation and annihilation operators for . Formally, this amounts to finding all operators that are linear combinations of , and satisfy
for some scalar . While the first condition is the usual definition of creation and annihilation operators, the second condition is less standard; the motivation for this condition is that commutes with (see Eq. 7). As a result, we can impose the requirement and we will still have enough quantum numbers to diagonalize since we can use the ’s in addition to the ’s.
Alternatively, there is another way to find creation and annihilation operators which is often more convenient: instead of looking for solutions to (10), one can look for solutions to
for some scalars with . Indeed, we show in appendix B.4 that every solution to (10) is also a solution to (11) and vice versa, so these two sets of equations are equivalent. In practice, it is easier to work with Eq. 11 than Eq. 10 because Eq. 11 is written in terms of , and thus it does not require us to work out the expression for .
The solutions to (10), or equivalently (11), can be divided into two classes: “annihilation operators” with , and “creation operators” with . Let denote a complete set of linearly independent annihilation operators. We will denote the corresponding ’s by and the creation operators by . The creation/annihilation operators should be normalized so that
We are now ready to discuss the second step of the recipe. This step involves searching for linear combinations of that have simple commutation relations with one another. The idea behind this step is that we ultimately need to construct a complete set of quantum numbers for labeling the eigenstates of . Some of these quantum numbers will necessarily involve the operators since these operators play a prominent role in the definition of the effective Hilbert space, . However, the ’s are unwieldy because they have complicated commutation relations with one another. Thus, it is natural to look for linear combinations of ’s that have simpler commutation relations.
With this motivation in mind, let be the matrix defined by
The matrix is integer and skew-symmetric, but otherwise arbitrary. Next, let
for some matrix and some vector . Then, where . The second step of the recipe is to find a matrix with integer entries and determinant , such that takes the simple form
Here is some integer with and denotes an matrix of zeros. In mathematical language, is an integer change of basis that puts into skew-normal form. It is known that such a change of basis always exists, though it is not unique.Newman (1972) After finding an appropriate , the offset should then be chosen so that
The reason for choosing in this way is that it ensures that for any , as can be easily seen from the Campbell-Baker-Hausdorff formula.
Once we perform these two steps, we can obtain the complete energy spectrum of with the help of a few results that we prove in appendix B.111These results rely on a small technical assumption, see Eq. 122. Our first result is that can always be written in the form
where is some (a priori unknown) quadratic function. Our second result (which is really just an observation) is that the following operators all commute with each other:
Furthermore, these operators commute with the occupation number operators . Therefore, we can simultaneously diagonalize (17) along with . We denote the simultaneous eigenstates by
or, in more abbreviated form, . Here the different quantum numbers are defined by
where , while is real valued and ranges over non-negative integers.
By construction the states form a complete basis for the Hilbert space . Our third result is that a subset of these states form a complete basis for the effective Hilbert space . This subset consists of all for which
for some integers .
We will denote this subset of eigenstates by .
We therefore have the full eigenspectrum of — up to the determination of the function . With a bit more work, one can go further and compute the function (see appendix B.3) but we will not discuss this issue here because in many cases of interest it is more convenient to find using problem-specific approaches.
One implication of Eq. 19 which is worth mentioning is that the energy is independent of the quantum numbers . Since ranges from , it follows that every eigenvalue of has a degeneracy of (at least)
In the special case where is non-degenerate (i.e. the case where ), this degeneracy can be conveniently written as
ii.4 Finite corrections
We now discuss our last major result. To understand this result, note that while gives the exact low energy spectrum of in the infinite limit, it only gives approximate results when is large but finite. Thus, to complete our picture we need to understand what types of corrections we need to add to to obtain an exact effective theory in the finite case.
It is instructive to start with a simple example: . As we discussed in section II.1, the low energy effective Hamiltonian in the infinite limit is while the low energy Hilbert space is spanned by position eigenstates where is an integer.
Let us consider this example in the case where is large but finite. In this case, we expect that there is some small amplitude for the system to tunnel from one cosine minima to another minima, . Clearly we need to add correction terms to that describe these tunneling processes. But what are these correction terms? It is not hard to see that the most general possible correction terms can be parameterized as
where is some unknown function which also depends on . Physically, each term describes a tunneling process since . The coefficient describes the amplitude for this process, which may depend on in general. (The one exception is the term, which does not describe tunneling at all, but rather describes corrections to the onsite energies for each minima).
Having developed our intuition with this example, we are now ready to describe our general result. Specifically, in the general case we show that the finite corrections can be written in the form
with the sum running over component integer vectors . Here, the are unknown functions of which also depend on . We give some examples of these results in section III. For a derivation of the finite corrections, see appendix C.
ii.5 Splitting of ground state degeneracy
One application of Eq. 23 is to determining how the ground state degeneracy of splits at finite . Indeed, according to standard perturbation theory, we can find the splitting of the ground state degeneracy by projecting the finite corrections onto the ground state subspace and then diagonalizing the resulting matrix. The details of this diagonalization problem are system dependent, so we cannot say much about it in general. However, we would like to mention a related result that is useful in this context. This result applies to any system in which the commutator matrix is non-degenerate. Before stating the result, we first need to define some notation: let be operators defined by
Note that, by construction, the operators obey the commutation relations
With this notation, our result is that
where are ground states and is some unknown proportionality constant. This result is useful because it is relatively easy to compute the matrix elements of ; hence the above relation allows us to compute the matrix elements of the finite corrections (up to the constants ) without much work. We derive this result in appendix C.
In this section, we illustrate our formalism with some concrete examples. These examples involve a class of two dimensional electron systems in which the spin-up and spin-down electrons form Laughlin states with opposite chiralities Bernevig and Zhang (2006). These states are known as “fractional quantum spin Hall insulators.” We will be primarily interested in the edges of fractional quantum spin Hall (FQSH) insulators Levin and Stern (2009). Since the edge of the Laughlin state can be modeled as a single chiral Luttinger liquid, the edge of a FQSH insulator consists of two chiral Luttinger liquids with opposite chiralities — one for each spin direction (Fig. 1).
The examples that follow will explore the physics of the FQSH edge in the presence of impurity-induced scattering. More specifically, in the first example, we consider a FQSH edge with a single magnetic impurity; in the second example we consider a FQSH edge with multiple magnetic impurities; in the last example we consider a FQSH edge with alternating magnetic and superconducting impurities. In all cases, we study the impurities in the infinite scattering limit, which corresponds to in (1). Then, in the last subsection we discuss how our results change when the scattering strength is large but finite.
We emphasize that the main purpose of these examples is to illustrate our formalism rather than to derive novel results. In particular, many of our findings regarding these examples are known previously in the literature in some form.
Of all the examples, the last one, involving magnetic and superconducting impurities, is perhaps most interesting: we find that this system has a ground state degeneracy that grows exponentially with the number of impurities. This ground state degeneracy is closely related to the previously known topological degeneracy that appears when a FQSH edge is proximity coupled to alternating ferromagnetic and superconducting strips Lindner et al. (2012); Cheng (2012); Barkeshli et al. (2013); Vaezi (2013); Clarke et al. (2013).
Before proceeding, we need to explain what we mean by “magnetic impurities” and “superconducting impurities.” At a formal level, a magnetic impurity is a localized object that scatters spin-up electrons into spin-down electrons. Likewise a superconducting impurity is a localized object that scatters spin-up electrons into spin-down holes. More physically, a magnetic impurity can be realized by placing the tip of a ferromagnetic needle in proximity to the edge while a superconducting impurity can be realized by placing the tip of a superconducting needle in proximity to the edge.
iii.1 Review of edge theory for clean system
As discussed above, the edge theory for the fractional quantum spin Hall state consists of two chiral Luttinger liquids with opposite chiralities — one for each spin direction (Fig. 1). The purpose of this section is to review the Hamiltonian formulation of this edge theory.Wen (1995, 2007); Levin and Stern (2009) More specifically we will discuss the edge theory for a disk geometry where the circumference of the disk has length . Since we will work in a Hamiltonian formulation, in order to define the edge theory we need to specify the Hamiltonian, the set of physical observables, and the canonical commutation relations.
We begin with the set of physical observables. The basic physical observables in the edge theory are a collection of operators along with two additional operators , where is an arbitrary, but fixed, point on the boundary of the disk. The operators can be thought of as the fundamental phase space operators in this system, i.e. the analogues of the operators in section II. Like , all other physical observables can be written as functions/functionals of . Two important examples are the operators and which are defined by
where the integral runs from to in the clockwise direction.
The physical meaning of these operators is as follows: the density of spin-up electrons at position is given by while the density of spin-down electron is . The total charge and total spin on the edge are given by and with
Finally, the spin-up and spin-down electron creation operators take the form
In the above discussion, we ignored an important subtlety: and are actually compact degrees of freedom which are only defined modulo . In other words, strictly speaking, and are not well-defined operators: only and are well-defined. (Of course the same also goes for and , in view of the above definition). Closely related to this fact, the conjugate “momenta” and are actually discrete degrees of freedom which can take only integer values.
The compactness of and discreteness of is inconvenient for us since the machinery discussed in section II is designed for systems in which all the phase space operators are real-valued, rather than systems in which some operators are angular valued and some are integer valued. To get around this issue, we will initially treat and and the conjugate momenta , as real valued operators. We will then use a trick (described in the next section) to dynamically generate the compactness of as well as the discreteness of .
Let us now discuss the commutation relations for the operators. Like the usual phase space operators , the commutators of are -numbers. More specifically, the basic commutation relations are
with the other commutators vanishing:
Using these basic commutation relations, together with the definition of (27), one can derive the more general relations
as well as
where the sgn function is defined by if and if , with the ordering defined in the clockwise direction. The latter commutation relations (29) and (30) will be particularly useful to us in the sections that follow.
Having defined the physical observables and their commutation relations, the last step is to define the Hamiltonian for the edge theory. The Hamiltonian for a perfectly clean, homogeneous edge is
where is the velocity of the edge modes.
At this point, the edge theory is complete except for one missing element: we have not given an explicit definition of the Hilbert space of the edge theory. There are two different (but equivalent) definitions that one can use. The first, more abstract, definition is that the Hilbert space is the unique irreducible representation of the operators and the commutation relations (28). (This is akin to defining the Hilbert space of the 1D harmonic oscillator as the irreducible representation of the Heisenberg algebra ). The second definition, which is more concrete but also more complicated, is that the Hilbert space is spanned by the complete orthonormal basis where the quantum numbers range over all integers 222Actually, range over arbitrary real numbers in our fictitious representation of the edge: as explained above, we initially pretend that are not quantized, and then introduce quantization later on using a trick. while range over all nonnegative integers for each value of . These basis states have a simple physical meaning: corresponds to a state with charge and on the two edge modes, and with and phonons with momentum on the two edge modes.
iii.2 Example 1: Single magnetic impurity
With this preparation, we now proceed to study a fractional quantum spin Hall edge with a single magnetic impurity in a disk geometry of circumference (Fig. 2a). We assume that the impurity, which is located at , generates a backscattering term of the form . Thus, in the bosonized language, the system with an impurity is described by the Hamiltonian
where is defined in Eq. 31. Here, we temporarily ignore the question of how we regularize the cosine term; we will come back to this point below.
Our goal is to find the low energy spectrum of in the strong backscattering limit, . We will accomplish this using the results from section II. Note that, in using these results, we implicitly assume that our formalism applies to systems with infinite dimensional phase spaces, even though we only derived it in the finite dimensional case.
First we describe a trick for correctly accounting for the compactness of and the quantization of . The idea is simple: we initially treat these variables as if they are real valued, and then we introduce compactness dynamically by adding two additional cosine terms to our Hamiltonian:
These additional cosine terms effectively force and to be quantized at low energies, thereby generating the compactness that we seek.333Strictly speaking we also need to add (infinitesimal) quadratic terms to the Hamiltonian of the form so that the matrix is non-degenerate. However, these terms play no role in our analysis so we will not include them explicitly. We will include all three cosine terms in our subsequent analysis.
The next step is to calculate the low energy effective Hamiltonian and low energy Hilbert space . Instead of working out the expressions in Eqs. 2, 5, we will skip this computation and proceed directly to finding creation and annihilation operators for using Eq. 11. (This approach works because equation (11) does not require us to find the explicit form of ).
According to Eq. 11, we can find the creation and annihilation operators for by finding all operators such that (1) is a linear combination of our fundamental phase space operators and (2) obeys
for some scalars with .
To proceed further, we note that the constraint , implies that cannot appear in the expression for . Hence, can be written in the general form
Substituting this expression into the first line of Eq. 34 we obtain the differential equations
(The terms drop out of these equations since commute with ). These differential equations can be solved straightforwardly. The most general solution takes the form
where , and
Here is the Heaviside step function defined as
We still have to impose one more condition on , namely . This condition leads to a second constraint on , but the derivation of this constraint is somewhat subtle. The problem is that if we simply substitute (35) into , we find
It is unclear how to translate this relation into one involving since are discontinuous at and hence are ill-defined. The origin of this puzzle is that the cosine term in Eq. 32 contains short-distance singularities and hence is not well-defined. To resolve this issue we regularize the argument of the cosine term, replacing with
where is a narrowly peaked function with . Here, we can think of as an approximation to a delta function. Note that effectively introduces a short-distance cutoff and thus makes the cosine term non-singular. After making this replacement, it is straightforward to repeat the above analysis and solve the differential equations for . In appendix G, we work out this exercise, and we find that with this regularization, the condition leads to the constraint
So far we have not imposed any restriction on the momentum . The momentum constraints come from the periodic boundary conditions on :
Using the explicit form of , these boundary conditions give
from which we deduce
Putting this all together, we see that the most general possible creation/annihilation operator for is given by
where is quantized as and . (Note that does not correspond to a legitimate creation/annihilation operator according to the definition given above, since we require ).
Following the conventions from section II.2, we will refer to the operators with — or equivalently — as “annihilation operators” and the other operators as “creation operators.” Also, we will choose the normalization constant so that