Physical interpretation of the 2d Thouless charge pump

Chapter IX: Higher-dimensional generalizations of the Thouless charge pump 87

9.6 Physical interpretation of the 2d Thouless charge pump

Our approach to defining higher-dimensional analogs of the Thouless charge pump was rather formal. In this section we are going to clarify their physical meaning in the case when the system is two-dimensional and the parameter space is a torus of dimension two. As proposed in the introduction, the physical interpretation involves making the parameters of the Hamiltonian slowly varying functions of both time and spatial coordinates.

As a warm-up, let us discuss an alternative interpretation of the usual Thouless charge pump for gapped 1d systems. As discussed in Section 2, the same term in the effective action gives rise to the Thouless charge pump and gives charge to 1d skyrmions. A skyrmion is defined as a topologically nontrivial map 𝜙 fromR to the parameter space ^M which approaches the same point both at 𝑥 = −∞ and 𝑥 = +∞. Such a map is topologically the same as a loop in the parameter space with a basepoint corresponding to the value of the parameters at𝑥 =±∞. It follows from eq. (9.4) that the topological invariantΔ𝑄 =∫

𝜙^∗𝜏attached to a loop can be interpreted in two different ways: as minus the net charge pumped through 𝑥 = 0 when the loop parameter is a slowly-varying function of time and as the charge of a skyrmion corresponding to the loop. From the point of view of lattice models, it is far from obvious that the same topological invariant controls both quantities. Our first goal is to show that this is indeed the case.

Let us consider the parameter space ^M given by a loop 𝑆¹ parameterized by a variable𝜆 ∈ [0,1] such that both 0 and 1 correspond to the basepoint. Thus we have a one-parameter family of gapped𝑈(1)-invariant Hamiltonians

𝐻(𝜆) =Õ

𝑝

𝐻_𝑝(𝜆) (9.39)

such that 𝐻𝑝(0) = 𝐻𝑝(1). We assume that all these Hamiltonians have a unique ground state and denote by 𝜉 the supremum of the correlation lengths of these ground states.

Let𝑔 :R→ [0,1] be a continuous function defined as follows:

𝑔(𝑥) =













0, 𝑥 < 𝐿

𝑥

𝐿 −1, 𝐿 ≤ 𝑥 ≤ 2𝐿 1, 𝑥 >2𝐿.

(9.40)

The "skyrmion" Hamiltonian𝐻^𝑠 is obtained by making𝜆depend on 𝑝:

𝐻^𝑠 =Õ

𝑝

𝐻_𝑝^𝑠 =Õ

𝑝

𝐻_𝑝(𝑔(𝑝)). (9.41)

Thus𝐻^𝑠_𝑝 =𝐻𝑝(0) if𝑝 < 𝐿or 𝑝 > 2𝐿.

To proceed, we need to make a technical assumption. Suppose we are given a family of gapped Hamiltonians depending on some parameters which live in a compact parameter spaceR. Suppose also that all these Hamiltonians have a unique ground state and let𝜉 be the supremum of the correlation lengths of all the ground states.

Now suppose we make the parameters slowly varying functions of coordinates. By a slow variation we mean that the parameters vary appreciably over a scale𝐿which is much larger than𝜉. Our technical assumption will be that the new Hamiltonian is still gapped, has a unique ground state, and its correlation length is of order 𝜉 and thus is still much smaller than𝐿. Essentially, this is the same as assuming that derivative expansion makes sense for gapped systems with a unique ground state.

Assuming this, we can apply the results of [60] on the insensitivity of the expectation values of local observables on the behavior of the Hamiltonian far from the support of the observable. In particular, the expectation values of observables supported at 𝑥 <0 or𝑥 >3𝐿in the ground state of𝐻^𝑠 are exponentially close to the expectation values of the same observables in the ground state of𝐻(0). Therefore we can define the skyrmion charge as follows:

𝑄^𝑠 = lim

𝐿→∞

Õ

𝑝

h𝑄_𝑝i𝑠− h𝑄_𝑝i0

, (9.42)

whereh. . .i^𝑠 andh. . .i0denote expectation values in ground states of𝐻^𝑠and𝐻(0), respectively. The sum over 𝑝 is converging exponentially fast away from 𝐿 < 𝑝 <

2𝐿, so we can write 𝑄^𝑠 =Õ

𝑝

h𝑄_𝑝i𝑠ℎ(𝑝) −Õ

𝑝

h𝑄_𝑝i0ℎ(𝑝) +𝑂(𝐿^−∞), (9.43) whereℎ(𝑥) =𝜃(𝑥) −𝜃(𝑥−3𝐿).

To compute the r.h.s. of eq. (9.43) we need a family of gapped Hamiltonians interpolating between𝐻(0)and𝐻^𝑠. Since the loop used to define𝐻^𝑠is assumed to be non-contractible, such an interpolation does not exist if we require the asymptotic behavior at 𝑥 = ±∞ to be fixed. But if we relax this constraint, the difficulty disappears. We are going to use the following one-parameter family:

𝐻˜(𝜇) =Õ

𝑝

𝐻˜_𝑝(𝜇) =Õ

𝑝

𝐻_𝑝(𝑔_𝜇(𝑝)), (9.44) where𝜇∈ [0,1] and the continuous function𝑔𝜇 :R→ [0,1] is defined as follows:

𝑔𝜇(𝑥) =

( 𝑔(𝑥), 𝑥 < 𝐿(1+𝜇),

𝜇, 𝑥 ≥ 𝐿(1+𝜇). (9.45)

Obviously, ˜𝐻(0) = 𝐻(0) and ˜𝐻(1) = 𝐻^𝑠. Also, if 𝑝 < 𝐿, then ˜𝐻𝑝(𝜇) = 𝐻𝑝(0) regardless of the value of 𝜇, while for 𝑝 > 2𝐿 𝐻˜𝑝(𝜇) = 𝐻𝑝(𝜇). By our basic

assumption, the Hamiltonian ˜𝐻(𝜇) is gapped for all 𝜇, has a unique ground state, and its correlation length is of order𝜉. We can write:

𝑄^𝑠 =Õ

𝑞

ℎ(𝑞)

∫ 𝜇 0 𝑑𝜇 𝑑

𝑑𝜇h𝑄_𝑞i^𝜇+𝑂(𝐿^−∞), (9.46) where h. . .i𝜇 denotes the expectation value in the ground-state corresponding to 𝐻˜(𝜇).

On the other hand, for the one-parameter family ˜𝐻(𝜇) we have an identity 𝑑h𝑄_𝑞i𝜇 =Õ

𝑝

𝑇˜_𝑝𝑞⁽¹⁾. (9.47)

Here the 1-form ˜𝑇_𝑝𝑞⁽¹⁾ on [0,1]is given by 𝑇˜_𝑝𝑞⁽¹⁾ =

∮ 𝑑𝑧

2𝜋𝑖T𝑟 𝐺𝑑˜ 𝐻˜_𝑝𝐺𝑄˜ _𝑞−𝐺𝑑˜ 𝐻˜_𝑞𝐺𝑄˜ _𝑝

, (9.48)

and ˜𝐺 = (𝑧−𝐻˜)⁻¹. It is skew-symmetric under the interchange of 𝑝, 𝑞and decays exponentially when|𝑝−𝑞|is large. Using this, we can re-write the skyrmion charge as follows:

𝑄^𝑠 = 1 2

∫ ₁

0

Õ

𝑝𝑞

(ℎ(𝑞) −ℎ(𝑝))𝑇˜_𝑝𝑞⁽¹⁾+𝑂(𝐿^−∞). (9.49) Now note that since the functionℎ(𝑥) =𝜃(𝑥) −𝜃(𝑥−3𝐿) is constant on the scale 𝜉 everywhere except near 𝑥 =0 and𝑥 =3𝐿, only the neighborhoods of 𝑝 =𝑞 =0 and 𝑝 =𝑞 = 3𝐿may contribute appreciably to the double sum over 𝑝, 𝑞. One can make this explicit by writing

𝑄^𝑠 =𝑄^𝑠₀−𝑄^𝑠_3𝐿 +𝑂(𝐿^−∞), (9.50) where𝑄^𝑠_𝑎is obtained from the r.h.s. of eq. (9.49) by replacing ℎ(𝑥) with𝜃(𝑥−𝑎). Now, since ˜𝐻𝑝(𝜇)is independent of𝜇for𝑝 < 𝐿, the 1-forms ˜𝑇_𝑝𝑞⁽¹⁾are identically zero when both 𝑝and𝑞are in the neighborhood of𝑥=0. Therefore𝑄^𝑠₀is exponentially small for𝐿 𝜉. Further, when both 𝑝and𝑞are in the neighborhood of𝑥=3𝐿, the 1-forms ˜𝑇_𝑝𝑞⁽¹⁾ are exponentially close to the 1-forms𝑇_𝑝𝑞⁽¹⁾ for the Hamiltonian𝐻(𝜇). This follows from the insensitivity of the correlators of the form

∮ 𝑑𝑧

2𝜋𝑖T𝑟(𝐺 𝐴𝐺 𝐵) (9.51)

to the Hamiltonian far from the support of 𝐴and𝐵[60]. Thus 𝑄^𝑠 =−1

2

∫ Õ

𝑝,𝑞

(𝜃(𝑞−3𝐿) −𝜃(𝑝−3𝐿))𝑇_𝑝𝑞⁽¹⁾ +𝑂(𝐿^−∞). (9.52)

Comparing with eq. (9.25) and taking the limit 𝐿 → ∞, we conclude that the skyrmion charge is minus the value of the Thouless charge pump for the corre- sponding loop.

Now we use the same approach to understand the physical meaning of the Thouless charge pump invariant for 2d systems. We start with a family of𝑈(1)-invariant gapped 2d Hamiltonians parameterized by𝑆¹×𝑆¹:

𝐻(𝜆, 𝜎) =Õ

𝑝

𝐻𝑝(𝜆, 𝜎). (9.53)

Here𝜆and𝜎are periodically identified with period 1. We are going to associate to this two-parameter family two one-parameter families. The first one is simply

𝐻⁰(𝜆) =Õ

𝑝

𝐻_𝑝(𝜆,0). (9.54)

To define the second one, we need to choose a stripSonR²of width 3𝐿much larger than the correlation length. We choose coordinates onR²so that the strip is given by the inequalities 0≤ 𝑦 ≤ 3𝐿. Let us also denote byS0the strip of width𝐿which in this coordinate system is given by𝐿 ≤ 𝑦 ≤ 2𝐿. Obviously,S0⊂ S.We define

𝐻^𝑠(𝜆) =Õ

𝑝

𝐻^𝑠_𝑝(𝜆) =Õ

𝑝

𝐻𝑝(𝜆, 𝑔(𝑦(𝑝))), (9.55) where𝑔is defined in (9.40). Thanks to the periodicity in𝜎, 𝐻^𝑠_𝑝(𝜆) coincides with 𝐻𝑝(𝜆,0)outside of the stripS0. Also, both families are𝑈(1)-invariant and periodic in𝜆with period 1. By our basic technical assumption, for sufficiently large 𝐿 the Hamiltonians𝐻^𝑠(𝜆) are gapped for all𝜆.

Now consider adiabatically varying𝜆as a function of time. As one varies𝜆from 0 to 1, the charge flows across the line𝑥 =0. The net charge transport across 𝑥 = 0 will be infinite both for𝐻⁰(𝜆) and𝐻^𝑠(𝜆). However, their difference is finite. This is because outside the stripS0the two Hamiltonians coincide, and thus outside the horizontal strip S correlators of the form (9.51) are the same up to terms which are exponentially suppressed far fromS. The difference of net charges transported across𝑥 =0 is

Δ𝑄^𝑠−Δ𝑄⁰ = 1 2

∫ ₁

0

Õ

𝑝𝑞

(𝑓(𝑞) − 𝑓(𝑝))

𝑇_𝑝𝑞^𝑠(1)(𝜆) −𝑇_𝑝𝑞⁽¹⁾(𝜆,0)

, (9.56) where 𝑓(𝑝) =𝜃(𝑥(𝑝)), and𝑇_𝑝𝑞^𝑠(1)(𝜆)and𝑇_𝑝𝑞⁽¹⁾(𝜆,0)are the 1-forms (9.22) on [0,1] for Hamiltonian families𝐻^𝑠(𝜆) and𝐻⁰(𝜆), respectively. The above expression for

Δ𝑄^𝑠−Δ𝑄⁰can be interpreted in more physical terms by stacking the family𝐻^𝑠(𝜆) with the time-reversal of the family 𝐻⁰(𝜆). This gives a family of 2d systems for which the Thouless charge pump across the line𝑥 = 0 is finite, because the charge transport cancels out outside the stripS.

Next we introduce a cut-off functionℎ(𝑝) =𝜃(𝑦(𝑝)) −𝜃(3𝐿−𝑦(𝑝))and write Δ𝑄^𝑠−Δ𝑄⁰= 1

2

∫ ₁

0

Õ

𝑝𝑞

(𝑓(𝑞) − 𝑓(𝑝))ℎ(𝑞)

𝑇_𝑝𝑞^𝑠(1)(𝜆) −𝑇_𝑝𝑞⁽¹⁾(𝜆,0)

+𝑂(𝐿^−∞). (9.57) Using the chain-cochain notation, this can also be written as

Δ𝑄^𝑠 −Δ𝑄⁰ =

∫ ₁

0 𝑇_𝑝𝑞^𝑠(1)(𝜆;𝛿 𝑓 ∪ℎ) −

∫ ₁

0 𝑇⁽¹⁾(𝜆,0;𝛿 𝑓 ∪ℎ) +𝑂(𝐿^−∞). (9.58) The advantage of introducing the cut-off function ℎ is that now both terms in eq.

(9.58) are separately well-defined.

To compute the r.h.s. of (9.58) we construct a two-parameter family of gapped Hamiltonians ˜𝐻(𝜆, 𝜇)which interpolates between the family 𝐻^𝑠(𝜆) and the family 𝐻⁰(𝜆). We define

𝐻˜(𝜆, 𝜇) =Õ

𝑝

𝐻˜_𝑝(𝜆, 𝜇) =Õ

𝑝

𝐻_𝑝(𝜆, 𝑔_𝜇(𝑦(𝑝))), (9.59) where𝑔𝜇 :R→ [0,1]is defined by eq. (9.45). After the same kind of manipulations that lead from (9.43) to (9.49) we get

Δ𝑄^𝑠 −Δ𝑄⁰=−

∫ 𝑇˜⁽²⁾(𝛿 𝑓 ∪𝛿ℎ) +𝑂(𝐿^−∞), (9.60) where the integration is over the square [0,1]²in the𝜆−𝜇plane. The contraction of a 2-chain with a 2-cochain involves a triple sum over 𝑝, 𝑞, 𝑟 ∈ Λ. Now we note that only the terms where all three points 𝑝, 𝑞, 𝑟 are close to the lines 𝑦 = 3𝐿 or 𝑦 = 0 contribute appreciably to the sum. The contribution of 𝑦 = 0 is of order 𝑂(𝐿^−∞), because ˜𝐻𝑝(𝜆, 𝜇) does not depend on𝜇there and thus the 2-form ˜𝑇_𝑝𝑞𝑟⁽²⁾ is exponentially small. When evaluating the contribution of𝑦 =3𝐿, one can replace 𝑇˜⁽²⁾ with𝑇⁽²⁾ while making an error of order𝑂(𝐿^−∞). Thus we get

Δ𝑄^𝑠−Δ𝑄⁰ =

∫

𝑇⁽²⁾(𝛿 𝑓 ∪𝛿ℎ_3𝐿) +𝑂(𝐿^−∞), (9.61) whereℎ₃𝐿(𝑝) =𝜃(𝑦(𝑝)−3𝐿). Taking the limit𝐿 → ∞we conclude thatΔ𝑄^𝑠−Δ𝑄⁰ is the topological invariant of the family𝐻(𝑡, 𝜎).