Canonical Drude Weight for Non-integrable Quantum Spin Chains

Abstract

The Drude weight is a central quantity for the transport properties of quantum spin chains. The canonical definition of Drude weight is directly related to Kubo formula of conductivity. However, the difficulty in the evaluation of such expression has led to several alternative formulations, accessible to different methods. In particular, the Euclidean, or imaginary-time, Drude weight can be studied via rigorous renormalization group. As a result, in the past years several universality results have been proven for such quantity at zero temperature; remarkably, the proofs work for both integrable and non-integrable quantum spin chains. Here we establish the equivalence of Euclidean and canonical Drude weights at zero temperature. Our proof is based on rigorous renormalization group methods, Ward identities, and complex analytic ideas.

Appendices

Appendix A: Formal Derivation of Kubo Formula

In this appendix we give formal derivation of Kubo formula for the conductivity of the quantum spin chain, Eq. (2.6). Equation (2.6) provides the linear response coefficient that allows to describe the variation of the average current density after introducing a weak external electric field, assumed to be uniform in space. The electric field is adiabatically switched on at $t=-\infty$, starting from the thermal state $\rho _{\beta }$ of the Hamiltonian $\mathcal {H}$.

In a rigorous derivation of Kubo formula, one has to take the thermodynamic limit before the limit of vanishing perturbation. Controlling the thermodynamic limit for a fixed external perturbation poses a technical challenge, that for interacting systems has only been solved for gapped systems, [3, 24, 32]. Instead, the models we are considering are gapless: the problem of deriving Kubo formula for this class of interacting quantum spin chains is wide open. We shall not try to study this interesting question here; instead, we will focus on the linear response of the current operator, formally neglecting higher order terms.

For simplicity, let us directly consider the case $L=\infty$: $\Lambda _{L} = \mathbb {Z}$. Let $\mathcal {X} = \sum _{x} x a^{+}_{x}a^{-}_{x}$ be the second quantization of the position operator. Consider the time-dependent Hamiltonian $\mathcal {H}(t) = \mathcal {H} - e^{\eta t} E \mathcal {X}$, for $t\in (-\infty , 0]$ and $\eta >0$, $E\in \mathbb {R}$. Let $\rho (t)$ denote the solution of the Schrödinger–von Neumann equation:

$$\begin{aligned} i\partial _{t} \rho (t) = [\mathcal {H}(t), \rho (t)],\qquad \rho (-\infty ) = \rho _{\beta }. \end{aligned}$$

(A.1)

We are interested in a formal expansion in E for the average current, in the limit $\eta \rightarrow 0^{+}$. An application of Duhamel formula gives:

$$\begin{aligned} \mathrm {Tr}\, j_{x} \rho (0)= & {} \mathrm {Tr}\, j_{x} \rho _{\beta } - i\int _{-\infty }^{0} dt\, e^{\eta t} \mathrm {Tr}\, j_{x} [ \rho _{\beta }, E \mathcal {X}(t) ] + o(E)\nonumber \\= & {} i\int _{-\infty }^{0} dt\, e^{\eta t} \mathrm {Tr}[ j_{x}(-t), E\mathcal {X} ]\rho _{\beta } + o(E), \end{aligned}$$

(A.2)

since $\mathrm {Tr}\, j_{x} \rho _{\beta } = 0$. Then,

$$\begin{aligned}{}[ j_{x}(-t), \mathcal {X} ]= & {} [ e^{-i \mathcal {H} t} j_{x} e^{i \mathcal {H} t}, \mathcal {X} ]\nonumber \\= & {} e^{-i \mathcal {H}t} [ j_x , \mathcal {X}] e^{i\mathcal {H}t} + [ e^{-i\mathcal {H}t}, \mathcal {X} ] j_x e^{i \mathcal {H}t} + e^{-i\mathcal {H}t} j_x [e^{i\mathcal {H}t}, \mathcal {X}],\qquad \end{aligned}$$

(A.3)

where:

$$\begin{aligned}{}[ j_x , \mathcal {X}] = -\frac{i\tau }{2} [a^{+}_{x+1} a^{-}_{x} + a^{+}_{x}a^{-}_{x+1}] \equiv i \Delta _{x}, \end{aligned}$$

(A.4)

and

$$\begin{aligned}{}[ e^{-i\mathcal {H}t}, \mathcal {X} ]= & {} e^{-i \mathcal {H} t} \mathcal {X} - \mathcal {X} e^{-i\mathcal {H} t} = \int _{t}^{0} ds\, \frac{d}{ds} e^{-i\mathcal {H}(t - s)} \mathcal {X} e^{-i\mathcal {H}s}\nonumber \\= & {} i\int _{t}^{0} ds\, e^{-i \mathcal {H}(t - s)} [ \mathcal {H}, \mathcal {X} ] e^{-i\mathcal {H}s}\nonumber \\ {}= & {} \int _{t}^{0}ds\, e^{-i\mathcal {H}t} \mathcal {J}(s), \end{aligned}$$

(A.5)

where we set $\mathcal {J} \equiv \sum _{x} j_{x}$. Also, $[ e^{i\mathcal {H}t}, \mathcal {X} ] = -[ e^{-i\mathcal {H}t}, \mathcal {X} ]^{*} = -\int _{t}^{0}ds\, \mathcal {J}(s) e^{i\mathcal {H}t}$. Therefore, plugging (A.3)–(A.5) into Eq. (A.2) we get:

$$\begin{aligned}&i\int _{-\infty }^{0} dt\, e^{\eta t} \mathrm {Tr}[ j_{x}(-t), \mathcal {X} ]\rho _{\beta } \nonumber \\&\quad = -\int _{-\infty }^{0} dt\, e^{\eta t} \mathrm {Tr}\, \Delta _{x} \rho _{\beta } + i \int _{-\infty }^{0} dt\, e^{\eta t} \int _{t}^{0} ds\, \mathrm {Tr}\, [ \mathcal {J}(s), j_{x} ] \rho _{\beta } \nonumber \\&\quad = -\frac{1}{\eta } \mathrm {Tr}\, \Delta _{x} \rho _{\beta } + i \int _{-\infty }^{0} ds\, \int _{-\infty }^{s} dt\, e^{\eta t} \mathrm {Tr}\, [ \mathcal {J}(s), j_{x} ] \rho _{\beta }\nonumber \\&\quad = \frac{1}{\eta } \Big [ -\mathrm {Tr}\, \Delta _{x} \rho _{\beta } + i\int _{-\infty }^{0} ds\, e^{\eta s} \mathrm {Tr}\, [ \mathcal {J}(s), j_{x} ] \rho _{\beta } \Big ]. \end{aligned}$$

(A.6)

By translation invariance, the above expression does not depend on x. Thus, the right-hand side of Eq. (A.6) reproduces Eq. (2.6), for $p = 0$ and $\eta \rightarrow 0^{+}$ (notice that $\hat{j}_{0}\equiv \mathcal {J}$). In general, $p\ne 0$ allows to take into account a space modulation of the external field.

Appendix B: On the Equivalence Between Thermal and Canonical Drude Weight

In this appendix we discuss the equivalence between a suitably regularized version of the thermal Drude weight, and the canonical Drude weight. Given two operators A, B, we define their Kubo scalar product as:

$$\begin{aligned} \langle A B\rangle ^{K}_{\beta ,L} := \int _{0}^{\beta } dx_{0}\, \langle A(-ix_{0}) B \rangle _{\beta ,L}. \end{aligned}$$

(B.1)

We then notice that:

$$\begin{aligned} i\partial _{t} \langle A(t) B \rangle ^{K}_{\beta ,L} = \langle [ A(t), B ] \rangle _{\beta ,L}. \end{aligned}$$

(B.2)

Therefore, one formally has:

$$\begin{aligned} \widetilde{D}^{\text {(Th)}} _\beta= & {} \lim _{t\rightarrow \infty } \lim _{L\rightarrow \infty } \frac{1}{L}\langle \mathcal {J}(t) \mathcal {J} \rangle ^{K}_{\beta , L} \nonumber \\= & {} \lim _{L\rightarrow \infty } \frac{1}{L}\langle \mathcal {J} \mathcal {J}\rangle _{\beta , L}^{K} - i \int _{0}^{\infty } ds\, \lim _{L\rightarrow \infty } \frac{1}{L}\langle [ \mathcal {J}(s), \mathcal {J} ] \rangle _{\beta ,L} \nonumber \\\equiv & {} \lim _{L\rightarrow \infty } \frac{1}{L}\langle \mathcal {J} \mathcal {J}\rangle _{\beta , L}^{K} + i \int _{-\infty }^{0} ds\, \lim _{L\rightarrow \infty } \frac{1}{L} \langle [ \mathcal {J}(s), \mathcal {J} ] \rangle _{\beta , L}.\nonumber \end{aligned}$$

The main problem in making sense of the above identity is the existence of the time integration: for nonintegrable systems, proving that the integral converges is a very hard open problem. Therefore, let us introduce the following regularized version of the thermal Drude weight, at positive and zero temperature:

$$\begin{aligned} \widehat{D}^{\text {(Th)}} _\beta= & {} \lim _{L\rightarrow \infty } \frac{1}{L}\langle \mathcal {J} \mathcal {J}\rangle ^{K}_{\beta , L} + \lim _{\eta \rightarrow 0^{+}}i \int _{-\infty }^{0} ds\, e^{\eta s} \lim _{L\rightarrow \infty } \frac{1}{L} \langle [ \mathcal {J}(s), \mathcal {J} ] \rangle _{\beta ,L}\nonumber \\ \widehat{D}^{\text {(Th)}} _\infty= & {} \lim _{\beta , L\rightarrow \infty }\frac{1}{L} \langle \mathcal {J} \mathcal {J}\rangle ^{K}_{\beta , L} + \lim _{\eta \rightarrow 0^{+}}i \int _{-\infty }^{0} ds\, e^{\eta s} \lim _{\beta , L\rightarrow \infty } \frac{1}{L}\langle [ \mathcal {J}(s), \mathcal {J} ] \rangle _{\beta ,L}. \end{aligned}$$

(B.3)

Of course, $\widehat{D}^{\text {(Th)}} = \widetilde{D}^{\text {(Th)}}$ if the real-time correlations decay fast enough. Such time regularization plays the same role of the adiabatic factor in the derivation of Kubo formula, see Appendix A. This regularized version of the thermal Drude weight matches the canonical formulation, Eq. (2.9), provided one can show that

$$\begin{aligned} \lim _{L\rightarrow \infty } \frac{1}{L}\langle \mathcal {J} \mathcal {J} \rangle ^{K}_{\beta , L} = - \lim _{L\rightarrow \infty } \langle \Delta _{x} \rangle _{\beta ,L}. \end{aligned}$$

(B.4)

To prove Eq. (B.4), we proceed as follows. Consider the lattice continuity equation:

$$\begin{aligned} \partial _{t} \rho _{x}(t) + d_{x} j_{x}(t) = 0, \end{aligned}$$

(B.5)

with $d_{x} j_{x} = j_{x} - j_{x-1}$ the discrete lattice derivative. In imaginary times, Eq. (B.5) reads:

$$\begin{aligned} i\partial _{x_{0}} \rho _{x}(-ix_{0}) + \partial _{x} j_{x}(-ix_{0}) = 0. \end{aligned}$$

(B.6)

Eq. (B.6) can be used to derive Ward identities for the current-current correlations, as follows. Let $j_{0,x} \equiv \rho _{x}$, $j_{1,x} \equiv j_{x}$. Recall:

$$\begin{aligned} \langle \mathbf{T}\, j_{0, x}(-ix_{0})\,; j_{\nu , 0}(-iy_{0}) \rangle _{\beta ,L}= & {} \theta (x_{0} - y_{0}) \langle j_{0, x}(-ix_{0})\,; j_{\nu , 0}(-iy_{0}) \rangle _{\beta ,L}\nonumber \\&+\,\theta (y_{0} - x_{0}) \langle j_{\nu , 0}(-iy_{0}) j_{0, x}(-ix_{0}) \rangle _{\beta ,L}\;; \end{aligned}$$

(B.7)

therefore, using the continuity equation Eq. (B.6):

$$\begin{aligned} i\partial _{x_{0}}\langle \mathbf{T}\, j_{0, x}(-ix_{0})\,; j_{\nu , 0}(-iy_{0}) \rangle _{\beta ,L}= & {} \langle \mathbf{T}\, i\partial _{x_{0}}j_{0, x}(-ix_{0})\,; j_{\nu , 0}(-iy_{0}) \rangle _{\beta ,L}\nonumber \\&+\,i\langle [ j_{0, x}\, , j_{\nu , 0} ] \rangle _{\beta ,L} \delta (x_{0} - y_{0})\nonumber \\\equiv & {} -\langle \mathbf{T}\, \partial _{x}j_{1, x}(-ix_{0})\,; j_{\nu , 0}(-iy_{0}) \rangle _{\beta ,L}\nonumber \\&+\,i\langle [ j_{0, x}\, , j_{\nu , 0} ] \rangle _{\beta ,L} \delta (x_{0} - y_{0}). \end{aligned}$$

(B.8)

Now, consider the Fourier transform:

$$\begin{aligned} \langle \mathbf{T}\, \hat{j}_{\mu ,\mathbf{p}}\,; \hat{j}_{\nu ,-\mathbf{p}} \rangle _{\beta , L}\equiv & {} \frac{1}{\beta } \int _{0}^{\beta } d x_{0} \int _{0}^{\beta } d y_{0}\,e^{-ip_{0}(x_{0} - y_{0})} \nonumber \\&\times \,\sum _{x = -L/2}^{L/2} e^{-ipx} \langle \mathbf{T}\, j_{\mu , x}(-i x_{0})\,; j_{\nu , 0}(-i y_{0}) \rangle _{\beta , L}, \end{aligned}$$

(B.9)

with $\mathbf{T}$ the fermionic time ordering, and $\mathbf{p}= (p_{0}, p) \in \frac{2\pi }{\beta } \mathbb {Z} \times \frac{2\pi }{L} \mathbb {Z}$. In Eq. (B.9), the $x_{0}, y_{0}$ integrations have to be understood as integrals over circles of length $\beta$ (the imaginary time evolution of the current operators is extended periodically outside of $[0, \beta )$). We have, integrating by parts the time-derivative:

$$\begin{aligned}&p_{0}\langle \mathbf{T}\, \hat{j}_{0,\mathbf{p}}\,; \hat{j}_{\nu ,-\mathbf{p}} \rangle _{\beta ,L} \\&\quad = \frac{1}{\beta } \int _{0}^{\beta } d x_{0} \int _{0}^{\beta } d y_{0}\, p_{0}e^{-ip_{0}(x_{0} - y_{0})}\sum _{x} e^{-ipx} \langle \mathbf{T}\, j_{0, x}(-i x_{0})\,; j_{\nu , 0}(-iy_{0}) \rangle _{\beta ,L}\\&\quad = \frac{1}{\beta } \int _{0}^{\beta } d x_{0}\, \int _{0}^{\beta } d y_{0}\, (i\partial _{x_{0}} e^{-ip_{0}(x_{0} - y_{0})})\sum _{x} e^{-ipx} \langle \mathbf{T}\, j_{0, x}(-ix_{0})\,; j_{\nu , 0}(-i y_{0}) \rangle _{\beta ,L}\\&\quad = -\frac{1}{\beta } \int _{0}^{\beta } dx_{0} \int _{0}^{\beta } dy_{0}\, e^{-ip_{0}(x_{0} - y_{0})} \sum _{x} e^{-ipx} i\partial _{x_{0}}\langle \mathbf{T}\, j_{0, x}(-i x_{0})\,; j_{\nu , 0}(-i y_{0}) \rangle _{\beta ,L}. \end{aligned}$$

Then, thanks to the continuity equation (B.5), we get:

$$\begin{aligned} p_{0}\langle \mathbf{T}\, \hat{j}_{0,\mathbf{p}}\,; \hat{j}_{\nu ,-\mathbf{p}} \rangle _{\beta ,L} = (1 - e^{-ip}) \langle \mathbf{T}\, \hat{j}_{1,\mathbf{p}}\,; \hat{j}_{\nu , -\mathbf{p}} \rangle _{\beta ,L} - i\sum _{x} e^{-ipx} \langle [ j_{0, x}\, , j_{\nu , 0} ] \rangle _{\beta ,L}. \end{aligned}$$

(B.10)

Let $\nu = 1$, and set $p_{0} = 0$, $p_{1}\ne 0$. Eq. (B.10) implies:

$$\begin{aligned} \begin{aligned} (1 - e^{-ip}) \langle \mathbf{T}\, \hat{j}_{1,(0,p)}\,; \hat{j}_{1, (0,-p)} \rangle _{\beta ,L}&= i\sum _{x} e^{-ipx} \langle [ j_{0, x}\, , j_{1, 0} ] \rangle _{\beta ,L} \\&= \frac{\tau }{2}(1 - e^{-ip}) \langle a^{+}_{1}a^-_{0} + a^{+}_{0}a^-_{1}\rangle _{\beta ,L}. \end{aligned} \end{aligned}$$

(B.11)

Now, notice that $\langle \mathbf{T}\, \hat{j}_{1,\mathbf{0}}\,; \hat{j}_{1,\mathbf{0}} \rangle _{\beta ,L} = L^{-1}\langle \mathcal {J} \mathcal {J}\rangle ^{K}_{\beta ,L}$. Hence, from Eq. (B.11), recalling Eq. (A.4), and setting $\langle \cdot \rangle _{\beta } \equiv \lim _{L\rightarrow \infty } \langle \cdot \rangle _{\beta ,L}$:

$$\begin{aligned} \lim _{L\rightarrow \infty } \frac{1}{L}\langle \mathcal {J} \mathcal {J}\rangle ^{K}_{\beta ,L} = \lim _{p\rightarrow 0}\langle \mathbf{T}\, \hat{j}_{1,(0,p)}\,; \hat{j}_{1, (0,-p)} \rangle _{\beta } = \frac{\tau }{2}\langle a^{+}_{1}a_{0} + a^{+}_{0}a_{1}\rangle _{\beta } \equiv -\langle \Delta _{x} \rangle _{\beta }, \end{aligned}$$

(B.12)

which proves Eq. (B.4).