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Kramers–Wannier Duality

Main result

For the isotropic 2D classical Ising model on a square lattice with reduced coupling $K = \beta J$, the high- and low-temperature expansions of the partition function map into each other under the Kramers–Wannier duality

\[\boxed{\; \tanh K^{*} \;=\; e^{-2K}, \qquad \sinh(2K)\,\sinh(2K^{*}) \;=\; 1. \;}\]

The self-dual fixed point $K = K^{*}$ gives the exact critical temperature of the 2D Ising model,

\[\boxed{\; K_{c} \;=\; \tfrac{1}{2}\ln\!\bigl(1 + \sqrt{2}\bigr), \qquad T_{c} \;=\; \frac{2 J}{\ln(1 + \sqrt{2})} \;\approx\; 2.269\,J. \;}\]

The 1D quantum TFIM $H = -J\sum \sigma^z_i \sigma^z_{i+1} - h\sum \sigma^x_i$ inherits an operator-level duality from the time- continuum limit of the 2D Ising transfer matrix: the ordered ($h < J$) and disordered ($h > J$) phases are interchanged by a unitary $U$ with

\[\boxed{\; U\,H(J, h)\,U^{\dagger} \;=\; H(h, J),\qquad U\,\sigma^z_i \sigma^z_{i+1}\,U^{\dagger} \;=\; \sigma^x_{i+1}, \qquad U\,\sigma^x_i\,U^{\dagger} \;=\; \sigma^z_{i-1}\sigma^z_{i}. \;}\]

The quantum critical point $h = J$ is the self-dual point of this operator duality.

Setup

2D classical Ising model

On an $L_x \times L_y$ square lattice with periodic boundary conditions, classical spins $s_v \in \{\pm 1\}$ live on vertices $v$ and the Hamiltonian is

\[H_{\rm cl} = -J\sum_{\langle u, v\rangle} s_u s_v, \qquad J > 0,\]

with the sum over nearest-neighbour bonds. The partition function is

\[Z = \sum_{\{s\}} e^{-\beta H_{\rm cl}} = \sum_{\{s\}} \prod_{\langle u, v\rangle} e^{K s_u s_v}, \qquad K \equiv \beta J.\]

1D quantum TFIM

The 1D transverse-field Ising model

\[H = -J\sum_{i=1}^{N}\sigma^z_i\sigma^z_{i+1} - h\sum_{i=1}^{N}\sigma^x_i\]

(PBC) is obtained from the 2D Ising model by taking the continuous- time limit of the row-to-row transfer matrix: the 2D partition function becomes $Z = \mathrm{Tr}\,e^{-\beta_{\rm eff} H}$ with $\beta_{\rm eff} \propto L_y$ and an anisotropic identification between $J, h$ and the two 2D couplings (Mattis §§3.4–3.5; Sachdev §5.2). The self-dual structure of 2D Ising therefore descends to an operator symmetry of the 1D TFIM.

Goal

Derive $\tanh K^{*} = e^{-2K}$ from first principles by computing both the high-$T$ and low-$T$ expansions of $Z$, identifying their common form, and reading off the duality relation. Then extract the critical temperature and translate the duality to the 1D TFIM operator algebra.

Calculation

Step 1 — High-temperature expansion

For a single bond $(u, v)$ with $s_u, s_v \in \{\pm 1\}$, the product $s_u s_v \in \{\pm 1\}$ too. Expanding the exponential,

\[e^{K s_u s_v} = \cosh K + s_u s_v \sinh K = \cosh K\,\bigl(1 + s_u s_v\,\tanh K\bigr),\]

using $\cosh K\pm\sinh K = e^{\pm K}$ and splitting by parity in $s_u s_v$. The full Boltzmann weight factorises:

\[e^{-\beta H_{\rm cl}} = (\cosh K)^{N_b}\, \prod_{\langle u, v\rangle}\bigl(1 + s_u s_v\,\tanh K\bigr),\]

where $N_b = L_x L_y \cdot 2$ is the total bond count (two bonds per vertex on the square lattice, PBC). Summing over spins,

\[Z = (\cosh K)^{N_b} \sum_{\{s\}}\prod_{\langle u, v\rangle} \bigl(1 + s_u s_v \tanh K\bigr).\]

Expand the product into $2^{N_b}$ terms, one for each subset $\Gamma \subseteq E$ of the bond set $E$. Each term looks like

\[(\tanh K)^{|\Gamma|}\cdot\prod_{(u,v)\in\Gamma} s_u s_v,\]

so

\[Z = (\cosh K)^{N_b} \sum_{\Gamma \subseteq E}(\tanh K)^{|\Gamma|} \sum_{\{s\}}\prod_{(u,v)\in\Gamma} s_u s_v.\]

The spin sum factorises over vertices. Each vertex $v$ appears in the product as $s_v^{n_v(\Gamma)}$ where $n_v(\Gamma)$ is the number of edges of $\Gamma$ incident to $v$. Because $s_v \in \{\pm 1\}$,

\[\sum_{s_v = \pm 1} s_v^{n_v} = \begin{cases} 2 & n_v\text{ even},\\ 0 & n_v\text{ odd}.\end{cases}\]

So $\sum_{\{s\}}\prod_{(u,v)\in\Gamma} s_u s_v$ vanishes unless every vertex has even $\Gamma$-degree. Such subsets $\Gamma$ are called closed graphs (each vertex is an even-degree endpoint, equivalent to a disjoint union of closed cycles on the lattice). If $\Gamma$ is closed, the sum equals $2^{N_v}$ where $N_v = L_x L_y$ is the number of vertices. Therefore

\[\boxed{\; Z = (\cosh K)^{N_b}\,2^{N_v}\, \sum_{\Gamma \in \mathcal{C}} (\tanh K)^{|\Gamma|}, \;} \tag{HT}\]

where $\mathcal{C}$ is the set of all closed subgraphs of the square lattice. Equation (HT) is the high-temperature expansion: the factor $\tanh K$ is small for $K \ll 1$, so the leading contributions come from $\Gamma = \emptyset$ (weight 1), then short closed loops (weight $\tanh^4 K$ for an elementary plaquette, etc.).

Step 2 — Low-temperature expansion

Now expand $Z$ around the ordered ground state $s_v = +1$ for all $v$, of energy $E_0 = -J N_b$. A configuration $\{s\}$ is specified by the set of domain walls: bonds $(u, v)$ across which $s_u s_v = -1$. Place a dual-lattice edge perpendicular to each domain wall; this dual edge lives on the dual square lattice $\Lambda^{*}$ (another square lattice, shifted by $(1/2, 1/2)$).

A closed domain-wall configuration is a subset $\Gamma^{*} \subseteq E^{*}$ of dual edges such that every dual vertex has even $\Gamma^{*}$-degree (domain walls are oriented only through the sign flip, but the boundary of an Ising-spin region is a closed cycle of dual edges). There is a 2-to-1 correspondence between $\{s\}$ configurations and $\Gamma^{*}$ configurations (flipping all spins gives the same $\Gamma^{*}$), so

\[\sum_{\{s\}}\dotsb = 2\sum_{\Gamma^{*} \in \mathcal{C}^{*}}\dotsb,\]

with $\mathcal{C}^{*}$ the set of closed subgraphs of the dual lattice.

Each domain wall costs an energy $2 J$ relative to the ordered reference (the bond $(u, v)$ had $s_u s_v = -1$ instead of $+1$), so for a configuration with $|\Gamma^{*}|$ broken bonds,

\[E(\{s\}) = E_0 + 2 J\,|\Gamma^{*}|,\qquad e^{-\beta E(\{s\})} = e^{-\beta E_0}\,e^{-2 K\,|\Gamma^{*}|}.\]

Hence

\[\boxed{\; Z = 2\,e^{-\beta E_0} \sum_{\Gamma^{*} \in \mathcal{C}^{*}} (e^{-2K})^{|\Gamma^{*}|} = 2\,e^{K N_b} \sum_{\Gamma^{*} \in \mathcal{C}^{*}} (e^{-2K})^{|\Gamma^{*}|}. \;} \tag{LT}\]

Equation (LT) is the low-temperature expansion: the factor $e^{-2K}$ is small for $K \gg 1$, so the leading contributions come from $\Gamma^{*} = \emptyset$ (all spins aligned), then single flipped spins (a rectangle of 4 dual edges), etc.

Step 3 — Matching (HT) and (LT)

The square lattice is self-dual: the dual of the square lattice is another square lattice (up to translation). So $\mathcal{C} = \mathcal{C}^{*}$ as abstract graph-combinatorial objects — both are "closed subgraphs of a square lattice with the same number of vertices and edges".

The HT expansion has weight $\tanh K$ per edge and prefactor $(\cosh K)^{N_b} 2^{N_v}$. The LT expansion has weight $e^{-2K^{*}}$ per edge (reading $K^{*}$ instead of $K$ in (LT), which corresponds to the dual coupling) and prefactor $2 e^{K^{*} N_b}$. Equating the two sums over $\Gamma$ term-by-term forces

\[\boxed{\;\tanh K^{*} \;=\; e^{-2K}.\;} \tag{1}\]

This is the Kramers–Wannier duality. Its geometric content is that the high-$T$ closed-loop expansion of the original lattice becomes the low-$T$ domain-wall expansion of the dual lattice, and on a self-dual lattice the two are structurally identical with coupling renamed.

Equivalent symmetric form. Taking $\sinh(2K) = 2\sinh K \cosh K$ and $\cosh(2K) = \cosh^2 K + \sinh^2 K$, one finds

\[\sinh(2K)\,\sinh(2K^{*}) = 2\,\frac{\sinh K}{\cosh K}\cdot\cosh^2 K \cdot 2\,\frac{\sinh K^{*}}{\cosh K^{*}}\cdot\cosh^2 K^{*} = 4\,\tanh K\,\tanh K^{*}\,\cosh^2 K\,\cosh^2 K^{*}.\]

Using (1) to substitute $\tanh K^{*} = e^{-2K}$ and $\tanh K = e^{-2K^{*}}$ (by symmetry of (1)):

\[4\tanh K\,\tanh K^{*}\,\cosh^2 K\,\cosh^2 K^{*} = 4\,e^{-2K^{*}}\,e^{-2K}\,\cosh^2 K\,\cosh^2 K^{*}.\]

Now $\cosh^2 K = \tfrac{1}{4}(e^K + e^{-K})^2 = \tfrac{1}{4} (e^{2K} + 2 + e^{-2K})$ and likewise for $K^{*}$, so $4\,e^{-2K^{*}}\cosh^2 K^{*} = e^{-2K^{*}}(e^{2K^{*}} + 2 + e^{-2K^{*}}) = 1 + 2 e^{-2K^{*}} + e^{-4K^{*}}$. Using $e^{-2K^{*}} = \tanh K$ this simplifies to $(1 + \tanh K)^2 = 4\cosh^2 K/(e^K + e^{-K})^2 \cdot (e^K)^2 / 1$ — which gets cumbersome. A cleaner route is

\[\sinh(2K^{*}) = \frac{2\tanh K^{*}}{1 - \tanh^2 K^{*}} = \frac{2\,e^{-2K}}{1 - e^{-4K}} = \frac{2\,e^{-2K}\cdot e^{2K}}{e^{2K} - e^{-2K}} = \frac{2}{2\sinh(2K)} = \frac{1}{\sinh(2K)}.\]

Hence

\[\sinh(2K)\,\sinh(2K^{*}) = 1, \tag{2}\]

the symmetric form of the duality.

Step 4 — Self-dual fixed point

Imposing $K = K^{*}$ in (2) gives $\sinh(2K_c)^2 = 1$, i.e. $\sinh(2K_c) = 1$ (positive root for physical $K > 0$). Inverting,

\[2 K_c = \ln\!\bigl(1 + \sqrt{2}\bigr) \quad\Longleftrightarrow\quad K_c = \tfrac{1}{2}\ln\!\bigl(1 + \sqrt{2}\bigr),\]

using $\sinh x = 1 \Leftrightarrow e^x - e^{-x} = 2 \Leftrightarrow e^{2x} - 2 e^x - 1 = 0 \Leftrightarrow e^x = 1 + \sqrt{2}$ (positive root). Converting to temperature $K_c = J/(k_B T_c)$ with $k_B = 1$,

\[T_c = \frac{J}{K_c} = \frac{2 J}{\ln(1 + \sqrt{2})} \;\approx\; 2.269\,J.\]

Onsager 1944 confirmed independently that this self-dual point is indeed the phase-transition temperature (not just a self-dual fixed point — the duality argument alone shows that if there is a unique critical point, it must sit at $K_c$, but not that $K_c$ is critical). The independent proof is in the yang-magnetization-toeplitz derivation, which exhibits the non-analyticity explicitly.

Step 5 — Operator duality for the 1D TFIM

The 1D TFIM inherits the 2D duality via the transfer-matrix identification. We derive the operator form directly by constructing dual Pauli operators on the bonds of the original chain.

Dual operators. Labelling the bonds by $b = 1, \dots, N$ (bond $b$ connects sites $b$ and $b+1$ with PBC identification $N+1 \equiv 1$), define

\[\tau^x_b \;\equiv\; \sigma^z_b\,\sigma^z_{b+1}, \qquad \tau^z_b \;\equiv\; \prod_{k \le b}\sigma^x_k. \tag{3}\]

Pauli algebra on each bond. Check $(\tau^x_b)^2 = (\sigma^z_b)^2 (\sigma^z_{b+1})^2 = 1$ and $(\tau^z_b)^2 = \prod_k (\sigma^x_k)^2 = 1$. Anticommutation on the same bond:

\[\tau^x_b\,\tau^z_b = \sigma^z_b\sigma^z_{b+1}\prod_{k\le b}\sigma^x_k = -\sigma^z_b\sigma^x_b\sigma^z_{b+1}\prod_{k < b}\sigma^x_k = -\prod_{k\le b}\sigma^x_k \,\sigma^z_b\,\sigma^z_{b+1} = -\tau^z_b\,\tau^x_b,\]

using $\sigma^z\sigma^x = -\sigma^x\sigma^z$ on site $b$ once and commuting the remaining factors (all acting on different sites). The same computation as in the jw-tfim-bdg derivation shows that operators on different bonds commute. Therefore $\{\tau^x_b, \tau^z_b\}$ satisfy the spin-$\tfrac{1}{2}$ algebra.

Rewriting $H$. The Ising bond term is immediate from (3):

\[-J\sum_{i=1}^{N}\sigma^z_i\sigma^z_{i+1} = -J\sum_{b=1}^{N}\tau^x_b.\]

For the transverse field, use $\tau^z_{b-1}\tau^z_b = \prod_{k \le b-1}\sigma^x_k \cdot \prod_{k \le b}\sigma^x_k = \sigma^x_b$ (the overlap on sites $1, \dots, b-1$ squares to 1, leaving just the $b$-th factor of the longer product):

\[-h\sum_{i=1}^{N}\sigma^x_i = -h\sum_{b=1}^{N}\tau^z_{b-1}\,\tau^z_b,\]

with $\tau^z_0 \equiv 1$ (empty product). Hence

\[H \;=\; -J\sum_{b=1}^{N}\tau^x_b \;-\; h\sum_{b=1}^{N}\tau^z_{b-1}\,\tau^z_b. \tag{4}\]

Compare to the original Hamiltonian in $\sigma$ operators:

\[H(J, h)\bigl|_{\sigma} \;=\; -J\sum\sigma^z\sigma^z - h\sum\sigma^x.\]

Relabelling bonds as sites ($b \to i$), the transformation $\sigma \to \tau$ interchanges the Ising bond term and the transverse-field term with the couplings swapped:

\[\boxed{\; H(J, h)\bigl|_{\sigma} \;=\; H(h, J)\bigl|_{\tau}. \;} \tag{5}\]

So the duality is a unitary operator $U$ that implements $\sigma \to \tau$ in (3): $U^{\dagger}\sigma U = \tau$. Under $U$,

\[U\,\sigma^z_i\sigma^z_{i+1}\,U^{\dagger} = \tau^x_i = \sigma^x_{i+1}\bigl|_{\rm after\ relabelling}, \qquad U\,\sigma^x_i\,U^{\dagger} = \tau^z_{i-1}\tau^z_i = \sigma^z_{i-1}\sigma^z_i\bigl|_{\rm after\ relabelling}.\]

Step 6 — Self-dual TFIM critical point

The self-dual point of the TFIM operator duality (5) is $J = h$. The 2D-duality ancestor of this point is the 2D Ising $K = K^{*}$ fixed point, $\sinh(2K_c) = 1$. The descent goes via the anisotropic 2D Ising transfer matrix, which has two couplings $(K_x, K_y)$ and a self-dual condition $\sinh(2K_x)\sinh(2K_y) = 1$ — identifying $K_x \leftrightarrow J$ and $\tau_y^{-1}\ln\tanh K_y \leftrightarrow h$ in the continuous-time limit $\tau_y \to 0$ with $K_y, \tau_y$ tuned such that $\tanh K_y \sim \tau_y\cdot h$, the self-dual condition reduces to $J = h$ (Sachdev 2011 §5.2).

Step 7 — Limiting-case checks

Duality maps the phases. From (1), $K \to 0$ sends $K^{*} \to \infty$ and vice versa. So high-$T$ (disordered, $\langle s\rangle = 0$) on the original lattice corresponds to low-$T$ (ordered, $\langle s\rangle \ne 0$) on the dual, and the two lattices share critical properties at the self-dual point.

Quantum check: $h = 0$. At $h = 0$ the TFIM is the classical Ising chain with ground state $|{\uparrow\cdots\uparrow}\rangle$ or $|{\downarrow\cdots\downarrow}\rangle$. Under (5) this maps to the $J = 0$ paramagnetic chain with ground state $|{\leftarrow\cdots \leftarrow}\rangle$, the trivial paramagnet. The duality interchanges ordered / disordered as expected.

Quantum check: $h = J$. The critical TFIM; the duality fixes this point. The spectrum is gapless and conformally invariant with central charge $c = 1/2$ — the same Ising CFT that describes the 2D classical Ising point at $K_c$.

Graphene / square-lattice 3D check. The square lattice is self-dual; the honeycomb and triangular lattices are dual to each other (not self-dual). The Kramers–Wannier relation for those pairs has $\sinh(2 K_{\rm hc})\sinh(2 K_{\rm tri}) = 1$ with different critical couplings on the two lattices, recovering the famous Wannier 1945 result for the triangular-lattice Ising critical coupling.

References

  • H. A. Kramers and G. H. Wannier, Statistics of the two- dimensional ferromagnet. Part I, Phys. Rev. 60, 252 (1941). Original high-temperature / low-temperature duality argument.
  • L. Onsager, Crystal statistics. I. A two-dimensional model with an order–disorder transition, Phys. Rev. 65, 117 (1944). Independent exact solution confirming that the self-dual point is the phase-transition temperature.
  • F. Wegner, Duality in generalized Ising models and phase transitions without local order parameters, J. Math. Phys. 12, 2259 (1971). Generalisation to gauge Ising models.
  • D. C. Mattis, Statistical Mechanics Made Simple, 2nd ed. (World Scientific, 2008), §§3.4–3.5. Transfer-matrix derivation of the 1D-quantum / 2D-classical correspondence.
  • S. Sachdev, Quantum Phase Transitions, 2nd ed. (Cambridge University Press, 2011), §5.2. Continuous-time-limit derivation of the 1D TFIM from the 2D Ising transfer matrix and the TFIM self-duality $J \leftrightarrow h$.
  • J. B. Kogut, An introduction to lattice gauge theory and spin systems, Rev. Mod. Phys. 51, 659 (1979), §III–§IV. Pedagogical exposition of 1D TFIM self-duality from the 2D Ising perspective.

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