Migdal-Kadanoff Bond-Moving Approximation

Real-space renormalization on a $d$ -dimensional lattice runs into a wall almost immediately: sum out a sublattice of Ising spins and you generate a proliferating mess of new couplings — diagonal bonds, four-spin terms, longer-range interactions — that no closed recursion can track. The Migdal–Kadanoff bond-moving approximation, introduced by Aleksandr Migdal in 1975 and reinterpreted geometrically by Leo Kadanoff in 1976, is the classic escape. Before decimating, one deliberately distorts the lattice: bonds are deleted and their strength "moved" onto neighbouring parallel bonds, until the spins slated for removal sit on isolated one-dimensional chains. Chain decimation is then exact, and the whole step closes on a single nearest-neighbour coupling $K=J/k_BT$ , giving a clean recursion $K'=R_b(K)$ . The trade is honest and explicit. Bond-moving discards the transverse fluctuations that separated the merged bonds, so the scheme is uncontrolled — there is no small parameter, and the answer depends on the order in which bonds are moved. What survives is the topology of the RG flow: the recursion is exact in $d=1$ (where there are no bonds to move and it reduces to ordinary decimation, correctly flowing to zero coupling with no transition), and for $d>1$ it produces a nontrivial unstable fixed point — a finite critical temperature — of the right qualitative character. It gets the lower critical dimension right and generalizes effortlessly to $O(n)$ , Potts, and disordered models, where it remains a workhorse for mapping phase diagrams. This article owns the bond-moving construction and its recursion, and it is candid about the numbers: for the 2D Ising model the simplest $b=2$ scheme places $T_c$ and the exponent $\nu$ within tens of percent of the exact Onsager values — qualitatively right, quantitatively approximate. It recalls the block-spin and decimation machinery (articles 1306, 1307) as its building blocks and the notion of a fixed point (1310) as its output. A deeper justification closes the circle: the MK recursions are the exact RG of a class of hierarchical (diamond) lattices, so every result is a genuine, thermodynamically consistent statistical-mechanics answer rather than an uncontrolled truncation.

Key takeaways

The Migdal–Kadanoff (MK) approximation is an approximate real-space RG built from two moves: bond-moving (shift and stack bonds so decimation becomes exact) and decimation (sum out the intervening spins).
Bond-moving converts renormalizing a $d$ -dimensional lattice into a sequence of one-dimensional decimations, giving a closed recursion $K'=R_b(K)$ for the coupling under rescaling by $b$ .
For the hypercubic Ising model the net map is $\tanh K'=[\tanh(b^{d-1}K)]^{b}$ : bond-moving multiplies $K\to b^{d-1}K$ ; series decimation of $b$ bonds gives the $\tanh^b$ .
It is exact in $d=1$ (flows to $K^\ast=0$ , no transition) and produces a nontrivial unstable fixed point (a finite $T_c$ ) for $d>1$ — but its exponents and $T_c$ are only approximate (tens of percent off Onsager in 2D).
The same recursions are the exact RG of hierarchical diamond lattices, giving MK a rigorous status: it is the exact answer for a different model, not a truncation of the real one.
Key idea: distort the lattice by moving bonds so the spins to be removed sit on 1D chains, then decimate exactly; the resulting closed recursion $\tanh K'=[\tanh(b^{d-1}K)]^b$ is exact in $d=1$ and on diamond lattices, and captures the qualitative RG flow — a fixed point for $d>1$ — everywhere else.

The core insight

Distort the lattice by moving bonds so the spins to be eliminated sit on one-dimensional chains, then decimate exactly. The resulting closed recursion, for the hypercubic Ising model $\tanh K'=[\tanh(b^{d-1}K)]^{b}$ , is exact in $d=1$ and on hierarchical diamond lattices; for $d>1$ it produces a nontrivial unstable fixed point (a finite critical temperature) of the right qualitative character but with only approximate exponents.

The physical picture

Picture a 2D square lattice and the task of removing every other spin. Do it directly and you generate diagonal and four-spin couplings — a closed recursion is impossible. Migdal and Kadanoff's fix is to cheat the geometry first: take the bonds attached to a soon-to-be-removed spin and slide them sideways onto the neighbouring line, doubling that line's coupling. After this "bond-moving," the spins you want gone lie on isolated horizontal chains, and a chain is the one lattice where decimation is elementary — the transfer-matrix trick sums out the middle spin exactly. You have swapped the true lattice for a topologically simpler one whose RG you can do by hand. The approximation lives entirely in that slide. Superimposing two bonds into one of double strength pretends they saw the same pair of end-spins, ignoring the transverse degrees of freedom that actually separated them. That neglect of fluctuations is why MK slightly overestimates the tendency to order — its $T_c$ comes out too high — and why its exponents are imperfect. But the shape of the flow survives: below the fixed point the coupling renormalizes to zero (disorder), above it the coupling grows (order), and the fixed point itself is the critical temperature. Dimensionality enters through two competing integers. Rescaling by a factor $b$ , each surviving bond absorbs $b^{d-1}$ parallel bonds from the transverse directions (a strengthening, $K\to b^{d-1}K$ ), while $b$ bonds in series are decimated along the longitudinal direction (a weakening, the $\tanh^b$ ). In $d=1$ there are no transverse directions, $b^{d-1}=1$ , nothing is moved, and the map is pure decimation that always flows to $K^\ast=0$ — no transition, exactly as the 1D Ising model demands. For $d>1$ the strengthening survives and the two effects balance at a nontrivial $K^\ast$ : a transition is born the instant $d$ exceeds one.

Figure

The MK recursion map K′(K) for d = 2: it crosses the diagonal at a nontrivial unstable fixed point K* (a T_c), absent for d = 1 where the map hugs below the diagonal

Figure

Bond-moving then decimation on a b = 2 block: transverse bonds are slid onto one strand (K → 2K), then two series bonds are summed out (tanh K′ = tanh²2K)

The core equations

The model is the nearest-neighbour Ising model with a single reduced coupling $K=J/k_BT$ , and the whole RG closes on that one number. Everything below tracks how $K$ transforms under one MK step of rescaling factor $b$ in dimension $d$ .

Z=\sum_{\{\sigma\}}\exp\!\Big(K\!\!\sum_{\langle ij\rangle}\!\sigma_i\sigma_j\Big),\qquad K=\frac{J}{k_BT}

where: $\sigma_i=\pm1$ is the Ising spin; $\langle ij\rangle$ runs over nearest-neighbour bonds. Large $K$ (low $T$ ) favours alignment; small $K$ (high $T$ ) favours disorder. The elementary exact step is decimation of a single spin in a chain: summing out a middle spin between two neighbours renormalizes the coupling. For a chain of $b$ bonds in series the composition gives the compact $\tanh$ rule.

\tanh K_{\rm dec}=(\tanh K)^{\,b} \qquad\text{(exact 1D decimation, }b\text{ bonds in series)}

where: $K_{\rm dec}$ is the effective coupling between the surviving end-spins. Because $|\tanh K|<1$ , series decimation always weakens the coupling — the source of disordering. Bond-moving supplies the opposite tendency. Rescaling by $b$ , the $b^{d-1}$ transverse bonds are stacked onto one strand, multiplying its coupling, before that strand is decimated. Composing the two moves gives the Migdal–Kadanoff recursion.

\boxed{\;\tanh K' = \big[\tanh\!\big(b^{\,d-1}K\big)\big]^{\,b}\;}

where: $b^{d-1}K$ is the bond-moved coupling (strengthening from the transverse directions), and the outer power $b$ is the series decimation (weakening). For $d=1$ , $b^{d-1}=1$ and the map is pure decimation flowing to $K^\ast=0$ ; for $d>1$ the two effects balance at a nontrivial $K^\ast$ . The critical exponent follows from linearizing about the fixed point: the thermal eigenvalue is the log of the map's slope, rescaled by $\ln b$ .

y_t=\frac{\ln\!\big(dK'/dK\big)\big|_{K^\ast}}{\ln b},\qquad \nu=\frac{1}{y_t}

where: $dK'/dK|_{K^\ast}>1$ signals an unstable fixed point (the critical surface); $\nu=1/y_t$ is the correlation-length exponent. For $b=2,\,d=2$ this yields $\nu\approx1.34$ , to be compared with the exact Onsager $\nu=1$ .

Derivation AThe Migdal-Kadanoff recursion from bond-moving plus decimation

The goal is to assemble the closed recursion $\tanh K'=[\tanh(b^{d-1}K)]^b$ from its two elementary moves and to make explicit why one strengthens and the other weakens the coupling. We work with the reduced coupling $K=J/k_BT$ and the $\tanh$ variable in which decimation composes multiplicatively, then combine the transverse bond-count with the longitudinal series-decimation to get the net map for a hypercubic lattice under rescaling by $b$ .

Step 1. Decimate one spin in a chain: $e^{K'\sigma_1\sigma_3}\propto\sum_{\sigma_2}e^{K(\sigma_1\sigma_2+\sigma_2\sigma_3)}$ . Evaluating for $\sigma_1\sigma_3=\pm1$ gives $\cosh 2K$ and $1$ respectively, and $e^{2K'}=\cosh 2K$ , i.e. $\tanh K'=\tanh^2 K$ for two bonds in series. Why $\tanh$ : the coupling composes multiplicatively in $\tanh K$ , so $b$ bonds in series give $\tanh K_{\rm dec}=(\tanh K)^b$ .

Step 2. Bond-move the transverse directions. Under a rescaling by $b$ , each surviving strand collects $b^{d-1}$ parallel bonds; parallel bonds add their couplings, so $K\to b^{d-1}K$ . Why this direction: stacking bonds strengthens the effective coupling, exactly opposing the decimation's weakening.

Step 3. Compose: first strengthen ( $K\to b^{d-1}K$ ), then decimate the resulting strand of $b$ series bonds, giving $\tanh K'=[\tanh(b^{d-1}K)]^b$ .

Step 4. Check $d=1$ : $b^{d-1}=1$ , so $\tanh K'=\tanh^b K<\tanh K$ — the coupling always shrinks and flows to $K^\ast=0$ . Why this is right: the 1D Ising model has no finite- $T$ transition, and MK reproduces that exactly (bond-moving is trivial in 1D). Verified in verify_aq-sm-migdal-kadanoff-bond-moving.py (the $d=1$ map flows to $K^\ast=0$ ; the $d=2,\,b=2$ map $\tanh K'=\tanh^2 2K$ has a nontrivial fixed point).

Go deeper · Derivation B · 1 minThe nontrivial fixed point, T_c, and the exponent ν in 2D

Derivation BThe nontrivial fixed point, T_c, and the exponent ν in 2D

The goal is to locate the finite-temperature fixed point of the 2D Ising MK map, read off the approximate critical temperature, and extract the correlation-length exponent by linearization — then compare honestly with the exact Onsager solution. We specialize to $b=2,\,d=2$ , so the recursion is $\tanh K'=\tanh^2(2K)$ , solve $K'=K$ numerically for the unstable root, and differentiate to get the thermal eigenvalue.

Step 1. Set $K'=K$ : the fixed-point condition is $\tanh K^\ast=\tanh^2(2K^\ast)$ . Small $K$ gives $K'<K$ (flows to disorder), large $K$ gives $K'>K$ (flows to order), so a single unstable root separates the phases. Numerically $K^\ast\approx0.305$ .

Step 2. Convert to a critical temperature: $k_BT_c/J=1/K^\ast\approx3.28$ . The honest comparison: the exact Onsager value is $2/\ln(1+\sqrt2)\approx2.27$ , so MK overestimates $T_c$ by about $45\%$ — the neglected transverse fluctuations make ordering look easier.

Step 3. Linearize: $dK'/dK|_{K^\ast}\approx1.68>1$ confirms an unstable fixed point. The thermal eigenvalue is $y_t=\ln(1.68)/\ln 2\approx0.75$ .

Step 4. Extract $\nu=1/y_t\approx1.34$ . The honest comparison: Onsager gives $\nu=1$ exactly, so MK is about $34\%$ high — qualitatively a diverging correlation length with a finite exponent, quantitatively approximate. Verified in verify_aq-sm-migdal-kadanoff-bond-moving.py (fixed point $K^\ast\approx0.305$ , $T_c\approx3.28$ vs Onsager $2.27$ , slope $\approx1.68$ , $\nu\approx1.34$ vs exact $1$ ).

Limiting cases

Case	Result	Meaning
$d=1$	$\tanh K'=\tanh^b K$ , $K^\ast=0$ only	No finite- $T$ transition — MK is exact here (nothing to move).
$d=2$ , $b=2$ (Ising)	$K^\ast\approx0.305$ , $T_c\approx3.28$ , $\nu\approx1.34$	A nontrivial unstable fixed point appears; numbers $\sim30\text{–}45\%$ off Onsager.
$d\to\infty$	Mean-field-like	Bond-moving multiplicity $b^{d-1}$ dominates; recovers the mean-field tendency.
Hierarchical (diamond) lattice	MK recursion exact	The MK map is the true RG of these lattices — no approximation, guaranteed consistent thermodynamics.

Common trapMK is an uncontrolled approximation of the real lattice — but an exact model in disguise

Bond-moving has no small parameter: its error is not systematically improvable, and the result depends on the (arbitrary) order in which bonds are moved, so different but "equivalent" schemes give slightly different $T_c$ and exponents. Do not read the MK numbers as precise — the 2D Ising $T_c$ and $\nu$ are only qualitatively right. The saving grace, and the correct way to think about it, is that the MK recursion is the exact RG of a hierarchical diamond lattice: it is not a truncation of the hypercubic model but the exact answer for a nearby model, which is why it never produces thermodynamic nonsense (no negative specific heats, no violated inequalities). Trust the flow topology and the lower critical dimension; distrust the third significant figure.

Worked examples

Example 1 — The 2D Ising fixed point and critical temperature. Use the $b=2$ recursion $\tanh K'=\tanh^2(2K)$ .

Step 1. Impose $K'=K$ : solve $\tanh K=\tanh^2(2K)$ for the nontrivial root. Numerically, $K^\ast=\mathbf{0.305}$ (bracketing between the disordered flow at small $K$ and the ordered flow at large $K$ ).

Step 2. Critical temperature $k_BT_c/J=1/K^\ast=1/0.305=\mathbf{3.28}$ .

Step 3. Compare with Onsager: exact $K_c=\tfrac12\ln(1+\sqrt2)=0.4407$ , $k_BT_c/J=2.269$ . MK overestimates $T_c$ by $(3.28-2.27)/2.27\approx\mathbf{45\%}$ — the expected sign and magnitude for a bond-moving scheme that suppresses fluctuations.

Example 2 — The correlation-length exponent by linearization. Differentiate the map at $K^\ast$ .

Step 1. $\dfrac{dK'}{dK}=\dfrac{d}{dK}\,\mathrm{arctanh}\!\big[\tanh^2(2K)\big]$ ; evaluated at $K^\ast=0.305$ this is $\mathbf{1.68}$ — greater than one, so the fixed point is unstable (it sits on the critical surface).

Step 2. Thermal eigenvalue $y_t=\ln(1.68)/\ln 2=\mathbf{0.747}$ .

Step 3. Correlation-length exponent $\nu=1/y_t=1/0.747=\mathbf{1.34}$ , versus the exact Onsager $\nu=1$ — a $34\%$ error, but the right qualitative physics: a divergent correlation length governed by a single relevant eigenvalue.

Origins · 1 minuniversality and how this was discovered

universality

shared critical behavior across microscopically distinct systems · The Migdal-Kadanoff scheme earns its keep as a cheap, reliable probe of universality — the fact that systems with the same dimension, symmetry, and interaction range share critical exponents regardless of microscopic detail. Because bond-moving preserves the RG flow's topology and the lower critical dimension while distorting only the quantitative exponents, it reproduces the universal features (which classes have transitions, in what dimensions) far better than it reproduces any single number. The recursion was introduced by Alexander A. Migdal (Sov. Phys. JETP 1975) as an approximate real-space RG for lattice spin systems and gauge theories, and given its bond-sliding geometric interpretation a year later by Leo Kadanoff, who also identified it as the exact RG of hierarchical diamond lattices. Both names are credited in the history below; this card portraits the universality concept the scheme is used to test rather than either person, whose portraits live with other articles. ::: :::history 1966 — Kadanoff's block-spin picture introduces real-space scaling near T_c 1971 — Wilson's momentum-space RG computes critical exponents systematically 1975 — Alexander A. Migdal introduces the bond-moving recursion relations 1976 — Kadanoff reinterprets bond-moving geometrically and links it to hierarchical lattices 1979 — Berker and Ostlund show the MK recursions are exact on diamond hierarchical lattices 1982 — Wilson awarded the Nobel Prize for the renormalization group theory of critical phenomena ::: ## Where this lives today Migdal–Kadanoff remains the go-to quick tool whenever a phase diagram is needed fast and exact solutions are out of reach. Its closed recursions extend with almost no extra work to

O(n)

and Potts models, to random-bond and random-field systems, and to spin glasses — where the MK recursion on a hierarchical lattice is a standard, numerically cheap surrogate for the intractable Edwards–Anderson model, capturing the existence and rough location of freezing transitions. In lattice gauge theory, Migdal's original motivation, bond-moving gives an analytic estimate of confinement and the deconfinement transition. The modern reframing — MK as the exact RG of a hierarchical diamond lattice — has made it a favourite testbed for RG ideas themselves. Because everything is exact on those lattices, one can study fixed-point structure, multifractality of correlations, and disorder-driven transitions with full rigor, then ask which features carry over to real lattices. It is the rare approximation that doubles as an exactly solvable model. :::quiz[What bond-moving buys you

Why does the Migdal–Kadanoff scheme move bonds before decimating? - ( ) To make the lattice larger - (✓) So the spins to be removed lie on 1D chains, where decimation is exact and closes on one coupling - ( ) To conserve energy - ( ) To add long-range interactions Direct decimation in $d>1$ generates uncontrolled new couplings; bond-moving reduces the step to exact 1D decimation.

Going further

References. Kardar, Statistical Physics of Fields, Ch. 9 (real-space RG; Migdal–Kadanoff) · Pathria & Beale, Statistical Mechanics, 3rd ed., Ch. 14 · Goldenfeld, Lectures on Phase Transitions and the Renormalization Group, Ch. 9 · Migdal, Sov. Phys. JETP 42, 743 (1975) · Kadanoff, Ann. Phys. 100, 359 (1976) · Berker & Ostlund, J. Phys. C 12, 4961 (1979).

Provenance. The recursion $\tanh K'=[\tanh(b^{d-1}K)]^b$ , its $d=1$ triviality, and the 2D fixed point are verified against Kardar Ch. 9 and Goldenfeld Ch. 9. verify_aq-sm-migdal-kadanoff-bond-moving.py checks that the $d=1$ map flows to $K^\ast=0$ , locates the $d=2,\,b=2$ fixed point $K^\ast\approx0.305$ ( $T_c\approx3.28$ vs the exact Onsager $2.269$ , a $45\%$ overestimate), and extracts $\nu\approx1.34$ from the linearized slope (vs exact $\nu=1$ ).

Builds on: Block-Spin (Real-Space) Renormalization Group · Decimation and the One-Dimensional Ising RG · RG Flow in Coupling Space · Fixed Points of the RG. Leads to: Linearization About a Fixed Point and Eigenvalue Spectrum · Critical Exponents from the RG · Relevant, Irrelevant, and Marginal Operators · The Wilson-Fisher Fixed Point.

Is Migdal–Kadanoff a controlled approximation?

No. Unlike the $\varepsilon$ -expansion, which is a systematic power series in $4-d$ , MK has no small parameter. Its error is not improvable by "going to higher order," and it even depends on the arbitrary choice of which bonds to move and in what order — different reasonable schemes give slightly different $T_c$ and exponents. What makes it trustworthy anyway is that the recursions are the exact RG of hierarchical diamond lattices, so the output is always a genuine statistical-mechanics answer. Use it for the flow topology, the lower critical dimension, and rough phase-diagram locations — not for precise exponents.

Why does MK overestimate the critical temperature?

Because bond-moving suppresses fluctuations. When you slide two bonds together into one of double strength, you pretend they coupled the same pair of spins, ignoring the transverse degrees of freedom that actually separated and decorrelated them. Killing those fluctuations makes ordering look easier than it is, so the transition happens at a higher temperature. That is why the 2D Ising $T_c$ comes out around $3.3$ in the simplest $b=2$ scheme rather than the exact Onsager $2.27$ — the sign of the error is a direct fingerprint of the approximation.

What is the connection to hierarchical (diamond) lattices?

A diamond hierarchical lattice is built by repeatedly replacing each bond with a motif of parallel branches, each subdivided into series bonds. On such a lattice the RG is exactly bond-move-then-decimate, because the lattice was constructed to make that operation exact — there are no transverse fluctuations to neglect. So the very recursion that is an approximation for the hypercubic lattice is the exact transformation for the diamond lattice. This reframing, due to Berker and Ostlund, is why MK never yields thermodynamic inconsistencies: it is solving a real model, just not quite the one you started with.

How does MK get the lower critical dimension right?

The lower critical dimension $d_\ell$ is where the finite- $T$ transition first disappears. In the MK map, the strengthening factor $b^{d-1}$ from bond-moving and the weakening from series decimation balance to give a nontrivial fixed point only when $d>d_\ell$ . For the Ising model the balance produces a fixed point for all $d>1$ and none at $d=1$ , so $d_\ell=1$ — correct. Extended to continuous-symmetry ( $O(n\ge2)$ ) models with the appropriate recursion, MK reproduces $d_\ell=2$ , matching the Mermin–Wagner result. Getting these thresholds right is one of the scheme's genuine strengths, independent of its exponent inaccuracy.

Can MK handle disorder and spin glasses?

Yes, and this is one of its most active uses. Because the recursion is a simple map on the coupling, replacing a single $K$ by a distribution of couplings (for random bonds) and iterating that distribution under the MK map is straightforward and numerically cheap. On hierarchical lattices this is exact, providing a controlled surrogate for the intractable Edwards–Anderson spin glass: one can locate the freezing transition, study the flow of the coupling distribution's width, and probe chaos in the spin-glass phase. Much of what is rigorously known about hierarchical spin glasses comes from exactly this MK-on-diamond-lattice construction.

Why work with tanh K rather than K directly?

Because decimation composes multiplicatively in $\tanh K$ , not in $K$ . Summing out a spin between two bonds of coupling $K$ gives an effective coupling whose $\tanh$ is $\tanh^2 K$ ; chaining $b$ bonds in series simply gives $(\tanh K)^b$ . In the raw variable $K$ this composition is a messy transcendental relation, but in $\tanh K$ it is a clean power. Bond-moving, meanwhile, is additive in $K$ (parallel couplings add), so the full MK step is naturally written by first adding in $K$ ( $K\to b^{d-1}K$ ) and then taking a power in $\tanh$ — exactly the form $\tanh K'=[\tanh(b^{d-1}K)]^b$ .