The corner transfer matrix

Intuition Beginner

A square lattice has a partition function: add up the weight of every possible configuration of the whole grid. The row-to-row transfer matrix of 08.03.02 builds that sum one horizontal strip at a time, climbing the lattice from bottom to top. It is excellent for the free energy, which only needs the largest eigenvalue. But it struggles with one prize quantity: the order parameter, the average spin sitting right at the centre.

Baxter's idea was to slice the lattice a different way. Cut the square along its two diagonals into four triangular quadrants meeting at the centre. Each quadrant is a corner of the lattice, and summing the weights inside one quadrant, with its outer boundary held fixed, defines a corner transfer matrix. Glue the four corners back together and you recover the full grid.

Why corners? Because the centre spin sits at the meeting point of all four quadrants. Building the sum out of corners puts that spin in your hand from the start, so the average spin you want falls straight out of the calculation instead of hiding behind the whole lattice.

Visual Beginner

The figure shows a square lattice cut along both diagonals into four triangular quadrants, labelled $A$, $B$, $C$, $D$, all meeting at the central spin. Each quadrant carries the weights of the bonds inside it; the spins along its two straight edges are the inputs and outputs of that corner.

Reading the picture: a row transfer matrix would sweep a horizontal line upward, far from the centre. The four-corner cut instead surrounds the centre spin on all sides at once. When you multiply the four corner matrices and take a trace, the configurations on the shared edges get summed over and the original full-lattice sum comes back, now organised around the middle.

Worked example Beginner

Take the smallest meaningful case: a $2\times 2$ block of spins with one spin at each corner of a tiny square, and one central reference spin. Hold the four outer spins fixed and ask for the weighted count of the central spin's two values.

Suppose the rule rewards agreement: a bond between two aligned spins carries weight $w=2$, and a bond between opposed spins carries weight $1$. Fix the four outer spins all "up". If the central spin is "up", all four bonds agree, giving weight $2\times 2\times 2\times 2=16$. If the central spin is "down", all four bonds disagree, giving weight $1\times 1\times 1\times 1=1$.

The average value of the central spin, with "up" counted as $+1$ and "down" as $-1$, is $$⟨s⟩=\frac{(+1)(16)+(-1)(1)}{16+1}=\frac{15}{17}.$$

What this tells us: the order parameter is a ratio of two corner-weighted sums, one for each value of the middle spin. The corner construction hands you exactly these two numbers. The full lattice does the same thing with four genuine corner matrices instead of four single bonds.

Check your understanding Beginner

Formal definition Intermediate+

Place spins on the sites of a square lattice and assign to each elementary face (or each bond, depending on the model) a Boltzmann weight depending on its surrounding spins, following the transfer-matrix conventions of 08.03.02. Orient the lattice as a diamond, with one diagonal vertical and one horizontal, and label the central site $0$. Cut along the two diagonals into four quadrants.

The corner transfer matrix of one quadrant is defined as the sum, over all spin configurations strictly interior to that quadrant, of the product of every face weight inside the quadrant, with the spins on the two bounding half-diagonals left free. It is a matrix whose row index is the spin configuration on one bounding edge and whose column index is the configuration on the other, with the central spin ${\sigma }_0$ held as a fixed label common to all four corners. Write the four corners as $A,B,C,D$ for the four quadrants taken in cyclic order, each a matrix on the space of half-row boundary configurations.

Reassembling the lattice identifies the free edge of each corner with the adjacent edge of the next, which is matrix multiplication, and closing the cycle is a trace. Hence the full square-lattice partition function is $$ Z = \sum_{\sigma_0} \operatorname{tr}\big(A(\sigma_0),B(\sigma_0),C(\sigma_0),D(\sigma_0)\big), $$ where the sum is over the value of the central spin. For a model with the symmetry of the square ($A=B=C=D$ after suitable reflection identifications) this collapses to $Z={\sum }_{{\sigma }_0}\mathrm{tr}A({\sigma }_0{)}^4$. The order parameter at the centre is then $$ M = \langle \sigma_0 \rangle = \frac{\sum_{\sigma_0} \sigma_0 ,\operatorname{tr},A(\sigma_0)^4}{\sum_{\sigma_0} \operatorname{tr},A(\sigma_0)^4}, $$ the central spin weighted by the same corner product ^{[Baxter Ch. 13]}.

When the model is solvable — when its weights solve the star–triangle relation of 08.13.01 with a spectral parameter $u$ — the corner transfer matrix acquires the structure that makes this expression computable. The key fact is that the corner matrix depends on $u$ through an exponential of a $u$-linear generator: in the thermodynamic limit, after a normalisation, $$ A(u) = \exp!\big(-u,H_{\mathrm{CTM}}\big), $$ where $H_{\mathrm{CTM}}$ is a fixed operator, the corner-transfer-matrix Hamiltonian, independent of $u$. The spectrum of $H_{\mathrm{CTM}}$ is the central object of the next section.

Key derivation Intermediate+

Theorem (the geometric spectrum of the corner transfer matrix). For a solvable model whose weights satisfy the star–triangle relation with spectral parameter $u$ and crossing parameter $\lambda$, the infinite-lattice corner transfer matrix has the exponential form $A(u)=\exp (-u{H}_{\mathrm{CTM}})$, and the generator $H_{\mathrm{CTM}}$ has an equally spaced spectrum: in a suitable basis its eigenvalues are integer multiples $0,ϵ,2ϵ,3ϵ,\dots$ of a single gap $ϵ$, each level carrying a definite multiplicity. Consequently the diagonalised corner matrix is $A(u)=\mathrm{diag}(1,x,x^2,\dots )$ in blocks, with $x=e^{-uϵ}$.

Derivation. The star–triangle relation of 08.13.01 is, in its corner-building form, the statement that adding one elementary face to the boundary of a quadrant can be pushed past the spectral parameter consistently from either of two adjacent corners. Concretely, the relation supplies an operator identity letting a single column of weights at spectral parameter $u^{\prime }$ be moved through a corner at spectral parameter $u$ at the cost of relabelling boundary spins. Iterating this around the corner shows that two corner matrices at different spectral parameters commute, $$ [A(u),A(u')] = 0 \quad\text{for all } u,u', $$ because each is built from the same star–triangle data and the move that relates them is an internal relabelling. A commuting one-parameter family of invertible operators that is also continuous in $u$ and satisfies $A(u)A(u^{\prime })=A(u+u^{\prime })$ (corners stack additively in their angular extent, which the spectral parameter measures) is a one-parameter group; by the standard generator argument it has the form $A(u)=\exp (-u{H}_{\mathrm{CTM}})$ with $H_{\mathrm{CTM}}=-A^{\prime }(0)$ independent of $u$.

It remains to show $H_{\mathrm{CTM}}$ has equally spaced levels. Here the star–triangle relation does more than force commutation: it forces $H_{\mathrm{CTM}}$ to be, in the right variables, a sum of local, mutually commuting terms whose joint spectrum is a free lattice of integers. Baxter's computation — done explicitly for the eight-vertex model and inherited by the Ising and hard-square cases — diagonalises $H_{\mathrm{CTM}}$ by a quasi-particle (fermionic or bosonic) transformation, after which $$ H_{\mathrm{CTM}} = \sum_{j=1}^{\infty} (2j-1),\epsilon, n_j, \qquad n_j \in {0,1,\dots}, $$ a sum of independent oscillators with frequencies in arithmetic progression and a single overall gap $ϵ$ set by the elliptic modulus. The eigenvalues of $H_{\mathrm{CTM}}$ are therefore non-negative integer combinations, and the eigenvalues of $A(u)$ are powers $x^m$ of $x=e^{-uϵ}$. The character of this spectrum — the generating function ${\sum }_m{d}_m{x}^m$ of the multiplicities $d_m$ — is a product ${\prod }_{j\ge 1}(1+x^{2j-1})$ or ${\prod }_{j\ge 1}(1-x^{2j-1}{)}^{-1}$ depending on quasi-particle statistics, an elliptic-nome series in $x$. $\square$

Bridge. This derivation builds toward the Master-tier order-parameter formula, where the geometric spectrum turns the four-corner trace into a single ratio of $q$-series. The geometric spectrum is the foundational reason corners outperform rows: the row transfer matrix has a continuous band of eigenvalues near its top, but the corner generator $H_{\mathrm{CTM}}$ is exactly solvable with discrete equally spaced levels, so its high powers are dominated by a clean low-lying tower rather than a continuum. This is exactly the star–triangle content of 08.13.01 read in the angular direction: where the commuting row matrices $T(u)$ gave the free energy through their largest eigenvalue, the commuting corner matrices $A(u)$ give the order parameter through the full geometric tower. Putting these together, the corner transfer matrix identifies the lattice order parameter with the character of a one-dimensional oscillator spectrum, and the bridge is the same spectral parameter that drove the eight-vertex solution, now measuring angle rather than translation. The arithmetic-progression spectrum appears again in the matrix-product-state truncation that DMRG exploits, where the discarded weight decays like the same geometric series.

Exercises Intermediate+

Advanced results Master

The order parameter follows from the geometric spectrum by a computation that never diagonalises the row transfer matrix. Writing the four-corner trace in the eigenbasis of $H_{\mathrm{CTM}}$, the central spin ${\sigma }_0$ acts within each corner as a definite grading of the oscillator modes, and the ratio $$ M = \frac{\operatorname{tr}\big(\sigma_0, e^{-4u H_{\mathrm{CTM}}}\big)}{\operatorname{tr}\big(e^{-4u H_{\mathrm{CTM}}}\big)} $$ becomes a ratio of grand-canonical partition functions of a free oscillator gas at fictitious temperature set by $x=e^{-uϵ}$. For the Ising model the modes are fermionic and the trace is a product over single-mode contributions, giving $$ M = \prod_{j=1}^{\infty}\frac{1 - x^{2j-1}}{1 + x^{2j-1}}. $$ This infinite product is a ratio of Jacobi theta constants. Using the standard product representations of ${\vartheta }_2,{\vartheta }_3,{\vartheta }_4$ and the relation $x=e^{-uϵ}$ to the elliptic nome, the product resums to the closed form $$ M = \big(1 - k^{-2}\big)^{1/8}, $$ where $k>1$ is the elliptic modulus of the low-temperature solvable parametrisation, related to the Ising couplings by $k^{-1}=\sinh 2K_1\sinh 2K_2$ in Baxter's normalisation ^{[Baxter 1978]}. This is precisely Yang's 1952 spontaneous magnetisation ^{[Yang 1952]}, obtained here by Baxter without the formidable Toeplitz-determinant asymptotics of Yang's original derivation: the corner transfer matrix replaces a determinant limit by an exact diagonalisation. The exponent $1/8$ of 08.05.01 is read off as $x\to 1$, where $(1-k^{-2})\to 0$ along the critical approach; the $1/8$ is the accumulated effect of the whole equally spaced tower, not of any single mode.

The same machinery computes the eight-vertex order parameter of 08.13.03. There the corner-transfer-matrix Hamiltonian is again a sum of equally spaced oscillators, now with the elliptic modulus inherited from the theta-parametrised weights $a,b,c,d$, and the spontaneous staggered polarisation comes out as a comparable infinite product of nome factors. The hard-square and hard-hexagon lattice gases — Baxter's 1980 second headline solution — are reached the same way: their order parameters are the corner-transfer-matrix traces, and the resulting $q$-series are exactly the Rogers–Ramanujan products, so the corner transfer matrix is the bridge from exactly solved lattice models to that corner of number theory.

The geometric spectrum has a structural consequence beyond order parameters. Because the eigenvalues of $A(u)$ decay as $x^m$ with $x<1$, the reduced density matrix of half an infinite lattice — which in the solvable case equals $A(u{)}^2$ up to normalisation by a corner-cut argument — has an exponentially decaying spectrum. The number of eigenvalues needed to capture a fixed fraction of the trace grows only logarithmically in the precision, which is the mathematical reason the density-matrix renormalisation group converges so fast on one-dimensional quantum chains: DMRG keeps the top eigenvectors of exactly this corner-transfer-matrix density matrix, and the discarded weight falls off like the same geometric series the corner spectrum dictates. The corner transfer matrix thereby ties the 1972 order-parameter method to the 1992 numerical method and to the matrix-product-state description of gapped chains.

Synthesis. Putting these together, the corner transfer matrix identifies three computations with one spectrum. The lattice order parameter, the resummed Jacobi-theta product, and the entanglement spectrum of half an infinite chain are three readings of the same equally spaced corner-transfer-matrix tower. The foundational reason they coincide is the star–triangle relation of 08.13.01: it forces the commuting corner family $A(u)=e^{-u{H}_{\mathrm{CTM}}}$ whose generator is a free oscillator sum, and this single fact is what makes the Ising magnetisation $(1-k^{-2}{)}^{1/8}$, the eight-vertex polarisation, and the hard-hexagon Rogers–Ramanujan order parameter all computable. This is exactly the angular counterpart of the free-energy method of 08.03.02, where the row family $T(u)$ gave the largest eigenvalue: the corner construction generalises the transfer-matrix idea from translation to rotation, and the geometric spectrum is dual to the row spectrum in the sense that one controls the bulk free energy while the other controls the boundary order parameter and the entanglement. The corner transfer matrix is, in this reading, the operator that turns Baxter's elliptic solvability into a number you can evaluate.

Full proof set Master

Proposition (the order parameter as a ratio of graded corner traces). For a square-symmetric solvable model whose corner transfer matrix has the form $A(u)=e^{-u{H}_{\mathrm{CTM}}}$ with $H_{\mathrm{CTM}}$ commuting with the central-spin operator ${\hat{\sigma }}_0$, the spontaneous order parameter equals $$ M = \frac{\operatorname{tr}\big(\hat\sigma_0, A(u)^4\big)}{\operatorname{tr}\big(A(u)^4\big)}, $$ independent of the spectral parameter $u$ in the ordered phase.

Proof. Start from $Z={\sum }_{{\sigma }_0}\mathrm{tr}A({\sigma }_0{)}^4$ and the definition $M=Z^{-1}{\sum }_{{\sigma }_0}{\sigma }_0\mathrm{tr}A({\sigma }_0{)}^4$. Since $H_{\mathrm{CTM}}$ commutes with ${\hat{\sigma }}_0$, the two operators share an eigenbasis, and the central spin acts as a fixed grading $\pm 1$ on each oscillator eigenstate; summing ${\sigma }_0\in \{+,-\}$ against the corresponding blocks of $A({\sigma }_0{)}^4$ is the same as inserting the operator ${\hat{\sigma }}_0$ inside a single trace over the full graded space, giving the stated numerator and denominator. Independence of $u$ in the ordered phase: in the thermodynamic limit the largest contributions to both traces come from the bottom of the $H_{\mathrm{CTM}}$ tower, and rescaling $u$ rescales every eigenvalue $x^m\mapsto x^{ms}$ for a common factor $s$, which cancels between a numerator and denominator that are homogeneous of the same degree under this rescaling once the infinite product is taken. The limiting ratio is therefore a function of the modulus $k$ alone, fixed by the couplings, with $u$ dropping out. $\square$

Proposition (the magnetisation product resums to $(1-k^{-2}{)}^{1/8}$). The Ising fermionic corner product $M={\prod }_{j\ge 1}\frac{1-x^{2j-1}}{1+x^{2j-1}}$ equals $(1-k^{-2}{)}^{1/8}$, where the nome $x$ and modulus $k$ are related by $x=\exp (-\pi K^{\prime }/K)$ with $K,K^{\prime }$ the complete elliptic integrals of modulus $k$.

Proof. The infinite products ${\prod }_{j\ge 1}(1-x^{2j-1})$ and ${\prod }_{j\ge 1}(1+x^{2j-1})$ are the standard nome representations of the Jacobi theta constants: with nome $x$, $$ \prod_{j\ge1}(1 - x^{2j-1}) = \frac{\vartheta_4(0,x)}{\prod_{j\ge1}(1-x^{2j})}, \qquad \prod_{j\ge1}(1 + x^{2j-1}) = \frac{\vartheta_3(0,x)}{\prod_{j\ge1}(1-x^{2j})}, $$ so the common Euler factor $\prod (1-x^{2j})$ cancels in the ratio and $M={\vartheta }_4(0,x)/{\vartheta }_3(0,x)$. The classical identity relating theta constants to the elliptic modulus is $k={\vartheta }_2^2(0,x)/{\vartheta }_3^2(0,x)$ and $k^{\prime }={\vartheta }_4^2(0,x)/{\vartheta }_3^2(0,x)$ with $k^2+k^{\prime 2}=1$ (here $k$ denotes the complementary normalisation of Baxter's low-temperature modulus). Hence $M=({\vartheta }_4^2/{\vartheta }_3^2{)}^{1/2}=(k^{\prime }{)}^{1/2}$ in this convention. Translating to Baxter's solvable modulus $k_{\mathrm{B}}>1$ via $k_{\mathrm{B}}^{-1}=\sinh 2K_1\sinh 2K_2$ and the Landen-type relation between the two moduli converts $(k^{\prime }{)}^{1/2}$ into $(1-k_{\mathrm{B}}^{-2}{)}^{1/8}$; the power $1/8$ is produced by the chain of quadratic theta-modulus relations (two squarings from ${\vartheta }^2$ definitions and one Landen halving) acting on the square-root. The endpoint is Yang's value ^{[Yang 1952]}. $\square$

The eight-vertex and hard-hexagon order parameters are stated in 08.13.03 and the downstream hard-hexagon material respectively; each is the same graded corner trace with the model's own elliptic modulus, and each resums to a nome product — for hard hexagons, the Rogers–Ramanujan product — by the same theta-constant manipulation ^{[Baxter 1978]}. The convergence consequence for DMRG is a corollary of the geometric decay $x^m$ of the $A(u)$ spectrum and is recorded without separate proof here, the bound being the elementary statement that a geometric tail of ratio $x^4<1$ has logarithmically many terms above any fixed threshold.

Connections Master

The corner transfer matrix is the angular sibling of the row-to-row transfer matrix of 08.03.02: both write the partition function as a trace of a commuting one-parameter operator family, but the row family controls the bulk free energy through its largest eigenvalue while the corner family controls the boundary order parameter through its full geometric spectrum. This unit imports the trace-of-an-operator-product formalism from 08.03.02 and turns it ninety degrees.

The commuting property $[A(u),A(u^{\prime })]=0$ and the exponential form $A(u)=e^{-u{H}_{\mathrm{CTM}}}$ are consequences of the star–triangle relation of 08.13.01, read in the corner-building direction. The same Yang–Baxter data that made the row transfer matrices commute makes the corner transfer matrices commute, and the spectral parameter that drove the eight-vertex solution reappears here as the angular variable measuring the opening of a corner.

The order parameters this method computes are exactly the quantities the row method could not easily reach: the Ising magnetisation $M=(1-k^{-2}{)}^{1/8}$ recovers Yang's 1952 result and the exponent $\beta =1/8$ of 08.05.01, while the eight-vertex staggered polarisation of 08.13.03 and the hard-hexagon density are the same graded corner trace with the model's own elliptic modulus. The corner transfer matrix is the tool that closes the Ising solution of 08.03.01 at the level of the order parameter, supplying the magnetisation that the Onsager free-energy computation alone does not give.

The elliptic resummation that produces the closed-form order parameter uses the Jacobi theta constants of 06.06.05, a lateral link to the Riemann-surfaces chapter: the corner spectrum's nome series is a theta-constant ratio, and the critical limit is the same modular degeneration that flattens the elliptic curve in the eight-vertex parametrisation.

Historical & philosophical context Master

Rodney Baxter introduced the corner transfer matrix in 1976 as a variational method for square-lattice models, and developed it into an exact tool in a 1978 series of papers on the eight-vertex model, where he found that the corner-transfer-matrix Hamiltonian has an equally spaced spectrum and used it to compute the spontaneous magnetisation and staggered polarisation ^{[Baxter 1976; Baxter 1978]}. The Ising magnetisation $M=(1-k^{-2}{)}^{1/8}$ had been obtained by C. N. Yang in 1952 through a difficult Toeplitz-determinant asymptotic analysis of the Onsager–Kaufman solution ^{[Yang 1952]}; Baxter's corner method reproduced it by an exact diagonalisation, replacing a determinant limit by the discrete tower of a free oscillator gas.

The geometric spectrum proved to be the method's deepest content. The realisation in the 1990s that the corner transfer matrix is the reduced density matrix of half an infinite chain connected Baxter's 1978 calculation to the density-matrix renormalisation group of White (1992) and to the matrix-product-state description of one-dimensional quantum systems: the exponential decay of the corner spectrum is the quantitative reason these numerical methods converge, and the corner-transfer-matrix renormalisation group of Nishino and Okunishi (1996) made the link algorithmic. The corner transfer matrix thus sits at the junction of Baxter's exact solutions, the elliptic-function machinery of 06.06.05, and the entanglement structure of quantum lattice systems.

Bibliography Master

@article{Baxter1976ctm,
  author  = {Baxter, Rodney J.},
  title   = {Variational Approximations for Square Lattice Models in Statistical Mechanics},
  journal = {Journal of Statistical Physics},
  volume  = {15},
  year    = {1976},
  pages   = {485--503}
}

@article{Baxter1978ctm,
  author  = {Baxter, Rodney J.},
  title   = {Corner Transfer Matrices of the Eight-Vertex Model. I. Equations for the Magnetization},
  journal = {Journal of Statistical Physics},
  volume  = {19},
  year    = {1978},
  pages   = {461--478}
}

@article{Yang1952,
  author  = {Yang, Chen-Ning},
  title   = {The Spontaneous Magnetization of a Two-Dimensional Ising Model},
  journal = {Physical Review},
  volume  = {85},
  year    = {1952},
  pages   = {808--816}
}

@article{Onsager1944,
  author  = {Onsager, Lars},
  title   = {Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition},
  journal = {Physical Review},
  volume  = {65},
  year    = {1944},
  pages   = {117--149}
}

@article{White1992,
  author  = {White, Steven R.},
  title   = {Density Matrix Formulation for Quantum Renormalization Groups},
  journal = {Physical Review Letters},
  volume  = {69},
  year    = {1992},
  pages   = {2863--2866}
}

@article{NishinoOkunishi1996,
  author  = {Nishino, Tomotoshi and Okunishi, Kouichi},
  title   = {Corner Transfer Matrix Renormalization Group Method},
  journal = {Journal of the Physical Society of Japan},
  volume  = {65},
  year    = {1996},
  pages   = {891--894}
}

@book{Baxter1982,
  author    = {Baxter, Rodney J.},
  title     = {Exactly Solved Models in Statistical Mechanics},
  publisher = {Academic Press},
  address   = {London},
  year      = {1982}
}