Octo as Part 2 left it (stripes along x, the yellow chi_zz question mark in the corner): today we build that chain. Two neighbours; between them three rods, x and y red, z white: the length of a rod is how strongly the pair cares about that axis, and with a positive coupling caring means preferring to point opposite (same way: higher; opposite: lower). Sum over every pair of neighbours: the Hamiltonian, one term per rod. The knob gamma (red, like the corners it will make) stretches the x rod to 1 + gamma and shortens the y rod to 1 - gamma by the same amount; the z rod stays at Delta_z: x and y tied with the knob off, x wins once it is on. One knob; which of its consequences does the meter care about? (Cue: say 'the knob is on', never a value.)
The Map: Delta_z across, gamma up. The paper lives in the blue strip |Delta_z| < 1, gamma > 0. Its bottom edge gamma = 0 is the free rotation of Part 2 (the tiny Octo turns by alpha, the product-state energy per bond does not move); the vertical line Delta_z = 0 is the XY line, free fermions, where the number under the question mark was known before (star at its foot, Delta_z = 0, gamma -> 0, where c = 1). The dots are the paper's iDMRG runs, columns at several Delta_z on both sides of the XY line; gold ring: home, on the repulsive side (a fainter ring round its own dot: the attractive repeat, its mirror image in Delta_z); grey ring: the hardest point, the longest memory of the paper's runs. Beyond Delta_z = 1 the map is grey: open, not today. Black box: Jordan-Wigner, XY line = free fermions (stated, not shown; two exact results feed 3.7 and 4.5). Cue: name the regions, never the coordinates.
Why does gamma matter so much? The x and y couplings drawn as a circle; two neighbours, pointing opposite, turned together by the purple alpha: with gamma off the gauge per bond stays pinned at its centre, -1 (a circle has no preferred direction). The RED knob turns gamma on: the circle becomes an ellipse, long along x (1+gamma by 1-gamma); the needles turn again and the marker swings between the low end -(1+gamma) along x and the high end -(1-gamma) along y: the energy knows where x is. Small print: a product state, not the ground state (the gauge follows the formula -1 - gamma cos 2 alpha; say 'lower along x, higher along y', no values).
Pictures before letters: a down arrow flipped up by sigma^+; the algebra sigma^x sigma^x - sigma^y sigma^y = 2(sigma^+ sigma^+ + sigma^- sigma^-), the left side being exactly what gamma multiplies; two neighbours flipped up together, two flipped down together. Octo in the side view with Part 2's counter, now Q for the whole chain (Q_A with the window removed): two downs become two ups, the count jumps by two (+2 flashes); two ups become two downs, down by two (-2). Gamma moves the charge in steps of two. [H,Q] != 0 <=> gamma != 0: with the knob off the circle and the pinned marker, with it on the ellipse and the swinging marker. A state whose charge is not fixed is a Grid with red corners between sectors (3.5 shows them on a real ground state): the red-corner factory. Caution: the gauge read a product state, not the ground state.
Two kinds of breaking, named apart. Explicit, the tilted table: the RED knob gamma tilts the table towards x and the marble rolls there; the rule itself prefers x, no choice of ours involved; [H, Q] != 0. Spontaneous, the falling pencil: the rule is even-handed between +x and -x (the mirror F = prod_j sigma^z_j, a half turn of every needle about z, leaves the energy unchanged), but a balanced pencil does not stay up: the infinite chain falls one way, stripes starting +x or their mirror image, same energy; [H, F] = 0, F|GS> != |GS>. Rule card: the 1/2 log ell growth needs the tilted table; the numerics sit in one fall of the pencil. Inset: a finite ring returns the cat, both falls at once; we pin one (Caveats: pinning = maximal overlap with the Neel-x product state).
The ladder rearranges as gamma turns on, on a small ring of spins in the computer (exact diagonalization; Delta_z fixed; energies in units of the couplings, never compared with M). At gamma = 0 the ground rung sits alone with a tied pair of rungs above it; the knob turns on: the pair splits, one rung sinks towards the ground rung, the other climbs; with the knob on the sinking rung sits just above the ground (small), the next one well above it (open) (two braces: 'two vacua, almost', 'the price of any error'). Side panel: E_1 - E_0 against the ring size L on a log axis is a straight line going down (closes exponentially: the two rungs become one pair of vacua, the two falls of the pencil); E_2 - E_0 stays open, the price of any error in the stripes. Formula: E_2 - E_0 > 0, E_1 - E_0 -> 0 (two vacua). Part 1 promised that a gap buys an area law; in a moment it buys a short memory too. Cue: no energy or ring size is quoted.
The pinned ground state of a small ring, unrolled and seen from above: needles alternating +x, -x with a length shorter than a full needle, the order parameter m_x ( = (-1)^j m_x, = 0; 0 < m_x < 1; exact diagonalization, pinned to one fall). Under each needle the z odds, one half and one half: along z every spin is a fair coin, the stripes live entirely in the red corners. A window and its charge-sorted Grid: every off-diagonal block red and hatched, in two kinds. Two apart (even): allowed by the symmetry gamma leaves behind, the mirror F; the charge moves in steps of two, so even the symmetric, unfallen ring state has them. One apart (odd): a needle along x is a corner between up and down, one spin at a time; these need the fall of the pencil. All blocks shaded; the one-apart blocks dominate (the weight bars: longest for |dq| = 0 and 1, short for 2, a sliver for 3). Caveats: pinned = the combination of the two lowest ring states with the largest overlap with the Neel-x product state. This is the state the meter will see. Cue: no weight or length is quoted.
Part 2's gold Meter beside the windows of the two pinned rings (a ring and a larger one, each drawn as its window plus six neighbours and an ellipsis): a short window, then a longer one; the Meter rises, then rises further (exact diagonalization, pinned to one fall). Axes ell against Delta S^(2)_A(ell): two point pairs, the reading grows with the window and the two rings all but coincide. Vote: turn the knob off (gamma = 0, same Delta_z), the symmetric ground state; what does the meter read? Still about a reading like before, exactly zero for every window, or does it depend on the ring size? Cue: no reading is quoted.
The answer: exactly zero, for every window and every ring. A separate column at the right edge, joined to the points by nothing (the knob is not swept): the symmetric ground state at gamma = 0, same Delta_z, reads 0 for the short window, the longer window, every window (exact: a definite charge makes rho_A block diagonal, so Part 2's rotation does nothing; checked on the unpinned ring). With the table flat the state has one definite charge, its windows have no red corners between sectors, and the meter has nothing to see. The meter cares about the tilted table; whether its reading keeps growing like half the log of ell is Part 4's test; rings this small cannot say.
A long row of spins in stripes; tag site 0 and ask whether spin r remembers its z. The bars of the connected correlation |_c| fall under the envelope e^{-2Mr} (sizes only; the signs alternate; the envelope is a cartoon with the exact M). At home the fading length 1/(2M) is a few sites (the 1/e crossing) and the kink size 1/M is twice that; M is an inverse length, never an energy, set by the couplings and known exactly. The knob turns down: halve gamma and the pair ruler more than doubles; halve it again and it more than doubles once more, the kink ruler running off the frame; at gamma = 0 the memory is infinite, the gapless edge of the Map. Cue: no length in sites, no value of M.
Two borrowed facts on the cards. Gap => exponential decay (Hastings-Koma 2006): a gapped ground state of a chain with neighbour couplings has connected correlations that fall off exponentially; the fading length is 1/(2M) on the repulsive side and set by the bound pair, 1/m_B1, on the attractive side (3.10b); the same ingredient gave the area law in 1.9. Baxter/JKM: M known exactly: Baxter (1972) and Johnson, Krinsky and McCoy (1973) solved this chain and M follows at any Delta_z, gamma of the strip |Delta_z| < 1, gamma > 0 (elliptic functions, Caveats; again in 4.8); on the XY line M = atanh gamma (the yellow line of the Map lights, its star at the foot, gamma = 0); every M in these talks is this exact value, never a fit. The knob is back home.
An error in the stripes: a single bond where two neighbours point the same way (the orange marker, 'a misprint in the stripes', shown large in an inset); left of it the stripes, right of it their mirror image (the F of 3.3): a wall between the two falls of the pencil, a kink. It slides: one site over, then bond by bond, and the stripes on both sides stay perfect. Zoom: the wall is not sharp; over about 1/M sites the needles shrink and turn over (tanh profile, cartoon), size ~ 1/M: at home the ruler 1/M, the second ruler of 3.7; beside it two walls at larger M, sharper and narrower (cartoon copies). The price: M is one over a length, an energy needs a speed; E_kink = v M, energy cost = v M, the speed v set by the couplings (on the XY line from the free-fermion dispersion, elsewhere from the Bethe ansatz: Caveats; no value is spoken). The kink is the chain's atom of error and v M is its price.
Can you make just one kink? Perfect stripes, a long row (the middle magnified below); a sigma^z stamp turns one needle half a turn about z, and the bad bonds are two, one on each side: a kink (orange) and an antikink (maroon), walls: 2. They fly apart, the stretch between them holds the mirror stripes, they meet and cancel, the stripes heal, walls: 0 (a cartoon at the home coupling; the walls' width is 1/M). The rule: anything local makes walls in pairs; far left and far right the stripes start with plus and nothing local changes what happens at infinity, so the number of walls in between is always even. At home, where walls repel, the lightest thing a local stamp can make is a pair, costing two v M (twice 3.8's price).
Two walls on one chain: do they feel each other? Only the white rod Delta_z decides today. The pair's energy against distance, in units of the single-wall cost vM: far apart it is two free walls (the dashed line). Delta_z > 0: the cartoon curve comes down to the line from above, pushing the walls together costs energy, they repel (home is here). Delta_z < 0: the curve dips below the line at a finite distance, the walls drift to it, they attract. Delta_z = 0: flat, they ignore each other: the free-fermion line of the Map. Shapes are cartoons; the sign is the physics; written on the knob: negative attracts, positive repels; home is on the repulsive side. Cue: the knob shows a sign, not a value.
On the attractive side (the white Delta_z knob reads Delta_z < 0) a kink and an antikink orbit each other: the bound pair, boxed in pink (cartoon). On the energy ladder in units of v M free pairs start at two single-wall costs (grey band); the bound pair is the pink rung below, its mass m_B1 = 2 M sin(pi xi/2) for xi < 1. Physicists relabel Delta_z as xi (strip: the free line Delta_z = 0 is xi = 1; attractive below, repulsive above; home sits on the repulsive side, xi > 1; the recipe is on the Caveats). The knob climbs towards the free line (readouts Delta_z < 0 then = 0, xi < 1 then = 1): the rung climbs and meets the band at xi = 1, where the pair is no longer bound (the rung row greys, the walls drift apart). Small print: scaling-limit law; on the lattice the ratio m_B1/2M sits a little below sin(pi xi/2); only the first bound state exists in the paper's range of xi, more appear only below xi = 1/2. Cue: no coupling, no rung height, no ratio is quoted.
Last picture: the stripes blur into a smooth string (the needles' angle) lying in a washboard of grooves; each groove is one fall of the pencil, a kink is the string stepping from one groove to the next, the antikink steps back. Black-box card (the continuum theory is named on it and nowhere else): as gamma -> 0 the stripes' angle becomes a smooth field in a cosine washboard, kinks its steps between grooves, the recipe Delta_z -> xi on the Caveats; the bound-pair law m_B1 = 2 M sin(pi xi/2) holds in that limit (on the lattice the ratio m_B1/2M sits a little below the law's sine and drifts down as gamma grows); Part 4 borrows from it the bound-pair law and what Lukyanov found for how a kink pair and the bound pair couple to the charge (4.7).
Part 2's question mark is filled. A window's charge, measured many times, is a bell; with a short memory its variance grows like a straight line in ell (each new site brings the same extra width); twice the slope is chi_zz (the yellow line of slope chi_zz/2 is the infinite chain at large ell, iDMRG; the coins' ell/4 is steeper). Second form: half the sum over every distance of the connected bars of 3.7 (the fan from one site, ell such fans, up to the window's edges). On a small ring the computer adds them: one half from the spin's own term, then a negative step from the neighbours, a small positive one, a smaller negative one, and the rest, landing well below one half (exact diagonalization, pinned): a toy, a small ring cannot hold the chain's memory. The infinite chain's value comes from iDMRG (next scene), a line drawn apart, never joined to the toy. Rule card: chi_zz is called a susceptibility; for a conserved charge the answer to a small field is the width per site divided by the temperature; here the charge is not conserved and chi_zz is strictly the width per site, at T = 0 (for this charge the field answer is the static Kubo moment, a different number).
The hand-off: the variance is a sum over excited states, one term per rung of the ladder, Var(Q_A) = sum over k != 0 of ||^2 (the letter k is not spoken). On the cartoon ladder the bound pair (pink, marked 'Delta_z < 0 only': it exists on the attractive side, not at the repulsive home point of the numbers on the right) and the kink pair (orange/maroon) are the lowest rungs and carry the longest weight bars; nothing is derived here, 4.6 opens the formula and shows why the lightest weigh most. The ring accumulator (a toy) and the iDMRG line stay on the right, apart; neither is quoted.
A wavefunction of L spins is a list of 2^L numbers: one more spin doubles it, twice the spins square it, exponential in L, hopeless. The trick is a different shape: one small table per site joined by wires with D numbers running through each wire (the train). Two moves, repeated: grow (two fresh sites slot into the middle, the chain settles; infinite = the pattern repeats) and compress (at a wire, the eigenvalue bars of Part 1 for that cut fall steeply; a gapped chain has few that matter, so keep the biggest D and drop the rest; L d D^2 << 2^L: polynomial against exponential). JPEG for wavefunctions. Is D big enough? Raise it and watch the value: at the hardest point of the Map (the longest memory of the paper's runs) chi_zz moves, then moves less, then stops moving (a schematic sketch; no value is quoted). Locked. Black box: iDMRG (everything inside): the train is a matrix-product state run with TeNPy; keeping the largest D bars at every cut is why it works for gapped chains (area law, 1.9); convergence is judged by raising D until the digits stop moving, and chi_zz is read from correlator sums in the converged state.
Three things to take home. Gamma tilts the table (explicit) and manufactures the red corners, the breaking the meter cares about; the stripes are one fall of the pencil (spontaneous). A gap buys a short memory: correlations fade over 1/(2M) sites, errors are kinks of size 1/M costing vM, made in pairs; M is exact (Baxter) and an inverse length. chi_zz is the width of the charge bell per site, twice the slope of the variance against ell or half the sum of the z correlations; computed at home by iDMRG (no value is quoted).
The Caveats, read by title only, page one: Jordan-Wigner on faith; the finite ring returns the cat and the pinned state is the max-overlap combination; ring energies are in units of the couplings and never compared with M; gap => exponential decay and Baxter's exact M on faith, the attractive fading length is 1/m_B1; a kink's energy is vM, and the speed v depends on the coupling (the free-fermion dispersion on the XY line, the Bethe ansatz elsewhere).
The Caveats, page two: a single sigma^z makes walls in pairs (topological charge); the xi recipe (the formula row below, left), home at xi > 1, only B_1 in the paper's range of xi; chi_zz at home is an iDMRG value and the ring accumulator (a toy) never joins it, the 3.11 word said once as a width per site at T = 0; "variance per site" is the talk's name for chi_zz, a deliberate simplification (strictly the variance per site is chi_zz/2); the matrix-product state (the formula row below, right) and its D, convergence judged at the hardest point of the Map.
Part 2's hole returns: Delta S^(2)_A = 1/2 log(4 pi V), V = ?. Two tokens slide towards it: chi_zz (measured by iDMRG, the infinite chain) and M (inverse size, exact, from Baxter), both at the home coupling; neither carries a value. The hole shows V ?= 1/2 chi_zz ell, the question mark pulsing: is V simply chi times ell over two, even for a mixed window? And can the chain's lightest particles pin chi over M from below, with nothing fitted? That is Part 4 (fourth dot pulsing).