Lecture 1: Quantum Matter, Emergence, and Effective Description

Quantum condensed matter asks how many quantum degrees of freedom organize into stable forms of matter whose low-energy laws can be simpler than their microscopic description. The many-body state space is enormous, yet metals, insulators, magnets, superconductors, and topological phases are reproducible and intelligible. This lecture introduces the ideas that make such intelligibility possible: locality, collective variables, large-system limits, phases, and matched effective descriptions. A solvable spin chain isolates these ideas in a minimal setting. The rest of the book develops the additional mathematical languages and microscopic mechanisms needed to make them quantitative in electrons, spins, collective modes, and transport.

A clean copper wire carries a steady current under a weak electric field. A piece of diamond does not. A ferromagnet can retain a macroscopic direction of magnetization after the aligning field is removed. A superconducting ring can sustain a macroscopic circulating current without an applied voltage. These are not minor quantitative variations of one behavior. They are different stable ways in which an enormous number of quantum degrees of freedom can organize.

All four systems obey quantum mechanics. Their electrons have the same charge and mass, and their nuclei interact with those electrons through the same electromagnetic law. Different atomic species, structures, densities, and interactions certainly matter, but listing those microscopic inputs does not yet explain why one resulting state conducts, another is rigidly insulating, another chooses a magnetic direction, and another locks its electrons into a single collective state that carries current without loss. The subject begins in the gap between an underlying law and an explanation of the realized form of matter.

The governing question is:

How can a quantum system with an astronomically large microscopic state space exhibit simple, stable, and qualitatively distinct forms of matter?

Before proceeding, choose which fact seems more surprising: that 100 two-state quantum degrees of freedom require roughly ${10^{30}}$ basis amplitudes, or that a macroscopic magnet can often be described by a direction, a stiffness, and a small family of excitations. The first fact creates the many-body problem. The second shows that the problem has physical structure. We return to this choice at the close of the section on emergence.

The Puzzle of Simple Matter

To see the gap concretely, write down the space the law acts on and then the law itself. For a fixed arrangement of nuclei, a microscopic electronic model lives on the antisymmetric ${N_{\mathrm e}}$ -electron Hilbert space

(1.1)

$\mathcal H_{\mathrm e} =\bigwedge^{N_{\mathrm e}} \left(L^2(\Omega)\otimes\mathbb C^2\right), \qquad \Omega\subset\mathbb R^3,$

where ${\Omega}$ is the sample region, ${L^2(\Omega)}$ is the space of square-integrable spatial wave functions, ${\mathbb C^2}$ is the spin space, and the exterior power enforces fermionic antisymmetry.

Three facts about this space are all the lecture needs, and none of them requires manipulating exterior powers: an ${N_{\mathrm e}}$ -electron state has ${N_{\mathrm e}}$ position arguments and spin labels, fermionic exchange symmetry is built into its mathematical habitat, the space on which it is defined, and the Hamiltonian acts on that habitat. The finite-site model of the next section replaces the continuum habitat with local finite-dimensional spaces so that the organizing ideas can be calculated exactly.

On a suitable dense domain ${\mathcal D(\hat H_{\mathrm e})\subset\mathcal H_{\mathrm e}}$ , a standard nonrelativistic Hamiltonian is

(1.2)

$\hat H_{\mathrm e} =\sum_{i=1}^{N_{\mathrm e}} \left[ -\frac{\hbar^2}{2m_{\mathrm e}}\nabla_i^2 +V_{\mathrm{ion}}(\mathbf r_i) \right] +\frac{1}{2}\sum_{i\ne j} \frac{e^2}{4\pi\epsilon_0|\mathbf r_i-\mathbf r_j|}.$

Here each ${\mathbf r_i\in\Omega}$ is a classical position coordinate used in the wave-function representation, ${V_{\mathrm{ion}}:\Omega\to\mathbb R}$ is the potential generated by the fixed nuclei, and ${\hat H_{\mathrm e}:\mathcal D(\hat H_{\mathrm e})\to\mathcal H_{\mathrm e}}$ is an operator with units of energy; its domain is the subspace on which these derivatives make sense, and Lecture 2 makes operator domains precise. Equation Equation (1.2) is already an approximation: the nuclei are fixed, relativistic and radiative effects are omitted, and the environment is absent. Each omission is acceptable when the requested accuracy and scale make it harmless.

The more important point is logical. Even an exact microscopic Hamiltonian would not label its eigenstates “metal,” “antiferromagnet,” or “superconductor.” Those are claims about organization: which state or family of states is realized, which quantities remain stable in large samples, what low-energy changes are possible, and how the system responds to an operation. The microscopic equation constrains every acceptable answer, but it does not replace the work needed to construct one.

In this book, quantum condensed matter means the study of extended systems with many coupled quantum degrees of freedom, with emphasis on their collective states, excitations, responses, and changes of organization. The word condensed covers more than rigid crystalline solids: quantum fluids, engineered lattices, magnetic insulators, superconducting circuits, and photonic or mechanical media raise the same structural questions.

The Microscopic State Space Grows Exponentially

We first isolate the size problem in a model of distinguishable sites with finite local state spaces, which needs neither indistinguishable electrons nor continuum coordinates.

Let ${\Lambda_N}$ be a finite set of ${N}$ sites. At site ${j\in\Lambda_N}$ , let ${\mathcal H_j}$ be a finite-dimensional complex Hilbert space. A finite quantum many-body model family is a size-indexed collection $\mathfrak M =\left\{ (\Lambda_N,{\{\mathcal H_j\}}_{j\in\Lambda_N}, \mathcal H_N,\hat H_N,\mathfrak b_N) \right\}_{N},$ where ${\mathcal H_N:=\bigotimes_{j\in\Lambda_N}\mathcal H_j}$ , ${\hat H_N:\mathcal H_N\to\mathcal H_N}$ is self-adjoint, and ${\mathfrak b_N}$ records the boundary condition and geometry. A single finite operator ${\hat H_N}$ is one member of the family; it is not by itself a thermodynamic phase.

The boundary record ${\mathfrak b_N}$ is part of the model because an open chain, a ring, and a finite patch with an edge need not have the same allowed excitations. The notation below usually suppresses it; its physical effect appears when the chain is opened in the next section.

Suppose every local space has the same dimension ${q}$ . Choose an orthonormal basis ${\{{|a\rangle}_j:a=1,\ldots,q\}}$ at each site. A product basis of ${\mathcal H_N}$ consists of the vectors

$|a_1,a_2,\ldots,a_N\rangle ={|a_1\rangle}_1\otimes{|a_2\rangle}_2\otimes\cdots\otimes{|a_N\rangle}_N.$

There are ${q}$ independent choices for each of the ${N}$ labels, and counting those choices gives the first structural fact.

For ${N}$ finite-dimensional spaces of common dimension ${q}$ , $\dim\mathcal H_N =\dim\left(\bigotimes_{j=1}^{N}\mathcal H_j\right) =\prod_{j=1}^{N}\dim\mathcal H_j =q^N.$

Take an open chain of ${N}$ two-state sites. How much information is needed to store a generic state, and how many terms specify a nearest-neighbor Hamiltonian ${\hat H=\sum_{j=1}^{N-1}\hat h_{j,j+1}}$ ? Evaluate the state dimension at ${N=10,50,100}$ . For ${N=100}$ , estimate storage at 16 bytes per complex coefficient and compare with the entry count of a dense operator. Distinguish these counts from the independent parameters of a normalized pure state.

State storage counts product-basis coefficients; Hamiltonian locality counts neighboring pairs, not matrix entries. The product rule just established gives

(1.3)

$2^{10}=1024, \qquad 2^{50}\simeq1.126\times10^{15}, \qquad 2^{100}\simeq1.268\times10^{30}.$

A generic state vector for 100 two-state sites therefore contains ${2^{100}}$ complex coefficients before normalization and global phase are accounted for. At 16 bytes per double-precision complex number, merely storing that coordinate vector would require about ${2.0\times10^{31}}$ bytes. A generic dense operator would contain ${{(2^{100})}^2=4^{100}\simeq1.61\times10^{60}}$ complex entries.

For ${N=2}$ , the four labels are ${(+,+),(+,-),(-,+),(-,-)}$ , but there is only one adjacent pair. For ${N=1}$ , there are two basis vectors and no pair term. In general, the nearest-neighbor Hamiltonian of the problem statement contains ${N-1}$ specified two-site terms, each described by a fixed-size matrix, whereas the generic state still has ${2^N}$ complex amplitudes. For a normalized pure state, normalization and the unobservable overall phase reduce the real parameter count to ${2(2^N)-2}$ ; they do not turn exponential growth into linear growth. All these counts are dimensionless; only the storage estimate carries bytes. The comparison therefore locates the structure in the dynamics.

Before inspecting Figure 1.1, predict which quantity is exponential and which one grows only linearly for a nearest-neighbor chain.

Figure 1.1. A logarithmic plot shows the exact exponential dimensions two to the N and three to the N, while a companion interaction graph shows only nearest-neighbor local terms on a chain and contrasts them with all possible site pairs.

In Figure 1.1, the dimension plot states the obstacle and the interaction graph begins the resolution. A physical Hamiltonian is assembled from terms constrained by spatial locality, symmetry, conservation laws, and microscopic mechanisms; it is generally not an arbitrary dense matrix specified entry by entry. These constraints do not make ${\mathcal H_N}$ small. They make the dynamics structured. Finite-range locality has a quantitative consequence: a local disturbance spreads through the lattice at a bounded effective speed. Lieb and Robinson showed that, in their finite-range quantum-spin setting, separated local observables satisfy an effective finite-velocity commutator bound. The bound has tails; it is not a sharp relativistic light cone [LR72]. In plain terms, a measurement far from the disturbance can depend on it only weakly at early times, and that bounded influence is the first quantitative sense in which “structured” replaces “small.”

Symmetry reduction leaves the same conclusion in place. Restricting ${N}$ spins to a fixed number ${n_\uparrow}$ of up spins leaves ${\binom{N}{n_\uparrow}}$ basis vectors. Near ${n_\uparrow=N/2}$ , that count is still exponential in ${N}$ . A good quantum number selects a sector and so makes a calculation possible; the sector itself remains a many-body problem.

This question refers to the accompanying figure.

Move the chain length and read the two counts together: the dimension ${q^N}$ with the storage it would demand at sixteen bytes a coefficient, and the number of two-site terms a nearest-neighbor Hamiltonian carries on the same chain. Find the length at which the storage passes a laboratory disk, and check that the term count has not yet reached three figures. The real parameter count ${2q^N-2}$ is shown beside the dimension: normalization and the overall phase remove two, and nothing else.

Locality Organizes the Many-Body Problem

We now make “structured” precise enough to calculate.

For a site set ${\Lambda_N}$ , an operator ${\hat O_X}$ has support in ${X\subseteq\Lambda_N}$ when it acts as the identity on every tensor factor outside ${X}$ . A Hamiltonian family is finite-range local when $\hat H_N=\sum_{X\subseteq\Lambda_N}\hat h_X,$ each ${\hat h_X}$ has support in ${X}$ , and there is a range ${R}$ independent of ${N}$ such that ${\hat h_X=0}$ whenever the diameter of ${X}$ exceeds ${R}$ . More general local families may allow interactions that decay with distance under a stated bound.

Local does not mean noninteracting, diagonal, classical, or weak. It says which tensor factors one term can couple directly. Repeated local operations can still generate long-range correlations and highly entangled states.

An exact local model

Let ${\Lambda_N=\{1,\ldots,N\}}$ be a ring with ${N\geq3}$ . At every site, ${\mathcal H_j=\mathbb C^2}$ has the orthonormal basis ${\{{|\uparrow\rangle}_j,{|\downarrow\rangle}_j\}}$ . Define ${\hat\sigma_j^z}$ as the Pauli ${z}$ operator on site ${j}$ tensored with the identity on all other sites. Thus

$\hat\sigma_j^z{|\uparrow\rangle}_j={|\uparrow\rangle}_j, \qquad \hat\sigma_j^z{|\downarrow\rangle}_j=-{|\downarrow\rangle}_j.$

The periodic ferromagnetic Ising Hamiltonian is the operator

(1.4)

$\hat H_N^{(\mathrm{per})} =-J\sum_{j=1}^{N}\hat\sigma_j^z\hat\sigma_{j+1}^z, \qquad \hat\sigma_{N+1}^z:=\hat\sigma_1^z, \qquad J>0,$

on ${\mathcal H_N={(\mathbb C^2)}^{\otimes N}}$ . The coupling ${J}$ has units of energy. Each summand has support on one nearest-neighbor bond.

The one-dimensional nearest-neighbor model entered the published literature through Ising’s 1925 paper, which the journal record identifies as an extract from his Hamburg dissertation [Isi25]. The notation, boundary convention, and ${4J}$ ring derivation that follows are course choices.

This model is exact and deliberately austere. It represents a strong easy-axis tendency and makes its product-state energies transparent. It does not yet include transverse quantum fluctuations, long-range dipolar fields, multiple orbitals, itinerant electrons, lattice motion, or a thermal ensemble. Real magnets can require any or all of them.

From microscopic labels to a collective defect count

Let ${\mathbf s=(s_1,\ldots,s_N)\in{\{-1,+1\}}^N}$ be a classical label vector and let

$|\mathbf s\rangle :={|s_1\rangle}_1\otimes\cdots\otimes{|s_N\rangle}_N \in\mathcal H_N, \qquad \hat\sigma_j^z|\mathbf s\rangle=s_j|\mathbf s\rangle.$

The function ${\mathbf s\mapsto|\mathbf s\rangle}$ maps a discrete label set to an orthonormal basis. The label vector is not the quantum state vector; it names one basis vector.

For one bond, ${s_js_{j+1}=+1}$ if the neighboring spins are aligned and ${-1}$ if they are opposite. A bond whose two spins are opposite is called a domain wall. Define the domain-wall count

(1.5)

$n_{\mathrm{dw}}(\mathbf s) :=\frac{1}{2}\sum_{j=1}^{N}\left(1-s_js_{j+1}\right) \in\{0,1,\ldots,N\}, \qquad s_{N+1}:=s_1.$

The map ${n_{\mathrm{dw}}:{\{-1,+1\}}^N\to\mathbb N_0}$ assigns to each configuration label the number of opposing bonds: each aligned bond contributes zero and each opposing bond contributes one.

Acting with Equation (1.4) on a product basis vector gives

$\hat H_N^{(\mathrm{per})}|\mathbf s\rangle =E_N^{(\mathrm{per})}(\mathbf s)|\mathbf s\rangle,$

where ${E_N^{(\mathrm{per})}:{\{-1,+1\}}^N\to\mathbb R}$ is the energy function. Because this Hamiltonian already acts diagonally on the product states, the energy of a configuration follows from a bond ledger, a count of bonds of each kind; no ${2^N}$ -dimensional matrix has to be diagonalized.

Try the bond count before continuing. On a six-site ring, start with all spins up and reverse only site 3, so that ${\mathbf s=(+1,+1,-1,+1,+1,+1)}$ . Write the six bond products, including the closing bond ${s_6s_1}$ . How many contributions change from ${-J}$ to ${+J}$ ? Keep the energy of the configuration separate from its energy above the aligned reference.

Answer. Two bonds change, the two that touch site 3; the closing bond $s_6s_1$ stays aligned. The configuration has $E=-2J$ against the aligned reference $E_0=-6J$ , so $E-E_0=4J$ , which is $2J$ for each of the two walls.

For the general configuration, there are ${N-n_{\mathrm{dw}}}$ aligned bonds and ${n_{\mathrm{dw}}}$ opposing bonds. Their contributions give

(1.6)

$\begin{aligned} E_N^{(\mathrm{per})}(\mathbf s) &=-J\sum_{j=1}^{N}s_js_{j+1}\\ &=-J\left[(N-n_{\mathrm{dw}})(+1)+n_{\mathrm{dw}}(-1)\right]\\ &=-J(N-n_{\mathrm{dw}})+Jn_{\mathrm{dw}}\\ &=-JN+2J n_{\mathrm{dw}}(\mathbf s). \end{aligned}$

For the six-site check, the products in bond order are ${(+1,-1,-1,+1,+1,+1)}$ . Four bonds contribute ${-J}$ and two contribute ${+J}$ , so ${E=-2J}$ , whereas the aligned reference has ${E_0=-6J}$ . Thus ${E-E_0=4J}$ . This direct sum checks the counting formula. It also shows why a wall costs ${2J}$ , not ${J}$ : it replaces ${-J}$ by ${+J}$ . The energies have units of ${J}$ ; the wall count is dimensionless.

For the Hamiltonian Equation (1.4), every product-basis energy is $E_N^{(\mathrm{per})}(\mathbf s) =-JN+2J n_{\mathrm{dw}}(\mathbf s).$ The two aligned configurations have ${n_{\mathrm{dw}}=0}$ and energy ${-JN}$ . A periodic configuration has an even number of domain walls, so the lowest nonzero defect count is two and the lowest defect energy is ${4J}$ above the ground energy.

Why must the count be even? Multiply all bond signs:

$\prod_{j=1}^{N}s_js_{j+1} =s_1^2s_2^2\cdots s_N^2=1.$

Every wall contributes one factor ${-1}$ , so the same product is ${{(-1)}^{n_{\mathrm{dw}}}}$ . Hence ${n_{\mathrm{dw}}}$ is even. This global constraint comes from closing the chain, not from the energy assigned to one bond.

A one-spin flip in an aligned ring reverses two neighboring bonds. The ring’s energy changes by ${2J}$ on each bond and therefore by ${4J}$ . The limit ${J\to0}$ collapses every difference, as it must when the interaction is removed.

Before viewing Figure 1.2, remove only the closing bond. Write whether the remaining chain can contain exactly one wall and predict its energy above its own aligned reference. Which step of the parity argument used the bond that has now been removed?

Answer. An open chain can hold exactly one wall, and it costs $2J$ above its own aligned reference $E_0=-J(N-1)$ . The even-wall rule on the ring came from multiplying all bond signs, a product that telescopes to $1$ only because the closing bond makes every spin appear twice; removing that bond removes the constraint, not the local cost.

Figure 1.2. Three aligned spin chains compare a fully aligned periodic ring, a periodic configuration with two domain walls, and an open configuration with one wall; aligned and frustrated bonds carry explicit energy contributions.

For an open chain the Hamiltonian is

$\hat H_N^{(\mathrm{open})} =-J\sum_{j=1}^{N-1}\hat\sigma_j^z\hat\sigma_{j+1}^z.$

The bond ledger now runs over the ${N-1}$ open-chain bonds, with wall count

$n_{\mathrm{dw}}^{(\mathrm{open})}(\mathbf s) :=\frac{1}{2}\sum_{j=1}^{N-1}\left(1-s_js_{j+1}\right) \in\{0,1,\ldots,N-1\},$

and no closure identification ${s_{N+1}:=s_1}$ . The ledger gives

(1.7)

$E_N^{(\mathrm{open})}(\mathbf s) =-J(N-1)+2J n_{\mathrm{dw}}^{(\mathrm{open})}(\mathbf s).$

There is no closure constraint. A configuration that is up on the left and down on the right has one wall and costs ${2J}$ . The local energy of a wall has not changed. The allowed number of walls has. Boundary conditions have therefore changed a qualitative low-energy statement.

For six sites with ${J>0}$ , compare the energy and excitation cost of ${\mathbf s=(+1,+1,+1,-1,-1,-1)}$ before and after restoring the closing bond in the Ising model, where each bond contributes ${-Js_is_j}$ . Use the ground energy of the corresponding boundary condition as the reference.

The strategy is to sum actual bonds and then subtract the ground energy of that same Hamiltonian. The five open-chain products are ${(+1,+1,-1,+1,+1)}$ , giving $E_{\mathrm{open}}=-J(4-1)=-3J,\qquad E_{0,\mathrm{open}}=-5J,\qquad E_{\mathrm{open}}-E_{0,\mathrm{open}}=2J.$ The closing product is ${s_6s_1=-1}$ , so it adds ${+J}$ to this state but ${-J}$ to the aligned reference: $E_{\mathrm{ring}}=-2J,\qquad E_{0,\mathrm{ring}}=-6J,\qquad E_{\mathrm{ring}}-E_{0,\mathrm{ring}}=4J.$ As an independent check on the sums, the wall-count formulas give one open wall and two ring walls, each costing ${2J}$ . The even ring count also satisfies the closure constraint, and both costs vanish as ${J\to0}$ . Energies have units of ${J}$ . Neither boundary choice changes the local cost per wall; the boundary changes which wall counts are allowed and the reference energy. Whether such a change matters for a chosen bulk observable is a separate size-scaling question.

Is every low-energy excitation of every magnet a domain wall? Before reading the interpretation, test that claim against either of the following two changes: a magnet whose moments can rotate continuously away from the chosen axis, or a metallic magnet with mobile electrons. Name the assumption of the Ising Hamiltonian that is no longer adequate. Does the bond ledger establish which excitation is lowest in that changed model?

The domain wall is a collective excitation: its position records a change in the ordered pattern across many microscopic labels, and its energy is most simply calculated by counting opposing (frustrated) bonds. It is not a new microscopic particle inserted into the chain. In a model with transverse spin dynamics, domain walls can move and occur in quantum superpositions. In a nearly isotropic magnet, long-wavelength spin waves may be the more appropriate low-energy objects. “Excitation” always refers to a state, operation, energy range, and physical model. The calculation therefore supports a claim about this specified Hamiltonian; the supplied alternatives change the assumptions needed to export that claim.

Close the notes and reconstruct the aligned ring, the ring with one reversed spin, and the open chain with one wall. Give the bond counts, total energy, reference energy, and excitation cost, then label which conclusion is exact for the Hamiltonian and which depends on its boundary condition. Do not infer an excited-level degeneracy merely by counting single-spin flips: distinct longer reversed domains can have the same two walls. Compare your ledger with the worked calculations, and repair the first incorrect bond count.

Reverse any spins of the chain and read its bond ledger: arrows show the spins; each bond is aligned or a wall and shows its contribution ${-Js_js_{j+1}}$ . The readout gives ${E}$ , the aligned reference ${E_0}$ , and ${E-E_0}$ beside ${2Jn_{\mathrm{dw}}}$ . Predict the cost of each flip before making it; the readout of the last flip checks the prediction. Beside the wall count the element states which counts the chosen boundary admits, so that the exact consequence of the Hamiltonian and the consequence of the boundary condition stay apart. Remove the closing bond to find the configurations with exactly one wall, and compare two reversed domains of different lengths that share the same two walls.

The energy ledger groups many microscopic configurations by their defect count. It does not distinguish the two oppositely aligned configurations, which have the same energy. To describe that difference we need an observable that records the collective orientation.

A collective variable

A collective observable for a model family is a size-dependent operator whose support grows with the system and whose normalization is declared. For the spin chain, the intensive magnetization operator is $\hat m_N:=\frac{1}{N}\sum_{j=1}^{N}\hat\sigma_j^z: \mathcal H_N\to\mathcal H_N.$ It is intensive because its eigenvalues remain in ${[-1,1]}$ as ${N}$ increases. The extensive total magnetization is ${N\hat m_N}$ .

The aligned basis vectors have ${\langle\Uparrow|\hat m_N|\Uparrow\rangle=+1}$ and ${\langle\Downarrow|\hat m_N|\Downarrow\rangle=-1}$ , where ${|\Uparrow\rangle:=|\uparrow\cdots\uparrow\rangle}$ and ${|\Downarrow\rangle:=|\downarrow\cdots\downarrow\rangle}$ . A symmetric superposition of those vectors can instead have zero expectation while still possessing long-range correlations. One number is therefore not a complete definition of an ordered phase. Before asking what a phase is, we must say in what sense a finite sample has a bulk at all; that is the next question.

Bulk Matter Is a Limiting Object

A laboratory sample is finite. Why, then, do we speak of bulk phases and sharp distinctions between them? The answer begins with scaling: a finite sample has a bulk, for a chosen observable, when the boundary’s share of that observable shrinks with size in a way that can be calculated and bounded.

Let ${\Lambda_L={\{1,\ldots,L\}}^d}$ be a ${d}$ -dimensional hypercubic set of sites, with ${L\geq3}$ and total count ${N=L^d}$ . Define its one-site boundary shell as the sites with at least one coordinate equal to ${1}$ or ${L}$ . The interior has ${{(L-2)}^d}$ sites, so

(1.8)

$N_{\partial}(L,d)=L^d-{(L-2)}^d, \qquad f_{\partial}(L,d) :=\frac{N_{\partial}(L,d)}{L^d} =1-\left(1-\frac{2}{L}\right)^d.$

Here ${f_{\partial}:\{3,4,\ldots\}\times\mathbb N\to[0,1]}$ is an exact dimensionless function. Expanding the final factor at fixed ${d}$ gives

(1.9)

$f_{\partial}(L,d) =\frac{2d}{L} -\frac{2d(d-1)}{L^2} +O(L^{-3}).$

The leading term shows that the boundary fraction vanishes as ${1/L}$ ; the second term is the first correction that the exact count keeps and the asymptote drops.

For a fixed-thickness boundary shell of a ${d}$ -dimensional hypercube, ${N_{\partial}/N=O(L^{-1})}$ as ${L\to\infty}$ . Consequently, a uniformly bounded boundary contribution divided by the volume can vanish in a bulk density. This conclusion does not make a boundary-supported observable or state physically negligible when the boundary itself is the target.

Suppose a dimensionless site observable differs from its interior value by at most one on each site of the one-site boundary shell and agrees with it elsewhere. Choose the smallest integer ${L\geq3}$ that guarantees a boundary error no greater than ${0.1}$ in its volume average, for a square and a cube of ${L}$ sites along each edge. Decide whether the same guarantee applies to an average over boundary sites alone.

Write ${\delta O_j}$ for the change at site ${j}$ and ${\delta\overline O}$ for the change in the volume average. The triangle inequality gives $|\delta\overline O| \leq\frac{1}{L^d}\sum_{j\in\partial\Lambda_L}|\delta O_j| \leq\frac{N_{\partial}}{L^d}=f_{\partial}(L,d).$ This bound is attained when every boundary-site change is ${+1}$ , so it is the worst-case error allowed by the assumptions. Thus the exact count, rather than its leading asymptote, decides the size: $N_{\partial}(L,2)=4L-4,\qquad N_{\partial}(L,3)=6L^2-12L+8.$ Writing the tolerance as ${\eta=0.1}$ , the inequality becomes $1-{(1-2/L)}^d\leq\eta \quad\Longleftrightarrow\quad L\geq\frac{2}{1-{(1-\eta)}^{1/d}}.$ Rounding upward gives ${L=39}$ for the square and ${L=58}$ for the cube. Check the neighboring integers: the square fractions are about ${0.1025}$ at 38 and ${0.09993}$ at 39; the cube fractions are about ${0.1016}$ at 57 and ${0.09992}$ at 58. At ${L=3}$ , direct counting gives 8 of 9 square sites and 26 of 27 cube sites on the boundary, agreeing with the exact formulas. As ${L\to\infty}$ , both fractions vanish.

The exact count also shows where the asymptote fails. For ${d=3}$ and ${L=10}$ , the leading estimate ${6/L=0.6}$ is poor; the exact boundary fraction is ${1-0.8^3=0.488}$ . At ${L=100}$ , the exact value is ${1-0.98^3\simeq0.0588}$ , close to ${0.06}$ . An asymptotic formula is not a license to use a large- ${L}$ conclusion at small ${L}$ .

The bound concerns a volume-normalized observable with a uniform site bound. If the measurement instead averages over boundary sites alone and every boundary-site change is ${+1}$ , its error is one at any ${L}$ . Thus the tolerance decision controls a stated bulk target; it does not justify discarding a surface measurement.

Before viewing Figure 1.3, predict whether the exact curve approaches the leading ${2d/L}$ estimate from above or below for ${d>1}$ .

Figure 1.3. A square lattice separates a one-site boundary shell from its interior, and exact boundary fractions in one, two, and three dimensions are compared with their leading large-size asymptotes on a declared range.

The exact fraction lies below ${2d/L}$ for ${d>1}$ because the leading estimate counts corners and edges more than once. The discrepancy decreases with ${L}$ . This expansion is an example of a controlled large-size statement: it identifies both its small parameter, ${1/L}$ , and its first omitted order.

This question refers to the accompanying figure.

Move the linear size and the dimension, and watch the exact shell fraction against the leading estimate ${2d/L}$ . Read off the size at which the exact share first falls below one half, and the size at which the leading estimate is within a tenth of the exact value. For ${d>1}$ , the relative error of that estimate falls like ${1/L}$ , which is the sense in which the expansion is controlled. For ${d=1}$ , the leading expression is already exact.

Choose a sequence of finite regions ${\Lambda_{N_1}\subset\Lambda_{N_2}\subset\cdots}$ whose linear size tends to infinity, together with boundary conditions and any fixed densities or couplings. A thermodynamic limit is the limit of declared intensive quantities or local-observable expectations along that sequence, when the limit exists. The object held fixed, the sequence, the normalization, and the state-selection prescription are part of the definition.

The definition assumes no statistical ensemble; it is a statement about a family of quantum systems and limiting observables. Statistical mechanics will add probability weights and thermal competition when those become necessary.

Boundary independence is a conclusion to prove for a chosen bulk target, not part of the word bulk. Surface spectroscopy, a boundary-localized state, and electrical current through contacts all make the boundary or interface part of the target. Later in the course, bulk and boundary information will constrain one another. The scaling calculation fixes the normalization in which the question of boundary independence is well posed.

Phases Describe Stable Organization

The word phase is often introduced through examples and left undefined. There is no single elementary definition that covers equilibrium thermal phases, gapless quantum matter, driven steady states, and open systems without additional hypotheses. We will therefore give one precise modern definition for an important restricted class and mark its boundary.

Consider two finite-range local Hamiltonian families ${{\{\hat H_N^{(0)}\}}_N}$ and ${{\{\hat H_N^{(1)}\}}_N}$ with the same local degrees of freedom and prescribed symmetry group ${G}$ . They are in the same zero-temperature gapped ground-state phase relative to ${G}$ when there is a continuous path ${s\mapsto\hat H_N(s)}$ , ${s\in[0,1]}$ , of uniformly local, ${G}$ -symmetric Hamiltonians joining them such that the bulk ground-state sector remains separated from bulk excitations by a positive gap in the large-system limit throughout the path.

For this definition, the bulk gap is tested with boundary conditions that do not confuse protected boundary levels with a bulk closing. The definition is restricted: it does not classify every gapless, thermal, driven, or open-system phase.

The parameter ${s}$ labels a path of Hamiltonians, not time evolution of one state. Continuity means that local coupling coefficients change continuously. The symmetry qualification matters: a path allowed to break a protecting symmetry can connect systems that are distinct when that symmetry must be preserved.

This definition captures stability rather than visual similarity. Adding a small, symmetry-preserving local term need not create a new phase: a gapped symmetric path can lead back to the unperturbed family. A qualitative reorganization requires an obstruction: for example, the bulk gap may close, the required symmetry may be broken, or the admissible local description may change. The definition also allows several microscopically different Hamiltonians to represent the same phase.

Two related ingredients motivate this restricted definition: quasi-adiabatic continuation preserves locality under stated gap and locality hypotheses, while local-unitary equivalence supplies a modern characterization of gapped ground-state phases within its own framework [HW05a; CGW10a]. Concretely, the first result shows that the ground states along a gapped path can be connected by operations that remain approximately local, and the second treats two gapped ground states as one phase when such local operations connect them.

The diagonal Ising ring illustrates both the power and the boundary of this language. Its two aligned ground configurations and finite domain-wall cost identify an ordered zero-temperature structure. A weak longitudinal field can select one orientation; the sign of that field as it is taken to zero must be stated. Finite-temperature order in a real magnet and the dynamics of the transverse-field Ising model require state-selection and dynamical arguments that this lecture has not developed. For the one-dimensional nearest-neighbor chain itself, no such order survives at nonzero temperature.

Effective Descriptions Are Matched Across Scales

Return to the six-site Ising ring. Its product basis contains 64 states, but suppose the question concerns only excitation energies below ${8J}$ . The bond ledger then selects the two aligned states and the 30 two-wall states. Wall positions and one reference spin orientation identify these 32 configurations. This is a smaller description of a specified part of the spectrum, rather than a replacement for every question about the magnet.

In this book, an effective description, or effective model, consists of selected states or variables in a declared space and rules for calculating particular properties of a system. It specifies what these variables mean physically and the conditions in which the calculation is meant to apply. Details that it does not track explicitly can still affect its parameters, interactions, or statistical rules.

This specifies the model’s intended use. A derivation or comparison with measurements must establish how accurately it answers the chosen question.

In the ring, the variables are wall positions and an orientation. Each wall contributes ${2J}$ to the excitation energy. No walls give zero, and two walls give ${4J}$ . The wall positions and one reference spin orientation let us reconstruct every spin label. Here the stated conditions are a periodic diagonal Ising chain and the energy window below ${8J}$ . The following construction checks exactly which states this rule covers.

For the periodic Ising chain with ${N\geq3}$ and ${J>0}$ , construct a retained low-energy description that reproduces every product-basis energy in the window ${0\leq E-E_0<8J}$ , where ${E_0=-NJ}$ . Specify its retained labels, count them, and state the boundary of its energy-matching claim.

The ring ledger is ${E-E_0=2J n_{\mathrm{dw}}}$ , with even ${n_{\mathrm{dw}}}$ . The exact ledger says that the only product-basis sectors in this window have zero or two domain walls. Label each retained configuration by its wall-bond set and one reference orientation ${s_1=\pm1}$ . Starting from ${s_1}$ , keep the sign across an aligned bond and reverse it across a wall; even wall parity closes the ring consistently. This rule reconstructs every spin label. A fixed unordered wall pair therefore has two orientations, not one state. Including the two no-wall states gives ${2+2\binom N2=2+N(N-1)}$ retained product-basis states. The retained quantum space is their span. The absolute retained energies are ${E_{\mathrm{eff}}=E_0+2J n_{\mathrm{dw}}}$ . Thus the excitation energy ${E_{\mathrm{eff}}-E_0}$ is zero for no walls and ${4J}$ for a wall pair, independent of orientation. The description is smaller only when ${N\geq4}$ .

For a concrete check, the six-site ring retains ${2+6\cdot5=32}$ basis states: 2 at zero excitation energy and 30 at ${4J}$ . For the reconstruction check from Example 1.2, choose ${W=\{3,6\}}$ and ${s_1=+1}$ . Keeping the sign except across those two bonds gives $(s_1,\ldots,s_6)=(+1,+1,+1,-1,-1,-1),\qquad E-E_0=2J\cdot2=4J.$ A second check is ${N=3}$ , where ${2+3\cdot2=8=2^3}$ : the retained labels exhaust the original basis. For the declared energy target the matching error is exactly zero. The label count and energy comparison test different parts of the proposed description.

This description is an exact relabeling of the diagonal model in the stated window. For ${N\geq4}$ , it becomes incomplete when the target includes ${E-E_0=8J}$ , where four-wall configurations enter. At ${N=3}$ there is no four-wall sector: the retained eight states already span the full model. The periodic sector choice also fails under a change to open boundaries, where one-wall states occur at ${2J}$ . Adding a transverse term ${-g\sum_j\hat\sigma_j^x}$ moves, creates, and annihilates walls, so the effective theory needs dynamics and quantum superpositions of defect positions. The matched energy ledger therefore earns a bounded low-energy claim; it does not yet supply dynamics for a perturbed magnet.

Why did excluding the higher-wall states leave these energies unchanged? The diagonal Hamiltonian does not mix them with the retained configurations. When a retained state couples to another state, its energy can shift even if we keep track only of the lower energy. Lecture 2 calculates such a shift in a two-level system. Choosing what to leave out therefore also requires deciding how to account for its influence on the quantities we calculate.

More generally, parameters may be derived from a microscopic model or calibrated against measurements. Exact agreement in a restricted sector, a controlled approximation, and empirical agreement over a tested range are different achievements. Fitting the calibration data alone does not test a new prediction.

To justify using an effective model for a particular question, specify:

the retained states or variables and their allowed values;
the law used to predict energies, motion, or statistics;
the correspondence to microscopic or measured quantities;
the target quantities or structural claims;
the parameter and scale domain ${D}$ ;
the evidence and accuracy of the comparison; and
known limitations and the first expected breakdown, when identified.

For a real scalar target ${Q}$ , one possible accuracy test is $\left|Q_{\mathrm{micro}}(x)-Q_{\mathrm{eff}}(x)\right| \leq\varepsilon_Q(x),\qquad x\in D.$ Here ${x}$ specifies the conditions being compared and ${\varepsilon_Q(x)\geq0}$ has the same units as ${Q}$ . A proved bound over ${D}$ differs from a tolerance supported only at tested conditions. Other targets require their own comparison, such as matrix elements or a stability statement. State unresolved limitations as unknown. Calling a model effective does not establish that its errors are controlled.

Before viewing Figure 1.4, identify which arrow would be absent if effective meant only simpler.

Figure 1.4. A scale-matching diagram follows a target and regime from microscopic variables to a retained description and effective law, tests the resulting predictions, and revisits the description when inadequate. Dotted paths compare stable consequences of microscopic families connected by an admissible path.

The upper left-to-right chain selects a description for a declared target: choose retained variables and build a law for them. The matching test compares its predictions with the microscopic model, an exact limit, or evidence. Dashed return paths express accountability: revise the description or revisit microscopic assumptions when the comparison is inadequate. The separate dotted branch compares zero-temperature gapped phases. An admissible path connects finite-range Hamiltonian families with the same local degrees of freedom. It varies couplings continuously, remains uniformly local, preserves the prescribed symmetry, and keeps a positive bulk gap in the large-system limit throughout the path. Boundary levels do not count as a bulk closing. The path varies Hamiltonians, not a state’s time evolution. Removing any one role produces an incomplete account of this microscopic construction. The figure illustrates that route; a phenomenological model can instead obtain its parameters from measurements and earn a bounded claim through further predictive tests.

Emergence Is a Claim About Stable Structure

An isolated spin can point up or down, but it has no domain wall. A wall marks a change between neighboring aligned regions of a chain. Once the interaction favors alignment, the wall becomes a useful collective defect with a calculable energy cost. We can now ask whether this organization survives a change in the individual bond strengths.

Relative to a specified microscopic description, we call a collective feature or law emergent when, in a stated regime, it describes an organization of the interacting system beyond the properties of its constituents considered separately, supports predictions in collective variables, and persists under a stated class of microscopic changes.

This is the book’s working meaning, not a universal philosophical definition. It permits derivation from microscopic laws. The collective structure can persist while its numerical parameters change.

Support the claim by answering four questions:

Scale: What collective feature appears, relative to which microscopic description, and in what energy, length, or system-size regime?
Stability: Which consequences persist under which physical changes, and which numerical details may change?
Explanatory autonomy: What can the collective variables predict, including the result of a changed condition?
Matching: What derivation or evidence relates the collective and microscopic descriptions, and what remains unresolved?

A change of labels or a difficult calculation alone does not answer these questions. An open reduction problem records missing knowledge, not evidence for emergence. A thermodynamic limit is needed only for claims that depend on that limit.

On a six-site periodic diagonal Ising chain, let the ferromagnetic bond energies satisfy ${0.8J\leq J_j\leq1.2J}$ , with ${J>0}$ . Show that the same 32 zero- and two-wall basis states describe the entire window ${0\leq E-E_0<6J}$ throughout this class. For ${J_2=0.8J}$ , ${J_5=1.1J}$ , and all other ${J_j=J}$ , calculate the cost of walls on bonds 2 and 5. What is robust and what has changed?

Use the same bond ledger as the domain-wall energy proposition, now with a different energy on each bond. Positivity still favors the two aligned configurations, with ${E_0=-\sum_{j=1}^{6}J_j}$ . For a wall-bond set ${W}$ , each affected bond changes from ${-J_j}$ to ${+J_j}$ , so $E-E_0=2\sum_{j\in W}J_j.$ Periodic closure still requires even wall number. Every wall pair costs between ${3.2J}$ and ${4.8J}$ , whereas four walls cost at least ${6.4J}$ . Higher wall counts cost still more. The new cutoff ${6J}$ therefore retains exactly the same ${2+2\binom62=32}$ basis states. Their labels remain wall positions and one reference orientation.

For the specified pair the cost is ${2(0.8+1.1)J=3.8J}$ . As a first check, setting all ${J_j=J}$ recovers ${4J}$ . As a second, every pair obeys the positive class lower bound ${3.2J}$ . The class bound ${3.2J}$ is not the exact gap of the specified chain: its two weakest distinct bonds have strengths ${0.8J}$ and ${J}$ , giving the smallest excitation ${3.6J}$ .

The aligned reference configurations, even parity, retained labels, and positive defect-energy scale persist. Individual energies, favored wall locations, and the uniform model’s excited-state degeneracy need not. The earlier cutoff ${8J}$ cannot be kept throughout this class, because four-wall states can occur below it: taking every ${J_j=0.8J}$ gives a four-wall excitation at ${6.4J}$ . Allowing two couplings to approach zero removes a positive class-wide lower bound on the two-wall gap. Opening the ring changes the parity rule; transverse terms require dynamics beyond this diagonal construction.

This comparison supplies a bounded emergence claim: the collective defect organizes the low-energy states and predicts their response to changed couplings. The mere relabeling of spin configurations would not establish that persistence. A finite six-site example does not establish a macroscopic phase or finite-temperature magnetic order.

Reduction and emergence are therefore not competing answers. Reduction asks how one description follows or matches another. Emergence asks which collective structures in the stated regime are stable, useful for a target, and shared across microscopic variations. A mature explanation often needs both.

This criterion is the book’s own synthesis, drawn from a longer lineage than the two works cited here. Anderson’s 1972 essay argues that broken symmetry and hierarchical organization make higher-level description indispensable, while Wilson and Kogut’s 1974 review develops the scale-dependent elimination and renormalization-group machinery behind universality [And72, pp. 393–396; WK74, pp. 75–199]. Neither source states the course’s definitions, seven-field adequacy audit, or four-question emergence test.

Return now to the choice posed before the first section. The size ${2^{100}}$ explains why a generic coordinate attack fails. It does not explain the magnet. Locality structures the interactions; the diagonal Ising ring sorts its product basis by exact defect energies. The large-system limit separates bulk from finite-size corrections; phase equivalence expresses stability; and effective variables compress the target while remaining matched to the microscopic theory. The surprising simplicity of matter is organized complexity.

Questions That Organize the Subject

When meeting an unfamiliar quantum material, begin with five questions:

Degrees of freedom: What microscopic and candidate low-energy objects are present, and what are their state spaces?
Structure: Which interactions are local, which symmetries or constraints organize the states, and which scales are separated?
Organization: What state or phase is realized, in what regime (and, where needed, under what limiting procedure), and which perturbations leave it stable?
Excitations and response: What operations create the accessible changes, what quantum numbers and energies do they carry, and which observations distinguish rival descriptions?
Scope: What was omitted, what accuracy is required, and which first correction can change the conclusion rather than only refine it?

Asked of the four systems that opened the lecture, these questions already separate them into distinct organizing problems. For the copper wire, the question is which low-energy excitations carry the current and why a weak field moves them freely. For diamond, it is why the many-electron ground state admits no low-energy charge motion at all. For the ferromagnet, it is which collective variable is selected and what sets the cost of changing it. For the superconducting ring, it is what stabilizes a macroscopic current against the weak local perturbations that dissipate a normal current. A topological phase, also named in the perspective that opened this lecture, poses a further kind of question: which of its properties is fixed by the organization of the whole rather than by any local order. Each answer is a claim about organization and must pass the five questions.

The questions are an audit of whether a physical explanation has connected its scales, and they can be asked of any account in this book in its own order and vocabulary.

We have used finite Hilbert spaces, operators, eigenstates, expectations, symmetry, and low-energy restriction while leaving some of their mathematical grammar at the level of remembered quantum mechanics. That is the precise need that Lecture 2 takes up: distinguish a state from its coordinates, an operator from its matrix, a projector from a basis label, and a controlled approximation from a plausible truncation.

The answer to the governing question is now bounded but substantive. Quantum matter is intelligible because its microscopic state space is not organized by an arbitrary operator. Locality, symmetry, and scale restrict the possible dynamics; low-energy and large-system limits reveal stable collective objects; and matched effective descriptions express their consequences in the variables appropriate to the question. The remainder of the book develops the mechanisms that realize this structure in actual quantum solids.

Problems

1. State counting inside a symmetry sector

Consider ${N}$ two-state sites with product labels ${\mathbf s\in{\{-1,+1\}}^N}$ . Let ${n_\uparrow}$ be the number of ${+1}$ labels.

Derive the full basis dimension and the dimension of the subspace with a fixed ${n_\uparrow}$ .
Evaluate both counts for ${N=20}$ and ${n_\uparrow=10}$ .
For even ${N}$ , use the supplied Stirling approximation

$n!\sim\sqrt{2\pi n}\left(\frac{n}{\mathrm e}\right)^n$

to show that ${\binom{N}{N/2}\sim2^N\sqrt{2/(\pi N)}}$ .

A classmate says, “Fixing a conserved quantity removes the exponential problem.” Diagnose the statement. Name one genuine advantage of the sector decomposition and one limitation that remains.

2. Domain walls, boundaries, and frustration

Let ${J>0}$ and ${N\geq4}$ .

Re-derive the periodic and open-chain energy formulas from the bond products. State the domain and codomain of each energy function.
List the allowed smallest nonzero domain-wall counts and excitation costs for each boundary condition. Explain why the local wall energy and the lowest system excitation are different statements.
Modify the periodic Hamiltonian by changing only the closing bond to antiferromagnetic:

$\hat H_N^{(\mathrm{fr})} =-J\sum_{j=1}^{N-1}\hat\sigma_j^z\hat\sigma_{j+1}^z +J\hat\sigma_N^z\hat\sigma_1^z.$

Find the minimum energy and ground-state degeneracy in the product basis. Give the answer for arbitrary ${N}$ , and explain why at least one bond must be unsatisfied.

Is the unsatisfied bond in part (c) a new microscopic particle? Answer in the language of the emergence criterion.

3. When is a boundary correction small?

For a one-site boundary shell of an ${L^d}$ hypercube, use the exact fraction ${f_{\partial}(L,d)}$ in Equation (1.8).

Derive explicit exact formulas for ${d=2}$ and ${d=3}$ .
Find the smallest integer ${L\geq3}$ for which ${f_{\partial}(L,2)<0.05}$ , and the smallest for which ${f_{\partial}(L,3)<0.05}$ . Use the exact formula and then compare the result with the asymptote.
At each answer from part (b), calculate the relative error of the leading estimate ${2d/L}$ .
Suppose an observable is the average energy per site and each boundary site changes the energy by at most a fixed constant independent of ${L}$ . Explain what the calculation permits you to conclude. Then give one condensed-matter target for which a vanishing boundary fraction does not make the boundary irrelevant.

4. Auditing a phase and an effective description

For the periodic Ising chain, a proposed low-energy description retains the two aligned states and every two-wall state for the target ${0\leq E-E_0<8J}$ .

Classify each of the following as a microscopic operator, collective operator, configuration function, retained variable, target domain, or error statement: ${\hat H_N^{(\mathrm{per})}}$ , ${\hat m_N}$ , ${n_{\mathrm{dw}}(\mathbf s)}$ , the two wall positions, ${0\leq E-E_0<8J}$ , and “the retained energies equal the exact energies.” More than one classification may be needed, but justify it.
Complete the seven-field adequacy audit from Criterion 1.1. State which claim is exact and which model change first invalidates it.
Two local Hamiltonian families with the same site spaces and symmetry group ${G}$ are joined by a continuous ${G}$ -symmetric local path whose bulk gap remains at least ${\Delta_0>0}$ . State exactly what Definition 1.5 permits you to conclude and name two claims it does not establish.
Change the chain from periodic to open, keeping the target window ${0\le E-E_0<8J}$ . Revise the retained variables and first failure to reflect the changed boundary condition, including the finite- ${N}$ exceptions.

References

[And72] Anderson, P. W. (1972). “More is different: Broken symmetry and the nature of the hierarchical structure of science.” Science 177 (4047), 393–396. https://doi.org/10.1126/science.177.4047.393.

[CGW10a] Chen, X., Gu, Z.-C., and Wen, X.-G. (2010). “Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order.” Physical Review B 82 (15), 155138. https://doi.org/10.1103/PhysRevB.82.155138.

[HW05a] Hastings, M. B., and Wen, X.-G. (2005). “Quasiadiabatic continuation of quantum states: The stability of topological ground-state degeneracy and emergent gauge invariance.” Physical Review B 72 (4), 045141. https://doi.org/10.1103/PhysRevB.72.045141.

[Isi25] Ising, E. (1925). “Beitrag zur theorie des ferromagnetismus.” Zeitschrift für Physik 31, 253–258. https://doi.org/10.1007/BF02980577.

[LR72] Lieb, E. H., and Robinson, D. W. (1972). “The finite group velocity of quantum spin systems.” Communications in Mathematical Physics 28 (3), 251–257. https://doi.org/10.1007/BF01645779.

[WK74] Wilson, K. G., and Kogut, J. (1974). “The renormalization group and the

\epsilon

expansion.” Physics Reports 12 (2), 75–199. https://doi.org/10.1016/0370-1573(74)90023-4.

Lecture 1: Quantum Matter, Emergence, and Effective Description

1.1 The Puzzle of Simple Matter

1.2 The Microscopic State Space Grows Exponentially

1.3 Locality Organizes the Many-Body Problem

1.3.1 An exact local model

1.3.2 From microscopic labels to a collective defect count

1.3.3 A collective variable

1.4 Bulk Matter Is a Limiting Object

1.5 Phases Describe Stable Organization

1.6 Effective Descriptions Are Matched Across Scales

1.7 Emergence Is a Claim About Stable Structure

1.8 Questions That Organize the Subject

1.9 Problems